On a Nonlinear Hyperbolic–Elliptic System Modeling Chemotaxis

Abstract

In this paper, we study a hyperbolic–elliptic system with $L^1$-initial data, which arises as a mathematical model for chemotaxis. In particular, after introducing the notion of entropy solutions in this setting, we prove results concerning the existence of solutions—either global or local in time—depending on how small the $L^1$-norm of the initial data is. Furthermore, we prove the uniqueness of the entropy solutions by adapting the standard doubling-variable method for hyperbolic conservation laws to the current framework.

1. Introduction

1.1. Statement of the Problem and Connection with the Literature

This paper deals with the following hyperbolic–elliptic system:

$$\left\{\begin{array}{cc}{u}_t^{(+)}+{\left[{\phi }^{(+)}(u^{(+)})\right]}_x={\mu }^{(+)}(g(v),{\left[g(v)\right]}_x,u^{(+)},u^{(-)})\hfill & \mathrm{in}S\hfill \\ \hfill & \\ u_t^{(-)}+{\left[{\phi }^{(-)}(u^{(-)})\right]}_x={\mu }^{(-)}(g(v),{\left[g(v)\right]}_x,u^{(+)},u^{(-)})\hfill & \mathrm{in}S\hfill \\ \hfill & \\ -{\left[g(v)\right]}_{xx}+v=\beta (u^{(+)},u^{(-)})\hfill & \mathrm{in}S,\hfill \end{array}\right.$$

(1)

where $S:=\mathbb{R}\times (0,T)$ and $T>0$, complemented by initial data

(H₀)

$$(u_0^{(+)},u_0^{(-)})\in {\left(L^1(\mathbb{R})\right)}^2.$$

We stress that, throughout the paper, square brackets $[·]$ are used as regular brackets. System (1) arises as a mathematical model describing chemosensitive movements of cells or other microorganisms. In this framework, $u^{(\pm )}$ represent the densities of cells moving towards the right or the left along the real axis, and v identifies the chemoattractant concentration produced by the cells themselves. Specific assumptions on data functions ${\phi }^{(\pm )}$, ${\mu }^{(\pm )}$, g, and $\beta$ are collected in Section 1.2 below.

Mathematical models for chemotaxis involving partial differential equations have been widely investigated in recent decades [1,2,3,4,5], starting from the Keller–Segel system [6], which typically combines a parabolic equation—describing the movement of individuals influenced by an external signal—with a parabolic or elliptic equation that determines the concentration of such a chemical nutrient. However, in order to represent a more realistic evolution process, avoiding in particular the effects of an infinitely fast propagation speed, models based on hyperbolic equations have also been proposed by several authors. In this context, we refer to [7,8], where chemosensitive movements in one space dimension have been simulated according to the Goldstein–Kac model [9,10]. Among the many other contributions (e.g., see [5,11,12,13,14,15] and the references therein), let us mention in particular the Greenberg–Alt system [1]

$$\left\{\begin{array}{c}{\displaystyle u_t^{(+)}+\lambda u_x^{(+)}=\frac{1}{2\lambda }\left(v_x-\chi \lambda \right)u^{(+)}+\frac{1}{2\lambda }\left(v_x+\chi \lambda \right)u^{(-)},}\hfill \\ \\ {\displaystyle u_t^{(-)}-\lambda u_x^{(-)}=-\frac{1}{2\lambda }\left(v_x-\chi \lambda \right)u^{(+)}-\frac{1}{2\lambda }\left(v_x+\chi \lambda \right)u^{(-)},}\hfill \\ \\ {\displaystyle -v_{xx}+v=\beta (u^{(+)}+u^{(-)})}\hfill \end{array}\right.$$

(2)

($\lambda ,\chi >0$), which typically fits with the class of systems considered in the present paper (refs. [7,14]; see also [16,17] for similar hyperbolic–elliptic or hyperbolic–parabolic systems on networks). It is worth observing that, by introducing the total cell density $u=u^{(+)}+u^{(-)}$ and the mean cell flux $w=\lambda (u^{(+)}-u^{(-)})$, problem (2) can be turned into the system

$$\left\{\begin{array}{c}{u}_t+w_x=0,\hfill \\ \\ w_t+{\lambda }^2{u}_x=u{v}_x-\chi w,\hfill \\ \\ -v_{xx}+v=\beta (u),\hfill \end{array}\right.$$

(3)

which has been studied in [1,13,15,18,19,20] (see also the references therein).

The first two equations in (2) can be seen as a particular case of a more general model where the elliptic equation

$$-v_{xx}+v=\beta (u^{(+)},u^{(-)})$$

(4)

(or its corresponding parabolic version) is complemented by the hyperbolic equations

$$u_t^{(+)}+\lambda u_x^{(+)}=-{\gamma }^{(+)}(v,v_x)u^{(+)}+{\gamma }^{(-)}(v,v_x)u^{(-)},$$

(5)

$$u_t^{(-)}-\lambda u_x^{(-)}={\gamma }^{(+)}(v,v_x)u^{(+)}-{\gamma }^{(-)}(v,v_x)u^{(-)}.$$

(6)

In this framework, the terms ${\gamma }^{(\pm )}(·,·)$ are called turning rates and control the probability of transition from $u^{(+)}$ (right-moving cells) to $u^{(-)}$ (left-moving cells) and vice versa. Furthermore, the function $\beta (u^{(+)},u^{(-)})$ provides the production (or degradation) of the chemoattractant v.

Systems (4)–(6) have been studied on the real line in [15], with sufficiently smooth nonnegative initial data and symmetric Lipschitz continuous turning rates ${\gamma }^{(\pm )}(\eta ,\widehat{\eta })$ satisfying

(A)

$${\gamma }^{(\pm )}:{\mathbb{R}}^2\to \mathbb{R}\mathrm{are} \mathrm{nonnegative} \mathrm{on} {\mathbb{R}}^2.$$

However, it is worth observing that the above condition does not tie in with the model case

$${\gamma }^{(+)}(\eta ,\widehat{\eta })=-{\displaystyle \frac{1}{2\lambda }}\left(\widehat{\eta }-\chi \lambda \right),{\gamma }^{(-)}(\eta ,\widehat{\eta })={\displaystyle \frac{1}{2\lambda }}\left(\widehat{\eta }+\chi \lambda \right)$$

(see (2)). For this reason, it will be dropped in the present paper (see Section 1.2 for more specific assumptions about system (1). From a mathematical point of view, in [7,15], the nonnegativity of the turning rates ensures nonnegativity of solutions to systems (4)–(6) for all nonnegative initial data $u_0^{(\pm )}$. As a consequence, the total particle $L^1$-norm is preserved, whence the boundedness of both v and $v_x$ follows from standard a priori estimates for elliptic equations, and global existence of solutions to (4)–(6) is guaranteed for all (sufficiently smooth) nonnegative initial data. Roughly speaking, these analytical results suggest a smooth long-term chemotactic evolution with no sharp aggregation or blow-up phenomena in the biological model.

Conversely, in order to study well-posedness of (1) with $L^1$-initial data, and to overcome the technical difficulties arising from the neglect of hypothesis (A) and the smoothness of the initial data functions $u_0^{(\pm )}$, we will use a two-step approximation procedure and passage to the limit (see Section 3 and Section 4), starting from a pseudoparabolic regularization of the hyperbolic equations in (1) (see problems $(P_{\epsilon ,\nu })$ in Section 3.1). In particular, this will yield existence of suitably defined entropy solutions to (1) (see Definition 2), for which uniqueness will follow from adapting the standard method of doubling variables for hyperbolic conservation laws ([21]; see also [22,23,24,25,26,27,28] and the references therein).

We would like to emphasize that our results concern both local and global existence of entropy solutions depending on the size of the $L^1$-norm of the initial data $U_0=(u_0^{(+)},u_0^{(-)})$ (see Theorem 2 below), a phenomenon that has already been observed in analogous chemotaxis models (e.g., see [13,29,30]). Indeed, small initial data (with respect to the $L^1$-norm) translates back to the intuitive idea of a small initial mass and an initial population that is only weakly aggregated. Consequently, we can expect a moderate cell chemotactic response, evolution of the chemical stimulus v without sharp gradients, and population dynamics without blow-up or infinite aggregation. Conversely, for larger initial $L^1$-norms, our analytical study suggests that cells may aggregate rapidly in response to steeper chemoattractant gradients and that solutions may blow up in a finite time. In this connection, it is worth mentioning the results in [31], where solutions to a coupled system analogous to (4)–(6) developing blow-up in a finite time have been explicitly constructed. Interestingly, in these examples, the turning rates ${\gamma }^{(\pm )}$ change sign near the time at which the blow-up occurs—a behavior that has not been observed in [7], where nonnegativity of the turning rates is assumed.

The plan of the paper is the following: the main assumptions and some preliminary notations are outlined in Section 1.2 and Section 1.3 below. In Section 2, we collect the main definitions and results about local/global existence and uniqueness of entropy solutions to (1) with $L^1$-initial data (see Theorems 1 and 2). Section 3 and Section 4 are devoted to the study of the approximating problems and include both a priori estimates (Section 3.2 and Section 4.1) and passages to the limit (Section 3.3 and Section 4.2). Finally, the proofs of the main results are the content of Section 5.

1.2. Assumptions

In this paragraph, we collect the main conditions on the data functions ${\phi }^{(\pm )}$, ${\mu }^{(\pm )}$, g, and $\beta$ in system (1). Specifically, we shall consider functions ${\mu }^{(\pm )}:{\mathbb{R}}^4\to \mathbb{R}$, ${\phi }^{(\pm )}:\mathbb{R}\to \mathbb{R}$, $g:\mathbb{R}\to \mathbb{R}$, and $\beta :{\mathbb{R}}^2\to \mathbb{R}$, satisfying the following hypotheses:

$(H_1)$

${\mu }^{(\pm )}\in C^1({\mathbb{R}}^4),{\mu }^{(\pm )}(\eta ,\widehat{\eta },0,0)=0;$

$(H_2)$

there exists $\widehat{\alpha }>0$ such that, for every $R>1$, there holds

$$|{\partial }_{{\xi }^{(+)}}{\mu }^{(\pm )}(\eta ,\widehat{\eta },{\xi }^{(+)},{\xi }^{(-)})|+|{\partial }_{{\xi }^{(-)}}{\mu }^{(\pm )}(\eta ,\widehat{\eta },{\xi }^{(+)},{\xi }^{(-)})|\le \widehat{\alpha }R,$$

(7)

$$|{\partial }_{\widehat{\eta }}{\mu }^{(\pm )}(\eta ,\widehat{\eta },{\xi }^{(+)},{\xi }^{(-)})|+|{\partial }_{\eta }{\mu }^{(\pm )}(\eta ,\widehat{\eta },{\xi }^{(+)},{\xi }^{(-)})|\le \widehat{\alpha }R(|{\xi }^{(+)}|+|{\xi }^{(-)}|)$$

(8)

for all $(\eta ,\widehat{\eta },{\xi }^{(+)},{\xi }^{(-)})\in {\mathbb{R}}^4$ such that $|\eta |,|\widehat{\eta }|\le R$;

$(H_3)$

${\phi }^{(\pm )}\in \mathrm{Lip}(\mathbb{R})$, ${\phi }^{(\pm )}(0)=0$.

$(H_4)$

$g\in C^1(\mathbb{R})\cap \mathrm{Lip}(\mathbb{R})$ is strictly increasing on $\mathbb{R}$, and $g(0)=0$;

$(H_5)$

$\beta \in \mathrm{Lip}({\mathbb{R}}^2)$, and $\beta (0,0)=0$.

We stress that all assumptions $(H_1)$–$(H_5)$ will always be assumed to hold throughout the paper.

It is worth mentioning some direct estimates that easily follow from assumptions $(H_1)$–$(H_2)$. Specifically, by the Lipschitz continuity of ${\mu }^{(\pm )}$, combined with inequalities (7) and (8) and the condition ${\mu }^{(\pm )}(\eta ,\widehat{\eta },0,0)=0$, for every $R>1$, there holds

$$|{\mu }^{(\pm )}(\eta ,\widehat{\eta },{\xi }^{(+)},{\xi }^{(-)})|\le \widehat{\alpha }R\left(|{\xi }^{(+)}|+|{\xi }^{(-)}|\right)$$

(9)

for all $(\eta ,\widehat{\eta },{\xi }^{(+)},{\xi }^{(-)})\in {\mathbb{R}}^4$ with $|\eta |,|\widehat{\eta }|\le R$, and

$$\begin{array}{ccc}\hfill & & |{\mu }^{(\pm )}({\eta }_2,{\widehat{\eta }}_2,{\xi }_2^{(+)},{\xi }_2^{(-)})-{\mu }^{(\pm )}({\eta }_1,{\widehat{\eta }}_1,{\xi }_1^{(+)},{\xi }_1^{(-)})|\hfill \\ & & \le \widehat{\alpha }R\left(\sum _{i=1}^2|{\xi }_i^{(+)}|+|{\xi }_i^{(-)}|\right)\left(|{\eta }_2-{\eta }_1|+|{\widehat{\eta }}_2-{\widehat{\eta }}_1|\right)+\widehat{\alpha }R\left(|{\xi }_2^{(+)}-{\xi }_1^{(+)}|+|{\xi }_2^{(-)}-{\xi }_1^{(-)}|\right)\hfill \end{array}$$

(10)

for all $({\eta }_i,{\widehat{\eta }}_i,{\xi }_i^{(+)},{\xi }_i^{(-)})\in {\mathbb{R}}^4$ with $|{\eta }_i|,|{\widehat{\eta }}_i|\le R$ ($i=1,2$).

1.3. Preliminaries

By $\mathcal{M}(\mathbb{R})$, we shall denote the space of finite (signed) Radon measures on $\mathbb{R}$. The total variation of $\mu \in \mathcal{M}(\mathbb{R})$ will be denoted by ${\parallel \mu \parallel }_{\mathcal{M}(\mathbb{R})}:=|\mu |(\mathbb{R}),$ whereas the Lebesgue measure by $|·|$. Furthermore, by a null set, we mean a Borel set E such that $|E|=0$; the expression “almost everywhere”, or shortly “a.e.”, “means that up to a set of zero Lebesgue measure”. We shall denote by $C_c(\mathbb{R})$ the space of continuous functions with compact support in $\mathbb{R}$, and by $〈\mu ,\zeta 〉={\int }_{\mathbb{R}}\zeta (x)d\mu (x)$ the duality between the space $\mathcal{M}(\mathbb{R})$ and $C_c(\mathbb{R})$. Similar notations will be used for the space of Radon measures on $S:=\mathbb{R}\times (0,T)$.

We recall that $L_{w*}^{\infty }(0,T;\mathcal{M}(\mathbb{R}))$ is the set of finite Radon measures $u\in \mathcal{M}(S)$ that satisfy the following property: for $a.e.$ $t\in (0,T)$, there exists a measure $u(t)\in \mathcal{M}(\mathbb{R})$ such that

$(i)$

for every $\zeta \in C_c(\overline{S})$, the map $t\to 〈u(t),\zeta (·,t)〉$ is Lebesgue measurable, and ${〈u,\zeta 〉}_S={\int }_0^T〈u(t),\zeta (·,t)〉dt;$

$(ii)$

there exists a constant $M>0$ such that ${\parallel u\parallel }_{L_{w*}^{\infty }(0,T;\mathcal{M}(\mathbb{R}))}:=\mathrm{ess}{sup}_{t\in (0,T)}{\parallel u(t)\parallel }_{\mathcal{M}(\mathbb{R})}\le M$.

Let us also mention that a sequence $\left\{u_n\right\}\subseteq L_{w*}^{\infty }(0,T;\mathcal{M}(\mathbb{R}))$*-weakly converges to $u\in L_{w*}^{\infty }(0,T;\mathcal{M}(\mathbb{R}))$ (written $u_n\stackrel{*}{\rightharpoonup }u$ in $L_{w*}^{\infty }(0,T;\mathcal{M}(\mathbb{R}))$) if

$${\int }_0^T〈u_n(t),\zeta (·,t)〉dxdt\to {\int }_0^T〈u(t),\zeta (·,t)〉dt.$$

for every $\zeta \in L^1(0,T;C_c(\mathbb{R}))$ (e.g., see Proposition 4.4.16 of [32]).

For any bounded open subset $\mathrm{\Omega }\subset \mathbb{R}$, we shall denote by $\mathcal{Y}(\mathrm{\Omega };\mathbb{R})$ the set of Young measures on $\mathrm{\Omega }\times \mathbb{R}$; i.e., positive Radon measures $\tau$ on $\mathrm{\Omega }\times \mathbb{R}$ such that $\tau (E\times \mathbb{R})=|E|$ for any Borel set $E\subseteq \mathrm{\Omega }$. If $u\in L^1(\mathrm{\Omega })$, the Young measure associated with u is identified by the equality $\tau (E\times F)=\left|(E\cap u^{-1}(F))\right|$ for any pair of Borel sets $E\subseteq \mathrm{\Omega },F\subseteq \mathbb{R}$. It is also worth recalling that, given a Young measure $\tau \in \mathcal{Y}(\mathrm{\Omega };\mathbb{R})$, for $a.e.$ $x\in \mathrm{\Omega }$, there exists a probability measure ${\tau }_{(x)}\in \mathcal{P}(\mathbb{R})$ such that, for any bounded Carathéodory integrand $\psi :\mathrm{\Omega }\times \mathbb{R}\to \mathbb{R}$, the map $x\to {\int }_{\mathbb{R}}\psi (x,\xi )d{\tau }_{(x)}(\xi )$ is Lebesgue measurable and ${\int }_{\mathrm{\Omega }\times \mathbb{R}}\psi d\tau ={\int }_{\mathrm{\Omega }}\left({\int }_{\mathbb{R}}\psi (x,\xi )d{\tau }_{(x)}(\xi )\right)dx.$ The family of probability measures ${\left\{{\tau }_{(x)}\right\}}_{x\in \mathrm{\Omega }}$ is called the disintegration of $\tau$. If $\tau$ is the Young measure associated with a function $u\in L^1(\mathrm{\Omega })$, then, for $a.e.$ $x\in \mathrm{\Omega }$, its disintegration is the Dirac measure ${\tau }_{(x)}={\delta }_{u(x)}$.

Finally, a sequence of Young measures $\left\{{\tau }_n\right\}$ on $\mathrm{\Omega }\times \mathbb{R}$ converges narrowly to a Young measure $\tau \in \mathcal{Y}(\mathrm{\Omega };\mathbb{R})$ if, for any bounded Carathéodory integrand $\psi :\mathrm{\Omega }\times \mathbb{R}\to \mathbb{R}$, there holds ${\int }_{\mathrm{\Omega }\times \mathbb{R}}\psi d{\tau }_n\to {\int }_{\mathrm{\Omega }\times \mathbb{R}}\psi d\tau$ as $n\to \infty$. Plus, if the sequence of Young measures $\left\{{\tau }_n\right\}$ is associated with a sequence of functions $\left\{u_n\right\}$ equibounded in $L^1(\mathrm{\Omega })$, then there exists a Young measure $\tau$ on $\mathrm{\Omega }\times \mathbb{R}$ such that (possibly up to a subsequence not relabeled) there holds ${\tau }_n\to \tau$ narrowly in $\mathrm{\Omega }\times \mathbb{R}$ (see [33,34]).

2. Results

Let us begin by the definition of weak solutions to (1) with $L^1$-initial data.

Definition 1.

For every ${\tau }_0\in (0,T]$, a weak solution of (1) in $S_{{\tau }_0}:=\mathbb{R}\times (0,{\tau }_0)$, with initial data $(u_0^{(+)},u_0^{(-)})\in {\left(L^1(\mathbb{R})\right)}^2$, is a triplet $(u^{(+)},u^{(-)},v)$, with

$$u^{(\pm )},v\in L^{\infty }(0,{\tau }_0;L^1(\mathbb{R})),$$

(11)

$$g(v)\in L^{\infty }(0,{\tau }_0;H^1(\mathbb{R}))\cap L^{\infty }(S_{{\tau }_0}),{\left[g(v)\right]}_x\in L^{\infty }(S_{{\tau }_0}),$$

(12)

such that

$(i)$

 for $a.e.$ $t\in (0,{\tau }_0)$ the equation

$$-{\left[g(v(·,t))\right]}_{xx}+v(·,t)=\beta (u^{(+)}(·,t),u^{(-)}(·,t))$$

(13)

is satisfied in $H^{-1}(\mathbb{R})$;

$(ii)$

 for all $\zeta \in C_c^1({\overline{S}}_{{\tau }_0})$, $\zeta (·,{\tau }_0)=0$, there holds

$$\begin{array}{ccc}\hfill & & {\int \int }_{S_{{\tau }_0}}\left\{u^{(\pm )}{\zeta }_t+{\phi }^{(\pm )}(u^{(\pm )}){\zeta }_x\right\}dxdt\hfill \\ & & +{\int \int }_{S_{{\tau }_0}}{\mu }^{(\pm )}(g(v),{\left[g(v)\right]}_x,u^{(+)},u^{(-)})\zeta dxdt=-{\int }_{\mathbb{R}}{u}_0^{(\pm )}\zeta (x,0)dx.\hfill \end{array}$$

(14)

Remark 1.

Let $(u^{(+)},u^{(-)},v)$ be a weak solution of (1) in $S_{{\tau }_0}$ with initial data $(u_0^{(+)},u_0^{(-)})\in {\left(L^1(\mathbb{R})\right)}^2$.

$(i)$

 Observe that the elliptic Equation (13) is well-posed since $\beta (u^{(+)}(·,t),u^{(-)}(·,t))\in L^1(\mathbb{R})$ for $a.e.$ $t\in (0,{\tau }_0)$ (see $(H_5)$ and recall that $u^{(\pm )}\in L^{\infty }(0,{\tau }_0;L^1(\mathbb{R}))$. Moreover, for $a.e.$ $t\in (0,{\tau }_0)$, it is uniquely solvable in $H^{-1}(\mathbb{R})$ within the class of functions $w\in L^1(\mathbb{R})$ with $g(w)\in H^1(\mathbb{R})$. Indeed, given any two solutions w and $\widehat{w}$ of (13) in this class, choosing the test function $g(w)-g(\widehat{w})$ in the equation $-{[g(w)-g(\widehat{w})]}_{xx}+w-\widehat{w}=0,$ yields

$${\int }_{\mathbb{R}}(w-\widehat{w})(g(w)-g(\widehat{w}))dx\le {\int }_{\mathbb{R}}{\left({[g(w)-g(\widehat{w})]}_x\right)}^{2}dx+{\int }_{\mathbb{R}}(w-\widehat{w})(g(w)-g(\widehat{w}))dx=0,$$

whence $w=\widehat{w}$ $a.e.$ in $\mathbb{R}$ (recall that the function g is increasing on $\mathbb{R}$).

$(ii)$

 The weak formulation (14) is also well-posed. This follows from observing that

$${\phi }^{(\pm )}(u^{(\pm )})\in L^{\infty }(0,{\tau }_0;L^1(\mathbb{R})),$$

(15)

$${\mu }^{(\pm )}(g(v),{\left[g(v)\right]}_x,u^{(+)},u^{(-)})\in L^{\infty }(0,{\tau }_0;L^1(\mathbb{R})).$$

(16)

The former condition is ensured by assumption $(H_3)$ since $u^{(\pm )}\in L^{\infty }(0,{\tau }_0;L^1(\mathbb{R}))$. Regarding (16), it is enough to notice that applying (9) with $R\ge {\parallel g(v)\parallel }_{L^{\infty }(S_{{\tau }_0})}+{\parallel {\left[g(v)\right]}_x\parallel }_{L^{\infty }(S_{{\tau }_0})}$ yields

$$|{\mu }^{(\pm )}(g(v),{\left[g(v)\right]}_x,u^{(+)},u^{(-)})|\le \widehat{\alpha }R\left(|u^{(+)}|+|u^{(-)}|\right)\in L^{\infty }(0,{\tau }_0,L^1(\mathbb{R})).$$

(17)

$(iii)$

 It is convenient to observe that any weak solution of (1) with initial data $(u_0^{(+)},u_0^{(-)})$ as above achieves the initial condition in the sense of the following convergence:

$$\mathrm{ess}\underset{\tau \to 0^{+}}{lim}{\int }_{\mathbb{R}}{u}^{(\pm )}(x,\tau )\rho (x)dx={\int }_{\mathbb{R}}{u}_0^{(\pm )}\rho (x)dxforall\rho \in C_c^1(\mathbb{R}).$$

(18)

To this purpose, fix any ρ as above, and, for $a.e.$ $\tau \in (0,{\tau }_0)$, choose $\zeta (x,t)=\rho (x)h_j(t)$, with

$$h_j(t)={\chi }_{[0,\tau -j^{-1})}(t)+j(\tau -t){\chi }_{[\tau -j^{-1},\tau ]}(t)(j\in \mathbb{N}largeenough),$$

(19)

as test function in (14). This yields

$$\begin{array}{ccc}& & j{\int }_{\tau -j^{-1}}^{\tau }{\int }_{\mathbb{R}}{u}^{(\pm )}\rho (x)dxdt={\int }_{\mathbb{R}}{u}_0^{(\pm )}(x)\rho (x)dx+\underset{S_{{\tau }_0}}{\int \int }{\phi }^{(\pm )}(u^{(\pm )}){\rho }^{\prime }(x)h_j(t)dxdt\hfill \\ & & +\underset{S_{{\tau }_0}}{\int \int }{\mu }^{(\pm )}(g(v),{\left[g(v)\right]}_x,u^{(+)},u^{(-)})\rho (x)h_j(t)dxdt.\hfill \end{array}$$

Plainly, setting $j\to \infty$ in the above equalities, for $a.e.$ $\tau \in (0,{\tau }_0)$, we get

$$\begin{array}{ccc}\hfill & & {\int }_{\mathbb{R}}{u}^{(\pm )}(x,\tau )\rho (x)dx={\int }_{\mathbb{R}}{u}_0^{(\pm )}(x)dx+\underset{S_{\tau }}{\int \int }{\phi }^{(\pm )}(u^{(\pm )}){\rho }^{\prime }(x)dxdt\hfill \\ & & +\underset{S_{\tau }}{\int \int }{\mu }^{(\pm )}(g(v),{\left[g(v)\right]}_x,u^{(+)},u^{(-)})\rho (x)dxdt\hfill \end{array}$$

(20)

$(S_{\tau }=\mathbb{R}\times (0,\tau )).$ Since by (15) and (16) there holds

$$\underset{\tau \to 0^{+}}{lim}\underset{S_{\tau }}{\int \int }{\phi }^{(\pm )}(u^{(\pm )}){\rho }^{\prime }(x)dxdt=0,\underset{\tau \to 0^{+}}{lim}\underset{S_{\tau }}{\int \int }{\mu }^{(\pm )}(g(v),{\left[g(v)\right]}_x,u^{(+)},u^{(-)})\rho (x)dxdt=0,$$

from (20) it follows that

$$\mathrm{ess}\underset{\tau \to 0^{+}}{lim}{\int }_{\mathbb{R}}{u}^{(\pm )}(x,\tau )\rho (x)dx={\int }_{\mathbb{R}}{u}_0^{(\pm )}\rho (x)dxforall\rho \in C_c^1(\mathbb{R}).$$

Let us now introduce the notion of entropy solution to (1).

Definition 2.

For every ${\tau }_0\in (0,T]$, by an entropy solution of (1) in $S_{{\tau }_0}:=\mathbb{R}\times (0,{\tau }_0)$, with initial data $(u_0^{(+)},u_0^{(-)})\in {\left(L^1(\mathbb{R})\right)}^2$, we mean a weak solution $(u^{(+)},u^{(-)},v)$ in the sense of Definition 1 such that the following entropy inequalities

$$\begin{array}{ccc}\hfill & & \underset{S_{{\tau }_0}}{\int \int }\left\{|u^{(\pm )}-k|{\zeta }_t+\mathrm{sgn}(u^{(\pm )}-k)[{\phi }^{(\pm )}(u^{(\pm )})-{\phi }^{(\pm )}(k)]{\zeta }_x\right\}dxdt\hfill \\ & & +\underset{S_{{\tau }_0}}{\int \int }\mathrm{sgn}(u^{(\pm )}-k){\mu }^{(\pm )}(g(v),{\left[g(v)\right]}_x,u^{(+)},u^{(-)})\zeta dxdt\ge -{\int }_{\mathbb{R}}|u_0^{(\pm )}-k|\zeta (x,0)dx\hfill \end{array}$$

(21)

are satisfied for all $k\in \mathbb{R}$ and $\zeta \in C_c^1({\overline{S}}_{{\tau }_0})$, with $\zeta \ge 0$ in $S_{{\tau }_0}$ and $\zeta (·,{\tau }_0)=0$.

By a global entropy solution of (1) with initial data $U_0=(u_0^{(+)},u_0^{(-)})\in {\left(L^1(\mathbb{R})\right)}^2$, we mean an entropy solution of (1) in $S=\mathbb{R}\times (0,T)$ with initial data $U_0$.

We explicitly observe that, in inequalities (21) with different upper notations “±”, we can choose different values of k.

Interestingly, entropy solutions of (1) assume initial data according to the $L_{\mathrm{loc}}^1$-strong convergence. This is the content of the following proposition.

Proposition 1.

Let $(u^{(+)},u^{(-)},v)$ be an entropy solution of (1) in $S_{{\tau }_0}$$({\tau }_0\in (0,T])$ with initial data $(u_0^{(+)},u_0^{(-)})\in {\left(L^1(\mathbb{R})\right)}^2$. Then, there exists a null subset ${\mathcal{N}}_0\subset (0,{\tau }_0)$ such that, for all sequences $\left\{{\tau }_n\right\}\subset (0,T)\setminus {\mathcal{N}}_0$ and ${\tau }_n\to 0^{+}$, there holds

$$u^{(\pm )}(·,{\tau }_n)\to u_0^{(\pm )}in{L}_{\mathrm{loc}}^1(\mathbb{R}).$$

(22)

Uniqueness of entropy solutions of (1) with $L^1$-initial data is the content of the following theorem.

Theorem 1.

Let $U_0=(u_0^{(+)},u_0^{(-)})\in {\left(L^1(\mathbb{R})\right)}^2$. Then, there exists at most one entropy solution of (1) in $S_{{\tau }_0}$ $({\tau }_0\in (0,T])$ with initial data $U_0$.

Finally, existence of local and global solutions to (1) (the latter for initial data with small enough $L^1$-norm) is addressed in the following theorem. We explicitly notice that these solutions are unique by Theorem 1.

Theorem 2.

For every $U_0=(u_0^{(+)},u_0^{(-)})\in {\left(L^1(\mathbb{R})\right)}^2$, set

$$T_0=min\left\{T;{\displaystyle \frac{1}{2\widehat{\alpha }{\widehat{C}}_1\left(\parallel u_0^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel u_0^{(-)}\parallel }_{L^1(\mathbb{R})}\right)}}\right\},$$

(23)

where $\widehat{\alpha }>0$ is the constant in assumption $(H_2)$,

$${\widehat{C}}_1=2L_{\beta }(L_g+1),$$

(24)

and $L_{\beta }$ and $L_g$ are the Lipschitz constants of β and g, respectively.

$(i)$

 For every ${\widehat{T}}_0\in (0,T_0)$, there exists an entropy solution of (1) in $S_{{\widehat{T}}_0}$ with initial data $U_0$.

$(ii)$

 Under the additional condition

$$\parallel u_0^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel u_0^{(-)}\parallel }_{L^1(\mathbb{R})}<{\displaystyle \frac{1}{2\widehat{\alpha }{\widehat{C}}_{1}T}},$$

(25)

there exists a global solution of (1) with initial data $U_0$.

3. Approximating Problems $({\mathit{P}}_{\mathit{\epsilon },\mathit{\nu }})$

3.1. Well-Posedness

For $\epsilon ,\nu \in (0,1)$, we shall consider the following regularization of system (1):

$$(P_{\epsilon ,\nu })\left\{\begin{array}{cc}{u}_t^{(+)}+{\left[{\phi }^{(+)}(u^{(+)})\right]}_x-\nu {\left[u^{(+)}\right]}_{xx}-\epsilon {\left[u^{(+)}\right]}_{txx}={\mu }_{\epsilon ,\nu }^{(+)}(g_{\nu }(v),{\left[g_{\nu }(v)\right]}_x,u^{(+)},u^{(-)})\hfill & \mathrm{in}S,\hfill \\ \\ u_t^{(-)}+{\left[{\phi }^{(-)}(u^{(-)})\right]}_x-\nu {\left[u^{(-)}\right]}_{xx}-\epsilon {\left[u^{(-)}\right]}_{txx}={\mu }_{\epsilon ,\nu }^{(-)}(g_{\nu }(v),{\left[g_{\nu }(v)\right]}_x,u^{(+)},u^{(-)})\hfill & \mathrm{in}S,\hfill \\ \\ -{\left[g_{\nu }(v)\right]}_{xx}+v=\beta (u^{(+)},u^{(-)})\hfill & \mathrm{in}S,\hfill \end{array}\right.$$

where we have set

$$g_{\nu }(s):=g(s)+\nu s(s\in \mathbb{R}),$$

(26)

and

$${\mu }_{\epsilon ,\nu }^{(\pm )}(\eta ,\widehat{\eta },{\xi }^{(+)},{\xi }^{(-)}):={\mu }^{(\pm )}\left(T_{\frac{1}{\nu }}(\eta ),T_{\frac{1}{\nu }}(\widehat{\eta }),T_{\frac{1}{\epsilon }}({\xi }^{(+)}),T_{\frac{1}{\epsilon }}({\xi }^{(-)})\right).$$

(27)

Here, for every $k>1$ ($k={\nu }^{-1}$ or $k={\epsilon }^{-1}$), we have denoted by

$$T_k(s)=max\{-k,min\{s,k\}\}(s\in \mathbb{R})$$

(28)

the usual truncation function (observe that $T_k(·)$ is a Lipschitz continuous function on $\mathbb{R}$, with Lipschitz constant $L=1$).

Remark 2.

In view of their definitions, the functions ${\mu }_{\epsilon ,\nu }^{(\pm )}$ are bounded on ${\mathbb{R}}^4$, and, from (9), we get

$$|{\mu }_{\epsilon ,\nu }^{(\pm )}(\eta ,\widehat{\eta },{\xi }^{(+)},{\xi }^{(-)})|\le {\displaystyle \frac{\widehat{\alpha }}{\nu }}\left(|T_{{\displaystyle \frac{1}{\epsilon }}}({\xi }^{(+)})|+|T_{{\displaystyle \frac{1}{\epsilon }}}({\xi }^{(-)})|\right)$$

(29)

for all $(\eta ,\widehat{\eta },{\xi }^{(+)},{\xi }^{(-)})\in {\mathbb{R}}^4$. Analogously, from (10), it follows that, for all $(\eta ,\widehat{\eta },{\xi }^{(+)},{\xi }^{(-)})\in {\mathbb{R}}^4$, there holds

$$\begin{array}{ccc}\hfill & & |{\mu }_{\epsilon ,\nu }^{(\pm )}({\eta }_2,{\widehat{\eta }}_2,{\xi }_2^{(+)},{\xi }_2^{(-)})-{\mu }_{\epsilon ,\nu }^{(\pm )}({\eta }_1,{\widehat{\eta }}_1,{\xi }_1^{(+)},{\xi }_1^{(-)})|\hfill \\ & & \le {\widehat{C}}_{\epsilon ,\nu }(|T_{\frac{1}{\nu }}({\eta }_2)-T_{\frac{1}{\nu }}({\eta }_1)|+|T_{\frac{1}{\nu }}({\widehat{\eta }}_2)-T_{\frac{1}{\nu }}({\widehat{\eta }}_1)|\hfill \\ & & +|T_{\frac{1}{\epsilon }}({\xi }_2^{(+)})-T_{\frac{1}{\epsilon }}({\xi }_1^{(+)})|+|T_{\frac{1}{\epsilon }}({\xi }_2^{(-)})-T_{\frac{1}{\epsilon }}({\xi }_1^{(-)})|)\hfill \\ & & \le {\widehat{C}}_{\epsilon ,\nu }\left(|{\eta }_2-{\eta }_1|+|{\widehat{\eta }}_2-{\widehat{\eta }}_1|+|{\xi }_2^{(+)}-{\xi }_1^{(+)}|+|{\xi }_2^{(-)}-{\xi }_1^{(-)}|\right),\hfill \end{array}$$

(30)

with

$${\widehat{C}}_{\epsilon ,\nu }={\displaystyle \frac{\widehat{\alpha }}{\nu }}\left(1+{\displaystyle \frac{4}{\epsilon }}\right).$$

(31)

In order to address well-posedness of problem $(P_{\epsilon ,\nu })$ with $H^1$-initial data, let us rephrase it as an abstract ODE in the Banach space $X={\left(H^1(\mathbb{R})\right)}^2$; i.e.,

$$U_t=\mathcal{L}(U),U:=(u^{(+)},u^{(-)})\in X.$$

(32)

Here, $\mathcal{L}:X\to X$ maps any $U=(u^{(+)},u^{(-)})\in X$ into the pair

$$\mathcal{L}(U)=({\widehat{u}}^{(+)},{\widehat{u}}^{(-)}),$$

(33)

where ${\widehat{u}}^{(\pm )}\in H^1(\mathbb{R})$ are the unique weak solutions of the elliptic equations

$${\widehat{u}}^{(\pm )}-\epsilon {\left[{\widehat{u}}^{(\pm )}\right]}_{xx}=-{\left[{\phi }^{(\pm )}(u^{(\pm )})\right]}_x+\nu {\left[u^{(\pm )}\right]}_{xx}+{\mu }_{\epsilon ,\nu }^{(\pm )}(g_{\nu }(v),{\left[g_{\nu }(v)\right]}_x,u^{(+)},u^{(-)})\mathrm{in}\mathbb{R},$$

(34)

with $v\in H^1(\mathbb{R})$ being the unique weak solution to the elliptic equation

$$-{\left[g_{\nu }(v)\right]}_{xx}+v=\beta (u^{(+)},u^{(-)})\mathrm{in}\mathbb{R}.$$

(35)

Remark 3.

We explicitly point out that the mapping $\mathcal{L}$ in (33)–(35) is well-posed. To this aim, notice that, for any $U=(u^{(+)},u^{(-)})\in X$, existence and uniqueness of $v\in H^1(\mathbb{R})$ follow from standard results on elliptic equations since $\beta (u^{(+)},u^{(-)})\in L^2(\mathbb{R})$ (this follows from $(H_5)$ and the condition $U\in X$), and $g_{\nu }^{\prime }(s)\ge \nu$ for all $s\in \mathbb{R}$ (see (26) and $(H_4)).$ Regarding Equation (34), it is enough to observe that, since $u^{(\pm )},v\in H^1(\mathbb{R})$, there holds

$$u_x^{(\pm )}\in H^{-1}(\mathbb{R}),{\left[{\phi }^{\pm }(u^{\pm })\right]}_x\in L^2(\mathbb{R}),{\mu }_{\epsilon ,\nu }^{(\pm )}(g_{\nu }(v),{\left[g_{\nu }(v)\right]}_x,u^{(+)},u^{(-)})\in L^2(\mathbb{R});$$

(36)

specifically, for the second claim, we have used $(H_3)$, whereas the third condition immediately follows from (29), which in turn implies

$$\begin{array}{ccc}& & |{\mu }_{\epsilon ,\nu }^{(\pm )}(g_{\nu }(v),{\left[g_{\nu }(v)\right]}_x,u^{(+)},u^{(-)})|\le {\displaystyle \frac{\widehat{\alpha }}{\nu }}(|u^{(+)}|+|u^{(-)}|)\in L^2(\mathbb{R}).\hfill \end{array}$$

Theorem 3.

For any $U_0=(u_0^{(+)},u_0^{(-)})\in X={\left(H^1(\mathbb{R})\right)}^2$, there exists a unique $(u^{(+)},u^{(-)},v)$, with

$$u^{(\pm )}\in C^1([0,T];H^1(\mathbb{R})),v\in C([0,T];H^1(\mathbb{R})),$$

$$u^{(\pm )}(·,0)=u_0^{(\pm )}a.e. in \mathbb{R},$$

(37)

such that, for all $t\in [0,T]$, the following equations

$$\begin{array}{ccc}\hfill & & u_t^{(\pm )}(·,t)+{\left[{\phi }^{(\pm )}(u^{(\pm )}(·,t))\right]}_x=\hfill \\ & & {\mu }_{\epsilon ,\nu }^{(\pm )}\left(g_{\nu }(v(·,t)),{\left[g_{\nu }(v(·,t))\right]}_x,u^{(+)}(·,t),u^{(-)}(·,t)\right)\hfill \\ & & +{\partial }_x\left(\nu {\left[u^{(\pm )}(·,t)\right]}_x+\epsilon {\left[u^{(\pm )}(·,t)\right]}_{tx}\right),\hfill \end{array}$$

(38)

$$-{\left[g_{\nu }(v(·,t))\right]}_{xx}+v(·,t)=\beta (u^{(+)}(·,t),u^{(-)}(·,t))$$

(39)

are satisfied in $H^{-1}(\mathbb{R})$.

Proof. 

The claim will easily follow if we prove that the mapping $\mathcal{L}:X\to X$ in (33)–(35) is Lipschitz continuous on X. To this aim, for any two $U_1=(u_1^{(+)},u_1^{(-)})\in X$ and $U_2=(u_2^{(+)},u_2^{(-)})\in X$, set

$$\mathcal{L}\left((u_i^{(+)},u_i^{(-)})\right)=({\widehat{u}}_i^{(+)},{\widehat{u}}_i^{(-)})$$

($i=1,2$). Accordingly, let $v_i\in H^1(\mathbb{R})$ be the weak solutions of the elliptic equations

$$-{\left[g_{\nu }(v_i)\right]}_{xx}+v_i=\beta (u_i^{(+)},u_i^{(-)})\mathrm{in}\mathbb{R}(i=1,2).$$

Multiplying the equation

$$-{[g_{\nu }(v_2)-g_{\nu }(v_1)]}_{xx}+[v_2-v_1]=\beta (u_2^{(+)},u_2^{(-)})-\beta (u_1^{(+)},u_1^{(-)})\mathrm{in}\mathbb{R}$$

(40)

by $g_{\nu }(v_2)-g_{\nu }(v_1)$ yields

$$\begin{array}{ccc}& & {\int }_{\mathbb{R}}{[g_{\nu }(v_2)-g_{\nu }(v_1)]}_x^{2}dx+{\int }_{\mathbb{R}}[v_2-v_1][g_{\nu }(v_2)-g_{\nu }(v_1)]dx\hfill \\ & & ={\int }_{\mathbb{R}}[\beta (u_2^{(+)},u_2^{(-)})-\beta (u_1^{(+)},u_1^{(-)})][g_{\nu }(v_2)-g_{\nu }(v_1)]dx\underset{(H_5)}{\underbrace{\le }}\hfill \\ & & L_{\beta }{\int }_{\mathbb{R}}\left(|u_2^{(+)}-u_1^{(+)}|+|u_2^{(-)}-u_1^{(-)}|\right)|g_{\nu }(v_2)-g_{\nu }(v_1)|dx\le \hfill \\ & & L_{\beta }\left(\parallel u_2^{(+)}-u_1^{(+)}{\parallel }_{L^2(\mathbb{R})}+{\parallel u_2^{(-)}-u_1^{(-)}\parallel }_{L^2(\mathbb{R})}\right)\parallel g_{\nu }(v_2)-g_{\nu }(v_1){\parallel }_{L^2(\mathbb{R})}\hfill \end{array}$$

($L_{\beta }>0$ being the Lipschitz constant of $\beta$). In view of the above inequality, and since

$$\begin{array}{ccc}& & {\int }_{\mathbb{R}}{[g_{\nu }(v_2)-g_{\nu }(v_1)]}^{2}dx\le \left(\nu +L_g\right){\int }_{\mathbb{R}}[v_2-v_1][g_{\nu }(v_2)-g_{\nu }(v_1)]dx\hfill \\ & & \le \left(1+L_g\right){\int }_{\mathbb{R}}[v_2-v_1][g_{\nu }(v_2)-g_{\nu }(v_1)]dx\hfill \end{array}$$

(here, $\nu +L_g>0$, with $\nu \in (0,1)$ is the Lipschitz constant of $g_{\nu }$ (see $(H_4)$ and (26)), we get

$$\begin{array}{ccc}& & {\int }_{\mathbb{R}}{[g_{\nu }(v_2)-g_{\nu }(v_1)]}_x^{2}dx+{\int }_{\mathbb{R}}{[g_{\nu }(v_2)-g_{\nu }(v_1)]}^{2}dx\hfill \\ & & \le C_1\left(\parallel u_2^{(+)}-u_1^{(+)}{\parallel }_{L^2(\mathbb{R})}+{\parallel u_2^{(-)}-u_1^{(-)}\parallel }_{L^2(\mathbb{R})}\right)\parallel g_{\nu }(v_2)-g_{\nu }(v_1){\parallel }_{L^2(\mathbb{R})},\hfill \end{array}$$

with $C_1=L_{\beta }\left(1+L_g\right)$. This proves that

$$\parallel g_{\nu }(v_2)-g_{\nu }(v_1){\parallel }_{H^1(\mathbb{R})}\le C_1\left(\parallel u_2^{(+)}-u_1^{(+)}{\parallel }_{L^2(\mathbb{R})}+{\parallel u_2^{(-)}-u_1^{(-)}\parallel }_{L^2(\mathbb{R})}\right).$$

(41)

Next, subtracting the elliptic Equation (34) satisfied by ${\widehat{u}}_i^{(\pm )}$ ($i=1,2$), we obtain

$$\begin{array}{ccc}\hfill & & [{\widehat{u}}_2^{(\pm )}-{\widehat{u}}_1^{(\pm )}]-\epsilon {[{\widehat{u}}_2^{(\pm )}-{\widehat{u}}_1^{(\pm )}]}_{xx}=\nu {[u_2^{(\pm )}-u_1^{(\pm )}]}_{xx}-{[{\phi }^{(\pm )}(u_2^{(\pm )})-{\phi }^{(\pm )}(u_1^{(\pm )})]}_x\hfill \\ & & +{\mu }_{\epsilon ,\nu }^{(\pm )}(g_{\nu }(v_2),{\left[g_{\nu }(v_2)\right]}_x,u_2^{(+)},u_2^{(-)})-{\mu }_{\epsilon ,\nu }^{(\pm )}(g_{\nu }(v_1),{\left[g_{\nu }(v_1)\right]}_x,u_1^{(+)},u_1^{(-)}).\hfill \end{array}$$

(42)

Since

$$|[{\phi }^{(\pm )}(u_2^{(\pm )})-{\phi }^{(\pm )}(u_1^{(\pm )})]|\le L_{{\phi }^{(\pm )}}|u_2^{(\pm )}-u_1^{(\pm )}|$$

(here, $L_{{\phi }^{(\pm )}}>0$ are the Lipschitz constants of ${\phi }^{(\pm )}$; see $(H_3)$), and

$$\begin{array}{ccc}& & |{\mu }_{\epsilon ,\nu }^{(\pm )}(g_{\nu }(v_2),{\left[g_{\nu }(v_2)\right]}_x,u_2^{(+)},u_2^{(-)})-{\mu }_{\epsilon ,\nu }^{(\pm )}(g_{\nu }(v_1),{\left[g_{\nu }(v_1)\right]}_x,u_1^{(+)},u_1^{(-)})|\hfill \\ & & \le {\widehat{C}}_{\epsilon ,\nu }\left(|g_{\nu }(v_2)-g_{\nu }(v_1)|+|{\left[g_{\nu }(v_2)\right]}_x-{\left[g_{\nu }(v_1)\right]}_x|+|u_2^{(+)}-u_1^{(+)}|+|u_2^{(-)}-u_1^{(-)}|\right)\hfill \end{array}$$

(see (30)), multiplying (42) by ${\widehat{u}}_2^{(\pm )}-{\widehat{u}}_1^{(\pm )}$ yields

$$\begin{array}{ccc}& & \epsilon {\int }_{\mathbb{R}}{\left({\widehat{u}}_2^{(\pm )}-{\widehat{u}}_1^{(\pm )}\right)}_x^{2}dx+{\int }_{\mathbb{R}}{\left({\widehat{u}}_2^{(\pm )}-{\widehat{u}}_1^{(\pm )}\right)}^{2}dx\hfill \\ & & \le \parallel {\widehat{u}}_2^{(\pm )}-{\widehat{u}}_1^{(\pm )}{\parallel }_{H^1(\mathbb{R})}\left(\nu \parallel u_2^{(\pm )}-u_1^{(\pm )}{\parallel }_{H^1(\mathbb{R})}+L_{{\phi }^{(\pm )}}{\parallel u_2^{(\pm )}-u_1^{(\pm )}\parallel }_{L^2(\mathbb{R})}\right)\hfill \\ & & +{\widehat{C}}_{\epsilon ,\nu }\parallel {\widehat{u}}_2^{(\pm )}-{\widehat{u}}_1^{(\pm )}{\parallel }_{H^1(\mathbb{R})}({\parallel g_{\nu }(v_2)-g_{\nu }(v_1)\parallel }_{L^2(\mathbb{R})}\hfill \\ & & +\parallel {\left[g_{\nu }(v_2)\right]}_x-{\left[g_{\nu }(v_1)\right]}_x{\parallel }_{L^2(\mathbb{R})}+\parallel u_2^{(+)}-u_1^{(+)}{\parallel }_{L^2(\mathbb{R})}+{\parallel u_2^{(-)}-u_1^{(-)}\parallel }_{L^2(\mathbb{R})}).\hfill \end{array}$$

Plainly, in view of (41), the above inequality implies the estimate

$$\begin{array}{ccc}& & {\int }_{\mathbb{R}}{\left({\widehat{u}}_2^{(\pm )}-{\widehat{u}}_1^{(\pm )}\right)}_x^{2}dx+{\int }_{\mathbb{R}}{\left({\widehat{u}}_2^{(\pm )}-{\widehat{u}}_1^{(\pm )}\right)}^{2}dx\hfill \\ & & \le {\tilde{C}}_{\epsilon ,\nu }\parallel {\widehat{u}}_2^{(\pm )}-{\widehat{u}}_1^{(\pm )}{\parallel }_{H^1(\mathbb{R})}({\parallel u_2^{(+)}-u_1^{(+)}\parallel }_{H^1(\mathbb{R})}+\left(\parallel u_2^{(-)}-u_1^{(-)}{\parallel }_{H^1(\mathbb{R})}\right),\hfill \end{array}$$

whence also

$$\begin{array}{ccc}& & \parallel \mathcal{L}(U_2)-\mathcal{L}(U_1){\parallel }_X=\parallel {\widehat{u}}_2^{(+)}-{\widehat{u}}_1^{(+)}{\parallel }_{H^1(\mathbb{R})}+{\parallel {\widehat{u}}_2^{(-)}-{\widehat{u}}_1^{(-)}\parallel }_{H^1(\mathbb{R})}\hfill \\ & & \le 2{\tilde{C}}_{\epsilon ,\nu }({\parallel u_2^{(+)}-u_1^{(+)}\parallel }_{H^1(\mathbb{R})}+\left(\parallel u_2^{(-)}-u_1^{(-)}{\parallel }_{H^1(\mathbb{R})}\right),\hfill \end{array}$$

for a suitable constant ${\tilde{C}}_{\epsilon ,\nu }>0$ independent of $U_1$ and $U_2$.

This proves the Lipschitz continuity of the map $\mathcal{L}$ on X. Thus, for any $U_0=(u_0^{(+)},u_0^{(-)})\in X={\left(H^1(\mathbb{R})\right)}^2$, there exists a unique $(u^{(+)},u^{(-)})\in C^1([0,T];X)$, satisfying (37) and Equations (38) and (39). Moreover, since $(u^{(+)},u^{(-)})\in C([0,T];X)$ and $\beta$ is Lipschitz continuous on ${\mathbb{R}}^2$ by assumption $(H_5)$, it can be easily proved that $\beta (u^{(+)},u^{(-)})\in C([0,T];L^2(\mathbb{R}))$. Hence, by standard elliptic estimates on Equation (39), we get that $v\in C([0,T];H^1(\mathbb{R}))$ (here, we have also held that $g_{\nu }^{\prime }(s)\ge \nu >0$ for all $s\in \mathbb{R}$). This concludes the proof. □

3.2. A Priori $\epsilon$-Independent Estimates for Problems $(P_{\epsilon ,\nu })$

Let us begin with the following a priori estimates.

Proposition 2.

Let $U_0=(u_0^{(+)},u_0^{(-)})\in {\left(H^1(\mathbb{R})\right)}^2$ and $(u_{\epsilon }^{(+)},u_{\epsilon }^{(-)},v_{\epsilon })$ be the solution of $(P_{\epsilon ,\nu })$. Then, there exists ${\overline{C}}_1>0$ independent of ε such that

$$\parallel u_{\epsilon }^{(\pm )}{\parallel }_{L^{\infty }(0,T;H^1(\mathbb{R}))}\le {\overline{C}}_1,$$

(43)

$$\parallel u_{\epsilon t}^{(\pm )}{\parallel }_{L^2(S)}\le {\overline{C}}_1,$$

(44)

$$\parallel \sqrt{\epsilon }{u}_{\epsilon tx}^{(\pm )}{\parallel }_{L^2(S)}\le {\overline{C}}_1.$$

(45)

Proof. 

For simplicity of notations, set $u^{(\pm )}\equiv u_{\epsilon }^{(\pm )}$ and $v\equiv v_{\epsilon }$. Fix any $\tau \in (0,T)$. Then, multiplying the first two equations in $(P_{\epsilon ,\nu })$ by $u^{(+)}$ and $u^{(-)}$, respectively, and integrating in $(0,\tau )$ yields

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{1}{2}}\left\{{\int }_{\mathbb{R}}{\left(u^{(\pm )}\right)}^2(x,\tau )dx-{\int }_{\mathbb{R}}{\left(u_0^{(\pm )}\right)}^{2}dx\right\}\hfill \\ & & +{\displaystyle \frac{\epsilon }{2}}\left\{{\int }_{\mathbb{R}}{\left(u_x^{(\pm )}\right)}^2(x,\tau )dx-{\int }_{\mathbb{R}}{\left(u_{0x}^{(\pm )}\right)}^{2}dx\right\}\hfill \\ & & +\nu {\int }_0^{\tau }{\int }_{\mathbb{R}}{\left(u_x^{(\pm )}\right)}^{2}dxdt+{\int }_0^{\tau }{\int }_{\mathbb{R}}{\left[{\phi }^{(\pm )}(u^{(\pm )})\right]}_x{u}^{(\pm )}dxdt\hfill \\ & & ={\int }_0^{\tau }{\int }_{\mathbb{R}}{u}^{(\pm )}{\mu }_{\epsilon ,\nu }^{(\pm )}(g_{\nu }(v),{\left[g_{\nu }(v)\right]}_x,u^{(+)},u^{(-)})dxdt.\hfill \end{array}$$

(46)

Since $u^{(\pm )}\in C([0,T];H^1(\mathbb{R}))$), it can be easily seen that

$${\int }_0^{\tau }{\int }_{\mathbb{R}}{\left[{\phi }^{(\pm )}(u^{(\pm )})\right]}_x{u}^{(\pm )}dxdt={\int }_0^{\tau }{\int }_{\mathbb{R}}{\left[{\psi }^{(\pm )}(u^{(\pm )})\right]}_{x}dxdt=0$$

(47)

(here, we have set ${\psi }^{(\pm )}(s)={\int }_0^s{\left({\phi }^{(\pm )}\right)}^{\prime }(\xi )\xi d\xi$ for every $s\in \mathbb{R}$). Moreover, by (29), the right-hand side of (46) can be estimated as follows:

$$\begin{array}{ccc}\hfill & & {\int }_0^{\tau }{\int }_{\mathbb{R}}{u}^{(\pm )}{\mu }_{\epsilon ,\nu }^{(\pm )}(g_{\nu }(v),{\left[g_{\nu }(v)\right]}_x,u^{(+)},u^{(-)})dxdt\hfill \\ & & \le {\displaystyle \frac{\widehat{\alpha }}{\nu }}{\int }_0^{\tau }{\int }_{\mathbb{R}}|u^{(\pm )}|\left(|T_{{\displaystyle \frac{1}{\epsilon }}}(u^{(+)})|+|T_{{\displaystyle \frac{1}{\epsilon }}}(u^{(-)})|\right)dxdt\le {\displaystyle \frac{\widehat{\alpha }}{\nu }}{\int }_0^{\tau }{\int }_{\mathbb{R}}|u^{(\pm )}|\left(|u^{(+)}|+|u^{(-)}|\right)dxdt.\hfill \end{array}$$

(48)

Combining (47) and (48) with (46) yields

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{1}{2}}{\int }_{\mathbb{R}}{\left(u^{(\pm )}\right)}^2(x,\tau )dx+{\displaystyle \frac{\epsilon }{2}}{\int }_{\mathbb{R}}{\left(u_x^{(\pm )}\right)}^2(x,\tau )dx+\nu {\int }_0^{\tau }{\int }_{\mathbb{R}}{\left(u_x^{(\pm )}\right)}^{2}dxdt\hfill \\ & & \le {\int }_{\mathbb{R}}{\left(u_0^{(\pm )}\right)}^{2}dx+{\int }_{\mathbb{R}}{\left(u_{0x}^{(\pm )}\right)}^{2}dx+{\displaystyle \frac{\widehat{\alpha }}{\nu }}{\int }_0^{\tau }{\int }_{\mathbb{R}}|u^{(\pm )}|\left(|u^{(+)}|+|u^{(-)}|\right)dxdt.\hfill \end{array}$$

(49)

Summing up the above inequalities with upper signs “+” and “−”, we get

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{1}{2}}{\int }_{\mathbb{R}}\left[{\left(u^{(+)}\right)}^2(x,\tau )+{\left(u^{(-)}\right)}^2(x,\tau )\right]dx\hfill \\ & & +{\displaystyle \frac{\epsilon }{2}}{\int }_{\mathbb{R}}\left[{\left(u_x^{(+)}\right)}^2(x,\tau )+{\left(u_x^{(-)}\right)}^2(x,\tau )\right]dx\hfill \\ & & +\nu {\int }_0^{\tau }{\int }_{\mathbb{R}}\left[{\left(u_x^{(+)}\right)}^2+{\left(u_x^{(-)}\right)}^2\right]dxdt\le {\int }_{\mathbb{R}}\left[{\left(u_0^{(+)}\right)}^2+{\left(u_0^{(-)}\right)}^2\right]dx\hfill \\ & & +{\int }_{\mathbb{R}}\left[{\left(u_{0x}^{(+)}\right)}^2+{\left(u_{0x}^{(-)}\right)}^2\right]dx+{\displaystyle \frac{\widehat{\alpha }}{\nu }}{\int }_0^{\tau }{\int }_{\mathbb{R}}{\left(|u^{(+)}|+|u^{(-)}|\right)}^{2}dxdt,\hfill \end{array}$$

(50)

whence also

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{1}{2}}{\int }_{\mathbb{R}}\left[{\left(u^{(+)}\right)}^2(x,\tau )+{\left(u^{(-)}\right)}^2(x,\tau )\right]dx\hfill \\ & & \le {\displaystyle \frac{2\widehat{\alpha }}{\nu }}{\int }_0^{\tau }{\int }_{\mathbb{R}}\left({\left(u^{(+)}\right)}^2+{\left(u^{(-)}\right)}^2\right)dxdt\hfill \\ & & +{\int }_{\mathbb{R}}\left[{\left(u_0^{(+)}\right)}^2+{\left(u_0^{(-)}\right)}^2\right]dx+{\int }_{\mathbb{R}}\left[{\left(u_{0x}^{(+)}\right)}^2+{\left(u_{0x}^{(-)}\right)}^2\right]dx.\hfill \end{array}$$

(51)

Applying Gronwall’s lemma to (51) plainly yields

$$\parallel u^{(\pm )}{\parallel }_{L^{\infty }(0,T;L^2(\mathbb{R}))}\le C_1,$$

(52)

for some $C_1>0$ independent of $\epsilon$.

Let us address (43)–(45). To this aim, using $u_t^{(\pm )}$ as test functions in the first two equalities of $(P_{\epsilon ,\nu })$ and integrating over $\mathbb{R}\times (0,\tau )$ (with $\tau \in (0,T)$), we obtain

$$\begin{array}{ccc}& & {\int }_0^{\tau }{\int }_{\mathbb{R}}{(u_t^{(\pm )})}^{2}dxdt+{\displaystyle \frac{\nu }{2}}{\int }_{\mathbb{R}}{(u_x^{(\pm )})}^2(x,\tau )dx\hfill \\ & & +\epsilon {\int }_0^{\tau }{\int }_{\mathbb{R}}{(u_{tx}^{(\pm )})}^{2}dxdt+{\int }_0^{\tau }{\int }_{\mathbb{R}}{\left[{\phi }^{(\pm )}(u^{(\pm )})\right]}_x{u}_t^{(\pm )}dxdt\hfill \\ & & ={\displaystyle \frac{\nu }{2}}{\int }_{\mathbb{R}}{(u_{0x}^{(\pm )})}^{2}dx+{\int }_0^{\tau }{\int }_{\mathbb{R}}{u}_t^{(\pm )}{\mu }_{\epsilon ,\nu }^{(\pm )}(g_{\nu }(v),{\left[g_{\nu }(v)\right]}_x,u^{(+)},u^{(-)})dxdt,\hfill \end{array}$$

whence (see (29))

$$\begin{array}{ccc}\hfill & & {\int }_0^{\tau }{\int }_{\mathbb{R}}{(u_t^{(\pm )})}^{2}dxdt+{\displaystyle \frac{\nu }{2}}{\int }_{\mathbb{R}}{(u_x^{(\pm )})}^2(x,\tau )dx+\epsilon {\int }_0^{\tau }{\int }_{\mathbb{R}}{(u_{tx}^{(\pm )})}^{2}dxdt\hfill \\ & & \le \parallel {\left({\phi }^{(\pm )}\right)}^{\prime }{\parallel }_{L^{\infty }(\mathbb{R})}{\int }_0^{\tau }{\int }_{\mathbb{R}}|u_x^{(\pm )}||u_t^{(\pm )}|dxdt+{\displaystyle \frac{\nu }{2}}{\int }_{\mathbb{R}}{(u_{0x}^{(\pm )})}^{2}dx\hfill \\ & & +{\displaystyle \frac{\widehat{\alpha }}{\nu }}{\int }_0^{\tau }{\int }_{\mathbb{R}}|u_t^{(\pm )}|\left(|u^{(+)}|+|u^{(-)}|\right)dxdt.\hfill \end{array}$$

(53)

As for the right-hand side of the above inequality, by Young’s inequality, there exist suitable constants $C_2,C_3>0$ independent of $\epsilon$ such that

$$\parallel {\left({\phi }^{(\pm )}\right)}^{\prime }{\parallel }_{L^{\infty }(\mathbb{R})}{\int }_0^{\tau }{\int }_{\mathbb{R}}|u_x^{(\pm )}||u_t^{(\pm )}|dxdt\le {\displaystyle \frac{1}{4}}{\int }_0^{\tau }{\int }_{\mathbb{R}}{(u_t^{(\pm )})}^{2}dxdt+C_2{\int }_0^{\tau }{\int }_{\mathbb{R}}{(u_x^{(\pm )})}^{2}dxdt,$$

and

$$\begin{array}{ccc}& & {\int }_0^{\tau }{\int }_{\mathbb{R}}|u_t^{(\pm )}|\left(|u^{(+)}|+|u^{(-)}|\right)dxdt\le {\displaystyle \frac{1}{4}}{\int }_0^{\tau }{\int }_{\mathbb{R}}{(u_t^{(\pm )})}^{2}dxdt\hfill \\ & & +C_3{\int }_0^{\tau }{\int }_{\mathbb{R}}\left(|u^{(+)}{|}^2+{|u^{(-)}|}^2\right)dxdt\le {\displaystyle \frac{1}{4}}{\int }_0^{\tau }{\int }_{\mathbb{R}}{(u_t^{(\pm )})}^{2}dxdt+C_4\hfill \end{array}$$

(see (52)). By the above estimates and (53), there exists $C_5>0$ independent of $\epsilon$ such that, for all $\tau \in (0,T)$, there holds

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{1}{2}}{\int }_0^{\tau }{\int }_{\mathbb{R}}{(u_t^{(\pm )})}^{2}dxdt+{\displaystyle \frac{\nu }{2}}{\int }_{\mathbb{R}}{(u_x^{(\pm )})}^2(x,\tau )dx+\epsilon {\int }_0^{\tau }{\int }_{\mathbb{R}}{(u_{tx}^{(\pm )})}^{2}dxdt\hfill \\ & & \le {\displaystyle \frac{\nu }{2}}{\int }_{\mathbb{R}}{(u_{0x}^{(\pm )})}^{2}dx+C_5\left({\int }_0^{\tau }{\int }_{\mathbb{R}}{(u_x^{(\pm )})}^{2}dxdt+1\right),\hfill \end{array}$$

(54)

whence also

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{\nu }{2}}{\int }_{\mathbb{R}}{(u_x^{(\pm )})}^2(x,\tau )dx\le {\displaystyle \frac{\nu }{2}}{\int }_{\mathbb{R}}{(u_{0x}^{(\pm )})}^{2}dx+C_5\left({\int }_0^{\tau }{\int }_{\mathbb{R}}{(u_x^{(\pm )})}^{2}dxdt+1\right).\hfill \end{array}$$

(55)

From the above estimate and Gronwall’s inequality, it follows that

$$\parallel u_x^{(\pm )}{\parallel }_{L^{\infty }(0,T;L^2(\mathbb{R}))}\le C_6,$$

(56)

for some $C_6>0$ independent of $\epsilon$. Combining (56) with (52) yields (43), whereas, from (56) and (54), both (44) and (45) follow at once. □

Proposition 3.

Let $U_0=(u_0^{(+)},u_0^{(-)})\in {\left(H^1(\mathbb{R})\right)}^2$ and $(u_{\epsilon }^{(+)},u_{\epsilon }^{(-)},v_{\epsilon })$ be the solution of $(P_{\epsilon ,\nu })$. Then, there exists ${\overline{C}}_2>0$, independent of ε, such that

$$\parallel v_{\epsilon }{\parallel }_{L^{\infty }(0,T;H^1(\mathbb{R}))}\le {\overline{C}}_2,$$

(57)

$$\parallel {\left[g_{\nu }(v_{\epsilon })\right]}_{xx}{\parallel }_{L^{\infty }(0,T;L^2(\mathbb{R}))}\le {\overline{C}}_2.$$

(58)

Proof. 

For simplicity of notations, set $v\equiv v_{\epsilon }$ and $u^{(\pm )}\equiv u_{\epsilon }^{(\pm )}$. For every $t\in (0,T)$, multiplying the elliptic equation

$$-{\left[g_{\nu }(v(·,t))\right]}_{xx}+v(·,t)=\beta (u^{(+)}(·,t),u^{(-)}(·,t))$$

(59)

by $v(·,t)$ and integrating over $\mathbb{R}$ yields (see also (26))

$$\begin{array}{ccc}& & {\int }_{\mathbb{R}}{g}^{\prime }(v(·,t)){(v_x(·,t))}^{2}dx+\nu {\int }_{\mathbb{R}}{(v_x(·,t))}^{2}dx\hfill \\ & & +{\int }_{\mathbb{R}}v(·,t)g_{\nu }(v(·,t))dx={\int }_{\mathbb{R}}\beta (u^{(+)}(·,t),u^{(-)}(·,t))v(·,t)dx.\hfill \end{array}$$

Since it can be easily seen that $\beta (u^{(+)}(·,t),u^{(-)}(·,t))$ is bounded in $L^2(\mathbb{R})$ by a constant $C_1>0$ independent of both $\epsilon$ and t (see (43) and assumption $(H_5)$), and $v(·,t)g_{\nu }(v(·,t))\ge \nu {(v(·,t))}^2$, from the above equality, we obtain

$$\nu {\int }_{\mathbb{R}}{(v_x(·,t))}^{2}dx+\nu {\int }_{\mathbb{R}}{(v(·,t))}^{2}dx\le C_1{\left({\int }_{\mathbb{R}}{(v(·,t))}^{2}dx\right)}^{{\displaystyle \frac{1}{2}}},$$

whence (57) follows at once. Plainly, (58) is a direct consequence of (57), (59), and the boundedness of the sequence $\{\beta (u_{\epsilon }^{(+)},u_{\epsilon }^{(-)}\}$ in $L^{\infty }(0,T;L^2(\mathbb{R}))$ (see (43) and $(H_5)$). □

3.3. Setting $\epsilon \to 0^{+}$ in Problems $(P_{\epsilon ,\nu })$

Fix any sequence $\left\{{\epsilon }_n\right\}\subset (0,1)$, ${\epsilon }_n\to 0^{+}$ as $n\to \infty$. Then, the following convergence results immediately follow from the a priori estimates (43)–(45).

Lemma 1.

Let $U_0=(u_0^{(+)},u_0^{(-)})\in {\left(H^1(\mathbb{R})\right)}^2$ and $(u_{{\epsilon }_n}^{(+)},u_{{\epsilon }_n}^{(-)},v_{{\epsilon }_n})$ be the solution of $(P_{{\epsilon }_n,\nu })$ with initial data $U_0$. Then, for every $\nu \in (0,1)$, there exist subsequences of $\left\{u_{{\epsilon }_n}^{(\pm )}\right\}$, not relabeled, and $u^{(\pm )}\in L^{\infty }(0,T;H^1(\mathbb{R}))\cap C([0,T];L^2(\mathbb{R}))$, with $u_t^{(\pm )}\in L^2(S)$, such that $u^{(\pm )}(·,0)=u_0^{(\pm )}$ $a.e.$ in $\mathbb{R}$, and

$$u_{{\epsilon }_n}^{(\pm )}\to u^{(\pm )}a.e. in S,$$

(60)

$$u_{{\epsilon }_n}^{(\pm )}\rightharpoonup u^{(\pm )}in L^2(S),$$

(61)

$$u_{{\epsilon }_n}^{(\pm )}\to u^{(\pm )}in C([0,T];L^2(\mathrm{\Omega }))$$

(62)

for every bounded open interval $\mathrm{\Omega }\subset \mathbb{R}$,

$$u_{{\epsilon }_{n}t}^{(\pm )}\rightharpoonup u_t^{(\pm )},u_{{\epsilon }_{n}x}^{(\pm )}\rightharpoonup u_x^{(\pm )}in L^2(S),$$

(63)

$${\epsilon }_n{u}_{{\epsilon }_{n}xt}^{(\pm )}\to 0in L^2(S).$$

(64)

Moreover, for all $t\in (0,T]$, there holds

$$u_{{\epsilon }_n}^{(\pm )}(·,t)\rightharpoonup u^{(\pm )}(·,t)in H^1(\mathbb{R}),$$

(65)

$$u_{{\epsilon }_n}^{(\pm )}(·,t)\to u^{(\pm )}(·,t)everywhere in \mathbb{R}.$$

(66)

Proof. 

The convergences in (61) and (62) follow from (43)–(45). Plainly, by the arbitrariness of $\mathrm{\Omega }$ in (62), we get (60) and the equality $u^{(\pm )}(·,0)=u_0^{(\pm )}$ $a.e.$ in $\mathbb{R}$ (recall that $u_{{\epsilon }_n}^{(\pm )}(·,0)=u_0^{(\pm )}$ $a.e.$ in $\mathbb{R}$).

Next, (65) follows from (43) and (62) (by the arbitrariness of $\mathrm{\Omega }$ in the latter). Finally, regarding (66), it suffices to observe that, by (65) for all $t\in (0,T]$, there holds $u_{{\epsilon }_n}^{(\pm )}(·,t)\to u^{(\pm )}$ in $C(\overline{\mathrm{\Omega }})$ for all bounded intervals $\mathrm{\Omega }\subset \mathbb{R}$. □

Lemma 2.

Let $U_0=(u_0^{(+)},u_0^{(-)})\in {\left(H^1(\mathbb{R})\right)}^2$. Let $\left\{(u_{{\epsilon }_n}^{(+)},u_{{\epsilon }_n}^{(-)},v_{{\epsilon }_n})\right\}$ and $u^{(\pm )}$ be the subsequences and the limiting functions given in Lemma 1. Then, for every $t\in (0,T]$, there holds

$$v_{{\epsilon }_n}(·,t)\rightharpoonup v(·,t)in{H}^1(\mathbb{R}),$$

(67)

$$v_{{\epsilon }_n}(·,t)\to v(·,t)everywhere in \mathbb{R},$$

(68)

where $v(·,t)\in H^1(\mathbb{R})$ is the unique weak solution of the elliptic equation

$$-{\left[g_{\nu }(v(·,t))\right]}_{xx}+v(·,t)=\beta (u^{(+)}(·,t),u^{(-)}(·,t))in\mathbb{R}.$$

(69)

Proof. 

To begin with, observe that uniqueness of weak solutions in $H^1(\mathbb{R})$ to (69) easily follows from standard elliptic arguments since $g_{\nu }\in C^1(\mathbb{R})$ with $g_{\nu }(0)=0$ and $g_{\nu }^{\prime }(s)\ge \nu >0$ for all $s\in \mathbb{R}$ and $\beta (u^{(+)}(·,t),u^{(-)}(·,t))\in L^2(\mathbb{R})$ (see $(H_5)$, and recall that $u^{(\pm )}(·,t)\in L^2(\mathbb{R})$ for all $t\in [0,T]$).

Next, for any fixed $t\in (0,T]$, let us take the limit as ${\epsilon }_n\to 0^{+}$ in the elliptic equation

$$-{\left[g_{\nu }(v_{{\epsilon }_n}(·,t))\right]}_{xx}+v_{{\epsilon }_n}(·,t)=\beta (u_{{\epsilon }_n}^{(+)}(·,t),u_{{\epsilon }_n}^{(-)}(·,t))\mathrm{in}\mathbb{R}.$$

(70)

To this aim, observe that, by (62) and $(H_5)$, there holds

$$\beta (u_{{\epsilon }_n}^{(+)}(·,t),u_{{\epsilon }_n}^{(-)}(·,t))\to \beta (u^{(+)}(·,t),u^{(-)}(·,t))\mathrm{in}{L}^2(\mathbb{R}).$$

(71)

Plus, by (57), there exists $w\in H^1(\mathbb{R})$ such that (possibly up to a subsequence, not relabeled) there holds

$$v_{{\epsilon }_n}(·,t)\rightharpoonup w\mathrm{in}{H}^1(\mathbb{R})\mathrm{and}\mathrm{everywhere}\mathrm{in}\mathbb{R},$$

(72)

whence also

$$g_{\nu }(v_{{\epsilon }_n}(·,t))\rightharpoonup g_{\nu }(w)\mathrm{in}{H}^1(\mathbb{R})$$

(73)

(notice that $g_{\nu }\in \mathrm{Lip}(\mathbb{R})$ and $g_{\nu }(0)=0$). In view of (71)–(73), setting ${\epsilon }_n\to 0^{+}$ in (70) proves that $w\in H^1(\mathbb{R})$ solves in the weak sense the elliptic equation $-{\left[g_{\nu }(w)\right]}_{xx}+w=\beta (u^{(+)}(·,t),u^{(-)}(·,t))$. By uniqueness of weak solutions to the above equation, this proves that $w=v(·,t)$, where $v(·,t)\in H^1(\mathbb{R})$ is the unique weak solution of (69). Thus, the convergences in (72) and (73) hold true along the whole sequence $\left\{v_{{\epsilon }_n}(·,t)\right\}$, and the conclusion follows. □

For every $\nu \in (0,1)$, set

$${\mu }_{\nu }^{(\pm )}(\eta ,\widehat{\eta },{\xi }^{(+)},{\xi }^{(-)}):={\mu }^{(\pm )}\left(T_{\frac{1}{\nu }}(\eta ),T_{\frac{1}{\nu }}(\widehat{\eta }),{\xi }^{(+)},{\xi }^{(-)}\right),$$

(74)

where $T_{\frac{1}{\nu }}$ are the functions defined in (28) (for $k=1/\nu$).

Theorem 4.

For every $U_0=(u_0^{(+)},u_0^{(-)})\in {\left(H^1(\mathbb{R})\right)}^2$ and $\nu \in (0,1)$, let $\left(u^{(+)},u^{(-)},v\right)$ be the limiting functions given in Lemmas 1 and 2. Then,

$(i)$

$u^{(\pm )}\in L^{\infty }(0,T;H^1(\mathbb{R}))\cap C([0,T];L^2(\mathbb{R}))$, $u_t\in L^2(S)$, and $v\in C([0,T];H^1(\mathbb{R}))$;

$(ii)$

for all $t\in (0,T)$, Equation (69) is satisfied in $H^{-1}(\mathbb{R})$, and the equalities

$$\begin{array}{ccc}\hfill & & \underset{S}{\int \int }{u}_t^{(\pm )}\zeta dxdt+\underset{S}{\int \int }{\left[{\phi }^{(\pm )}(u^{(\pm )})\right]}_x\zeta dxdt+\nu \underset{S}{\int \int }{u}_x^{(\pm )}{\zeta }_{x}dxdt\hfill \\ & & =\underset{S}{\int \int }{\mu }_{\nu }^{(\pm )}\left(g_{\nu }(v),{\left[g_{\nu }(v)\right]}_x,u^{(+)},u^{(-)}\right)\zeta dxdt\hfill \end{array}$$

(75)

are satisfied for all $\zeta \in L^2(0,T;H^1(\mathbb{R}))$.

Proof. 

Claim $(i)$ follows from Lemmas 1 and 2. We only point out that $v\in C([0,T];H^1(\mathbb{R}))$ by standard continuous dependence results in weak solutions to elliptic equations since $g_{\nu }^{\prime }(s)\ge \nu >0$ for all $s\in \mathbb{R}$ and $\beta (u^{(+)},u^{(-)})\in C([0,T];L^2(\mathbb{R}))$ (this follows from $(H_5)$ as $u^{(\pm )}\in C([0,T];L^2(\mathbb{R}))$).

In view of Lemma 2, claim $(ii)$ will follow if we prove that $\left(u^{(+)},u^{(-)},v\right)$ satisfies equalities (75) for every $\zeta \in L^2(0,T;H^1(\mathbb{R}))$. To this aim, let $\left\{\left(u_{{\epsilon }_n}^{(+)},u_{{\epsilon }_n}^{(-)},v_{{\epsilon }_n}\right)\right\}$ be the subsequence given in Lemmas 1 and 2. Then, choosing $\zeta$ as above as test function in (38) yields

$$\begin{array}{ccc}\hfill & & \underset{S}{\int \int }{u}_{{\epsilon }_{n}t}^{(\pm )}\zeta dxdt+\underset{S}{\int \int }{\left[{\phi }^{(\pm )}(u_{{\epsilon }_n}^{(\pm )})\right]}_x\zeta dxdt+\nu \underset{S}{\int \int }{u}_{{\epsilon }_{n}x}^{(\pm )}{\zeta }_{x}dxdt\hfill \\ & & +{\epsilon }_n\underset{S}{\int \int }{u}_{{\epsilon }_{n}tx}^{(\pm )}{\zeta }_{x}dxdt=\underset{S}{\int \int }{\mu }_{{\epsilon }_n,\nu }^{(\pm )}\left(g_{\nu }(v_{{\epsilon }_n}),{\left[g_{\nu }(v_{{\epsilon }_n})\right]}_x,u_{{\epsilon }_n}^{(+)},u_{{\epsilon }_n}^{(-)}\right)\zeta dxdt.\hfill \end{array}$$

(76)

Let us consider the limit as ${\epsilon }_n\to 0^{+}$ on the left-hand side of the above equality. To this purpose, observe that, by (43), the sequences $\left\{{\left[{\phi }^{(\pm )}(u_{{\epsilon }_n}^{(\pm )})\right]}_x\right\}$ are bounded in $L^{\infty }(0,T;L^2(\mathbb{R}))$ (recall that ${\phi }^{(\pm )}\in \mathrm{Lip}(\mathbb{R})$; see $(H_3)$). Since there also holds ${\phi }^{(\pm )}(u_{{\epsilon }_n}^{(\pm )})\to {\phi }^{(\pm )}(u^{(\pm )})$ $a.e.$ in S and in $L^2(S)$ (see (60) and (61)), a routine proof shows that

$${\left[{\phi }^{(\pm )}(u_{{\epsilon }_n}^{(\pm )})\right]}_x\rightharpoonup {\left[{\phi }^{(\pm )}(u^{(\pm )})\right]}_x\mathrm{in}{L}^2(S).$$

(77)

From the previous convergence and (63) and (64), we get

$$\begin{array}{ccc}\hfill & & \underset{{\epsilon }_n\to 0^{+}}{lim}\left({\int \int }_S{u}_{{\epsilon }_{n}t}^{(\pm )}\zeta dxdt+{\int \int }_S{\left[{\phi }^{(\pm )}(u_{{\epsilon }_n}^{(\pm )})\right]}_x\zeta dxdt+\nu {\int \int }_S{u}_{{\epsilon }_{n}x}^{(\pm )}{\zeta }_{x}dxdt\right.\hfill \\ & & \left.+{\epsilon }_n{\int \int }_S{u}_{{\epsilon }_{n}tx}^{(\pm )}{\zeta }_{x}dxdt\right)={\int \int }_S{u}_t^{(\pm )}\zeta dxdt+{\int \int }_S{\left[{\phi }^{(\pm )}(u^{(\pm )})\right]}_x\zeta dxdt+\nu {\int \int }_S{u}_x^{(\pm )}{\zeta }_{x}dxdt\hfill \end{array}$$

(78)

for all $\zeta \in L^2(0,T;H^1(\mathbb{R}))$. Regarding the right-hand side of (76), observe that, by (29), there holds $\left|{\mu }_{{\epsilon }_n,\nu }^{(\pm )}\left(g_{\nu }(v_{{\epsilon }_n}),{\left[g_{\nu }(v_{{\epsilon }_n})\right]}_x,u_{{\epsilon }_n}^{(+)},u_{{\epsilon }_n}^{(-)}\right)\right|\le {\displaystyle \frac{\widehat{\alpha }}{\nu }}\left(|u_{{\epsilon }_n}^{(+)}|+|u_{{\epsilon }_n}^{(-)}|\right)$, whence (see also (43))

$$\left\{{\mu }_{{\epsilon }_n,\nu }^{(\pm )}\left(g_{\nu }(v_{{\epsilon }_n}),{\left[g_{\nu }(v_{{\epsilon }_n})\right]}_x,u_{{\epsilon }_n}^{(+)},u_{{\epsilon }_n}^{(-)}\right)\right\} \mathrm{is} \mathrm{bounded} \mathrm{in} L^{\infty }(0,T;L^2(\mathbb{R})).$$

(79)

Moreover, from (58) and (73), it follows that, for all $t\in (0,T)$, there holds ${\left[g_{\nu }(v(·,t))\right]}_{xx}\in L^2(\mathbb{R})$ and ${\left[g_{\nu }(v_{{\epsilon }_n}(·,t))\right]}_x\rightharpoonup {\left[g_{\nu }(v(·,t))\right]}_x$ in $H^1(\mathbb{R})$. From this convergence and (58), we get that ${\left[g_{\nu }(v_{{\epsilon }_n}(·,t))\right]}_x\to {\left[g_{\nu }(v(·,t))\right]}_x$ in $C(\overline{\mathrm{\Omega }})$ for every bounded interval $\mathrm{\Omega }\subset \mathbb{R}$ and for all $t\in (0,T)$. Plainly, by the arbitrariness of $\mathrm{\Omega }$, we have

$${\left[g_{\nu }(v_{{\epsilon }_n})\right]}_x\to {\left[g_{\nu }(v)\right]}_x\mathrm{everywhere} \mathrm{in} S.$$

(80)

Similarly, by (68), there holds

$$g_{\nu }(v_{{\epsilon }_n})\to g_{\nu }(v)\mathrm{everywhere} \mathrm{in} S.$$

(81)

From (60), (80) and (81), it is easily seen that, as ${\epsilon }_n\to 0^{+}$, there holds (see also (27))

$${\mu }_{{\epsilon }_n,\nu }^{(\pm )}\left(g_{\nu }(v_{{\epsilon }_n}),{\left[g_{\nu }(v_{{\epsilon }_n})\right]}_x,u_{{\epsilon }_n}^{(+)},u_{{\epsilon }_n}^{(-)}\right)\to {\mu }_{\nu }^{(\pm )}\left(g_{\nu }(v),{\left[g_{\nu }(v)\right]}_x,u^{(+)},u^{(-)}\right)a.e. \mathrm{in} S.$$

In view of the above convergence and (79), a routine proof shows that

$${\mu }_{{\epsilon }_n,\nu }^{(\pm )}\left(g_{\nu }(v_{{\epsilon }_n}),{\left[g_{\nu }(v_{{\epsilon }_n})\right]}_x,u_{{\epsilon }_n}^{(+)},u_{{\epsilon }_n}^{(-)}\right)\rightharpoonup {\mu }_{\nu }^{(\pm )}\left(g_{\nu }(v),{\left[g_{\nu }(v)\right]}_x,u^{(+)},u^{(-)}\right)\mathrm{in} L^2(S),$$

(82)

whence

$$\underset{{\epsilon }_n\to 0^{+}}{lim}\underset{S}{\int \int }{\mu }_{{\epsilon }_n,\nu }^{(\pm )}\left(g_{\nu }(v_{{\epsilon }_n}),{\left[g_{\nu }(v_{{\epsilon }_n})\right]}_x,u_{{\epsilon }_n}^{(+)},u_{{\epsilon }_n}^{(-)}\right)\zeta dxdt=\underset{S}{\int \int }{\mu }_{\nu }^{(\pm )}\left(g_{\nu }(v),{\left[g_{\nu }(v)\right]}_x,u^{(+)},u^{(-)}\right)\zeta dxdt$$

for all $\zeta \in L^2(0,T;H^1(\mathbb{R}))$. Plainly, by the above convergence and (78), setting ${\epsilon }_n\to 0^{+}$ in (76) yields (75). □

4. The Approximating Problems for $\mathit{\epsilon }=\mathbf{0}$ and $\mathit{\nu }\in (\mathbf{0},\mathbf{1})$

4.1. A Priori Estimates

Given any $U_0=(u_0^{(+)},u_0^{(-)})\in {\left(L^1(\mathbb{R})\right)}^2$, let $\left\{u_{0,\nu }^{(\pm )}\right\}\subseteq H^1(\mathbb{R})\cap L^1(\mathbb{R})$ ($\nu \in (0,1)$) be chosen so that

$$\parallel u_{0,\nu }^{(\pm )}{\parallel }_{L^1(\mathbb{R})}\le {\parallel u_0^{(\pm )}\parallel }_{L^1(\mathbb{R})},$$

(83)

$$u_{0,\nu }^{(\pm )}\to u_0^{(\pm )}\mathrm{in} L^1(\mathbb{R}) \mathrm{and} a.e. \mathrm{in} \mathbb{R} \mathrm{as} \nu \to 0^{+}.$$

(84)

For every $\nu \in (0,1)$, we shall consider the following system:

$$\left\{\begin{array}{cc}{u}_t^{(+)}+{\left[{\phi }^{(+)}(u^{(+)})\right]}_x-\nu {\left[u^{(+)}\right]}_{xx}={\mu }_{\nu }^{(+)}(g_{\nu }(v),{\left[g_{\nu }(v)\right]}_x,u^{(+)},u^{(-)})\hfill & \mathrm{in}S,\hfill \\ \\ u_t^{(-)}+{\left[{\phi }^{(-)}(u^{(-)})\right]}_x-\nu {\left[u^{(-)}\right]}_{xx}={\mu }_{\nu }^{(-)}(g_{\nu }(v),{\left[g_{\nu }(v)\right]}_x,u^{(+)},u^{(-)})\hfill & \mathrm{in}S,\hfill \\ \\ -{\left[g_{\nu }(v)\right]}_{xx}+v=\beta (u^{(+)},u^{(-)})\hfill & \mathrm{in}S,\hfill \end{array}\right.$$

(85)

with initial condition

$$u_{\nu }^{(\pm )}(·,0)=u_{0,\nu }^{(\pm )}\mathrm{in}\mathbb{R}.$$

(86)

Here, $g_{\nu }$ is defined in (26), and ${\mu }_{\nu }^{(\pm )}$ are the functions in (74).

Plainly, existence of solutions to (85) and (86), in the sense of the following definition, is ensured by Theorem 4.

Definition 3.

For every $\nu \in (0,1)$, by a solution to (85) and (86), we mean a triplet $\left(u_{\nu }^{(+)},u_{\nu }^{(-)},v_{\nu }\right)$ such that

$(i)$

 $u_{\nu }^{(\pm )}\in L^{\infty }(0,T;H^1(\mathbb{R}))\cap C([0,T];L^2(\mathbb{R}))$, $u_{\nu t}^{(\pm )}\in L^2(S)$, and $v_{\nu }\in C([0,T];H^1(\mathbb{R}))$.

$(ii)$

 Equalities (75) are satisfied for all $\zeta \in L^2(0,T;H^1(\mathbb{R}))$.

$(iii)$

 For all $t\in (0,T)$, Equations (69) are satisfied in $H^{-1}(\mathbb{R})$.

Remark 4.

It is worth considering some direct consequences of inequalities (9) and (10). Specifically, from (9) and the very definition of ${\mu }_{\nu }^{(\pm )}$, it follows that

$$|{\mu }_{\nu }^{(\pm )}(\eta ,\widehat{\eta },{\xi }^{(+)},{\xi }^{(-)})|\le {\displaystyle \frac{\widehat{\alpha }}{\nu }}\left(|{\xi }^{(+)}|+|{\xi }^{(-)}|\right)$$

(87)

for all $(\eta ,\widehat{\eta },{\xi }^{(+)},{\xi }^{(-)})\in {\mathbb{R}}^4$. Similarly, by (10) for all $({\eta }_i,{\widehat{\eta }}_i,{\xi }_i^{(+)},{\xi }_i^{(-)})\in {\mathbb{R}}^4$ ($i=1,2$), there holds

$$\begin{array}{ccc}\hfill & & |{\mu }_{\nu }^{(\pm )}({\eta }_2,{\widehat{\eta }}_2,{\xi }_2^{(+)},{\xi }_2^{(-)})-{\mu }_{\nu }^{(\pm )}({\eta }_1,{\widehat{\eta }}_1,{\xi }_1^{(+)},{\xi }_1^{(-)})|\hfill \\ & & \le {\displaystyle \frac{\widehat{\alpha }}{\nu }}\left(\sum _{i=1}^2|{\xi }_i^{(+)}|+|{\xi }_i^{(-)}|\right)\left(|T_{\frac{1}{\nu }}({\eta }_2)-T_{\frac{1}{\nu }}({\eta }_1)|+|T_{\frac{1}{\nu }}({\widehat{\eta }}_2)-T_{\frac{1}{\nu }}({\widehat{\eta }}_1)|\right)\hfill \\ & & +{\displaystyle \frac{\widehat{\alpha }}{\nu }}\left(|{\xi }_2^{(+)}-{\xi }_1^{(+)}|+|{\xi }_2^{(-)}-{\xi }_1^{(-)}|\right)\hfill \\ & & \le {\displaystyle \frac{\widehat{\alpha }}{\nu }}\left(\sum _{i=1}^2|{\xi }_i^{(+)}|+|{\xi }_i^{(-)}|\right)\left(|{\eta }_2-{\eta }_1|+|{\widehat{\eta }}_2-{\widehat{\eta }}_1|\right)+{\displaystyle \frac{\widehat{\alpha }}{\nu }}\left(|{\xi }_2^{(+)}-{\xi }_1^{(+)}|+|{\xi }_2^{(-)}-{\xi }_1^{(-)}|\right).\hfill \end{array}$$

(88)

We begin with the following lemma.

Lemma 3.

For every $\nu \in (0,1)$, let $(u_{\nu }^{(+)},u_{\nu }^{(-)},v_{\nu })$ be a solution of (85) and (86).

$(i)$

 For every $\tau \in (0,T)$ and $k\in \mathbb{R}$, the entropy inequalities

$$\begin{array}{ccc}\hfill & & {\int }_{\mathbb{R}}|u_{\nu }^{(\pm )}(x,\tau )-k|\zeta (x,\tau )dx-{\int }_{\mathbb{R}}|u_{0,\nu }^{(\pm )}-k|\zeta (x,0)dx\hfill \\ & & \le \nu {\int }_0^{\tau }{\int }_{\mathbb{R}}|u_{\nu }^{(\pm )}-k|{\zeta }_{xx}dxdt+{\int }_0^{\tau }{\int }_{\mathbb{R}}|u_{\nu }^{(\pm )}(x,\tau )-k|{\zeta }_{t}dxdt\hfill \\ & & +{\int }_0^{\tau }{\int }_{\mathbb{R}}\mathrm{sgn}(u_{\nu }^{(\pm )}-k)({\phi }^{(\pm )}(u_{\nu }^{(\pm )})-{\phi }^{(\pm )}(k)){\zeta }_{x}dxdt\hfill \\ & & +{\int }_0^{\tau }{\int }_{\mathbb{R}}\mathrm{sgn}(u_{\nu }^{(\pm )}-k)\left[{\mu }_{\nu }^{(\pm )}(g_{\nu }(v_{\nu }),{\left[g_{\nu }(v_{\nu })\right]}_x,u_{\nu }^{(+)},u_{\nu }^{(-)})\right]\zeta dxdt\hfill \end{array}$$

(89)

are satisfied for all $\zeta \in C_c^2(\overline{S})$, $\zeta \ge 0$.

$(ii)$

 $u_{\nu }^{(\pm )}\in L^{\infty }(0,T;L^1(\mathbb{R}))$, and, for every $\tau \in (0,T)$, there holds

$$\begin{array}{ccc}\hfill & & {\int }_{\mathbb{R}}|u_{\nu }^{(\pm )}(x,\tau )|dx\le {\int }_{\mathbb{R}}|u_0^{(\pm )}|dx+{\int }_0^{\tau }{\int }_{\mathbb{R}}|{{\mu }^{(\pm )}}_{\nu }(g_{\nu }(v_{\nu }),{\left[g_{\nu }(v_{\nu })\right]}_x,u_{\nu }^{(+)},u_{\nu }^{(-)})|dxdt.\hfill \end{array}$$

(90)

Proof. 

$(i)$ For every $j\in \mathbb{N}$, set

$$H_j(s):={\int }_0^s{h}_j(y)dy,h_j(y)=j{T}_{\frac{1}{j}}(s)(s,y\in \mathbb{R})$$

(here, $T_{\frac{1}{j}}(s)$ is the truncation function $T_k(s)$ with $k=\frac{1}{j}$). Let $\zeta \in C_c^2(\overline{S})$, $\zeta \ge 0$ be fixed arbitrarily. Multiplying the first two equations in (85) by $h_j(u_{\nu }^{(\pm )}-k)\zeta$, respectively, and integrating in $\mathbb{R}\times (0,\tau )$ yields

$$\begin{array}{ccc}& & {\int }_{\mathbb{R}}{H}_j(u_{\nu }^{(\pm )}(x,\tau )-k)\zeta (x,\tau )dx-{\int }_{\mathbb{R}}{H}_j(u_{0,\nu }^{(\pm )}-k)\zeta (x,0)dx\hfill \\ & & +\nu {\int }_0^{\tau }{\int }_{\mathbb{R}}{h}_j^{\prime }(u_{\nu }^{(\pm )}-k){\left(u_{\nu x}^{(\pm )}\right)}^2\zeta dxdt={\int }_0^{\tau }{\int }_{\mathbb{R}}{H}_j(u_{\nu }^{(\pm )}-k){\zeta }_{t}dxdt\hfill \\ & & +{\int }_0^{\tau }{\int }_{\mathbb{R}}\left({\int }_0^{u_{\nu }^{(\pm )}(x,t)-k}{\left({\phi }^{(\pm )}\right)}^{\prime }(s+k)h_j(s)ds\right){\zeta }_{x}dxdt+\nu {\int }_0^{\tau }{\int }_{\mathbb{R}}{H}_j(u_{\nu }^{(\pm )}-k){\zeta }_{xx}dxdt\hfill \\ & & +{\int }_0^{\tau }{\int }_{\mathbb{R}}{h}_j(u_{\nu }^{(\pm )}-k){\mu }_{\nu }^{(\pm )}(g_{\nu }(v_{\nu }),{\left[g_{\nu }(v_{\nu })\right]}_x,u_{\nu }^{(+)},u_{\nu }^{(-)})\zeta ,dxdt,\hfill \end{array}$$

whence

$$\begin{array}{ccc}\hfill & & {\int }_{\mathbb{R}}{H}_j(u_{\nu }^{(\pm )}(x,\tau )-k)\zeta (x,\tau )dx-{\int }_{\mathbb{R}}{H}_j(u_{0,\nu }^{(\pm )}-k)\zeta (x,0)dx\hfill \\ & & \le {\int }_0^{\tau }{\int }_{\mathbb{R}}{H}_j(u_{\nu }^{(\pm )}-k){\zeta }_{t}dxdt+\nu {\int }_0^{\tau }{\int }_{\mathbb{R}}{H}_j(u_{\nu }^{(\pm )}-k){\zeta }_{xx}dxdt\hfill \\ & & +{\int }_0^{\tau }{\int }_{\mathbb{R}}\left({\int }_0^{u_{\nu }^{(\pm )}(x,t)-k}{\left({\phi }^{(\pm )}\right)}^{\prime }(s+k)h_j(s)ds\right){\zeta }_{x}dxdt\hfill \\ & & +{\int }_0^{\tau }{\int }_{\mathbb{R}}{h}_j(u_{\nu }^{(\pm )}-k)\left[{\mu }_{\nu }^{(\pm )}(g_{\nu }(v_{\nu }),{\left[g_{\nu }(v_{\nu })\right]}_x,u_{\nu }^{(+)},u_{\nu }^{(-)})\right]\zeta ,dxdt.\hfill \end{array}$$

(91)

Since $h_j(s)\to \mathrm{sgn}(s)$ and $H_j(s)\to |s|$ as $j\to \infty$ ($s\in \mathbb{R}$), a routine proof shows that inequality (89) follows from setting $j\to \infty$ in (91).

$(ii)$

 Let $\rho \in C_c^1((-2,2))$, $0\le \rho \le 1$ be any function such that $\rho (x)=1$ for $x\in [-1,1]$. For every $n\in \mathbb{N}$ and $x\in \mathbb{R}$, set ${\rho }_n(x):=\rho \left({\displaystyle \frac{x}{n}}\right)$. Choosing $\zeta (x,t)={\rho }_n(x)$ and $k=0$ in (89) yields (recall that ${\phi }^{(\pm )}(0)=0$; see $(H_3)$)

$$\begin{array}{ccc}\hfill & & {\int }_{\mathbb{R}}|u_{\nu }^{(\pm )}(x,\tau )|{\rho }_n(x)dx-{\int }_{\mathbb{R}}|u_{0,\nu }^{(\pm )}|{\rho }_n(x)dx\hfill \\ & & \le {\displaystyle \frac{\nu }{n^2}}{\int }_0^{\tau }{\int }_{\mathbb{R}}|u_{\nu }^{(\pm )}|{\rho }^{″}\left({\displaystyle \frac{x}{n}}\right)dxdt+{\displaystyle \frac{1}{n}}{\int }_0^{\tau }{\int }_{\mathbb{R}}|{\phi }^{(\pm )}(u_{\nu }^{(\pm )})||{\rho }^{\prime }|\left({\displaystyle \frac{x}{n}}\right)dxdt\hfill \\ & & +{\int }_0^{\tau }{\int }_{\mathbb{R}}|{\mu }_{\nu }^{(\pm )}(g_{\nu }(v_{\nu }),{\left[g_{\nu }(v_{\nu })\right]}_x,u_{\nu }^{(+)},u_{\nu }^{(-)})|{\rho }_n(x)dxdt,\hfill \end{array}$$

(92)

whence (see also (87))

$$\begin{array}{ccc}& & {\int }_{\mathbb{R}}|u_{\nu }^{(\pm )}(x,\tau )|{\rho }_n(x)dx-{\int }_{\mathbb{R}}|u_{0,\nu }^{(\pm )}|{\rho }_n(x)dx\hfill \\ & & \le {\displaystyle \frac{\nu }{n^2}}{\int }_0^{\tau }{\int }_{\mathbb{R}}|u_{\nu }^{(\pm )}||{\rho }^{″}|\left({\displaystyle \frac{x}{n}}\right)dxdt+{\displaystyle \frac{1}{n}}{\int }_0^{\tau }{\int }_{\mathbb{R}}|{\phi }^{(\pm )}(u_{\nu }^{(\pm )})||{\rho }^{\prime }|\left({\displaystyle \frac{x}{n}}\right)dxdt\hfill \\ & & +{\displaystyle \frac{\widehat{\alpha }}{\nu }}{\int }_0^{\tau }{\int }_{\mathbb{R}}\left(|u_{\nu }^{(+)}|+|u_{\nu }^{(-)})|\right){\rho }_n(x)dxdt.\hfill \end{array}$$

Summing up the above inequalities with upper signs “+” and “−”, we get

$$\begin{array}{ccc}& & {\int }_{\mathbb{R}}\left(|u_{\nu }^{(+)}(x,\tau )|+|u_{\nu }^{(-)}(x,\tau )|\right){\rho }_n(x)dx\le {\int }_{\mathbb{R}}\left(|u_{0,\nu }^{(+)}|+|u_{0,\nu }^{(-)}|\right){\rho }_n(x)dx\hfill \\ & & +{\displaystyle \frac{\nu }{n^2}}{\int }_0^{\tau }{\int }_{\mathbb{R}}\left(|u_{\nu }^{(+)}|+|u_{\nu }^{(-)}|\right)|{\rho }^{″}|\left({\displaystyle \frac{x}{n}}\right)dxdt\hfill \\ & & +{\displaystyle \frac{1}{n}}{\int }_0^{\tau }{\int }_{\mathbb{R}}\left(|{\phi }^{(+)}(u_{\nu }^{(+)})|+|{\phi }^{(-)}(u_{\nu }^{(-)})|\right)|{\rho }^{\prime }|\left({\displaystyle \frac{x}{n}}\right)dxdt\hfill \\ & & +{\displaystyle \frac{2\widehat{\alpha }}{\nu }}{\int }_0^{\tau }{\int }_{\mathbb{R}}\left(|u_{\nu }^{(+)}|+|u_{\nu }^{(-)})|\right){\rho }_n(x)dxdt.\hfill \end{array}$$

Thus, by Gronwall’s inequality, there holds

$$\begin{array}{ccc}\hfill & & {\int }_{\mathbb{R}}\left(|u_{\nu }^{(+)}(x,\tau )|+|u_{\nu }^{(-)}(x,\tau )|\right){\rho }_n(x)dx\le e^{{\displaystyle \frac{2\tau \widehat{\alpha }}{\nu }}}\left({\displaystyle \frac{\nu }{n^2}}{\int }_0^{\tau }{\int }_{\mathbb{R}}\left(|u_{\nu }^{(+)}|+|u_{\nu }^{(-)}|\right)|{\rho }^{″}|\left({\displaystyle \frac{x}{n}}\right)dxdt\right.\hfill \\ & & \left.+{\int }_{\mathbb{R}}\left(|u_{0,\nu }^{(+)}|+|u_{0,\nu }^{(-)}|\right){\rho }_n(x)dx+{\displaystyle \frac{1}{n}}{\int }_0^{\tau }{\int }_{\mathbb{R}}\left(|{\phi }^{(+)}(u_{\nu }^{(+)})|+|{\phi }^{(-)}(u_{\nu }^{(-)})|\right)|{\rho }^{\prime }|\left({\displaystyle \frac{x}{n}}\right)dxdt\right)\hfill \end{array}$$

(93)

for all $\tau \in (0,T)$. Since $|{\phi }^{(\pm )}(u^{(\pm )})|\le L_{{\phi }^{(\pm )}}|u^{(\pm )}|$ (see $(H_3)$) and $u^{(\pm )}\in L^{\infty }(0,T;H^1(\mathbb{R}))$, it can be easily seen that

$$\begin{array}{ccc}& & \underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}{\int }_0^{\tau }{\int }_{\mathbb{R}}\left(|{\phi }^{(+)}(u_{\nu }^{(+)})|+|{\phi }^{(-)}(u_{\nu }^{(-)})|\right)|{\rho }^{\prime }|\left({\displaystyle \frac{x}{n}}\right)dxdt\hfill \\ & & =\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}{\int }_0^{\tau }{\int }_{\{n\le |x|\le 2n\}}\left(|{\phi }^{(+)}(u_{\nu }^{(+)})|+|{\phi }^{(-)}(u_{\nu }^{(-)})|\right)|{\rho }^{\prime }|\left({\displaystyle \frac{x}{n}}\right)dxdt=0,\hfill \end{array}$$

and

$$\underset{n\to \infty }{lim}{\displaystyle \frac{\nu }{n^2}}{\int }_0^{\tau }{\int }_{\mathbb{R}}\left(|u_{\nu }^{(+)}|+|u_{\nu }^{(-)}|\right)|{\rho }^{″}|\left({\displaystyle \frac{x}{n}}\right)dxdt=\underset{n\to \infty }{lim}{\displaystyle \frac{\nu }{n^2}}{\int }_0^{\tau }{\int }_{\{n\le |x|\le 2n\}}\left(|u_{\nu }^{(+)}|+|u_{\nu }^{(-)}|\right)|{\rho }^{″}|\left({\displaystyle \frac{x}{n}}\right)dxdt=0.$$

In view of the previous convergences, since ${\rho }_n(x)\to 1$ for all $x\in \mathbb{R}$ and $u_{0,\nu }^{(\pm )}\in L^1(\mathbb{R})$, setting $n\to \infty$ in (93), it easily follows that $u^{(\pm )}\in L^{\infty }(0,T;L^1(\mathbb{R}))$. Moreover, since

$$|{\mu }_{\nu }^{(\pm )}(g_{\nu }(v_{\nu }),{\left[g_{\nu }(v_{\nu })\right]}_x,u_{\nu }^{(+)},u_{\nu }^{(-)})|\le {\displaystyle \frac{\widehat{\alpha }}{\nu }}\left(|u_{\nu }^{(+)}|+|u_{\nu }^{(-)}|\right)$$

(see (87)), it can be easily seen that the functions ${\mu }_{\nu }^{(\pm )}(g_{\nu }(v_{\nu }),{\left[g_{\nu }(v_{\nu })\right]}_x,u_{\nu }^{(+)},u_{\nu }^{(-)})$ belong to $L^{\infty }(0,T;L^1(\mathbb{R}))$ as well. Therefore, inequality (90) can be easily obtained by setting $n\to \infty$ in (92) and using (83). □

Lemma 4.

For every $\nu \in (0,1)$, let $(u_{\nu }^{(+)},u_{\nu }^{(-)},v_{\nu })$ be a solution of (85) and (86). Then, for every $\tau \in (0,T]$, there holds

$${∥{\left[g_{\nu }(v_{\nu }(·,\tau ))\right]}_{xx}∥}_{L^1(\mathbb{R})}\le 2L_{\beta }\left(\parallel u_{\nu }^{(+)}{(·,\tau )\parallel }_{L^1(\mathbb{R})}+{\parallel u_{\nu }^{(-)}(·,\tau )\parallel }_{L^1(\mathbb{R})}\right),$$

(94)

$$\parallel v_{\nu }{(·,\tau )\parallel }_{L^1(\mathbb{R})}\le L_{\beta }\left(\parallel u_{\nu }^{(+)}{(·,\tau )\parallel }_{L^1(\mathbb{R})}+{\parallel u_{\nu }^{(-)}(·,\tau )\parallel }_{L^1(\mathbb{R})}\right),$$

(95)

$${∥{\left[g_{\nu }(v_{\nu }(·,\tau ))\right]}_x∥}_{L^{\infty }(\mathbb{R})}\le 2L_{\beta }\left(\parallel u_{\nu }^{(+)}{(·,\tau )\parallel }_{L^1(\mathbb{R})}+{\parallel u_{\nu }^{(-)}(·,\tau )\parallel }_{L^1(\mathbb{R})}\right),$$

(96)

$${∥g_{\nu }(v_{\nu }(·,\tau ))∥}_{L^{\infty }(\mathbb{R})}\le {\widehat{C}}_1\left(\parallel u_{\nu }^{(+)}{(·,\tau )\parallel }_{L^1(\mathbb{R})}+{\parallel u_{\nu }^{(-)}(·,\tau )\parallel }_{L^1(\mathbb{R})}\right),$$

(97)

$${∥g_{\nu }(v_{\nu }(·,\tau ))∥}_{H^1(\mathbb{R})}\le {\widehat{C}}_1\left(\parallel u_{\nu }^{(+)}{(·,\tau )\parallel }_{L^1(\mathbb{R})}+{\parallel u_{\nu }^{(-)}(·,\tau )\parallel }_{L^1(\mathbb{R})}\right),$$

(98)

where ${\widehat{C}}_1$ is defined in (24).

Proof. 

For simplicity of notations, set $v\equiv v_{\nu }(·,\tau )$ and $u^{(\pm )}\equiv u_{\nu }^{(\pm )}(·,\tau )$ for $\tau \in (0,T]$ arbitrarily fixed. For every $K>0$, set

$$h_K(s)={\displaystyle \frac{1}{K}}{T}_K(s),$$

(99)

where $T_K(s)$ is the usual truncation function ($s\in \mathbb{R}$). Choosing $h_K\left(g_{\nu }(v)\right)$ as test function in the elliptic equation

$$-{\left[g_{\nu }(v)\right]}_{xx}+v=\beta (u^{(+)},u^{(-)}),$$

(100)

we get

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{K}}{\int }_{\{|g_{\nu }(v)|<K\}}{\left[g_{\nu }(v)\right]}_x^{2}dx+{\displaystyle \frac{1}{K}}{\int }_{\mathbb{R}}v{h}_K(g_{\nu }(v))dx\hfill \\ & & ={\int }_{\mathbb{R}}\beta (u^{(+)},u^{(-)})h_K(g_{\nu }(v))dx\le L_{\beta }\left(\parallel u^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel u^{(-)}\parallel }_{L^1(\mathbb{R})}\right),\hfill \end{array}$$

where $L_{\beta }>0$ is the Lipschitz constant of $\beta$ (see $(H_5)$). Since $h_K(g_{\nu }(v))v\to |v|$ as $K\to 0^{+}$, setting $K\to 0^{+}$ in the above inequality, we get (95). Plainly, (94) immediately follows from (95) and Equation (100), whereas (96) is a direct consequence of (94). Next, multiplying (100) by $g_{\nu }(v)$ yields

$$\begin{array}{ccc}\hfill & & {\int }_{\mathbb{R}}{\left[g_{\nu }(v)\right]}_x^{2}dx+{\int }_{\mathbb{R}}v{g}_{\nu }(v)dx={\int }_{\mathbb{R}}\beta (u^{(+)},u^{(-)})g_{\nu }(v)dx\hfill \\ & & \le L_{\beta }{\parallel g_{\nu }(v)\parallel }_{L^{\infty }(\mathbb{R})}\left(\parallel u^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel u^{(-)}\parallel }_{L^1(\mathbb{R})}\right).\hfill \end{array}$$

(101)

Since $\parallel g_{\nu }{(v)\parallel }_{L^{\infty }(\mathbb{R})}\le \sqrt{2}{\parallel g_{\nu }(v)\parallel }_{H^1(\mathbb{R})}$ by general inequalities in Sobolev spaces [35], and

$${\int }_{\mathbb{R}}v{g}_{\nu }(v)\ge {\displaystyle \frac{1}{L_g+\nu }}{\int }_{\mathbb{R}}{(g_{\nu }(v))}^{2}dx\ge {\displaystyle \frac{1}{L_g+1}}{\int }_{\mathbb{R}}{(g_{\nu }(v))}^{2}dx$$

(recall that $0<\nu <1$ and $L_g>0$ is the Lipschitz constant of g; see $(H_4)$), by (101), we have

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{L_g+1}}{\parallel g_{\nu }(v)\parallel }_{H^1(\mathbb{R})}^2={\displaystyle \frac{1}{L_g+1}}\left\{{\int }_{\mathbb{R}}{\left[g_{\nu }(v)\right]}_x^{2}dx+{\int }_{\mathbb{R}}{\left(g_{\nu }(v)\right)}^{2}dx\right\}\hfill \\ & & \le \sqrt{2}{L}_{\beta }{\parallel g_{\nu }(v)\parallel }_{H^1(\mathbb{R})}\left(\parallel u^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel u^{(-)}\parallel }_{L^1(\mathbb{R})}\right),\hfill \end{array}$$

whence

$$\parallel g_{\nu }{(v)\parallel }_{H^1(\mathbb{R})}\le \sqrt{2}(L_g+1)L_{\beta }\left(\parallel u^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel u^{(-)}\parallel }_{L^1(\mathbb{R})}\right),$$

(102)

$$\parallel g_{\nu }{(v)\parallel }_{L^{\infty }(\mathbb{R})}\le \sqrt{2}{\parallel g_{\nu }(v)\parallel }_{H^1(\mathbb{R})}\le 2(L_g+1)L_{\beta }\left(\parallel u^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel u^{(-)}\parallel }_{L^1(\mathbb{R})}\right).$$

(103)

Plainly, from (102) and (103), inequalities (97) and (98) follow at once. □

Proposition 4.

For every $\nu \in (0,1)$, let $(u_{\nu }^{(+)},u_{\nu }^{(-)},v_{\nu })$ be a solution of (85) and (86). Then, for all $0<\tau <T_0$, there holds

$$\begin{array}{ccc}\hfill & & \parallel u_{\nu }^{(+)}{(·,\tau )\parallel }_{L^1(\mathbb{R})}+{\parallel u_{\nu }^{(-)}(·,\tau )\parallel }_{L^1(\mathbb{R})}\hfill \\ & & \le {\displaystyle \frac{\parallel u_0^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel u_0^{(-)}\parallel }_{L^1(\mathbb{R})}}{1-2\tau \widehat{\alpha }{\widehat{C}}_1\left(\parallel u_0^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel u_0^{(-)}\parallel }_{L^1(\mathbb{R})}\right)}},\hfill \end{array}$$

(104)

where $\widehat{\alpha }>0$ is the constant in assumption $(H_2)$, and $T_0$ is defined in (23).

Proof. 

By (96) and (97), for every $t\in (0,T)$, we have

$$\parallel g_{\nu }{(v(·,t))\parallel }_{L^{\infty }(\mathbb{R})},{\parallel {\left[g_{\nu }(v(·,t))\right]}_x\parallel }_{L^{\infty }(\mathbb{R})}\le {\widehat{C}}_1\left(\parallel u_{\nu }^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel u_{\nu }^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right).$$

(105)

Plus, observe that, by (74), there holds

$$\begin{array}{ccc}\hfill & & |{\mu }_{\nu }^{(\pm )}(g_{\nu }(v_{\nu }(·,t)),{\left[g_{\nu }(v_{\nu }(·,t))\right]}_x,u_{\nu }^{(+)}(·,t),u_{\nu }^{(-)}(·,t))|\hfill \\ & & =|{\mu }^{(\pm )}(T_{\frac{1}{\nu }}(g_{\nu }(v_{\nu }(·,t))),T_{\frac{1}{\nu }}({\left[g_{\nu }(v_{\nu }(·,t))\right]}_x),u_{\nu }^{(+)}(·,t),u_{\nu }^{(-)}(·,t))|.\hfill \end{array}$$

(106)

Since $|T_{\frac{1}{\nu }}(g_{\nu }(v_{\nu }(·,t)))|\le |g_{\nu }(v_{\nu }(·,t))|$ and $|T_{\frac{1}{\nu }}({\left[g_{\nu }(v_{\nu }(·,t))\right]}_x)|\le |{\left[g_{\nu }(v_{\nu }(·,t))\right]}_x|$, in view of (105) and applying (9) to the right-hand side of (106) with

$$R=R(t)={\widehat{C}}_1\left(\parallel u_{\nu }^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel u_{\nu }^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right),$$

we obtain

$$\begin{array}{ccc}& & |{\mu }_{\nu }^{(\pm )}(g_{\nu }(v_{\nu }(·,t)),{\left[g_{\nu }(v_{\nu }(·,t))\right]}_x,u_{\nu }^{(+)}(·,t),u_{\nu }^{(-)}(·,t))|\hfill \\ & & \le \widehat{\alpha }{\widehat{C}}_1\left(\parallel u_{\nu }^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel u_{\nu }^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right)\left(|u_{\nu }^{(+)}(·,t)|+|u_{\nu }^{(-)}(·,t)|\right).\hfill \end{array}$$

Plugging the previous inequality into (90) yields

$$\begin{array}{ccc}& & {\int }_{\mathbb{R}}|u_{\nu }^{(\pm )}(x,\tau )|dx\le {\int }_{\mathbb{R}}|u_{0,\nu }^{(\pm )}|dx+\widehat{\alpha }{\widehat{C}}_1{\int }_0^{\tau }{\left(\parallel u_{\nu }^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel u_{\nu }^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right)}^{2}dt\hfill \\ & & \le {\int }_{\mathbb{R}}|u_0^{(\pm )}|dx+\widehat{\alpha }{\widehat{C}}_1{\int }_0^{\tau }{\left(\parallel u_{\nu }^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel u_{\nu }^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right)}^{2}dt\hfill \end{array}$$

for every $\tau \in (0,T]$ (see also (83)). Summing up the above inequalities with upper signs “+” and “−” yields

$$\begin{array}{c}\hfill F(\tau )\le F_0+2\widehat{\alpha }{\widehat{C}}_1{\int }_0^{\tau }{\left(F(t)\right)}^{2}dt,\end{array}$$

where $F(t):=\parallel u_{\nu }^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel u_{\nu }^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}$ and $F_0:=\parallel u_0^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel u_0^{(-)}\parallel }_{L^1(\mathbb{R})}$. By a nonlinear Gronwall-type inequality (Theorem 25 of [36]), the above inequality ensures that

$$F(\tau )\le {\displaystyle \frac{F_0}{1-2\tau \widehat{\alpha }{\widehat{C}}_1{F}_0}}\mathrm{for}\mathrm{all}0\le \tau <{\displaystyle \frac{1}{2\widehat{\alpha }{\widehat{C}}_1{F}_0}},$$

which in turn yields (104) for all $\tau \in (0,T_0)$. □

Remark 5.

Let $T_0$ be as in (23). Then, from Lemma 4 and Proposition 4, it follows that, for all $\tau \in (0,T_0)$, there holds

$${∥{\left[g_{\nu }(v_{\nu }(·,\tau ))\right]}_{xx}∥}_{L^1(\mathbb{R})}\le {\displaystyle \frac{2L_{\beta }\left(\parallel u_0^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel u_0^{(-)}\parallel }_{L^1(\mathbb{R})}\right)}{1-2\tau \widehat{\alpha }{\widehat{C}}_1\left(\parallel u_0^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel u_0^{(-)}\parallel }_{L^1(\mathbb{R})}\right)}},$$

(107)

$$\parallel v_{\nu }{(·,\tau )\parallel }_{L^1(\mathbb{R})}\le {\displaystyle \frac{L_{\beta }\left(\parallel u_0^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel u_0^{(-)}\parallel }_{L^1(\mathbb{R})}\right)}{1-2\tau \widehat{\alpha }{\widehat{C}}_1\left(\parallel u_0^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel u_0^{(-)}\parallel }_{L^1(\mathbb{R})}\right)}},$$

(108)

$${∥{\left[g_{\nu }(v_{\nu }(·,\tau ))\right]}_x∥}_{L^{\infty }(\mathbb{R})}\le {\displaystyle \frac{2L_{\beta }\left(\parallel u_0^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel u_0^{(-)}\parallel }_{L^1(\mathbb{R})}\right)}{1-2\tau \widehat{\alpha }{\widehat{C}}_1\left(\parallel u_0^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel u_0^{(-)}\parallel }_{L^1(\mathbb{R})}\right)}},$$

(109)

$${∥g_{\nu }(v_{\nu }(·,\tau ))∥}_{L^{\infty }(\mathbb{R})}\le {\displaystyle \frac{{\widehat{C}}_1\left(\parallel u_0^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel u_0^{(-)}\parallel }_{L^1(\mathbb{R})}\right)}{1-2\tau \widehat{\alpha }{\widehat{C}}_1\left(\parallel u_0^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel u_0^{(-)}\parallel }_{L^1(\mathbb{R})}\right)}},$$

(110)

$${∥g_{\nu }(v_{\nu }(·,\tau ))∥}_{H^1(\mathbb{R})}\le {\displaystyle \frac{{\widehat{C}}_1\left(\parallel u_0^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel u_0^{(-)}\parallel }_{L^1(\mathbb{R})}\right)}{1-2\tau \widehat{\alpha }{\widehat{C}}_1\left(\parallel u_0^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel u_0^{(-)}\parallel }_{L^1(\mathbb{R})}\right)}}.$$

(111)

4.2. Setting $\nu \to 0^{+}$ in Problems (85) and (86)

Let us begin with the following lemma.

Lemma 5.

Let $U_0=(u_0^{(+)},u_0^{(-)})\in {\left(L^1(\mathbb{R})\right)}^2$ and $T_0\in (0,T]$ be as in (23). For every ${\widehat{T}}_0\in (0,T_0)$, set

$${\nu }_{{\widehat{T}}_0}:={\displaystyle \frac{1-2{\widehat{T}}_0\widehat{\alpha }{\widehat{C}}_1\left(\parallel u_0^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel u_0^{(-)}\parallel }_{L^1(\mathbb{R})}\right)}{{\widehat{C}}_1\left(\parallel u_0^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel u_0^{(-)}\parallel }_{L^1(\mathbb{R})}\right)}},$$

(112)

where $\widehat{\alpha }>0$ and ${\widehat{C}}_1$ are the constants in $(H_2)$ and (24), respectively. Let $(u_{\nu }^{(+)},u_{\nu }^{(-)},v_{\nu })$ be a solution of (85) and (86) for $\nu \in (0,1)$.

$(i)$

 For every $0<\nu <min\{1;{\nu }_{{\widehat{T}}_0}\}$, the triplet $(u_{\nu }^{(+)},u_{\nu }^{(-)},v_{\nu })$ is a solution in $S_{{\widehat{T}}_0}=\mathbb{R}\times (0,{\widehat{T}}_0)$ of

$$\left\{\begin{array}{cc}{u}_t^{(+)}+{\left[{\phi }^{(+)}(u^{(+)})\right]}_x-\nu {\left[u^{(+)}\right]}_{xx}={\mu }^{(+)}(g_{\nu }(v),{\left[g_{\nu }(v)\right]}_x,u^{(+)},u^{(-)})\hfill & in{S}_{{\widehat{T}}_0},\hfill \\ \\ u_t^{(-)}+{\left[{\phi }^{(-)}(u^{(-)})\right]}_x-\nu {\left[u^{(-)}\right]}_{xx}={\mu }^{(-)}(g_{\nu }(v),{\left[g_{\nu }(v)\right]}_x,u^{(+)},u^{(-)})\hfill & in{S}_{{\widehat{T}}_0},\hfill \\ \\ -{\left[g_{\nu }(v)\right]}_{xx}+v=\beta (u^{(+)},u^{(-)})\hfill & in{S}_{{\widehat{T}}_0}.\hfill \end{array}\right.$$

(113)

$(ii)$

 For every $0<\nu <min\{1;{\nu }_{{\widehat{T}}_0}\}$ and $\tau \in (0,{\widehat{T}}_0)$, there holds

$$\begin{array}{ccc}\hfill & & \parallel u_{\nu }^{(+)}{(·,\tau )\parallel }_{L^1(\mathbb{R})}+{\parallel u_{\nu }^{(-)}(·,\tau )\parallel }_{L^1(\mathbb{R})}\le {\displaystyle \frac{1}{{\widehat{C}}_1{\nu }_{{\widehat{T}}_0}}},\hfill \end{array}$$

(114)

$${∥{\left[g_{\nu }(v_{\nu }(·,\tau ))\right]}_{xx}∥}_{L^1(\mathbb{R})}\le {\displaystyle \frac{1}{{\nu }_{{\widehat{T}}_0}}},$$

(115)

$$\parallel v_{\nu }{(·,\tau )\parallel }_{L^1(\mathbb{R})}\le {\displaystyle \frac{1}{{\nu }_{{\widehat{T}}_0}}},$$

(116)

$${∥{\left[g_{\nu }(v_{\nu }(·,\tau ))\right]}_x∥}_{L^{\infty }(\mathbb{R})}\le {\displaystyle \frac{1}{{\nu }_{{\widehat{T}}_0}}},$$

(117)

$${∥g_{\nu }(v_{\nu }(·,\tau ))∥}_{L^{\infty }(\mathbb{R})}\le {\displaystyle \frac{1}{{\nu }_{{\widehat{T}}_0}}},$$

(118)

$${∥g_{\nu }(v_{\nu }(·,\tau ))∥}_{H^1(\mathbb{R})}\le {\displaystyle \frac{1}{{\nu }_{{\widehat{T}}_0}}}.$$

(119)

$(iii)$

 The families $\left\{{\mu }^{(\pm )}(g_{\nu }(v_{\nu }),{\left[g_{\nu }(v_{\nu })\right]}_x,u_{\nu }^{(+)},u_{\nu }^{(-)})\right\}$, $\left\{{\phi }^{(\pm )}(u_{\nu }^{(\pm )})\right\}$, and $\left\{\beta (u_{\nu }^{(+)},u_{\nu }^{(-)})\right\}$ are bounded in $L^{\infty }(0,{\widehat{T}}_0;L^1(\mathbb{R}))$.

Proof. 

$(i)$

 Since $2L_{\beta }<{\widehat{C}}_1$ (see (24)), by (109) and (110), there holds ${∥g_{\nu }(v_{\nu })∥}_{L^{\infty }(S_{{\widehat{T}}_0})}\le {\displaystyle \frac{1}{{\nu }_{{\widehat{T}}_0}}}<{\displaystyle \frac{1}{\nu }}$ and ${∥{\left[g_{\nu }(v_{\nu })\right]}_x∥}_{L^{\infty }(S_{{\widehat{T}}_0})}\le {\displaystyle \frac{1}{{\nu }_{{\widehat{T}}_0}}}<{\displaystyle \frac{1}{\nu }}$ for all $0<\nu <min\{1;{\nu }_{{\widehat{T}}_0}\}$. Then, the conclusion easily follows from the previous inequalities and the definition of ${\mu }_{\nu }^{(\pm )}$ in (74).

$(ii)$

 The a priori estimates in (114)–(119) immediately follow from (104) and (107)–(111).

$(iii)$

 Combining (9), (117) and (118) yields

$$|{\mu }^{(\pm )}(g_{\nu }(v),{\left[g_{\nu }(v)\right]}_x,u^{(+)},u^{(-)})|\le {\displaystyle \frac{\widehat{\alpha }}{{\nu }_{{\widehat{T}}_0}}}\left(|u_{\nu }^{(+)}|+|u_{\nu }^{(-)}|\right)a.e. \mathrm{in} S_{{\widehat{T}}_0}.$$

(120)

Then, the boundedness in $L^{\infty }(0,{\widehat{T}}_0;L^1(\mathbb{R}))$ of the families $\left\{{\mu }^{(\pm )}(g_{\nu }(v_{\nu }),{\left[g_{\nu }(v_{\nu })\right]}_x,u_{\nu }^{(+)},u_{\nu }^{(-)})\right\}$ follows from the above inequality and (114). Similarly, the boundedness in $L^{\infty }(0,{\widehat{T}}_0;L^1(\mathbb{R}))$ of $\left\{{\phi }^{(\pm )}(u_{\nu }^{(\pm )})\right\}$ and $\left\{\beta (u_{\nu }^{(+)},u_{\nu }^{(-)})\right\}$ is a direct consequence of (114) by the inequalities $|{\phi }^{(\pm )}({\xi }^{(\pm )})|\le L_{{\phi }^{(\pm )}}|{\xi }^{(\pm )}|$ and $|\beta ({\xi }^{(+)},{\xi }^{(-)})|\le L_{\beta }(|{\xi }^{(+)}|+|{\xi }^{(-)}|)$ for all ${\xi }^{(\pm )}\in \mathbb{R}$ (see assumptions $(H_3)$ and $(H_5)$). □

Lemma 6.

Let $U_0=(u_0^{(+)},u_0^{(-)})\in {\left(L^1(\mathbb{R})\right)}^2$ and $T_0\in (0,T]$ be as in (23). Let $\left\{{\nu }_n\right\}\subseteq (0,1)$, ${\nu }_n\to 0^{+}$, and let $(u_{{\nu }_n}^{(+)},u_{{\nu }_n}^{(-)},v_{{\nu }_n})$ be a solution of (85) and (86) with $\nu ={\nu }_n$. Then, for every ${\widehat{T}}_0\in (0,T_0)$, there exist $u^{(\pm )}\in L_{w*}^{\infty }(0,{\widehat{T}}_0;\mathcal{M}(\mathbb{R}))$, ${\phi }_{*}^{(\pm )}\in L_{w*}^{\infty }(0,{\widehat{T}}_0;\mathcal{M}(\mathbb{R}))$, ${\mu }_{*}^{(\pm )}\in L_{w*}^{\infty }(0,{\widehat{T}}_0;\mathcal{M}(\mathbb{R}))$, and ${\beta }_{*}\in L_{w*}^{\infty }(0,{\widehat{T}}_0;\mathcal{M}(\mathbb{R}))$ such that, possibly up to subsequences (not relabeled), there holds

$$u_{{\nu }_n}^{(\pm )}\stackrel{*}{\rightharpoonup }{u}^{(\pm )}in{L}_{w*}^{\infty }(0,{\widehat{T}}_0;\mathcal{M}(\mathbb{R})),$$

(121)

$${\phi }^{(\pm )}(u_{{\nu }_n}^{(\pm )})\stackrel{*}{\rightharpoonup }{\phi }_{*}^{(\pm )}in{L}_{w*}^{\infty }(0,{\widehat{T}}_0;\mathcal{M}(\mathbb{R})),$$

(122)

$${\mu }^{(\pm )}(g_{{\nu }_n}(v_{{\nu }_n}),{\left[g_{{\nu }_n}(v_{{\nu }_n})\right]}_x,u_{{\nu }_n}^{(+)},u_{{\nu }_n}^{(-)})\stackrel{*}{\rightharpoonup }{\mu }_{*}^{(\pm )}in{L}_{w*}^{\infty }(0,{\widehat{T}}_0;\mathcal{M}(\mathbb{R})),$$

(123)

$$\beta (u_{{\nu }_n}^{(+)},u_{{\nu }_n}^{(-)})\stackrel{*}{\rightharpoonup }{\beta }_{*}in{L}_{w*}^{\infty }(0,{\widehat{T}}_0;\mathcal{M}(\mathbb{R})).$$

(124)

Proof. 

The convergences in (121)–(124) follow from the a priori estimate (114) and claim $(iii)$ in Lemma 5. □

Lemma 7.

Let $U_0=(u_0^{(+)},u_0^{(-)})\in {\left(L^1(\mathbb{R})\right)}^2$ and $T_0\in (0,T]$ be as in (23). For every ${\widehat{T}}_0\in (0,T_0)$, let $\left\{(u_{{\nu }_n}^{(+)},u_{{\nu }_n}^{(-)},v_{{\nu }_n})\right\}$, $u^{(\pm )}\in L_{w*}^{\infty }(0,{\widehat{T}}_0;\mathcal{M}(\mathbb{R}))$ be the subsequences and the limiting measures given in Lemma 6. Then, for $a.e.$ $\tau \in (0,{\widehat{T}}_0)$, there holds

$$u_{{\nu }_n}^{(\pm )}(·,\tau )\stackrel{*}{\rightharpoonup }{u}^{(\pm )}(\tau )in\mathcal{M}(\mathbb{R}).$$

(125)

Proof. 

Let $\rho \in C_c^2(\mathbb{R})$ and $h\in C^1([0,{\widehat{T}}_0])$ be fixed arbitrarily. For all $\tau \in (0,{\widehat{T}}_0]$ and n large enough (such that ${\nu }_n<{\nu }_{{\widehat{T}}_0}$; see (112)), multiplying the first two equations of (113) by $\zeta (x,t)=\rho (x)h(t)$ yields

$$\begin{array}{ccc}\hfill & & {\int }_{\mathbb{R}}{u}_{{\nu }_n}^{(\pm )}(x,\tau )\rho (x)h(\tau )dx-{\int }_{\mathbb{R}}{u}_{0,{\nu }_n}^{(\pm )}(x)\rho (x)h(0)dx\hfill \\ & & ={\nu }_n{\int }_0^{\tau }{\int }_{\mathbb{R}}{u}_{{\nu }_n}^{(\pm )}{\rho }^{″}(x)h(t)dxdt\hfill \\ & & +{\int }_0^{\tau }{\int }_{\mathbb{R}}\left\{u_{{\nu }_n}^{(\pm )}\rho (x)h^{\prime }(t)+{\phi }^{(\pm )}(u_{{\nu }_n}^{(\pm )}){\rho }^{\prime }(x)h(t)\right\}dxdt\hfill \\ & & +{\int }_0^{\tau }{\int }_{\mathbb{R}}{\mu }^{(\pm )}(g_{{\nu }_n}(v_{{\nu }_n}),{\left[g_{{\nu }_n}(v_{{\nu }_n})\right]}_x,u_{{\nu }_n}^{(+)},u_{{\nu }_n}^{(-)})\rho (x)h(t)dxdt.\hfill \end{array}$$

(126)

By (84), (121) and (122), setting $n\to \infty$ in (126) (under the extra condition $h(\tau )=0$), we get

$$\begin{array}{ccc}& & -{\int }_0^{\tau }〈u^{(\pm )}(t),\rho 〉h^{\prime }(t)dt-h(0){\int }_{\mathbb{R}}{u}_0^{(\pm )}(x)\rho (x)dx\hfill \\ & & ={\int }_0^{\tau }〈{\phi }_{*}^{(\pm )}(t),{\rho }^{\prime }〉h(t)dt+{\int }_0^{\tau }〈{\mu }_{*}^{(\pm )}(t),\rho 〉h(t)dt.\hfill \end{array}$$

Choosing in the above equality

$$h(t)=h_j(t)={\chi }_{[0,\tau -j^{-1}]}(t)+j(\tau -t){\chi }_{(\tau -j^{-1},\tau ]}(t)$$

($j\in \mathbb{N}$ large enough), and taking the limit as $j\to \infty$, for $a.e.$ $\tau$ as above, we get

$$〈u^{(\pm )}(\tau ),\rho 〉={\int }_{\mathbb{R}}{u}_0^{(\pm )}(x)\rho (x)dx+{\int }_0^{\tau }〈{\phi }_{*}^{(\pm )}(t),{\rho }^{\prime }〉dt+{\int }_0^{\tau }〈{\mu }_{*}^{(\pm )}(t),\rho 〉dt.$$

(127)

On the other hand, choosing $h(t)=1$ in (126) and setting $j\to \infty$ yields (see also (122) and (123))

$$\begin{array}{ccc}& & \underset{n\to \infty }{lim}{\int }_{\mathbb{R}}{u}_{{\nu }_n}^{(\pm )}(x,\tau )\rho (x)dx=\underset{n\to \infty }{lim}\left\{{\int }_{\mathbb{R}}{u}_{0,{\nu }_n}^{(\pm )}(x)\rho (x)dx\right.\hfill \\ & & +{\nu }_n{\int }_0^{\tau }{\int }_{\mathbb{R}}{u}_{{\nu }_n}^{(\pm )}{\rho }^{″}(x)dxdt+{\int }_0^{\tau }{\int }_{\mathbb{R}}{\phi }^{(\pm )}(u_{{\nu }_n}^{(\pm )}){\rho }^{\prime }(x)dxdt\hfill \\ & & \left.+{\int }_0^{\tau }{\int }_{\mathbb{R}}{\mu }^{(\pm )}(g_{{\nu }_n}(v_{{\nu }_n}),{\left[g_{{\nu }_n}(v_{{\nu }_n})\right]}_x,u_{{\nu }_n}^{(+)},u_{{\nu }_n}^{(-)})\rho (x)dxdt\right\}\hfill \\ & & ={\int }_{\mathbb{R}}{u}_0^{(\pm )}(x)\rho (x)dx+{\int }_0^{\tau }〈{\phi }_{*}^{(\pm )}(t),{\rho }^{\prime }〉dt+{\int }_0^{\tau }〈{\mu }_{*}^{(\pm )}(t),\rho 〉dt,\hfill \end{array}$$

whence

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{lim}{\int }_{\mathbb{R}}{u}_{{\nu }_n}^{(\pm )}(x,\tau )\rho (x)dx& =& {\int }_{\mathbb{R}}{u}_0^{(\pm )}(x)\rho (x)dx+{\int }_0^{\tau }〈{\phi }_{*}^{(\pm )}(t),{\rho }^{\prime }〉dt+\hfill \\ & & {\int }_0^{\tau }〈{\mu }_{*}^{(\pm )}(t),\rho 〉dt\hfill \end{array}$$

for all $\rho \in C_c^2(\mathbb{R})$. By the above equation and (127), we get

$$\underset{n\to \infty }{lim}{\int }_{\mathbb{R}}{u}_{{\nu }_n}^{(\pm )}(x,\tau )\rho (x)dx=〈u^{(\pm )}(\tau ),\rho 〉.$$

Since for all $\tau \in (0,{\widehat{T}}_0)$ the sequences $\left\{u_{{\nu }_n}^{(\pm )}(·,\tau )\right\}$ are bounded in $L^1(\mathbb{R})$ by (114), the previous convergence implies (125). □

Proposition 5.

Let $U_0=(u_0^{(+)},u_0^{(-)})\in {\left(L^1(\mathbb{R})\right)}^2$ and $T_0\in (0,T]$ be as in (23). For every ${\widehat{T}}_0\in (0,T_0)$, let $\left\{(u_{{\nu }_n}^{(+)},u_{{\nu }_n}^{(-)},v_{{\nu }_n})\right\}$, $u^{(\pm )}\in L_{w*}^{\infty }(0,{\widehat{T}}_0;\mathcal{M}(\mathbb{R}))$ be the subsequence and the limiting measures given in Lemma 6. Then,

$$u^{(\pm )}\in L^{\infty }(0,{\widehat{T}}_0;L^1(\mathbb{R})),$$

(128)

and, for $a.e.$ $\tau \in (0,{\widehat{T}}_0)$, there holds

$$u_{{\nu }_n}^{(\pm )}(·,\tau )\to u^{(\pm )}(·,\tau )in{L}^1(\mathbb{R}).$$

(129)

Moreover,

$$u_{{\nu }_n}^{(\pm )}\to u^{(\pm )}in{L}^1(S_{{\widehat{T}}_0}).$$

(130)

Proof. 

Since ${\nu }_n\to 0^{+}$, there exists $n_0\in \mathbb{N}$ such that ${\nu }_n<{\nu }_{{\widehat{T}}_0}$ for all $n>n_0$ (see (112) for the definition of ${\nu }_{{\widehat{T}}_0}$). For any $n>n_0$, $h\in \mathbb{R}$ and $t\in (0,{\widehat{T}}_0)$, for simplicity of notations, set

$${\widehat{u}}_{{\nu }_n}^{(\pm )}(x,t)=u_{{\nu }_n}^{(\pm )}(x+h,t),{\widehat{v}}_{{\nu }_n}(x,t)=v_{{\nu }_n}(x+h,t)$$

($(x,t)\in S_{{\widehat{T}}_0}$). Let us proceed in four steps.

• Step 1. For every $K>0$, let $h_K(s)$ be the function defined in (99) ($s\in \mathbb{R}$). Let us choose $h_K\left(g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n})\right)$ as test function in the elliptic equation

  $$-{[g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n})]}_{xx}+{\widehat{v}}_{{\nu }_n}-v_{{\nu }_n}=\beta ({\widehat{u}}_{{\nu }_n}^{(+)},{\widehat{u}}_{{\nu }_n}^{(-)})-\beta (u_{{\nu }_n}^{(+)},u_{{\nu }_n}^{(-)}).$$

  (131)

  By the nondecreasing character of $h_K$ and the Lipschitz continuity of $\beta$ (see $(H_5)$), we get

  $$\begin{array}{ccc}& & {\int }_{\mathbb{R}}({\widehat{v}}_{{\nu }_n}-v_{{\nu }_n})h_K(g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n}))dx\le L_{\beta }\left(\parallel {\widehat{u}}_{{\nu }_n}^{(+)}-u_{{\nu }_n}^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel {\widehat{u}}_{{\nu }_n}^{(-)}-u_{{\nu }_n}^{(-)}\parallel }_{L^1(\mathbb{R})}\right)\hfill \end{array}$$

  (here, use of the estimate $h_K(s)\le 1$ for all $s\in \mathbb{R}$ has occurred). Since by the (strictly) increasing character of $g_{\nu }$ there holds $({\widehat{v}}_{{\nu }_n}-v_{{\nu }_n})h_K(g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n}))\to |{\widehat{v}}_{{\nu }_n}-v_{{\nu }_n}|$ as $K\to 0^{+}$, setting $K\to 0^{+}$ in the above inequality yields

  $$\parallel {\widehat{v}}_{{\nu }_n}-v_{{\nu }_n}{\parallel }_{L^1(\mathbb{R})}\le L_{\beta }\left(\parallel {\widehat{u}}_{{\nu }_n}^{(+)}-u_{{\nu }_n}^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel {\widehat{u}}_{{\nu }_n}^{(-)}-u_{{\nu }_n}^{(-)}\parallel }_{L^1(\mathbb{R})}\right),$$

  (132)

  whence (see (131))

  $$\parallel {[g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n})]}_{xx}{\parallel }_{L^1(\mathbb{R})}\le 2L_{\beta }\left(\parallel {\widehat{u}}_{{\nu }_n}^{(+)}-u_{{\nu }_n}^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel {\widehat{u}}_{{\nu }_n}^{(-)}-u_{{\nu }_n}^{(-)}\parallel }_{L^1(\mathbb{R})}\right),$$

  (133)

  and

  $$\parallel {[g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n})]}_x{\parallel }_{L^{\infty }(\mathbb{R})}\le 2L_{\beta }\left(\parallel {\widehat{u}}_{{\nu }_n}^{(+)}-u_{{\nu }_n}^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel {\widehat{u}}_{{\nu }_n}^{(-)}-u_{{\nu }_n}^{(-)}\parallel }_{L^1(\mathbb{R})}\right).$$

  (134)

  Let us prove that

  $$\parallel g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n}){\parallel }_{L^{\infty }(\mathbb{R})}\le 2(L_g+1)L_{\beta }\left(\parallel {\widehat{u}}_{{\nu }_n}^{(+)}-u_{{\nu }_n}^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel {\widehat{u}}_{{\nu }_n}^{(-)}-u_{{\nu }_n}^{(-)}\parallel }_{L^1(\mathbb{R})}\right),$$

  (135)

  where $L_g>0$ is the Lipschitz constant of g.

To this aim, observe that choosing $g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n})$ as test function in (131) yields

$$\begin{array}{ccc}\hfill & & {\int }_{\mathbb{R}}{[g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n})]}_x^{2}dx+{\int }_{\mathbb{R}}({\widehat{v}}_{{\nu }_n}-v_{{\nu }_n})(g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n}))dx\hfill \\ & & \le L_{\beta }{\parallel g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n})\parallel }_{L^{\infty }(\mathbb{R})}\left(\parallel {\widehat{u}}_{{\nu }_n}^{(+)}-u_{{\nu }_n}^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel {\widehat{u}}_{{\nu }_n}^{(-)}-u_{{\nu }_n}^{(-)}\parallel }_{L^1(\mathbb{R})}\right).\hfill \end{array}$$

(136)

Observing that $\parallel g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n}){\parallel }_{L^{\infty }(\mathbb{R})}\le \sqrt{2}{\parallel g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n})\parallel }_{H^1(\mathbb{R})}$ and

$$\begin{array}{ccc}& & {\int }_{\mathbb{R}}({\widehat{v}}_{{\nu }_n}-v_{{\nu }_n})(g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n}))\ge {\displaystyle \frac{1}{L_g+{\nu }_n}}{\int }_{\mathbb{R}}{(g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n}))}^{2}dx\hfill \\ & & \ge {\displaystyle \frac{1}{L_g+1}}{\int }_{\mathbb{R}}{(g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n}))}^{2}dx\hfill \end{array}$$

(recall that $0<{\nu }_n<1$), by inequality (136), there holds

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{L_g+1}}\left\{{\int }_{\mathbb{R}}{[g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n})]}_x^{2}dx+{\int }_{\mathbb{R}}{\left(g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n})\right)}^{2}dx\right\}\hfill \\ & & \le \sqrt{2}{L}_{\beta }{\parallel g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n})\parallel }_{H^1(\mathbb{R})}\left(\parallel {\widehat{u}}_{{\nu }_n}^{(+)}-u_{{\nu }_n}^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel {\widehat{u}}_{{\nu }_n}^{(-)}-u_{{\nu }_n}^{(-)}\parallel }_{L^1(\mathbb{R})}\right),\hfill \end{array}$$

whence

$$\parallel g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n}){\parallel }_{H^1(\mathbb{R})}\le \sqrt{2}(L_g+1)L_{\beta }\left(\parallel {\widehat{u}}_{{\nu }_n}^{(+)}-u_{{\nu }_n}^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel {\widehat{u}}_{{\nu }_n}^{(-)}-u_{{\nu }_n}^{(-)}\parallel }_{L^1(\mathbb{R})}\right).$$

(137)

Since $\parallel g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n}){\parallel }_{L^{\infty }(\mathbb{R})}\le \sqrt{2}{\parallel g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n})\parallel }_{H^1(\mathbb{R})}$, inequality (135) follows from (137).

• Step 2. By Lemma 5-$(ii)$ for every $t\in (0,{\widehat{T}}_0)$ and $n>n_0$, there holds

  $$\parallel g_{{\nu }_n}(v_{{\nu }_n}(·,t)){\parallel }_{L^{\infty }(\mathbb{R})},{\parallel {\left[g_{{\nu }_n}(v_{{\nu }_n}(·,t))\right]}_x\parallel }_{L^{\infty }(\mathbb{R})}\le {\displaystyle \frac{1}{{\nu }_{{\widehat{T}}_0}}},$$

  (138)

  $$\parallel g_{{\nu }_n}({\widehat{v}}_{{\nu }_n}(·,t)){\parallel }_{L^{\infty }(\mathbb{R})},{\parallel {\left[g_{{\nu }_n}({\widehat{v}}_{{\nu }_n}(·,t))\right]}_x\parallel }_{L^{\infty }(\mathbb{R})}\le {\displaystyle \frac{1}{{\nu }_{{\widehat{T}}_0}}},$$

  (139)

  $$\parallel u_{{\nu }_n}^{(+)}{(·,t))\parallel }_{L^1(\mathbb{R})}+{\parallel u_{{\nu }_n}^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\le {\displaystyle \frac{1}{{\widehat{C}}_1{\nu }_{{\widehat{T}}_0}}},$$

  (140)

  $$\parallel {\widehat{u}}_{{\nu }_n}^{(+)}{(·,t))\parallel }_{L^1(\mathbb{R})}+{\parallel {\widehat{u}}_{{\nu }_n}^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\le {\displaystyle \frac{1}{{\widehat{C}}_1{\nu }_{{\widehat{T}}_0}}},$$

  (141)

  where ${\widehat{C}}_1>0$ is the constant in (24). Then, for every $\tau \in (0,{\widehat{T}}_0)$, by (10) and the above estimates, there holds

  $$\begin{array}{ccc}& & {\int }_0^{\tau }{\int }_{\mathbb{R}}|{\mu }^{(\pm )}(g_{{\nu }_n}({\widehat{v}}_{{\nu }_n}),{\left[g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})\right]}_x,{\widehat{u}}_{{\nu }_n}^{(+)},{\widehat{u}}_{{\nu }_n}^{(-)})-{\mu }^{(\pm )}(g_{{\nu }_n}(v_{{\nu }_n}),{\left[g_{{\nu }_n}(v_{{\nu }_n})\right]}_x,u_{{\nu }_n}^{(+)},u_{{\nu }_n}^{(-)})|dxdt\hfill \\ & & \le {\displaystyle \frac{\widehat{\alpha }}{{\nu }_{{\widehat{T}}_0}}}{\int }_0^{\tau }{\int }_{\mathbb{R}}\left(|{\widehat{u}}_{{\nu }_n}^{(+)}|+|{\widehat{u}}_{{\nu }_n}^{(-)}|+|u_{{\nu }_n}^{(+)}|+|u_{{\nu }_n}^{(-)}|\right)(|g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n})|\hfill \\ & & +|{\left[g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})\right]}_x-{\left[g_{{\nu }_n}(v_{{\nu }_n})\right]}_x|)dxdt+{\displaystyle \frac{\widehat{\alpha }}{{\nu }_{{\widehat{T}}_0}}}{\int }_0^{\tau }{\int }_{\mathbb{R}}\left(|{\widehat{u}}_{{\nu }_n}^{(+)}-u_{{\nu }_n}^{(+)}|+|{\widehat{u}}_{{\nu }_n}^{(-)}-u_{{\nu }_n}^{(-)}|\right)dxdt\hfill \\ & & \le {\displaystyle \frac{\widehat{\alpha }}{{\nu }_{{\widehat{T}}_0}}}{\int }_0^{\tau }\left(\parallel g_{{\nu }_n}({\widehat{v}}_{{\nu }_n}(·,t))-g_{{\nu }_n}(v_{{\nu }_n}(·,t)){\parallel }_{L^{\infty }(\mathbb{R})}+{\parallel {\left[g_{{\nu }_n}({\widehat{v}}_{{\nu }_n}(·,t))\right]}_x-{\left[g_{{\nu }_n}(v_{{\nu }_n}(·,t))\right]}_x\parallel }_{L^{\infty }(\mathbb{R})}\right)\hfill \\ & & \times \left(\parallel {\widehat{u}}_{{\nu }_n}^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+\parallel u_{{\nu }_n}^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+\parallel {\widehat{u}}_{{\nu }_n}^{(-)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel u_{{\nu }_n}^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right)dt\hfill \\ & & +{\displaystyle \frac{\widehat{\alpha }}{{\nu }_{{\widehat{T}}_0}}}{\int }_0^{\tau }\left(\parallel {\widehat{u}}_{{\nu }_n}^{(+)}(·,t)-u_{{\nu }_n}^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel {\widehat{u}}_{{\nu }_n}^{(-)}(·,t)-u_{{\nu }_n}^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right)dt\hfill \\ & & \le {\displaystyle \frac{\widehat{\alpha }}{{\nu }_{{\widehat{T}}_0}}}\left(1+{\displaystyle \frac{2}{{\widehat{C}}_1{\nu }_{{\widehat{T}}_0}}}\right)\left\{{\int }_0^{\tau }\left(\parallel {\widehat{u}}_{{\nu }_n}^{(+)}(·,t)-u_{{\nu }_n}^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel {\widehat{u}}_{{\nu }_n}^{(-)}(·,t)-u_{{\nu }_n}^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right)dt\right.\hfill \\ & & \left.+{\int }_0^{\tau }\left(\parallel g_{{\nu }_n}({\widehat{v}}_{{\nu }_n}(·,t))-g_{{\nu }_n}(v_{{\nu }_n}(·,t)){\parallel }_{L^{\infty }(\mathbb{R})}+{\parallel {[g_{{\nu }_n}({\widehat{v}}_{{\nu }_n}(·,t))-g_{{\nu }_n}(v_{{\nu }_n}(·,t))]}_x\parallel }_{L^{\infty }(\mathbb{R})}\right)dt\right\}.\hfill \end{array}$$

  By the previous inequality and (134) and (135), we obtain

  $$\begin{array}{ccc}\hfill & & {\int }_0^{\tau }{\int }_{\mathbb{R}}|{\mu }^{(\pm )}(g_{{\nu }_n}({\widehat{v}}_{{\nu }_n}),{\left[g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})\right]}_x,{\widehat{u}}_{{\nu }_n}^{(+)},{\widehat{u}}_{{\nu }_n}^{(-)})-{\mu }^{(\pm )}(g_{{\nu }_n}(v_{{\nu }_n}),{\left[g_{{\nu }_n}(v_{{\nu }_n})\right]}_x,u_{{\nu }_n}^{(+)},u_{{\nu }_n}^{(-)})|dxdt\hfill \\ & & \le {\widehat{C}}_{2,{\nu }_{{\widehat{T}}_0}}{\int }_0^{\tau }\left(\parallel {\widehat{u}}_{{\nu }_n}^{(+)}(·,t)-u_{{\nu }_n}^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel {\widehat{u}}_{{\nu }_n}^{(-)}(·,t)-u_{{\nu }_n}^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right)dt,\hfill \end{array}$$

  (142)

  with

  $${\widehat{C}}_{2,{\nu }_{{\widehat{T}}_0}}:={\displaystyle \frac{\widehat{\alpha }}{{\nu }_{{\widehat{T}}_0}}}\left(1+{\displaystyle \frac{2}{{\widehat{C}}_1{\nu }_{{\widehat{T}}_0}}}\right)\left[1+4L_{\beta }+2L_{\beta }{L}_g\right].$$

  (143)

• Step 3. Fix any $\tau \in (0,{\widehat{T}}_0)$. By the first two equations in (113) (written for $u_{{\nu }_n}^{(\pm )}$ and ${\widehat{u}}_{{\nu }_n}^{(\pm )}$, respectively), we get

  $$\begin{array}{ccc}& & {[{\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )}]}_t+{[{\phi }^{(\pm )}({\widehat{u}}_{{\nu }_n}^{(\pm )})-{\phi }^{(\pm )}(u_{{\nu }_n}^{(\pm )})]}_x-{\nu }_n{[{\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )}]}_{xx}\hfill \\ & & ={\mu }^{(\pm )}(g_{{\nu }_n}({\widehat{v}}_{{\nu }_n}),{\left[g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})\right]}_x,{\widehat{u}}_{{\nu }_n}^{(+)},{\widehat{u}}_{{\nu }_n}^{(-)})-{\mu }^{(\pm )}(g_{{\nu }_n}(v_{{\nu }_n}),{\left[g_{{\nu }_n}(v_{{\nu }_n})\right]}_x,u_{{\nu }_n}^{(+)},{\widehat{u}}_{{\nu }_n}^{(-)}).\hfill \end{array}$$

  Multiplying the above equality by $h_K({\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )})$ (see (99) for the definition of $h_K(s)$; $s\in \mathbb{R}$) and integrating in $\mathbb{R}\times (0,\tau )$ yields

  $$\begin{array}{ccc}& & {\int }_0^{\tau }{\int }_{\mathbb{R}}{h}_K({\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )}){({\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )})}_{t}dxdt+{\displaystyle \frac{{\nu }_n}{K}}{\int }_0^{\tau }{\int }_{\mathbb{R}}{\left({({\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )})}_x\right)}^2{\chi }_{\{|{\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )}|<K\}}dxdt\hfill \\ & & ={\displaystyle \frac{1}{K}}{\int }_0^{\tau }{\int }_{\mathbb{R}}({\phi }^{(\pm )}({\widehat{u}}_{{\nu }_n}^{(\pm )})-{\phi }^{(\pm )}(u_{{\nu }_n}^{(\pm )})){({\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )})}_x{\chi }_{\{|{\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )}|<K\}}dxdt\hfill \\ & & +{\int }_0^{\tau }{\int }_{\mathbb{R}}\left({\mu }^{(\pm )}(g_{{\nu }_n}({\widehat{v}}_{{\nu }_n}),{\left[g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})\right]}_x,{\widehat{u}}_{{\nu }_n}^{(+)},{\widehat{u}}_{{\nu }_n}^{(-)})-{\mu }^{(\pm )}(g_{{\nu }_n}(v_{{\nu }_n}),{\left[g_{{\nu }_n}(v_{{\nu }_n})\right]}_x,u_{{\nu }_n}^{(+)},u_{{\nu }_n}^{(-)})\right)\hfill \\ & & \times h_K({\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )})dxdt\le {\displaystyle \frac{L_{{\phi }^{(\pm )}}}{K}}{\int }_0^{\tau }{\int }_{\mathbb{R}}|{\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )}||{({\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )})}_x|{\chi }_{\{|{\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )}|<K\}}dxdt\hfill \\ & & +{\int }_0^{\tau }{\int }_{\mathbb{R}}|{\mu }^{(\pm )}(g_{{\nu }_n}({\widehat{v}}_{{\nu }_n}),{\left[g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})\right]}_x,{\widehat{u}}_{{\nu }_n}^{(+)},{\widehat{u}}_{{\nu }_n}^{(-)})-{\mu }^{(\pm )}(g_{{\nu }_n}(v_{{\nu }_n}),{\left[g_{{\nu }_n}(v_{{\nu }_n})\right]}_x,u_{{\nu }_n}^{(+)},u_{{\nu }_n}^{(-)})|dxdt\hfill \\ & & \le {\displaystyle \frac{{\nu }_n}{2K}}{\int }_0^{\tau }{\int }_{\mathbb{R}}{\left({({\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )})}_x\right)}^2{\chi }_{\{|{\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )}|<K\}}dxdt\hfill \\ & & +{\displaystyle \frac{{(L_{{\phi }^{(\pm )}})}^2}{2{\nu }_{n}K}}{\int }_0^{\tau }{\int }_{\mathbb{R}}{[{\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )}]}^2{\chi }_{\{|{\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )}|<K\}}dxdt\hfill \\ & & +{\widehat{C}}_{2,{\nu }_{{\widehat{T}}_0}}{\int }_0^{\tau }\left(\parallel {\widehat{u}}_{{\nu }_n}^{(+)}(·,t)-u_{{\nu }_n}^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel {\widehat{u}}_{{\nu }_n}^{(-)}(·,t)-u_{{\nu }_n}^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right)dt\hfill \end{array}$$

  (here, we have used (142) and the Lipschitz continuity of ${\phi }^{(\pm )}$). From the above inequality, it follows that

  $$\begin{array}{ccc}\hfill & & {\int }_0^{\tau }{\int }_{\mathbb{R}}{h}_K({\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )}){({\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )})}_{t}dxdt\hfill \\ & & +{\displaystyle \frac{{\nu }_n}{2K}}{\int }_0^{\tau }{\int }_{\mathbb{R}}{\left({({\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )})}_x\right)}^2{\chi }_{\{|{\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )}|<K\}}dxdt\hfill \\ & & \le {\displaystyle \frac{{(L_{{\phi }^{(\pm )}})}^2}{2{\nu }_n}}{\int }_0^{\tau }{\int }_{\mathbb{R}}|{\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )}|{\chi }_{\{|{\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )}|<K\}}dxdt\hfill \\ & & +{\widehat{C}}_{2,{\nu }_{{\widehat{T}}_0}}{\int }_0^{\tau }\left(\parallel {\widehat{u}}_{{\nu }_n}^{(+)}(·,t)-u_{{\nu }_n}^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel {\widehat{u}}_{{\nu }_n}^{(-)}(·,t)-u_{{\nu }_n}^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right)dt.\hfill \end{array}$$

  (144)

  In order to take the limit as $K\to 0^{+}$ in (144), observe that

  $$\begin{array}{ccc}\hfill \underset{K\to 0^{+}}{lim}{\int }_0^{\tau }{\int }_{\mathbb{R}}{h}_K({\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )}){({\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )})}_{t}dxdt& =& {\int }_{\mathbb{R}}|{\widehat{u}}_{{\nu }_n}^{(\pm )}(x,\tau )-u_{{\nu }_n}^{(\pm )}(x,\tau )|dx\hfill \\ & & -{\int }_{\mathbb{R}}|u_{0,{\nu }_n}^{(\pm )}(x+h)-u_{0,{\nu }_n}^{(\pm )}(x)|dx,\hfill \end{array}$$

  and ${lim}_{K\to 0^{+}}{\int }_0^{\tau }{\int }_{\mathbb{R}}|{\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )}|{\chi }_{\{|{\widehat{u}}_{{\nu }_n}^{(\pm )}-u_{{\nu }_n}^{(\pm )}|<K\}}dxdt=0$. In view of the above convergences, setting $K\to 0^{+}$ in (144) yields

  $$\begin{array}{ccc}& & {\int }_{\mathbb{R}}|{\widehat{u}}_{{\nu }_n}^{(\pm )}(x,\tau )-u_{{\nu }_n}^{(\pm )}(x,\tau )|dx\le {\int }_{\mathbb{R}}|u_{0,{\nu }_n}^{(\pm )}(x+h)-u_{0,{\nu }_n}^{(\pm )}(x)|dx\hfill \\ & & +{\widehat{C}}_{2,{\nu }_{{\widehat{T}}_0}}{\int }_0^{\tau }\left(\parallel {\widehat{u}}_{{\nu }_n}^{(+)}(·,t)-u_{{\nu }_n}^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel {\widehat{u}}_{{\nu }_n}^{(-)}(·,t)-u_{{\nu }_n}^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right)dt\hfill \end{array}$$

  for all $\tau \in (0,{\widehat{T}}_0)$. Plainly, by summing up the previous inequalities with upper signs “+” and “−”, and using Gronwall’s lemma, for every such $\tau$, we get

  $$\begin{array}{ccc}\hfill & & {\int }_{\mathbb{R}}|{\widehat{u}}_{{\nu }_n}^{(+)}(x,\tau )-u_{{\nu }_n}^{(+)}(x,\tau )|dx+{\int }_{\mathbb{R}}|{\widehat{u}}_{{\nu }_n}^{(-)}(x,\tau )-u_{{\nu }_n}^{(-)}(x,\tau )|dx\hfill \\ & & \le e^{2{\widehat{T}}_0{\widehat{C}}_{2,{\nu }_{{\widehat{T}}_0}}}\left({\int }_{\mathbb{R}}|u_{0,{\nu }_n}^{(+)}(x+h)-u_{0,{\nu }_n}^{(+)}(x)|dx+{\int }_{\mathbb{R}}|u_{0,{\nu }_n}^{(-)}(x+h)-u_{0,{\nu }_n}^{(-)}(x)|dx\right).\hfill \end{array}$$

  (145)

• Step 4. In view of the Frechet–Kolmogorov compactness theorem, by (84) and (145), for every $\tau \in (0,{\widehat{T}}_0)$, it will follow that the sequences $\left\{u_{{\nu }_n}^{(\pm )}(·,\tau )\right\}$ are relatively compact in $L^1(\mathbb{R})$ if we prove that

  $$\underset{r\to \infty }{lim}{\int }_{\{|x|>r\}}|u_{{\nu }_n}^{(\pm )}(x,\tau )|dx=0\mathrm{uniformly}\mathrm{for}n>n_0.$$

  (146)

  Let us postpone the proof of (146) and conclude the proof. To this aim, for $a.e.$ $\tau \in (0,{\widehat{T}}_0)$, it is enough to observe that the sequences $\left\{u_{{\nu }_n}^{(\pm )}(·,\tau )\right\}$ are relatively compact in $L^1(\mathbb{R})$ and satisfy the convergences in (125). Plainly, this implies that the limiting measures $u^{(\pm )}(\tau )$ in (125) are absolutely continuous with respect to the Lebesgue measure for $a.e.$ $\tau \in (0,{\widehat{T}}_0)$—whence (128) immediately follows—and, in addition, the whole sequences $\left\{u_{{\nu }_n}^{(\pm )}(·,\tau )\right\}$ strongly converge to $u^{(\pm )}(·,\tau )$ in $L^1(\mathbb{R})$ for $a.e.$ $\tau$ as above. This proves (129). Finally, we have

  $$\underset{n\to \infty }{lim}\parallel u_{{\nu }_n}^{(\pm )}-u^{(\pm )}{\parallel }_{L^1(S_{{\widehat{T}}_0})}=\underset{n\to \infty }{lim}{\int }_0^{{\widehat{T}}_0}{\parallel u_{{\nu }_n}^{(\pm )}(·,t)-u^{(\pm )}(·,t)\parallel }_{L^1(\mathbb{R})}dt=0;$$

  regarding the limit on the right-hand side, we have used the dominated convergence theorem since, for $a.e.$ t, there holds $\parallel u_{{\nu }_n}^{(\pm )}(·,t)-u^{(\pm )}{(·,t)\parallel }_{L^1(\mathbb{R})}\to 0$ and $\parallel u_{{\nu }_n}^{(\pm )}(·,t)-u^{(\pm )}{(·,t)\parallel }_{L^1(\mathbb{R})}\le {\displaystyle \frac{2}{{\widehat{C}}_1{\nu }_{{\widehat{T}}_0}}}$ (see (114)).

Let us address (146). To this aim, fix any $\rho \in C^2(\mathbb{R})$, $0\le \rho \le 1$ such that $\rho (x)=1$ for $|x|\ge 1$ and $\rho (x)=0$ for $|x|\le 1/2$. For every $r>0$, set

$${\rho }_r(x)=\rho \left({\displaystyle \frac{x}{r}}\right)(x\in \mathbb{R}).$$

(147)

Choosing in (89) $k=0$, $\zeta (x,t)\equiv {\rho }_r(x)$, $\nu ={\nu }_n$ (for $n>n_0$), and using (120), we get

$$\begin{array}{ccc}& & {\int }_{\mathbb{R}}|u_{{\nu }_n}^{(\pm )}(x,\tau )|{\rho }_r(x)dx-{\int }_{\mathbb{R}}|u_{0,{\nu }_n}^{(\pm )}|{\rho }_r(x)dx\le {\displaystyle \frac{{\nu }_n}{r^2}}{\int }_0^{\tau }{\int }_{\mathbb{R}}|u_{{\nu }_n}^{(\pm )}|\left|{\rho }^{″}\right|\left({\displaystyle \frac{x}{r}}\right)dxdt\hfill \\ & & +{\displaystyle \frac{L_{{\phi }^{(\pm )}}}{r}}{\int }_0^{\tau }{\int }_{\mathbb{R}}|u_{{\nu }_n}^{(\pm )}||{\rho }^{\prime }|\left({\displaystyle \frac{x}{r}}\right)dxdt+{\displaystyle \frac{\widehat{\alpha }}{{\nu }_{{\widehat{T}}_0}}}{\int }_0^{\tau }{\int }_{\mathbb{R}}\left(|u_{{\nu }_n}^{(+)}+|u_{{\nu }_n}^{(-)}|\right){\rho }_r(x)dxdt\hfill \end{array}$$

(here, use of the Lipschtiz continuity of ${\phi }^{(\pm )}$ has occurred). Combining the above estimate and (114), it follows that there exists a constant $C>0$ (depending on ${\widehat{T}}_0$ and ${\nu }_{{\widehat{T}}_0}$) such that

$$\begin{array}{ccc}& & {\int }_{\mathbb{R}}|u_{{\nu }_n}^{(\pm )}(x,\tau )|{\rho }_r(x)dx-{\int }_{\mathbb{R}}|u_{0,{\nu }_n}^{(\pm )}|{\rho }_r(x)dx\le {\displaystyle \frac{C}{r^2}}\parallel {\rho }^{″}{\parallel }_{\infty }+{\displaystyle \frac{C}{r}}{\parallel {\rho }^{\prime }\parallel }_{\infty }\hfill \\ & & +{\displaystyle \frac{\widehat{\alpha }}{{\nu }_{{\widehat{T}}_0}}}{\int }_0^{\tau }{\int }_{\mathbb{R}}\left(|u_{{\nu }_n}^{(+)}+|u_{{\nu }_n}^{(-)}|\right){\rho }_r(x)dxdt,\hfill \end{array}$$

whence

$$\begin{array}{ccc}& & {\int }_{\mathbb{R}}\left(|u_{{\nu }_n}^{(+)}(x,\tau )|+|u_{{\nu }_n}^{(-)}(x,\tau )|\right){\rho }_r(x)dx\le {\int }_{\mathbb{R}}\left(|u_{0,{\nu }_n}^{(+)}|+|u_{0,{\nu }_n}^{(-)}|\right){\rho }_r(x)dx\hfill \\ & & +{\displaystyle \frac{2C}{r^2}}\parallel {\rho }^{″}{\parallel }_{\infty }+{\displaystyle \frac{2C}{r}}{\parallel {\rho }^{\prime }\parallel }_{\infty }+{\displaystyle \frac{2\widehat{\alpha }}{{\nu }_{{\widehat{T}}_0}}}{\int }_0^{\tau }{\int }_{\mathbb{R}}\left(|u_{{\nu }_n}^{(+)}+|u_{{\nu }_n}^{(-)}|\right){\rho }_r(x)dxdt.\hfill \end{array}$$

Applying Gronwall’s inequality, we have

$$\begin{array}{ccc}\hfill & & {\int }_{\mathbb{R}}\left(|u_{{\nu }_n}^{(+)}(x,\tau )|+|u_{{\nu }_n}^{(-)}(x,\tau )|\right){\rho }_r(x)dx\hfill \\ & & \le e^{{\displaystyle \frac{2{\widehat{T}}_0\widehat{\alpha }}{{\nu }_{{\widehat{T}}_0}}}}\left({\int }_{\mathbb{R}}\left(|u_{0,{\nu }_n}^{(+)}|+|u_{0,{\nu }_n}^{(-)}|\right){\rho }_r(x)dx+{\displaystyle \frac{2C}{r^2}}\parallel {\rho }^{″}{\parallel }_{\infty }+{\displaystyle \frac{2C}{r}}{\parallel {\rho }^{\prime }\parallel }_{\infty }\right).\hfill \end{array}$$

(148)

Then, the conclusion follows from the above inequality and the properties of ${\rho }_r(x)$ since

$$\underset{r\to \infty }{lim}\left({\int }_{\mathbb{R}}\left(|u_{0,{\nu }_n}^{(+)}|+|u_{0,{\nu }_n}^{(-)}|\right){\rho }_r(x)dx+{\displaystyle \frac{2C}{r^2}}\parallel {\rho }^{″}{\parallel }_{\infty }+{\displaystyle \frac{2C}{r}}{\parallel {\rho }^{\prime }\parallel }_{\infty }\right)=0$$

uniformly in n (here, we have also used (84)). □

Proposition 6.

Let $U_0=(u_0^{(+)},u_0^{(-)})\in {\left(L^1(\mathbb{R})\right)}^2$ and $T_0\in (0,T]$ be as in (23). For every ${\widehat{T}}_0\in (0,T_0)$, let $\left\{(u_{{\nu }_n}^{(+)},u_{{\nu }_n}^{(-)},v_{{\nu }_n})\right\}$, $u^{(\pm )}\in L^{\infty }(0,{\widehat{T}}_0;L^1(\mathbb{R}))$ be the subsequence and the limiting functions given in Lemma 6 and Proposition 5. Then, there exists $v\in L^{\infty }(0,{\widehat{T}}_0;L^1(\mathbb{R}))$, with $g(v)\in L^{\infty }(0,{\widehat{T}}_0;H^1(\mathbb{R}))$ and ${\left[g(v)\right]}_x\in L^{\infty }(S_{{\widehat{T}}_0})$, such that possibly up to a subsequence (not relabeled) there holds

$$v_{{\nu }_n}\to va.e. in S_{{\widehat{T}}_0},$$

(149)

$$g_{{\nu }_n}(v_{{\nu }_n})\to g(v),{\left[g_{{\nu }_n}(v_{{\nu }_n})\right]}_x\to {\left[g(v)\right]}_{x}a.e. in S_{{\widehat{T}}_0}.$$

(150)

Moreover, for $a.e.$ $\tau \in (0,{\widehat{T}}_0)$, we have

$$g_{{\nu }_n}(v_{{\nu }_n}(·,\tau ))\rightharpoonup g(v(·,\tau ))in{H}^1(\mathbb{R}).$$

(151)

Proof. 

Since ${\nu }_n\to 0^{+}$, there exists $n_0\in \mathbb{N}$ such that, for all $n>n_0$, there holds ${\nu }_n<{\nu }_{{\widehat{T}}_0}$, ${\nu }_{{\widehat{T}}_0}>0$ being defined in (112).

By (116), there exists $v\in L_{w*}^{\infty }(0,{\widehat{T}}_0;\mathcal{M}(\mathbb{R}))$ such that, possibly up to a subsequence (not relabeled), there holds

$$v_{{\nu }_n}\stackrel{*}{\rightharpoonup }v\mathrm{in}{L}_{w*}^{\infty }(0,{\widehat{T}}_0;\mathcal{M}(\mathbb{R})).$$

(152)

Let us prove that $v\in L^{\infty }(0,{\widehat{T}}_0;L^1(\mathbb{R}))$ and the convergences in (149)–(151) hold true. To this aim, let $k_0>0$ be arbitrarily fixed. For every $n>n_0$ and $(h,k)\in {\mathbb{R}}^2$, with $|k|<k_0$, set

$${\widehat{v}}_{{\nu }_n}(x,t):=v_{{\nu }_n}(x+h,t+k),{\widehat{u}}_{{\nu }_n}^{(\pm )}(x,t):=u_{{\nu }_n}^{(\pm )}(x+h,t+k)$$

($(x,t)\in S^{(k_0)}=\mathbb{R}\times (k_0,{\widehat{T}}_0-k_0)$). It can be easily seen that, arguing as in step 1 of the proof of Proposition 5, we obtain inequalities (132) and (136); i.e.,

$$\parallel {\widehat{v}}_{{\nu }_n}-v_{{\nu }_n}{\parallel }_{L^1(\mathbb{R})}\le L_{\beta }\left(\parallel {\widehat{u}}_{{\nu }_n}^{(+)}-u_{{\nu }_n}^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel {\widehat{u}}_{{\nu }_n}^{(-)}-u_{{\nu }_n}^{(-)}\parallel }_{L^1(\mathbb{R})}\right),$$

(153)

and

$$\begin{array}{ccc}\hfill & & {\int }_{\mathbb{R}}{[g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n})]}_x^{2}dx\le L_{\beta }{\parallel g_{{\nu }_n}({\widehat{v}}_{{\nu }_n})-g_{{\nu }_n}(v_{{\nu }_n})\parallel }_{L^{\infty }(\mathbb{R})}\hfill \\ & & \times \left(\parallel {\widehat{u}}_{{\nu }_n}^{(+)}-u_{{\nu }_n}^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel {\widehat{u}}_{{\nu }_n}^{(-)}-u_{{\nu }_n}^{(-)}\parallel }_{L^1(\mathbb{R})}\right)\hfill \\ & & \le {\displaystyle \frac{2L_{\beta }}{{\nu }_{{\widehat{T}}_0}}}\left(\parallel {\widehat{u}}_{{\nu }_n}^{(+)}-u_{{\nu }_n}^{(+)}{\parallel }_{L^1(\mathbb{R})}+{\parallel {\widehat{u}}_{{\nu }_n}^{(-)}-u_{{\nu }_n}^{(-)}\parallel }_{L^1(\mathbb{R})}\right)\hfill \end{array}$$

(154)

for $a.e.$ $t\in (k_0,{\widehat{T}}_0-k_0)$ (see also (118)). Integrating in $(k_0,{\widehat{T}}_0-k_0)$ the above inequalities yields

$$\begin{array}{ccc}\hfill & & {\int }_{k_0}^{{\widehat{T}}_0-k_0}{\int }_{\mathbb{R}}|v_{{\nu }_n}(x+h,t+k)-v_{{\nu }_n}(x,t)|dxdt={\int }_{k_0}^{{\widehat{T}}_0-k_0}{\parallel {\widehat{v}}_{{\nu }_n}-v_{{\nu }_n}\parallel }_{L^1(\mathbb{R})}dt\hfill \\ & & \le L_{\beta }{\int }_{k_0}^{{\widehat{T}}_0-k_0}{\int }_{\mathbb{R}}|u_{{\nu }_n}^{(+)}(x+h,t+k)-u_{{\nu }_n}^{(+)}(x,t)|dxdt\hfill \\ & & +L_{\beta }{\int }_{k_0}^{{\widehat{T}}_0-k_0}{\int }_{\mathbb{R}}|u_{{\nu }_n}^{(-)}(x+h,t+k))-u_{{\nu }_n}^{(-)}(x,t)|dxdt,\hfill \end{array}$$

(155)

and

$$\begin{array}{ccc}\hfill & & {\int }_{k_0}^{{\widehat{T}}_0-k_0}{\int }_{\mathbb{R}}{\left({\left[g_{{\nu }_n}(v_{{\nu }_n})\right]}_x(x+h,t+k)-{\left[g_{{\nu }_n}(v_{{\nu }_n})\right]}_x(x,t)\right)}^{2}dxdt\hfill \\ & & \le {\displaystyle \frac{2L_{\beta }}{{\nu }_{{\widehat{T}}_0}}}\left\{{\int }_{k_0}^{{\widehat{T}}_0-k_0}{\int }_{\mathbb{R}}|u_{{\nu }_n}^{(+)}(x+h,t+k)-u_{{\nu }_n}^{(+)}(x,t)|dxdt\right.\hfill \\ & & \left.+{\int }_{k_0}^{{\widehat{T}}_0-k_0}{\int }_{\mathbb{R}}|u_{{\nu }_n}^{(-)}(x+h,t+k)-u_{{\nu }_n}^{(-)}(x,t)|dxdt\right\}.\hfill \end{array}$$

(156)

By the above inequalities and (130), the following limits

$$\underset{(h,k)\to (0,0)}{lim}{\int }_{k_0}^{{\widehat{T}}_0-k_0}{\int }_{\mathbb{R}}|v_{{\nu }_n}(x+h,t+k)-v_{{\nu }_n}(x,t)|dxdt=0,$$

(157)

$$\underset{(h,k)\to (0,0)}{lim}{\int }_{k_0}^{{\widehat{T}}_0-k_0}{\int }_{\mathbb{R}}{\left({\left[g_{{\nu }_n}(v_{{\nu }_n})\right]}_x(x+h,t+k)-{\left[g_{{\nu }_n}(v_{{\nu }_n})\right]}_x(x,t)\right)}^{2}dxdt=0$$

(158)

hold uniformly for $n>n_0$. Moreover, it is worth recalling that the sequences $\left\{v_{{\nu }_n}\right\}$ and $\left\{{\left[g_{{\nu }_n}(v_{{\nu }_n})\right]}_x\right\}$ are bounded in $L^{\infty }(0,{\widehat{T}}_0;L^1(\mathbb{R}))$ and $L^{\infty }(S_{{\widehat{T}}_0})$, respectively (see (116) and (117)). Thus, by the Frechet–Kolmogorov compactness theorem, it follows that the sequences $\left\{v_{{\nu }_n}\right\}$ and $\left\{{\left[g_{{\nu }_n}(v_{{\nu }_n})\right]}_x\right\}$ are relatively compact in $L_{loc}^1(S^{(k_0)})$ and $L_{loc}^2(S^{(k_0)})$, respectively. As a consequence, by the arbitrariness of $k_0$, it can be easily proven that the limiting measure v in (152) belongs to $L^{\infty }(0,{\widehat{T}}_0;L^1(\mathbb{R}))$, and

$$v_{{\nu }_n}\to v\mathrm{in}{L}_{loc}^1(S_{{\widehat{T}}_0}),$$

(159)

$${\left[g_{{\nu }_n}(v_{{\nu }_n})\right]}_x\to {\left[g(v)\right]}_x\mathrm{in}{L}_{loc}^2(S_{{\widehat{T}}_0}),$$

(160)

whence also

$$g_{{\nu }_n}(v_{{\nu }_n})\to g(v)\mathrm{in}{L}_{loc}^1(S_{{\widehat{T}}_0})$$

(161)

(here, we have also used the Lipschitz continuity of g and the very definition of $g_{{\nu }_n}$; see (26)). Plainly, possibly extracting another subsequence, the above convergences imply (149) and (150).

Regarding (151), it suffices to observe that, by (150), there exists a null set $\mathcal{N}\subset (0,T)$ such that, for all $\tau \in (0,T)\setminus \mathcal{N}$, there holds $g_{{\nu }_n}(v_{{\nu }_n}(·,\tau ))\to g(v(·,\tau ))a.e. \mathrm{in} \mathbb{R}.$ Thus, the conclusion follows immediately from this convergence and the a priori estimate in (119). Finally, observe that, from the convergences in (150) and the a priori estimates (117) and (118), it easily follows that $g(v),{\left[g(v)\right]}_x\in L^{\infty }(S_{{\widehat{T}}_0})$ □

Proposition 7.

Let $U_0=(u_0^{(+)},u_0^{(-)})\in {\left(L^1(\mathbb{R})\right)}^2$ and $T_0\in (0,T]$ be as in (23). For every ${\widehat{T}}_0\in (0,T_0)$, let $(u_{{\nu }_n}^{(+)},u_{{\nu }_n}^{(-)},v_{{\nu }_n})$, $u^{(\pm )}$, v be the subsequences and the limiting functions given in Lemma 6 and Propositions 5 and 6. Then, there holds

$${\mu }_{{\nu }_n}^{(\pm )}(g_{{\nu }_n}(v_{{\nu }_n}),{\left[g_{{\nu }_n}(v_{{\nu }_n})\right]}_x,u_{{\nu }_n}^{(+)},u_{{\nu }_n}^{(-)})\to {\mu }^{(\pm )}(g(v),{\left[g(v)\right]}_x,u^{(+)},u^{(-)})$$

(162)

in $L^1(S_{{\widehat{T}}_0})$ and $a.e.$ in $S_{{\widehat{T}}_0}$.

Proof. 

Let ${\nu }_n\to 0^{+}$ be any sequence as in Lemma 6. By (130), possibly up to a subsequence (not relabeled), we have

$$u_{{\nu }_n}^{(\pm )}\to u^{(\pm )}a.e. \mathrm{in} S_{{\widehat{T}}_0}.$$

(163)

From the above convergence and (150), it follows that

$${\mu }_{{\nu }_n}^{(\pm )}(g_{{\nu }_n}(v_{{\nu }_n}),{\left[g_{{\nu }_n}(v_{{\nu }_n})\right]}_x,u_{{\nu }_n}^{(+)},u_{{\nu }_n}^{(-)})\to {\mu }^{(\pm )}(g(v),{\left[g(v)\right]}_x,u^{(+)},u^{(-)})a.e. \mathrm{in} S_{{\widehat{T}}_0};$$

(164)

here, we also hold that, for all sufficiently large n, there holds ${\nu }_n<{\nu }_{{\widehat{T}}_0}$ (see (112)); hence,

$${\mu }_{{\nu }_n}^{(\pm )}(g_{{\nu }_n}(v_{{\nu }_n}),{\left[g_{{\nu }_n}(v_{{\nu }_n})\right]}_x,u_{{\nu }_n}^{(+)},u_{{\nu }_n}^{(-)})={\mu }^{(\pm )}(g_{{\nu }_n}(v_{{\nu }_n}),{\left[g_{{\nu }_n}(v_{{\nu }_n})\right]}_x,u_{{\nu }_n}^{(+)},u_{{\nu }_n}^{(-)})$$

(165)

$a.e.$ in $S_{{\widehat{T}}_0}$ (see (74), (117) and (118)).

Let us address (162). For all sufficiently large n, by (9) (applied with $R={\displaystyle \frac{1}{{\nu }_{{\widehat{T}}_0}}}$), (117), (118) and (165), we have

$$\left|{\mu }_{{\nu }_n}^{(\pm )}(g_{{\nu }_n}(v_{{\nu }_n}),{\left[g_{{\nu }_n}(v_{{\nu }_n})\right]}_x,u_{{\nu }_n}^{(+)},u_{{\nu }_n}^{(-)})\right|\le {\displaystyle \frac{\widehat{\alpha }}{{\nu }_{{\widehat{T}}_0}}}\left(|u_{{\nu }_n}^{(+)}|+|u_{{\nu }_n}^{(-)}|\right)a.e. \mathrm{in} S_{{\widehat{T}}_0}.$$

(166)

Thus, the strong convergence in (162) follows from the generalized dominated convergence theorem, by the above inequality and the $a.e.$ convergence in (164), since the sequences $\left\{u_{{\nu }_n}^{(\pm )}\right\}$ are convergent in $L^1(S_{{\widehat{T}}_0})$ (see (130)). □

5. Main Results: Proofs

Proof of Proposition 1.

For every nonnegative $\rho \in C_c^1(\mathbb{R})$, choosing in (21) the test function $\zeta (x,t)=\rho (x)h_j(t)$, with $h_j$ as in (19), and arguing as in Remark 1-$(iii)$ yields

$$\begin{array}{ccc}\hfill & & {\int }_{\mathbb{R}}|u^{(\pm )}(x,\tau )-k|\rho (x)dx\hfill \\ & & \le {\int \int }_{S_{\tau }}\mathrm{sgn}(u^{(\pm )}-k)[{\phi }^{(\pm )}(u^{(\pm )})-{\phi }^{(\pm )}(k)]{\rho }^{\prime }(x)dxdt\hfill \\ & & +{\int \int }_{S_{\tau }}|{\mu }^{(\pm )}(\left[g(v)\right],{\left[g(v)\right]}_x,u^{(+)},u^{(-)})|\rho (x)dxdt+{\int }_{\mathbb{R}}|u_0^{(\pm )}-k|\rho (x)dx\hfill \end{array}$$

(167)

for all $\tau \in (0,{\tau }_0)\setminus {\mathcal{N}}_0$ for a suitable null set ${\mathcal{N}}_0\subset (0,{\tau }_0)$. Observe that, by separability arguments, the choice of ${\mathcal{N}}_0$ can be made independent of the test function $\rho \in C_c^1(\mathbb{R})$.

Fix any $\left\{{\tau }_n\right\}\subset (0,{\tau }_0)\setminus {\mathcal{N}}_0$, ${\tau }_n\to 0^{+}$. Let $\mathrm{\Omega }\subset \mathbb{R}$ be a bounded open interval fixed arbitrarily. For every $\rho \in C_c^1(\mathrm{\Omega })$, $\rho \ge 0$, choosing $\tau ={\tau }_n$ in (167) and setting $n\to \infty$, we get

$$\underset{n\to \infty }{lim\; sup}{\int }_{\mathbb{R}}|u^{(\pm )}(·,{\tau }_n)-k|\rho (x)dx\le {\int }_{\mathbb{R}}|u_0^{(\pm )}-k|\rho (x)dx$$

(168)

for all $k\in \mathbb{R}$. In the proof of the above convergence, we hold that

$$\underset{n\to \infty }{lim}{\int \int }_{S_{{\tau }_n}}\mathrm{sgn}(u^{(\pm )}-k)[{\phi }^{(\pm )}(u^{(\pm )})-{\phi }^{(\pm )}(k)]{\rho }^{\prime }(x)dxdt=0,$$

$$\underset{n\to \infty }{lim}{\int \int }_{S_{{\tau }_n}}|{\mu }^{(\pm )}(\left[g(v)\right],{\left[g(v)\right]}_x,u^{(+)},u^{(-)})|\rho (x)dxdt=0,$$

since ${\tau }_n\to 0^{+}$ and ${\phi }^{(\pm )}(u^{(\pm )}),{\mu }^{(\pm )}(\left[g(v)\right],{\left[g(v)\right]}_x,u^{(+)},u^{(-)})$ belong to $L^{\infty }(0,{\tau }_0;L^1(\mathbb{R}))$.

Let us now consider the sequences $\left\{{\nu }_n^{(\pm )}\right\}\subseteq \mathcal{Y}(\mathrm{\Omega }\times \mathbb{R})$ of Young measures associated with $\left\{u^{(\pm )}(·,{\tau }_n)\right\}$, respectively (e.g., see [33,34]). Since the latter are bounded in $L^1(\mathbb{R})$ (recall that $u^{(\pm )}\in L^{\infty }(0,{\tau }_0;L^1(\mathbb{R}))$), by the fundamental theorem for Young measures, there exist Young measures ${\nu }^{(\pm )}\in \mathcal{Y}(\mathrm{\Omega }\times \mathbb{R})$ such that, up to a subsequence (not relabeled), there holds

$${\nu }_n^{(\pm )}\to {\nu }^{(\pm )}\mathrm{narrowly} \mathrm{in} \mathrm{\Omega }\times \mathbb{R}$$

[33,34]. By the above convergences and standard lower-semicontinuity results on the narrow topology (see Theorem 6 of [34]), for any $k\in \mathbb{R}$ and nonnegative $\rho$ as above, there holds

$${\int }_{\mathrm{\Omega }}\left({\int }_{\mathbb{R}}|\xi -k|d{\nu }_{(x)}^{(\pm )}(\xi )\right)\rho (x)dx\le \underset{n\to \infty }{lim\; inf}{\int }_{\mathrm{\Omega }}|u^{(\pm )}(x,{\tau }_n)-k|\rho (x)dx;$$

here, for $a.e.$ $x\in \mathrm{\Omega }$, we have denoted by ${\nu }_{(x)}^{(\pm )}$ the disintegration of ${\nu }^{(\pm )}$ at x. Combining the above inequality with (168), we get

$${\int }_{\mathrm{\Omega }}\left({\int }_{\mathbb{R}}|\xi -k|d{\nu }_{(x)}^{(\pm )}(\xi )\right)\rho (x)dx\le {\int }_{\mathrm{\Omega }}|u_0^{(\pm )}-k|\rho (x)dx$$

for all $k\in \mathbb{R}$ and nonnegative $\rho \in C_c^1(\mathrm{\Omega })$. Thus, there exists a null set $E\subset \mathrm{\Omega }$ (independent of $k\in \mathbb{R}$ by separability arguments) such that ${\int }_{\mathbb{R}}|\xi -k|d{\nu }_{(x)}^{(\pm )}(\xi )\le |u_0^{(\pm )}(x)-k|$ for all $x\in \mathrm{\Omega }\setminus E$ and $k\in \mathbb{R}$. For any fixed x as above, choosing $k=u_0^{(\pm )}(x)$ in the previous inequalities yields ${\int }_{\mathbb{R}}|\xi -u_0^{(\pm )}(x)|d{\nu }_{(x)}^{(\pm )}(\xi )\le 0$, whence for $a.e.$ $x\in \mathrm{\Omega }$ the probability measure ${\nu }_{(x)}^{(\pm )}$ turns out to be the delta measure concentrated at the point $u_0^{(\pm )}(x)$. This implies that the sequences $\left\{u^{(\pm )}(·,{\tau }_n)\right\}$ converge to $u_0^{(\pm )}$ in measure (Proposition 1 of [34]), whence

$$u^{(\pm )}(·,{\tau }_n)\to u_0^{(\pm )}a.e. \mathrm{in} \mathrm{\Omega }$$

(169)

possibly up to a subsequence (not relabeled). Therefore, by the Fatou Lemma, we have

$${\int }_{\mathrm{\Omega }}|u_0^{(\pm )}(x)|\rho (x)dx\le \underset{n\to \infty }{lim\; inf}{\int }_{\mathrm{\Omega }}|u^{(\pm )}(x,{\tau }_n)|\rho (x)dx$$

for all nonnegative $\rho \in C_c(\mathbb{R})$. Combining this convergence and (168) (with $k=0$), it easily follows that

$$\underset{n\to \infty }{lim}{\int }_{\mathrm{\Omega }}|u^{(\pm )}(x,{\tau }_n)|\rho (x)dx={\int }_{\mathrm{\Omega }}|u_0^{(\pm )}(x)|\rho (x)dx$$

(170)

for every $\rho \in C_c^1(\mathrm{\Omega })$, $\rho \ge 0$. Plainly, relying on the $a.e.$ convergence in (169) and the convergence of the $L^1$-norm in (170), there holds

$$u^{(\pm )}(x,{\tau }_n)\rho \to u_0^{(\pm )}\rho \mathrm{in}{L}^1(\mathrm{\Omega })$$

(171)

for all $\rho$ as above. Moreover, since the limits $u_0^{(\pm )}$ are determined, it is easily seen that the above convergences hold true along the whole sequences $\left\{u^{(\pm )}(x,{\tau }_n)\right\}$. Then, (22) immediately follows from (171) by the arbitrariness of $\rho$ and $\mathrm{\Omega }$. □

Proof of Theorem 1.

Let $(u_i^{(+)},u_i^{(-)},v_i)$ be two entropy solutions of problem (1) in $S_{{\tau }_0}$ ($i=1,2$), with the same initial data $(u_0^{(+)},u_0^{(-)})\in {\left(L^1(\mathbb{R})\right)}^2$.

Let us apply the Kružkov method of doubling variables ([21]). To this aim, let $\zeta =\zeta (x,t,y,s)$ be defined in $S_{{\tau }_0}\times S_{{\tau }_0}$, $\zeta \ge 0$, and assume that $\zeta (·,·,y,s)\in C_c^1(S_{{\tau }_0})$ for every $(y,s)\in S_{{\tau }_0}$ and $\zeta (x,t,·,·)\in C_c^1(S_{{\tau }_0})$ for every $(x,t)\in S_{{\tau }_0}$.

We choose $k=u_2^{(+)}(y,s)$ and $k=u_2^{(-)}(y,s)$ in the entropy inequality (21) for $u_1^{(+)}$ and $u_1^{(-)}$, respectively. This yields

$$\begin{array}{ccc}\hfill & & {\int \int }_{S_{{\tau }_0}}\left\{|u_1^{(+)}(x,t)-u_2^{(+)}(y,s)|{\zeta }_t+\mathrm{sgn}(u_1^{(+)}(x,t)-u_2^{(+)}(y,s))\right.\hfill \\ & & \left.\times [{\phi }^{(+)}(u_1^{(+)}(x,t))-{\phi }^{(+)}(u_2^{(+)}(y,s))]{\zeta }_x\right\}dxdt\hfill \\ & & +{\int \int }_{S_{{\tau }_0}}{\mu }^{(+)}(\left[g(v_1)\right](x,t),{\left[g(v_1)\right]}_x(x,t),u_1^{(+)}(x,t),u_1^{(-)}(x,t))\hfill \\ & & \times \mathrm{sgn}(u_1^{(+)}(x,t)-u_2^{(+)}(y,s))\zeta dxdt\ge 0,\hfill \end{array}$$

(172)

and

$$\begin{array}{ccc}\hfill & & {\int \int }_{S_{{\tau }_0}}\left\{|u_1^{(-)}(x,t)-u_2^{(-)}(y,s)|{\zeta }_t+\mathrm{sgn}(u_1^{(-)}(x,t)-u_2^{(-)}(y,s))\right.\hfill \\ & & \left.\times [{\phi }^{(-)}(u_1^{(-)}(x,t))-{\phi }^{(-)}(u_2^{(-)}(y,s))]{\zeta }_x\right\}dxdt\hfill \\ & & +{\int \int }_{S_{{\tau }_0}}{\mu }^{(-)}(\left[g(v_1)\right](x,t),{\left[g(v_1)\right]}_x(x,t),u_1^{(+)}(x,t),u_1^{(-)}(x,t))\hfill \\ & & \times \mathrm{sgn}(u_1^{(-)}(x,t)-u_2^{(-)}(y,s))\zeta dxdt\ge 0.\hfill \end{array}$$

(173)

Similarly, choosing $k=u_1^{(+)}(x,t)$ and $k=u_1^{(-)}(x,t)$ in the entropy inequality (21) for $u_2^{(+)}$ and $u_2^{(-)}$, respectively, yields

$$\begin{array}{ccc}\hfill & & {\int \int }_{S_{{\tau }_0}}\left\{|u_2^{(+)}(y,s)-u_1^{(+)}(x,t)|{\zeta }_s+\mathrm{sgn}(u_2^{(+)}(y,s)-u_1^{(+)}(x,t))\right.\hfill \\ & & \left.\times [{\phi }^{(+)}(u_2^{(+)}(y,s))-{\phi }^{(+)}(u_1^{(+)}(x,t))]{\zeta }_y\right\}dyds\hfill \\ & +& {\int \int }_{S_{{\tau }_0}}{\mu }^{(+)}(\left[g(v_2)\right](y,s),{\left[g(v_2)\right]}_y(y,s),u_2^{(+)}(y,s),u_2^{(-)}(y,s))\hfill \\ & & \times \mathrm{sgn}(u_2^{(+)}(y,s)-u_1^{(+)}(x,t)))\zeta dyds\ge 0,\hfill \end{array}$$

(174)

and

$$\begin{array}{ccc}\hfill & & {\int \int }_{S_{{\tau }_0}}\left\{|u_2^{(-)}(y,s)-u_1^{(-)}(x,t)|{\zeta }_s+\mathrm{sgn}(u_2^{(-)}(y,s)-u_1^{(-)}(x,t))\right.\hfill \\ & & \left.\times [{\phi }^{(-)}(u_2^{(-)}(y,s))-{\phi }^{(-)}(u_1^{(-)}(x,t))]{\zeta }_y\right\}dyds\hfill \\ & +& {\int \int }_{S_{{\tau }_0}}{\mu }^{(-)}(\left[g(v_2)\right](y,s),{\left[g(v_2)\right]}_y(y,s),u_2^{(+)}(y,s),u_2^{(-)}(y,s))\hfill \\ & & \times \mathrm{sgn}(u_2^{(-)}(y,s)-u_1^{(-)}(x,t)))\zeta dyds\ge 0.\hfill \end{array}$$

Let ${\rho }_{ϵ}$ $(ϵ>0)$ be a symmetric mollifier in $\mathbb{R}$, and set in the previous inequalities

$$\zeta (x,t,y,s)=\eta \left({\displaystyle \frac{x+y}{2}},{\displaystyle \frac{t+s}{2}}\right){\rho }_{ϵ}(x-y){\rho }_{ϵ}(t-s)$$

(175)

with $\eta \in C_c^1(S_{{\tau }_0})$, $\eta \ge 0$. Summing up the above inequalities with the same upper signs “+” and “−”, respectively, yields (for all $ϵ>0$ small enough)

$$\begin{array}{ccc}\hfill & & {\int \int }_{S_{{\tau }_0}}{\int \int }_{S_{{\tau }_0}}{\rho }_{ϵ}(x-y){\rho }_{ϵ}(t-s)|u_1^{(\pm )}(x,t)-u_2^{(\pm )}(y,s)|{\eta }_t\left({\displaystyle \frac{x+y}{2}},{\displaystyle \frac{t+s}{2}}\right)dxdtdyds\hfill \\ & +& {\int \int }_{S_{{\tau }_0}}{\int \int }_{S_{{\tau }_0}}\mathrm{sgn}(u_1^{(\pm )}(x,t)-u_2^{(\pm )}(y,s))[{\phi }^{(\pm )}(u_1^{(\pm )}(x,t))-{\phi }^{(\pm )}(u_2^{(\pm )}(y,s))]\hfill \\ & & \times {\eta }_x\left({\displaystyle \frac{x+y}{2}},{\displaystyle \frac{t+s}{2}}\right){\rho }_{ϵ}(x-y){\rho }_{ϵ}(t-s)dxdtdyds\hfill \\ & +& {\int \int }_{S_{{\tau }_0}}{\int \int }_{S_{{\tau }_0}}\mathrm{sgn}(u_1^{(\pm )}(x,t)-u_2^{(\pm )}(y,s))\left({\mu }_1^{(\pm )}(x,t)-{\mu }_2^{(\pm )}(y,s)\right)\hfill \\ & & \times \eta \left({\displaystyle \frac{x+y}{2}},{\displaystyle \frac{t+s}{2}}\right){\rho }_{ϵ}(x-y){\rho }_{ϵ}(t-s)dxdtdyds\ge 0.\hfill \end{array}$$

(176)

Here, for simplicity of notations, we have set

$${\mu }_1^{(\pm )}(x,t)={\mu }^{(\pm )}(\left[g(v_1)\right](x,t),{\left[g(v_1)\right]}_x(x,t),u_1^{(+)}(x,t),u_1^{(-)}(x,t)),$$

$${\mu }_2^{(\pm )}(y,s)={\mu }^{(\pm )}(\left[g(v_2)\right](y,s),{\left[g(v_2)\right]}_y(y,s),u_2^{(+)}(y,s),u_2^{(-)}(y,s)).$$

By well-known properties of mollifiers, setting $ϵ\to 0^{+}$ in the above inequalities yields

$$\begin{array}{ccc}\hfill & & {\int \int }_{S_{{\tau }_0}}\left\{|u_1^{(\pm )}-u_2^{(\pm )}|{\eta }_t+\mathrm{sgn}(u_1^{(\pm )}-u_2^{(\pm )})[{\phi }^{(\pm )}(u_1^{(\pm )})-{\phi }^{(\pm )}(u_2^{(\pm )})]{\eta }_x\right\}dxdt\hfill \\ & +& {\int \int }_{S_{{\tau }_0}}\mathrm{sgn}(u_1^{(\pm )}-u_2^{(\pm )})\left({\mu }_1^{(\pm )}-{\mu }_2^{(\pm )}\right)\eta dxdt\ge 0,\hfill \end{array}$$

(177)

where

$${\mu }_i^{(\pm )}(x,t):={\mu }^{(\pm )}(\left[g(v_i)\right](x,t),{\left[g(v_i)\right]}_x(x,t),u_i^{(+)}(x,t),u_i^{(-)}(x,t))\mathrm{for}a.e.(x,t)\in S_{{\tau }_0}$$

($i=1,2$). Let

$$\widehat{R}\ge \sum _{i=1}^2\left(\parallel g(v_i){\parallel }_{L^{\infty }(S_{{\tau }_0})}+{\parallel {\left[g(v_i)\right]}_x\parallel }_{L^{\infty }(S_{{\tau }_0})}\right).$$

Applying (10) with $R=\widehat{R}$, for $a.e.$ $t\in (0,{\tau }_0)$, we get

$$\begin{array}{ccc}\hfill & & |{\mu }_1^{(\pm )}(·,t)-{\mu }_2^{(\pm )}(·,t)|\le \widehat{\alpha }\widehat{R}\left(|u_1^{(+)}(·,t)-u_2^{(+)}(·,t)|+|u_1^{(-)}(·,t)-u_2^{(-)}(·,t)|\right)\hfill \\ & & +\widehat{\alpha }\widehat{R}\left(\sum _{i=1}^2|u_i^{(+)}(·,t)|+|u_i^{(-)}(·,t)|\right)\hfill \\ & & \times \left(\parallel g(v_1(·,t))-g(v_2(·,t)){\parallel }_{L^{\infty }(\mathbb{R})}+{\parallel {\left[g(v_1(·,t))\right]}_x-{\left[g(v_2(·,t))\right]}_x\parallel }_{L^{\infty }(\mathbb{R})}\right)\hfill \end{array}$$

(178)

$a.e.$ in $\mathbb{R}$. For $a.e.$ $t\in (0,{\tau }_0)$, set $v_1=v_1(·,t)$, $v_2=v_2(·,t)$. Then, there holds $g(v_i)\in H^1(\mathbb{R})$, $v_i\in L^1(\mathbb{R})$ $(i=1,2$), and

$$-{[g(v_1)-g(v_2)]}_{xx}+(v_1-v_2)=\beta (u_1^{(+)}(·,t),u_1^{(-)}(·,t))-\beta (u_2^{(+)}(·,t),u_2^{(-)}(·,t))$$

in $H^{-1}(\mathbb{R})$. Plainly, arguing as in step 1 of the proof of Proposition 5, we get (see the proof of (134) and (135))

$$\begin{array}{ccc}\hfill & & \parallel {\left[g(v_1)\right]}_x-{\left[g(v_2)\right]}_x{\parallel }_{L^{\infty }(\mathbb{R})}\hfill \\ & & \le 2L_{\beta }\left(\parallel u_1^{(+)}(·,t)-u_2^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel u_1^{(-)}(·,t)-u_2^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right),\hfill \end{array}$$

(179)

$$\begin{array}{ccc}\hfill & & \parallel g(v_1)-g(v_2){\parallel }_{L^{\infty }(\mathbb{R})}\hfill \\ & & \le 2L_{\beta }(L_g+1)\left(\parallel u_1^{(+)}(·,t)-u_2^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel u_1^{(-)}(·,t)-u_2^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right).\hfill \end{array}$$

(180)

Using (179) and (180) in (178) yields

$$\begin{array}{ccc}\hfill & & |{\mu }_1^{(\pm )}(·,t)-{\mu }_2^{(\pm )}(·,t)|\le \widehat{\alpha }\widehat{R}\left(|u_1^{(+)}(·,t)-u_2^{(+)}(·,t)|+|u_1^{(-)}(·,t)-u_2^{(-)}(·,t)|\right)\hfill \\ & & +\widehat{\alpha }\widehat{R}\left(\sum _{i=1}^2|u_i^{(+)}(·,t)|+|u_i^{(-)}(·,t)|\right)\hfill \\ & & \times 2L_{\beta }(L_g+2)\left(\parallel u_1^{(+)}(·,t)-u_2^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel u_1^{(-)}(·,t)-u_2^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right)\hfill \end{array}$$

(181)

$a.e.$ in $\mathbb{R}$. Therefore, for every $\tau \in (0,{\tau }_0)$, there holds

$$\begin{array}{ccc}\hfill & & {\int \int }_{S_{\tau }}|{\mu }_1^{(\pm )}-{\mu }_2^{(\pm )}|dxdt\hfill \\ & & \le \widehat{\alpha }\widehat{R}{\int }_0^{\tau }\left(\parallel u_1^{(+)}(·,t)-u_2^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel u_1^{(-)}(·,t)-u_2^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right)dt\hfill \\ & & +\widehat{\alpha }\widehat{R}2L_{\beta }(L_g+2){\int }_0^{\tau }\left(\sum _{i=1}^2\parallel u_i^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel u_i^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right)\hfill \\ & & \times \left(\parallel u_1^{(+)}(·,t)-u_2^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel u_1^{(-)}(·,t)-u_2^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right)dt\hfill \\ & & \le C_1\left\{{\int }_0^{\tau }\left(\parallel u_1^{(+)}(·,t)-u_2^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel u_1^{(-)}(·,t)-u_2^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right)dt\right\};\hfill \end{array}$$

(182)

here, we have set $S_{\tau }:=\mathbb{R}\times (0,\tau )$ and

$$C_1:=\widehat{\alpha }\widehat{R}\left\{1+2L_{\beta }(L_g+2)\left(\sum _{i=1}^2\parallel u_i^{(+)}{\parallel }_{L^{\infty }(0,{\tau }_0;L^1(\mathbb{R}))}+{\parallel u_i^{(-)}\parallel }_{L^{\infty }(0,{\tau }_0;L^1(\mathbb{R}))}\right)\right\}.$$

Fix any $\tau$ as above and observe that, for all $\eta \in C_c^1(S_{\tau })$, $0\le \eta \le 1$, by (182), inequality (177) reads as

$$\begin{array}{ccc}\hfill & & -{\int \int }_{S_{\tau }}|u_1^{(\pm )}-u_2^{(\pm )}|{\eta }_{t}dxdt\le {\int \int }_{S_{\tau }}\mathrm{sgn}(u_1^{(\pm )}-u_2^{(\pm )})[{\phi }^{(\pm )}(u_1^{(\pm )})-{\phi }^{(\pm )}(u_2^{(\pm )})]{\eta }_{x}dxdt\hfill \\ & & +C_1\left\{{\int }_0^{\tau }\left(\parallel u_1^{(+)}(·,t)-u_2^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel u_1^{(-)}(·,t)-u_2^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right)dt\right\}.\hfill \end{array}$$

(183)

Next, for any $a,b\in \mathbb{R}$, with $a+L_{{\phi }^{(\pm )}}{\tau }_0<b-L_{{\phi }^{(\pm )}}{\tau }_0$ and $k\in \mathbb{N}$ large enough, set

$${\psi }_k^{(\pm )}(x,t)=\left\{\begin{array}{cc}k(x-a-t{L}_{{\phi }^{(\pm )}}+k^{-1})\hfill & \mathrm{for}a+t{L}_{{\phi }^{(\pm )}}-k^{-1}\le x\le a+t{L}_{{\phi }^{(\pm )}},\hfill \\ 1\hfill & \mathrm{for}a+t{L}_{{\phi }^{(\pm )}}<x<b-t{L}_{{\phi }^{(\pm )}},\hfill \\ k(b-t{L}_{{\phi }^{(\pm )}}+k^{-1}-x)\hfill & \mathrm{for}b-t{L}_{{\phi }^{(\pm )}}\le x\le b-t{L}_{{\phi }^{(\pm )}}+k^{-1},\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(here, $L_{{\phi }^{(\pm )}}$ are the Lipschitz constants of ${\phi }^{(\pm )}$, respectively). Given any $h\in C_0^1([0,\tau ])$, using $\eta (x,t)=h(t){\psi }_k^{(\pm )}(x,t)$ as test function in (183) (with upper signs “+” and “−”, respectively), we obtain

$$\begin{array}{ccc}& & -{\int \int }_{S_{\tau }}|u_1^{(\pm )}-u_2^{(\pm )}|h^{\prime }(t){\psi }_k^{(\pm )}(x,t)dxdt+k{L}_{{\phi }^{(\pm )}}{\int }_0^{\tau }dt{\int }_{a+t{L}_{{\phi }^{(\pm )}}-k^{-1}}^{a+t{L}_{{\phi }^{(\pm )}}}|u_1^{(\pm )}-u_2^{(\pm )}|h(t)dx\hfill \\ & & +k{L}_{{\phi }^{(\pm )}}{\int }_0^{\tau }dt{\int }_{b-t{L}_{{\phi }^{(\pm )}}}^{b-t{L}_{{\phi }^{(\pm )}}+k^{-1}}|u_1^{(\pm )}-u_2^{(\pm )}|h(t)dx\hfill \\ & & \le k{\int }_0^{\tau }dt{\int }_{a+t{L}_{{\phi }^{(\pm )}}-k^{-1}}^{a+t{L}_{{\phi }^{(\pm )}}}\mathrm{sgn}(u_1^{(\pm )}-u_2^{(\pm )})[{\phi }^{(\pm )}(u_1^{(\pm )})-{\phi }^{(\pm )}(u_2^{(\pm )})]h(t)dxdt\hfill \\ & & -k{\int }_0^{\tau }dt{\int }_{b-t{L}_{{\phi }^{(\pm )}}}^{b-t{L}_{{\phi }^{(\pm )}}+k^{-1}}\mathrm{sgn}(u_1^{(\pm )}-u_2^{(\pm )})[{\phi }^{(\pm )}(u_1^{(\pm )})-{\phi }^{(\pm )}(u_2^{(\pm )})]h(t)dxdt\hfill \\ & & +C_1\left\{{\int }_0^{\tau }\left(\parallel u_1^{(+)}(·,t)-u_2^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel u_1^{(-)}(·,t)-u_2^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right)dt\right\},\hfill \end{array}$$

whence (using the Lipschitz continuity of ${\phi }^{(\pm )}$ with Lipschitz constants $L_{{\phi }^{(\pm )}}$, respectively)

$$\begin{array}{ccc}& & -{\int \int }_{S_{\tau }}|u_1^{(\pm )}-u_2^{(\pm )}|h^{\prime }(t){\psi }_k^{(\pm )}(x,t)dxdt\hfill \\ & & \le C_1\left\{{\int }_0^{\tau }\left(\parallel u_1^{(+)}(·,t)-u_2^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel u_1^{(-)}(·,t)-u_2^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right)dt\right\}.\hfill \end{array}$$

Then, setting $k\to \infty$ in the above inequality, there holds

$$\begin{array}{ccc}\hfill & & -{\int }_0^{\tau }{\int }_{a+t{L}_{{\phi }^{(\pm )}}}^{b-t{L}_{{\phi }^{(\pm )}}}|u_1^{(\pm )}-u_2^{(\pm )}|h^{\prime }(t)dxdt\hfill \\ & & \le C_1\left\{{\int }_0^{\tau }\left(\parallel u_1^{(+)}(·,t)-u_2^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel u_1^{(-)}(·,t)-u_2^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right)dt\right\}.\hfill \end{array}$$

(184)

Next, for every $j\in \mathbb{N}$ large enough, set

$$h_j(t)=jt{\chi }_{[0,j^{-1})}(t)+{\chi }_{[j^{-1},\tau -j^{-1}]}(t)+j(\tau -t){\chi }_{[\tau -j^{-1},\tau ]}(t)(t\in [0,\tau ]).$$

Plainly, for $h(t)=h_j(t)$, inequality (184) can be rephrased as follows:

$$\begin{array}{c}j{\int }_{\tau -j^{-1}}^{\tau }\left({\int }_{a+\tau L_{{\phi }^{(\pm )}}}^{b-\tau L_{{\phi }^{(\pm )}}}|u_1^{(\pm )}(x,t)-u_2^{(\pm )}(x,t)|dx\right)dt\hfill \\ \le j{\int }_{\tau -j^{-1}}^{\tau }\left({\int }_{a+t{L}_{{\phi }^{(\pm )}}}^{b-t{L}_{{\phi }^{(\pm )}}}|u_1^{(\pm )}(x,t)-u_2^{(\pm )}(x,t)|dx\right)dt\hfill \\ \le j{\int }_0^{j^{-1}}\left({\int }_{a+t{L}_{{\phi }^{(\pm )}}}^{b-t{L}_{{\phi }^{(\pm )}}}|u_1^{(\pm )}(x,t)-u_2^{(\pm )}(x,t)|dx\right)dt\hfill \\ +C_1\left\{{\int }_0^{\tau }\left(\parallel u_1^{(+)}(·,t)-u_2^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel u_1^{(-)}(·,t)-u_2^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right)dt\right\}.\hfill \end{array}$$

(185)

Since $u_i^{(\pm )}\in L^{\infty }(0,{\tau }_0;L^1(\mathbb{R}))$ and satisfy the initial condition (22) with the same initial data, setting $j\to \infty$ in (185) yields

$$\begin{array}{ccc}& & {\int }_{a+\tau L_{{\phi }^{(\pm )}}}^{b-\tau L_{{\phi }^{(\pm )}}}|u_1^{(\pm )}(x,\tau )-u_2^{(\pm )}(x,\tau )|dx\hfill \\ & & \le C_1\left\{{\int }_0^{\tau }\left(\parallel u_1^{(+)}(·,t)-u_2^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel u_1^{(-)}(·,t)-u_2^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right)dt\right\}\hfill \end{array}$$

for $a.e.$ $\tau \in (0,{\tau }_0)$, whence (for $a\to -\infty$ and $b\to +\infty$)

$$\begin{array}{c}\parallel u_1^{(\pm )}(·,\tau )-u_2^{(\pm )}{(·,\tau )\parallel }_{L^1(\mathbb{R})}\hfill \\ \le C_1\left\{{\int }_0^{\tau }\left(\parallel u_1^{(+)}(·,t)-u_2^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel u_1^{(-)}(·,t)-u_2^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right)dt\right\}.\hfill \end{array}$$

(186)

Finally, summing up the above inequalities with upper signs “+” and “−” yields

$$\begin{array}{ccc}& & \parallel u_1^{(+)}(·,\tau )-u_2^{(+)}{(·,\tau )\parallel }_{L^1(\mathbb{R})}+{\parallel u_1^{(-)}(·,\tau )-u_2^{(-)}(·,\tau )\parallel }_{L^1(\mathbb{R})}\hfill \\ & & \le 2C_1\left\{{\int }_0^{\tau }\left(\parallel u_1^{(+)}(·,t)-u_2^{(+)}{(·,t)\parallel }_{L^1(\mathbb{R})}+{\parallel u_1^{(-)}(·,t)-u_2^{(-)}(·,t)\parallel }_{L^1(\mathbb{R})}\right)dt\right\}\hfill \end{array}$$

for $a.e.$ $\tau \in (0,{\tau }_0)$. Applying Gronwall’s lemma to the previous inequality, we get

$$\parallel u_1^{(+)}(·,\tau )-u_2^{(+)}{(·,\tau )\parallel }_{L^1(\mathbb{R})}+{\parallel u_1^{(-)}(·,\tau )-u_2^{(-)}(·,\tau )\parallel }_{L^1(\mathbb{R})}=0$$

for $a.e.$ $\tau \in (0,{\tau }_0)$, whence ${\int \int }_{S_{{\tau }_0}}\left(|u_1^{(+)}-u_2^{(+)}|+|u_1^{(-)}-u_2^{(-)}|\right)dxdt=0$. Clearly, this implies that $u_1^{(\pm )}=u_2^{(\pm )}$ $a.e.$ in $S_{{\tau }_0}$. Therefore, for $a.e.$ $t\in (0,{\tau }_0)$, the functions $v_1(·,t)$ and $v_2(·,t)$ satisfy the elliptic Equation (13) with the same data $\beta (u_1^{(+)}(·,t),u_1^{(-)}(·,t))=\beta (u_2^{(+)}(·,t),u_2^{(-)}(·,t))\in L^1(\mathbb{R})$ (also recall that both $v_i(·,t)$ belong to $L^1(\mathbb{R})$, with $g(v_i(·,t))\in H^1(\mathbb{R})$; $i=1,2$). Then, $g(v_1)(·,t)=g(v_2)(·,t)$ everywhere in $\mathbb{R}$ (see Remark 1-$(i)$), whence it easily follows that $v_1=v_2$ $a.e.$ in $S_{{\tau }_0}$ (recall that g is strictly increasing). This concludes the proof. □

Proof of Theorem 2.

$(i)$ Let $U_0=(u_0^{(+)},u_0^{(-)})\in {\left(L^1(\mathbb{R})\right)}^2$ and $T_0\in (0,T]$ be as in (23). Let ${\widehat{T}}_0\in (0,T_0)$ and $(u_{{\nu }_n}^{(+)},u_{{\nu }_n}^{(-)},v_{{\nu }_n})$, $u^{(\pm )}$, v be the subsequences and the limiting functions given in Lemma 6 and Propositions 5 and 6.

Observe for future reference that, by (130) and the Lipschitz continuity of ${\phi }^{(\pm )}$ and $\beta$, there holds

$${\phi }^{(\pm )}(u_{{\nu }_n}^{(\pm )})\to {\phi }^{(\pm )}(u^{(\pm )}),\beta (u_{{\nu }_n}^{(+)},u_{{\nu }_n}^{(-)})\to \beta (u^{(+)},u^{(-)})\mathrm{in}{L}^1(S_{{\widehat{T}}_0}),$$

(187)

whereas, from (129), we have

$$\beta (u_{{\nu }_n}^{(+)}(·,t),u_{{\nu }_n}^{(-)}(·,t))\to \beta (u^{(+)}(·,t),u^{(-)}(·,t))\mathrm{in}{L}^1(\mathbb{R})$$

(188)

for $a.e.$ $t\in (0,{\widehat{T}}_0)$. Moreover, in view of (159), it can be easily checked that, for every bounded interval $\mathrm{\Omega }\subset \mathbb{R}$ and for $a.e.$ t, there holds (possibly up to a subsequence not relabeled)

$$v_{{\nu }_n}(·,t)\to v(·,t)\mathrm{in}{L}_{loc}^1(\mathbb{R}).$$

(189)

Let us prove that $(u^{(+)},u^{(-)},v)$ is a weak solution of (1) in $S_{{\widehat{T}}_0}$ with initial data $U_0$. To this aim, observe that (11) and (12) follow from Propositions 5 and 6. Next, by the convergences in (84), (130), (162) and (187), setting $\nu ={\nu }_n\to 0^{+}$ in (75) (written with $\zeta \in C_c^2({\overline{S}}_{{\widehat{T}}_0})$, $\zeta (·,{\widehat{T}}_0)=0$) yields equalities (14). Furthermore, by (151), (188) and (189), for $a.e.$ $t\in (0,{\widehat{T}}_0)$, passing to the limit as $\nu ={\nu }_n\to 0^{+}$ in (69) proves that Equation (13) is satisfied in $\mathcal{D}(\mathbb{R})$ (hence in $H^{-1}(\mathbb{R})$).

Thus, in order to prove that $(u^{(+)},u^{(-)},v)$ is an entropy solution of (1) in $S_{{\widehat{T}}_0}$, with initial data $U_0$, it only remains to check inequality (21). The latter easily follows for every nonnegative $\zeta \in C_c^1({\overline{S}}_{{\widehat{T}}_0})$, with $\zeta (·,{\widehat{T}}_0)=0$, setting $\nu ={\nu }_n\to 0^{+}$ in (89) (written with $\tau ={\widehat{T}}_0$, $u_{\nu }^{(\pm )}=u_{{\nu }_n}^{(\pm )}$, $v_{\nu }=v_{{\nu }_n}$) by means of the convergences in (84), (130), (162) and (187).

$(ii)$

Let $U_0=(u_0^{(+)},u_0^{(-)})\in {\left(L^1(\mathbb{R})\right)}^2$ satisfy (25). This implies that $T_0=T$ (see (23)). Hence, by $(i)$, for every ${\widehat{T}}_0\in (0,T)$, there exists an entropy solution $({\widehat{u}}^{(+)},{\widehat{u}}^{(-)},\widehat{v})$ of (1) in $S_{{\widehat{T}}_0}$ with initial data $U_0$. In order to prove the existence of an entropy solution of (1) in $S=\mathbb{R}\times (0,T)$, it is enough to notice that, by construction, the above entropy solutions $({\widehat{u}}^{(+)},{\widehat{u}}^{(-)},\widehat{v})$ satisfy all the estimates in (114)–(119), with a constant on the right-hand side satisfying

$${\displaystyle \frac{1}{{\nu }_{{\widehat{T}}_0}}}\le {\displaystyle \frac{{\widehat{C}}_1(\parallel u_0^{(+)}{\parallel }_{L^1(\mathbb{R})}+\parallel u_0^{(-)}{\parallel }_{L^1(\mathbb{R})})}{1-2T\widehat{\alpha }{\widehat{C}}_1(\parallel u_0^{(+)}{\parallel }_{L^1(\mathbb{R})}+\parallel u_0^{(-)}{\parallel }_{L^1(\mathbb{R})})}}.$$

Since the right-hand side in the above inequalities is independent of ${\widehat{T}}_0\in (0,T)$ and uniqueness of entropy solutions to (1) in every $S_{{\widehat{T}}_0}$ is ensured by Theorem 1, the conclusion easily follows from standard prolongation techniques (we omit the details). □

6. Conclusions

In this paper, we investigate the well-posedness of a coupled hyperbolic–elliptic system as in (1) with $L^1$-initial data. Using a two-step approximation procedure involving pseudoparabolic regularization of the hyperbolic equations (see Section 3.1) and a suitable approximation of the initial data (see Section 4.1), we are able to overcome the analytical challenges arising from the lack of smoothness of the initial conditions and remove specific structural assumptions on the data functions ${\mu }^{(\pm )}(·,·,·,·)$. Regarding the latter, in the case where the elliptic Equation (4) is complemented with hyperbolic equations as in (5) and (6), we neglect in particular assumption $(A)$ on the nonnegativity of the turning rates${\gamma }^{(\pm )}(·,·)$, which does not fall within the model case considered in system (2) (see [1]). For initial data $U_0\in {(L^1(\mathbb{R}))}^2$, our approach enables us to establish the existence of either local or global entropy solutions to problem (1) (in the sense of Definition 2) depending on the “smallness” of the initial norm $\parallel U_0{\parallel }_{{(L^1(\mathbb{R}))}^2}$ (in this regard, see Theorem 2). Furthermore, the uniqueness of entropy solutions (see Theorem 1) follows from an adaptation of the classical doubling-variable technique for scalar conservation laws ([21]; see also [22,23,24,25,26,27,28] and the references therein). Our results provide an analytical foundation for basic issues (existence and uniqueness) regarding entropy solutions to coupled systems such as (1), which typically arise in biological contexts. This paves the way for a deeper understanding of their qualitative properties, such as the mechanism behind their global existence and blow-up in finite time.