Asymptotic Behavior of Solutions of Two-Species Chemotaxis System with Strong Competition

Abstract

This paper is concerned with a chemotaxis-competition system modeling the spatiotemporal evolution of two species that proliferate and compete according to Lotka–Volterra-type kinetics. We study the asymptotic behavior of solutions in the case of strong competition and show that they spatially segregate as the competition rate tends to infinity. Moreover, using a blow-up method, we obtain the uniform Hölder continuity of the solutions.

1. Introduction

In this paper, we consider the initial-boundary value problem

$$\left\{\begin{array}{cc}{u}_t=d_1▵u-{\chi }_1\nabla ·(u\nabla w)+{\mu }_{1}u(1-u-a_{1}v)\hfill & \mathrm{in}Q,\hfill \\ v_t=d_2▵v-{\chi }_2\nabla ·(v\nabla w)+{\mu }_{2}v(1-v-a_{2}u)\hfill & \mathrm{in}Q,\hfill \\ \tau w_t=d_3▵w-\gamma w+\alpha u+\beta v\hfill & \mathrm{in}Q,\hfill \\ {\displaystyle \frac{\partial u}{\partial \nu }}={\displaystyle \frac{\partial v}{\partial \nu }}={\displaystyle \frac{\partial w}{\partial \nu }}=0\hfill & \mathrm{on}\partial \Omega \times (0,+\infty ),\hfill \\ u(x,0)=u_0\left(x\right),v(x,0)=v_0\left(x\right),\tau w(x,0)=\tau w_0\left(x\right)\hfill & \mathrm{in}\Omega ,\hfill \end{array}\right.$$

(1)

where $Q=\Omega \times (0,+\infty )$, and $\Omega \subset {\mathbb{R}}^N(N\ge 1)$ is a bounded domain with smooth boundary $\partial \Omega$, and ${\displaystyle \frac{\partial }{\partial \nu }}$ denotes the outward normal derivative on $\partial \Omega$, $\tau \in \{0,1\}$. The parameters $d_1,d_2,d_3,{\chi }_1,{\chi }_2,{\mu }_1,{\mu }_2,\alpha ,\beta$ and $\gamma$ are positive, $u_0,v_0$ and $w_0$ are given non-negative functions.

System (1) arises in mathematical biology as a model for the spatiotemporal evolution of two populations that proliferate and compete according to Lotka–Volterra-type kinetics, and in which individuals disperse according to both random diffusion and chemotaxis toward a signal jointly produced by themselves. In this setting, $u=u(x,t)$ and $v=v(x,t)$ represent the respective densities of the two populations, and $w=w(x,t)$ denotes the concentration of the chemical. When $\tau =0$, system (1) represents a two-species chemotaxis-competition system of parabolic–parabolic–elliptic type; while $\tau =1$, system (1) becomes fully parabolic.

System (1) was first proposed by Tello–Winkler [1] and has been studied by many researchers (see, for example, [2,3,4,5,6,7,8,9]), and various existing results for system (1) have been developed. In particular, when ${\mu }_1={\mu }_2=0$, global existence and boundedness was obtained in [4,6,8], and the phenomenon of finite-time blow-up has been detected in [10,11,12,13,14]. The above results imply that without kinetic terms, system (1) inherits some important properties from the original Keller–Segel model for single-species chemotaxis. We refer to [15,16] for the discussions of single-species models, and [17] for a broader survey. When ${\mu }_1·{\mu }_2\ne 0$, it is proved that the conditions for global existence and boundedness of solutions to system (1) strongly depend on $a_1,a_2$, and have been obtained in the literature, see [1,5,18,19] for the parabolic–parabolic–elliptic case, and [17,20] for the fully parabolic case.

A further related problem is the study of the large time behavior of solutions to system (1). For the case of a parabolic–parabolic–elliptic system, it is proved in [18] that in the large time limit, ($u,v$) approaches to the constant vector (${\displaystyle \frac{1-a_1}{1-a_1{a}_2}},{\displaystyle \frac{1-a_2}{1-a_1{a}_2}}$) when $a_1,a_2\in (0,1)$, which implies that two competing species can coexist under weak competition. When $a_1>1>a_2$, or $a_1,a_2\ge 1$, the phenomenon of competitive exclusion occurs, i.e., $(u,v)\to (0,1)$ as $t\to \infty$, we refer to [17,20] and references therein for more details.

In this paper, we consider a different situation from the one obtained above, namely the case that both competitive effects in (1) are very strong in the sense that $a_1\gg 1$ and $a_2\gg 1$. To study this situation, it is convenient to rewrite (1) as

$$\left\{\begin{array}{cc}{u}_t=d_1▵u-{\chi }_1\nabla ·(u\nabla w)+{\mu }_{1}u(1-u)-kuv\hfill & \mathrm{in}Q,\hfill \\ v_t=d_2▵v-{\chi }_2\nabla ·(v\nabla w)+{\mu }_{2}v(1-v)-kbuv\hfill & \mathrm{in}Q,\hfill \\ \tau w_t=d_3▵w-\gamma w+\alpha u+\beta v\hfill & \mathrm{in}Q,\hfill \\ {\displaystyle \frac{\partial u}{\partial \nu }}={\displaystyle \frac{\partial v}{\partial \nu }}={\displaystyle \frac{\partial w}{\partial \nu }}=0\hfill & \mathrm{on}\partial \Omega \times (0,+\infty ),\hfill \\ u(x,0)=u_0\left(x\right),v(x,0)=v_0\left(x\right),\hfill & \mathrm{in}\Omega .\hfill \end{array}\right.$$

(2)

where k and b are positive constants. We assume that k is the only parameter, which is large, and all the other parameters are fixed constants. According to Gause’s principle of competitive exclusion, two competing species cannot coexist under strong competition. The migration or the spatial distribution changes the situation, and then all the species survive, but have disjoint habits, which is called spatial segregation [21]. For the case when ${\chi }_1={\chi }_2=0$ and $w=0$, system (2) is the classic Lotka–Volterra competition model:

$$\left\{\begin{array}{cc}{u}_t=d_1▵u+{\mu }_{1}u(1-u)-kuv\hfill & \mathrm{in}Q,\hfill \\ v_t=d_2▵v+{\mu }_{2}v(1-v)-kbuv\hfill & \mathrm{in}Q,\hfill \\ {\displaystyle \frac{\partial u}{\partial \nu }}={\displaystyle \frac{\partial v}{\partial \nu }}=0\hfill & \mathrm{on}\partial \Omega \times (0,+\infty ),\hfill \\ u(x,0)=u_0\left(x\right),v(x,0)=v_0\left(x\right),\hfill & \mathrm{in}\Omega .\hfill \end{array}\right.$$

(3)

Dancer et al. [22] show that as the interspecific competition rate k gets large, system (3) with non-negative initial data is expected to approach a limiting configuration where spatial segregation occurs. More precisely, they proved that for any $T>0$, there exist subsequences $u_{km}$ and $v_{km}$ of the k-dependent non-negative solutions converging in $L^2(\Omega \times (0,T))$ to a bounded state with disjoint support and solving a limiting free boundary problem. In the case of equal diffusion coefficients, Dancer and Zhang [23] proved that, for large k, the solutions approach stationary states as t tends to infinity. For more qualitative studies triggered by strong competition, we refer to [24,25,26,27,28] and references therein.

The purpose of this paper is to study the limit behavior of solutions in the case where the competition rate k tends to infinity. For simplicity, we let $d_i=1(i=1,2,3)$, $\alpha =\beta =\gamma =1$, and consider the parabolic–parabolic–elliptic case of (2):

$$\left\{\begin{array}{cc}{u}_t=▵u-{\chi }_1\nabla ·(u\nabla w)+{\mu }_{1}u(1-u)-kuv\hfill & \mathrm{in}Q,\hfill \\ v_t=▵v-{\chi }_2\nabla ·(v\nabla w)+{\mu }_{2}v(1-v)-kbuv\hfill & \mathrm{in}Q,\hfill \\ 0=▵w-w+u+v\hfill & \mathrm{in}Q,\hfill \\ {\displaystyle \frac{\partial u}{\partial \nu }}={\displaystyle \frac{\partial v}{\partial \nu }}={\displaystyle \frac{\partial w}{\partial \nu }}=0\hfill & \mathrm{on}\partial \Omega \times (0,+\infty ),\hfill \\ u(x,0)=u_0\left(x\right),v(x,0)=v_0\left(x\right)\hfill & \mathrm{in}\Omega ,\hfill \end{array}\right.$$

(4)

where we assume

$$\begin{array}{c}0\le u_0\in C\left(\overline{\Omega }\right)\setminus \left\{0\right\},0\le v_0\in C\left(\overline{\Omega }\right)\setminus \left\{0\right\}.\hfill \end{array}$$

(5)

By a k-dependent solution of (4), we mean a pair of non-negative functions ($u_k,v_k,w_k$) such that $(u_k,v_k,w_k)\in C(\overline{\Omega }\times [0,\infty ))\cap C^{2,1}(\overline{\Omega }\times (0,\infty ))$ and satisfy system (4). We assume further that there exists constant $M>0$, independent of k, such that

$${\parallel (u_k,v_k)\parallel }_{L^{\infty }\left(Q\right)}\le M.$$

(6)

This assumption is an a priori uniform $L^{\infty }$ bound, which can be obtained, for instance, under smallness conditions on the chemotactic sensitivities (${\chi }_i<{\mu }_i$, $i=1,2$) or when the logistic damping is sufficiently strong, as shown in the literature [5,20].

In comparison with the existing literature on chemotaxis-competition systems, our work tackles system (4) from a new perspective, i.e., the transient behavior of solutions driven by strong competition. Our main result is

Theorem 1.

Let $\left\{(u_k,v_k,w_k)\right\}$ be a family of non-negative solutions to (4). Then for every $T>0$, there exist functions $u,v\in L^{\infty }(\Omega \times (0,T))\cap L^2(0,T;H^1(\Omega ))$ and $w\in L^{\infty }(0,T;H^1(\Omega ))\cap L^2(0,T;H^2(\Omega ))$ such that, up to a subsequence,

$$u_k\to u,v_k\to v,w_k\to w\mathit{strongly}\mathit{in}{L}^2(0,T;H^1(\Omega )).$$

and, the limit $u,v$ satisfy a separation condition

$$uv=0a.e.\mathit{in}{Q}_T\mathit{and}u=y^{+},v=b{y}^{-},$$

where $(y,w)$ is a weak solution of the following problem

$$\left\{\begin{array}{cc}{y}_t=\nabla ·\left(\nabla y-h\left(y\right)\nabla w\right)+{\mu }_1{y}^{+}(1-y^{+})-{\mu }_2{y}^{-}(1-b{y}^{-})\hfill & \mathit{in}{Q}_T,\hfill \\ 0=▵w-w+y^{+}+b{y}^{-}\hfill & \mathit{in}{Q}_T,\hfill \\ {\displaystyle \frac{\partial y}{\partial \nu }}={\displaystyle \frac{\partial w}{\partial \nu }}=0\hfill & \mathit{on}\partial \Omega \times (0,T),\hfill \\ y(x,0)=u_0-{\displaystyle \frac{v_0}{b}}\hfill & \mathit{in}\Omega ,\hfill \end{array}\right.$$

(7)

with

$$\begin{array}{c}\hfill h\left(y\right)=\left\{\begin{array}{c}{\chi }_{1}y,y\ge 0,\hfill \\ {\chi }_{2}y,y<0.\hfill \end{array}\right.\end{array}$$

Moreover, $y\in C^{\lambda ,\frac{\lambda }{2}}(\Omega \times (0,T)),w\in C^{2+\lambda ,\frac{\lambda }{2}}(\Omega \times (0,T))$ for all $\lambda \in (0,1)$.

We point out that while spatial segregation limits have been established for the classical Lotka–Volterra competition system [22], the presence of chemotactic terms introduces additional nonlinear drift that complicates the derivation of uniform a priori estimates and the characterization of the limit problem. Precisely, we introduce auxiliary function $y_k=u_k-v_k/b$ to cancel the competition terms, and its limit y satisfies a scalar equation whose positive and negative parts correspond to the limiting densities of $u_k$ and $v_k$. The chemotactic effects persist in the limit through a drift coefficient $h\left(y\right)$ that is piecewise constant. This formulation is analogous to the classical segregation limit for the Lotka–Volterra competition system, but with an additional chemotactic flux. The biological insight is that the segregation property $uv=0$ means that under strong competition, the two species cannot coexist in the same location, and the limit equation for $(y,w)$ describes the evolution of the difference in densities, which determines the interface between the species. The chemotactic drift term biases the movement of each species toward the chemical signal, which in the limit affects the segregation interface.

However, from a rigorous mathematical point of view, the nature of this limit (not necessarily continuous) and the spatial separation have not been well understood so far. Using a blow-up method, developed by Terracini and her coauthors [24,29], where they give a uniform Hölder estimates for sequences of solutions to some singularly perturbed elliptic systems (see also Dancer, Wang and Zhang [30] for the corresponding parabolic case), we prove that $L^{\infty }$ boundedness implies $C^{\alpha }$ boundedness for each $\alpha \in (0,1)$, uniformly as $k\to +\infty$.

Theorem 2.

Let $\left\{(u_k,v_k,w_k)\right\}$ be any non-negative solutions of (4) and (6) holds. Then for every compact set $Q^{\prime }⋐Q$, $\alpha \in (0,1)$, there exists $C>0$, independent of k, such that

$$max\left\{{\parallel u_k\parallel }_{C^{\alpha ,\frac{\alpha }{2}}\left(Q^{\prime }\right)},{\parallel v_k\parallel }_{C^{\alpha ,\frac{\alpha }{2}}\left(Q^{\prime }\right)}\right\}\le C\mathrm{and}{\parallel w_k\parallel }_{C^{2+\alpha ,\frac{\alpha }{2}}\left(Q^{\prime }\right)}⩽C.$$

We note that the long-time behavior of (4) for k large (or its limit at $k=+\infty$) is not discussed in this paper. As mentioned above, when k is fixed, system (4) has three possible constant steady states, i.e., the extinction state $(0,0)$, semi-trivial steady states $(0,1)$ or $(1,0)$, and the co-existence state (${\displaystyle \frac{1-a_1}{1-a_1{a}_2}},{\displaystyle \frac{1-a_2}{1-a_1{a}_2}}$). A question which naturally arises is whether the solutions of (4) stabilize towards a stationary segregated state along some subsequences $t_j\to +\infty$ and $k_j\to +\infty$. In fact, in the Lotka–Volterra competition model (3), the long-time behavior was studied, and its analysis is difficult and requires some regularity results of the common zero set $\mathsf{\Gamma }=\{u=v=0\}$, see for example [23,25]. However, the long-time behavior of (4) both for k sufficiently large and $k\to +\infty$ remains a challenge; this will be the objective of a forthcoming paper.

We also note that the study of strong competition limits of elliptic or parabolic systems is of interest not only for questions of spatial segregation and co-existence in population dynamics, as here and the papers quoted above, but is also key to the understanding of phase separation in Hartree–Fock type approximations of systems of modeling Bose–Einstrin condensates, see [31,32,33,34] and the references therein.

Notations. We use the following notations throughout this paper.

• We write C or $C_1,C_2,\cdots$ for a generic constant.
• We use $Q_T$ to denote the cylindrical domain $\Omega \times (0,T]$.
• In ${\mathbb{R}}^N$, we let $B_R\left(x\right)$ denote the open ball centered at x with radius R, we abbreviate $B_R\left(0\right)$ to $B_R$.
• For $X=(x,t),Y=(y,s)\in {\mathbb{R}}^{N+1}$, the parabolic distance between X and Y is denote by

  $$d(X,Y):={max\left\{\right|x-y|}^2{,|t-s|\}}^{\frac{1}{2}},$$

  and $d(x,t)=d(X,0)$ is the distance to the origin. For $\gamma >0$, we define the parabolic dilating as $\gamma X=(\gamma x,{\gamma }^{2}t)$.
• We denote the parabolic cylinder $Q_R(x,t):=B_R\left(x\right)\times (t-R^2,t)$, and $Q_R:=B_R\left(0\right)\times (t-R^2,0)$.

Throughout this paper, we always assume that (5) and (6) hold. The rest of the paper is organized as follows: Section 2 is devoted to giving the a priori estimates. Section 3 deals with the limit problem as $k\to +\infty$. In Section 4, we prove the uniform Hölder bounds.

2. Some Preliminary Results

In this section, we will establish a priori bounds for the solution $(u_k,v_k,w_k)$ of the system (4) which are uniform with respect to the parameter k in the equations. This will enable us to study the asymptotic properties of the family of solutions $(u_k,v_k,w_k)$ for large values of k.

Lemma 1.

There exists $C>0$ independent of k such that if $(u_k,v_k,w_k)$ is a non-negative solution of (4) for some $k\in \mathbb{N}$, then for every $T>0$

$${\parallel u_k\parallel }_{L^2(0,T;H^1(\Omega ))}\le C,{\parallel v_k\parallel }_{L^2(0,T;H^1(\Omega ))}\le C and {\parallel w_k\parallel }_{L^{\infty }(0,T;H^1(\Omega ))}\le C.$$

Proof. 

We multiply the first equation in (4) by $u_k$ and integrate it over $\Omega$, obtaining

$$\begin{array}{ccc}& & k{\int }_{\Omega }{u}_k^2{v}_{k}dx+{\int }_{\Omega }{|\nabla u_k|}^{2}dx\hfill \\ & =& {\int }_{\Omega }-{\chi }_1(u_k\nabla u_k·\nabla w_k+u_k^2▵w_k)dx+{\int }_{\Omega }{\mu }_1{u}_k^2(1-u_k)dx-{\displaystyle \frac{1}{2}}{\int }_{\Omega }{\displaystyle \frac{d}{dt}}{u}_k^{2}dx\hfill \\ & =& -{\displaystyle \frac{1}{2}}{\int }_{\Omega }{\chi }_1{u}_k^2▵w_{k}dx+{\int }_{\Omega }{\mu }_1{u}_k^2(1-u_k)dx-{\displaystyle \frac{1}{2}}{\int }_{\Omega }{\displaystyle \frac{d}{dt}}{u}_k^{2}dx.\hfill \end{array}$$

Hence, we have

$${\int }_{\Omega }{|\nabla u_k|}^{2}dx\le {\int }_{\Omega }-{\displaystyle \frac{{\chi }_1}{2}}{u}_k^2(w_k-u_k-v_k)+{\mu }_1{u}_k^2(1-u_k)dx-{\displaystyle \frac{1}{2}}{\int }_{\Omega }{\displaystyle \frac{d}{dt}}{u}_k^{2}dx.$$

(8)

Integrating (8) over $(0,T)$ gives

$$\begin{array}{ccc}\hfill {\int }_0^T{\int }_{\Omega }{|\nabla u_k|}^{2}dxdt& \le & {\int }_0^T{\int }_{\Omega }-{\displaystyle \frac{{\chi }_1}{2}}{u}_k^2(w_k-u_k-v_k)+{\mu }_1{u}_k^2(1-u_k)dxdt\hfill \\ & & -{\displaystyle \frac{1}{2}}{\int }_{\Omega }{u}_k^2(x,T)-u_k^2(x,0)dx\le C,\hfill \end{array}$$

where we have used the $L^{\infty }$-boundedness of $u_k$ and $v_k$. The estimate for $v_k$ follows similarly.

Now, multiplying the equation for $w_k$ in (4) by $w_k$ and integrating it over $\Omega$ yields

$$\begin{array}{ccc}\hfill {\int }_{\Omega }{|\nabla w_k|}^2+w_k^{2}dx& =& {\int }_{\Omega }{w}_k{u}_k+w_k{v}_{k}dx\hfill \\ & \le & {\displaystyle \frac{1}{2}}{\int }_{\Omega }{w}_k^{2}dx+{\int }_{\Omega }{u}_k^2+v_k^{2}dx.\hfill \end{array}$$

Again, by the $L^{\infty }$-boundedness of $u_k$ and $v_k$, we have

$${\displaystyle \frac{1}{2}}{\int }_{\Omega }{|\nabla w_k|}^2+w_k^{2}dx\le C,$$

and we complete the proof of Lemma 1. □

Remark 1.

For all $p\in (1,+\infty )$, the standard elliptic regularity argument (see e.g., [35], Theorem 19.1) leads to the existence of a constant $C>0$, independent of k, such that

$$\begin{array}{c}\hfill {\parallel w(·,t)\parallel }_{W^{2,p}(\Omega )}\le C({\parallel u(·,t)\parallel }_{L^p(\Omega )}+{\parallel v(·,t)\parallel }_{L^p(\Omega )}),\end{array}$$

(9)

for all $t\in (0,T)$. Therefore, a combination of (9) with Lemma 1 yields that $w_k$ is uniformly bounded in $L^2(0,T;H^2(\Omega ))\cap L^{\infty }(0,T;H^1(\Omega ))$. Moreover, from the Sobolev embedding theorem, $w_k(·,t)$ is uniformly bounded in $C^{1+\lambda }(\Omega )$, for every $\lambda \in (0,1)$ and $t\in (0,T)$.

Lemma 2.

There exists $C>0$ independent of k such that if $(u_k,v_k,w_k)$ is a non-negative solution of (4) for some $k\in \mathbb{N}$, then

$$k{\int }_0^T{\int }_{\Omega }{u}_k{v}_{k}dxdt\le C.$$

Proof. 

Integrating the first equation in (4) over $\Omega \times [0,T]$ and using the homogeneous Neumann boundary conditions for $u_k$ and $w_k$ yield

$$\begin{array}{ccc}\hfill k{\int }_0^T{\int }_{\Omega }{u}_k{v}_{k}dxdt& =& {\int }_0^T{\int }_{\partial \Omega }{\displaystyle \frac{\partial u_k}{\partial \nu }}dSdt-{\chi }_1{\int }_0^T{\int }_{\partial \Omega }{u}_k{\displaystyle \frac{\partial w_k}{\partial \nu }}dSdt\hfill \\ & & +{\int }_0^T{\int }_{\Omega }{\mu }_1{u}_k(1-u_k)dxdt-{\int }_{\Omega }{u}_k(x,T)dx+{\int }_{\Omega }{u}_k(x,0)dx\hfill \\ & =& {\int }_0^T{\int }_{\Omega }{\mu }_1{u}_k(1-u_k)dxdt-{\int }_{\Omega }{u}_k(x,T)dx+{\int }_{\Omega }{u}_k(x,0)dx\hfill \\ & \le & C,\hfill \end{array}$$

(10)

where we have used the $L^{\infty }$-boundedness of $u_k$. □

Next, we consider the function

$$y_k:=u_k-{\displaystyle \frac{v_k}{b}},$$

which appears when we eliminate the terms involving k from (4). It satisfies

$$\left\{\begin{array}{cc}{\displaystyle \frac{\partial y_k}{\partial t}}=▵y_k-{\chi }_1\nabla ·(u_k\nabla w_k)+{\chi }_2\nabla ·({\displaystyle \frac{v_k}{b}}\nabla w_k)\hfill & \\ +{\mu }_1{u}_k(1-u_k)-{\displaystyle \frac{{\mu }_2}{b}}{v}_k(1-v_k)\hfill & \mathrm{in}Q,\hfill \\ 0=▵w_k-w_k+u_k+v_k\hfill & \mathrm{in}Q,\hfill \\ {\displaystyle \frac{\partial y_k}{\partial \nu }}={\displaystyle \frac{\partial w_k}{\partial \nu }}=0\hfill & \mathrm{on}\partial \Omega \times (0,+\infty ),\hfill \\ y_k(x,0)=u_0-{\displaystyle \frac{v_0}{b}}\hfill & \mathrm{in}\Omega .\hfill \end{array}\right.$$

(11)

Lemma 3.

The sequence $\left\{{\displaystyle \frac{\partial y_k}{\partial t}}\right\}$ is bounded in $L^2(0,T;{\left(H^1(\Omega )\right)}^{\prime })$, uniformly with respect to k.

Proof. 

Multiplying the equation for $y_k$ by $\xi \in L^2(0,T;H^1(\Omega ))$ and integrating it over $Q_T:=\Omega \times (0,T)$, we obtain

$$\begin{array}{ccc}\hfill {\int }_0^T\langle {\displaystyle \frac{\partial y_k}{\partial t}},\xi \rangle dt& =& {\int }_0^T{\int }_{\Omega }\left[-{\chi }_1\nabla ·(u_k\nabla w_k)+{\mu }_1{u}_k(1-u_k)\right]\xi dxdt\hfill \\ & & +{\int }_0^T{\int }_{\Omega }\left[{\displaystyle \frac{{\chi }_2}{b}}\nabla ·(v_k\nabla w_k)-{\displaystyle \frac{{\mu }_2}{b}}{v}_k(1-v_k)\right]\xi dxdt\hfill \\ & & +{\int }_0^T{\int }_{\Omega }\nabla y_k·\nabla \xi dxdt,\hfill \end{array}$$

where $\langle ·,·\rangle$ is the duality product between the space $H^1(\Omega )$ and ${\left(H^1(\Omega )\right)}^{\prime }$. Hence, by Lemma 2,

$$\left|{\int }_0^T\langle {\displaystyle \frac{\partial y_k}{\partial t}},\xi \rangle dt\right|\le C{\parallel \xi \parallel }_{L^2(0,T;H^1(\Omega ))},$$

in which C is a positive constant, which does not depend on k or $\xi$. This means that

$${∥{\displaystyle \frac{\partial y_k}{\partial t}}∥}_{L^2(0,T;H^{-1}(\Omega ))}\le C,$$

and we complete the proof of the lemma. □

3. Characterization of the $\mathit{k}\mathbf{\to }\mathbf{\infty }$ Limit

We deduce from Lemmas 1 and 3 and Remark 1 that the families $\left\{u_k\right\},\left\{v_k\right\}$ are bounded in $L^2(0,T;H^1(\Omega ))$, $\left\{w_k\right\}$ is bounded in $L^{\infty }(0,T;H^1(\Omega ))\cap L^2(0,T;H^2(\Omega ))$ and that the family $\left\{y_k\right\}$ is precompact in $L^2\left(Q_T\right)$. Thus, there exist functions $u,v\in L^2(0,T;H^1(\Omega ))$ and $w\in L^{\infty }(0,T;H^1(\Omega ))\cap L^2(0,T;H^2(\Omega ))$ such that, up to a subsequence,

$$u_k\rightharpoonup u,v_k\rightharpoonup v\mathrm{in}{L}^2(0,T;H^1(\Omega ))\mathrm{and}\mathrm{strongly}\mathrm{in}{L}^2(0,T;L^2(\Omega ))$$

(12)

and

$$w_k\rightharpoonup w\mathrm{weakly}\mathrm{in}{L}^2(0,T;H^2(\Omega ))\mathrm{and}\mathrm{strongly}\mathrm{in}{L}^2(0,T;H^1(\Omega )).$$

(13)

Moreover,

$$y_k=u_k-{\displaystyle \frac{v_k}{b}}\to y\mathrm{in}{L}^2(0,T;L^2(\Omega ))\mathrm{and}\mathrm{a}.\mathrm{e}.\mathrm{in}{Q}_T.$$

(14)

Combining (12) and (14) with Lemma 2, we have a basic segregation result:

$$uv=0\mathrm{a}.\mathrm{e}.\mathrm{in}{Q}_T\mathrm{and}u=y^{+},v=b{y}^{-},$$

(15)

where $y^{+}:=max\{y,0\}$, $y^{-}:=max\{0,-y\}$.

Under the previous notations, we aim to derive the limit problem for the singular limit and complete the proof of Theorem 1. We note that the auxiliary function $y_k=u_k-v_k/b$ introduced in Section 2 cancels the competition terms. From (15) we have $y=u-{\displaystyle \frac{v}{b}}$, whose positive and negative parts correspond to the limiting densities of $u_k$ and $v_k$. Moreover, we can expect from (11) that the limit function pair $(y,w)$ satisfies the following limit problem:

$$\left\{\begin{array}{cc}{y}_t=\nabla ·\left(\nabla y-h\left(y\right)\nabla w\right)+{\mu }_1{y}^{+}(1-y^{+})-{\mu }_2{y}^{-}(1-b{y}^{-})\hfill & \mathrm{in}{Q}_T,\hfill \\ 0=▵w-w+y^{+}+b{y}^{-}\hfill & \mathrm{in}{Q}_T,\hfill \\ {\displaystyle \frac{\partial y}{\partial \nu }}={\displaystyle \frac{\partial w}{\partial \nu }}=0\hfill & \mathrm{on}\partial \Omega \times (0,T),\hfill \\ y(x,0)=u_0-{\displaystyle \frac{v_0}{b}}\hfill & \mathrm{in}\Omega ,\hfill \end{array}\right.$$

(16)

where

$$\begin{array}{c}\hfill h\left(y\right)=\left\{\begin{array}{c}{\chi }_{1}y,y\ge 0,\hfill \\ {\chi }_{2}y,y<0.\hfill \end{array}\right.\end{array}$$

We will show in this section that the above expectation is true. To begin with, we define a weak solution of (16).

Definition 1.

Let $T>0$ be arbitrary. A function pair $(y,w)$ is a weak solution of (16) if

(i) $y\in L^{\infty }\left(Q_T\right)\cap L^2(0,T;H^1(\Omega ))\cap C([0,T];L^2(\Omega ))$ and $w\in L^2(0,T;H^1(\Omega ))$;

(ii) there hold

$$\begin{array}{ccc}& & -\int {\int }_{Q_T}y{\phi }_{t}dxdt-{\int }_{\Omega }{y}_0\phi (x,0)dx\hfill \\ & =& \int {\int }_{Q_T}-\left(\nabla y-h\left(y\right)\nabla w\right)·\nabla \phi +\left[{\mu }_1{y}^{+}(1-y^{+})-{\mu }_2{y}^{-}(1-b{y}^{-})\right]\phi dxdt,\hfill \end{array}$$

and

$$\int {\int }_{Q_T}\nabla w·\nabla \phi dxdt=\int {\int }_{Q_T}(-w+y^{+}+b{y}^{-})\phi dxdt,$$

for all $\phi \in {\mathcal{F}}_T$ where

$${\mathcal{F}}_T:=\left\{\phi \in C^{2,1}(\overline{\Omega }\times [0,T])|\phi (·,T)=0\mathit{in}\Omega \right\}.$$

The proof of Theorem 1 is based on several lemmas.

Lemma 4.

The function pair $(y,w)$ is a weak solution of (16). Moreover,

$$y\in L^2(0,T;H^2(\Omega ))\cap H^1(0,T;L^2(\Omega )),w\in L^2(0,T;H^4(\Omega ))\cap H^1(0,T;H^2(\Omega )).$$

(17)

Proof. 

Since $y^{+}=u$ and $y^{-}={\displaystyle \frac{v}{b}}$, it then follows from Lemma 2 that $y\in L^{\infty }\left(Q_T\right)\cap L^2(0,T;H^1(\Omega ))$ and from Lemma 3 that $y_t\in L^2(0,T;{\left(H^1(\Omega )\right)}^{\prime })$. A standard regularity result asserts that $y\in C([0,T];L^2(\Omega ))$.

Multiplying the equations for $y_k,w_k$ in (11) by a testing function $\phi \in {\mathcal{F}}_T$, and integrating by parts, we obtain the equalities

$$\begin{array}{ccc}\hfill -\int {\int }_{Q_T}{y}_k{\phi }_t-\nabla y_k·\nabla \phi dxdt& =& \int {\int }_{Q_T}[-{\chi }_1(\nabla u_k·\nabla w_k+u_k▵w_k)\hfill \\ & & +{\displaystyle \frac{{\chi }_2}{b}}(\nabla v_k·\nabla w_k+v_k▵w_k)+{\mu }_1{u}_k(1-u_k)\hfill \\ & & -{\displaystyle \frac{{\mu }_2}{b}}{v}_k(1-v_k)]\phi dxdt+{\int }_{\Omega }{y}_k(x,0)\phi (x,0)dx.\hfill \end{array}$$

And

$$\int {\int }_{Q_T}\nabla w_k·\nabla \phi dxdt=\int {\int }_{Q_T}-w_k\phi +u_k\phi +v_k\phi dxdt.$$

Let $k\to +\infty$,

$$\begin{array}{ccc}\hfill -\int {\int }_{Q_T}y{\phi }_t-\nabla y·\nabla \phi dxdt& =& \int {\int }_{Q_T}[-{\chi }_1(\nabla u·\nabla w+u▵w)\hfill \\ & & +{\displaystyle \frac{{\chi }_2}{b}}(\nabla v·\nabla w+v▵w)+{\mu }_{1}u(1-u)\hfill \\ & & -{\displaystyle \frac{{\mu }_2}{b}}v(1-v)]\phi dxdt+{\int }_{\Omega }{y}_0\phi (x,0)dx,\hfill \end{array}$$

(18)

and

$$\int {\int }_{Q_T}\nabla w·\nabla \phi dxdt=\int {\int }_{Q_T}-w\phi +u\phi +v\phi dxdt.$$

Hence, (16) follows by replacing u and v with $y^{+}$ and $b{y}^{-}$.

Moreover, we observe that $(y,w)$ satisfies the differential equations in (15) as well as the homogeneous Neumann boundary condition and the initial condition in the sense of distributions, and we complete the proof of the first part of Lemma 4. The right hand side of the differential equation about y belongs to $L^2(0,T;L^2(\Omega ))$, which implies that $y\in L^2(0,T;H^2(\Omega ))\cap H^1(0,T;L^2(\Omega ))\cap C([0,T];L^2(\Omega ))$ and $w\in L^2(0,T;H^4(\Omega ))\cap H^1(0,T;H^2(\Omega ))$. □

Theorem 3.

Let $\left\{u_k\right\},\left\{v_k\right\},\left\{y_k\right\},\left\{w_k\right\}$ and $u,v,y,w$ be as the statements in (12)–(14). Then, up to a subsequence,

$$\nabla u_k\to \nabla u,\nabla v_k\to \nabla vin{L}^2(0,T;L^2(\Omega )).$$

Proof. 

By Lemma 4 we have

$$y\in H^1(0,T;L^2(\Omega ))\cap C([0,T];L^2(\Omega )).$$

Hence,

$$u=y^{+}={\displaystyle \frac{y+\left|y\right|}{2}}\in H^1(0,T;L^2(\Omega ))\cap C([0,T];L^2(\Omega )),$$

and it follows from Lemma 7.7 in [36] that

$${\displaystyle \frac{\partial u}{\partial t}}={\displaystyle \frac{\partial \left(y^{+}\right)}{\partial t}}\in L^2(0,T;L^2(\Omega )).$$

Note that it follows from $uv=0$ a.e. in $Q_T$ and [36] (Lemma 7.7) that

$$\begin{array}{c}\hfill \nabla u·\nabla v=0a.e.\mathrm{in}{Q}_T.\end{array}$$

Now, multiplying the equation containing $▵v^k$ in (4) by the limit u and integration it over $\Omega \times (0,\tau ),\tau \in (0,T)$, we have

$${\int }_0^{\tau }{\int }_{\Omega }{\displaystyle \frac{\partial v_k}{\partial t}}u-▵v_{k}u+{\chi }_2\nabla ·(v_k\nabla w_k)u-{\mu }_2{v}_k(1-v_k)u+kb{u}_k{v}_{k}udxdt=0.$$

An integrating by parts asserts

$$\begin{array}{c}\hfill {\int }_0^{\tau }{\int }_{\Omega }-v_k{\displaystyle \frac{\partial u}{\partial t}}+\nabla v_k·\nabla u+{\chi }_2\nabla ·(v_k\nabla w_k)u-{\mu }_2{v}_k(1-v_k)u+kb{u}_k{v}_{k}udxdt\\ \hfill +{\int }_{\Omega }{v}_k(x,\tau )u(x,\tau )-v_k(x,0)u_{0}dx=0.\end{array}$$

(19)

Integrating (19) with respect to $\tau$ over (0, T) gives

$$\begin{array}{c}\hfill {\int }_0^T{\int }_0^{\tau }{\int }_{\Omega }-v_k{\displaystyle \frac{\partial u}{\partial t}}+\nabla v_k·\nabla u+{\chi }_2\nabla ·(v_k\nabla w_k)u-{\mu }_2{v}_k(1-v_k)u+kb{u}_k{v}_{k}udxdtd\tau \\ \hfill +{\int }_0^T{\int }_{\Omega }{v}_k(x,\tau )u(x,\tau )-v_k(x,0)u_{0}dxd\tau =0.\end{array}$$

It follows from (12) that, as $k\to \infty$,

$${\int }_0^T{\int }_{\Omega }{v}_k(x,\tau )u(x,\tau )dxd\tau \to {\int }_0^T{\int }_{\Omega }v(x,\tau )u(x,\tau )dxd\tau =0,$$

and

$${\int }_0^T{\int }_{\Omega }{v}_k(x,0)u(x,0)dxd\tau \to {\int }_0^T{\int }_{\Omega }{v}_0{u}_{0}dxd\tau =0,$$

by condition (5). Since ${\displaystyle \frac{\partial u}{\partial t}}$ is bounded in $L^2(0,T;\Omega )$ and $\nabla u·\nabla v=0$ in $Q_T$, we apply the Fubini theorem to obtain, as $k\to \infty$,

$$\begin{array}{ccc}\hfill {\int }_0^T{\int }_0^{\tau }{\int }_{\Omega }{v}_k{\displaystyle \frac{\partial u}{\partial t}}dxdtd\tau & =& {\int }_0^T(T-t){\int }_{\Omega }{v}_k{\displaystyle \frac{\partial u}{\partial t}}dxdt\hfill \\ & ⩽& T\left|{\int }_0^T{\int }_{\Omega }{v}_k{\displaystyle \frac{\partial u}{\partial t}}dxdt\right|\hfill \\ & \to & T\left|{\int }_0^T{\int }_{\Omega }v{\displaystyle \frac{\partial u}{\partial t}}dxdt\right|=0.\hfill \end{array}$$

Similarly, we have

$${\int }_0^T{\int }_0^{\tau }{\int }_{\Omega }\nabla v_k·\nabla udxdtd\tau \to 0,$$

$${\int }_0^T{\int }_0^{\tau }{\int }_{\Omega }{\chi }_2\nabla ·(v_k\nabla w_k)udxdtd\tau \to 0,$$

and

$${\int }_0^T{\int }_0^{\tau }{\int }_{\Omega }{\mu }_2{v}_k(1-v_k)udxdtd\tau \to 0.$$

So that, as $k\to \infty$,

$$\begin{array}{c}\hfill k{\int }_0^T{\int }_0^{\tau }{\int }_{\Omega }{u}_k{v}_{k}udxdtd\tau \to 0,\end{array}$$

(20)

from which it follows that we also have

$$k{\int }_0^T{\int }_0^{\tau }{\int }_{\Omega }{u}_k{v}_{k}vdxdtd\tau \to 0.$$

Then by multiplying the equation containing $u_k$ in (4) by the limit u and integration it over $\Omega \times (0,\tau )$, we have

$$\begin{array}{c}\hfill {\int }_0^{\tau }{\int }_{\Omega }-u_k{\displaystyle \frac{\partial u}{\partial t}}+\nabla u_k·\nabla udxdt+{\int }_{\Omega }{u}_k(x,\tau )u(x,\tau )-u_k(x,0)u_{0}dx\\ \hfill ={\int }_0^{\tau }{\int }_{\Omega }{\mu }_1{u}_k(1-u_k)u-{\chi }_1\nabla ·(u_k\nabla w_k)u-k{u}_k{v}_{k}udxdt.\end{array}$$

(21)

Integrating (21) with respect to $\tau$ over $(0,T)$ and passing to the limit as $k\to \infty$ yield

$$\begin{array}{ccc}\hfill {\int }_0^T{\int }_0^{\tau }{\int }_{\Omega }{|\nabla u|}^{2}dxdtd\tau & =& {\int }_0^T{\int }_0^{\tau }{\int }_{\Omega }{\mu }_1{u}^2(1-u)-{\chi }_1\nabla ·(u\nabla w)udxdtd\tau \hfill \\ & & -{\displaystyle \frac{1}{2}}{\int }_0^T{\int }_{\Omega }{u}^2(x,\tau )-u_0^{2}dxd\tau ,\hfill \end{array}$$

(22)

in which we have used (20).

Finally, multiplying the equation containing $▵u_k$ in (4) by $u_k$ and integration it over $\Omega \times (0,T)$, we have

$${\int }_0^{\tau }{\int }_{\Omega }{\displaystyle \frac{1}{2}}{\displaystyle \frac{\partial }{\partial t}}{u}_k^2+{|\nabla u_k|}^2+k{u}_k^2{v}_{k}dxdt={\int }_0^{\tau }{\int }_{\Omega }{\mu }_1{u}_k^2(1-u_k)-{\chi }_1\nabla ·(u_k\nabla w_k)u_{k}dxdt.$$

Therefore we have

$$\begin{array}{ccc}\hfill {\int }_0^T{\int }_0^{\tau }{\int }_{\Omega }{|\nabla u_k|}^{2}dxdtd\tau & ⩽& {\int }_0^T{\int }_0^{\tau }{\int }_{\Omega }{\mu }_1{u}_k^2(1-u_k)-{\chi }_1\nabla ·(u_k\nabla w_k)u_{k}dxdtd\tau \hfill \\ & & -{\displaystyle \frac{1}{2}}{\int }_0^T{\int }_{\Omega }{u}^2(x,\tau )-u_0^{2}dxd\tau \hfill \\ & \to & {\int }_0^T{\int }_0^{\tau }{\int }_{\Omega }{\mu }_1{u}^2(1-u)-{\chi }_1\nabla ·(u\nabla w)udxdtd\tau \hfill \\ & & -{\displaystyle \frac{1}{2}}{\int }_0^T{\int }_{\Omega }{u}^2(x,\tau )-u_0^{2}dxd\tau \hfill \\ & =& {\int }_0^T{\int }_0^{\tau }{\int }_{\Omega }{|\nabla u|}^{2}dxdtd\tau ,\hfill \end{array}$$

(23)

by (22). Since (12) and weak lower semi-continuity yields

$${\int }_0^{\tau }{\int }_{\Omega }{|\nabla u|}^{2}dxdt⩽\underset{k\to \infty }{lim\; inf}{\int }_0^{\tau }{\int }_{\Omega }{|\nabla u_k|}^{2}dxdt.$$

By Fatou’s lemma, we have

$${\int }_0^T{\int }_0^{\tau }{\int }_{\Omega }{|\nabla u|}^{2}dxdtd\tau ⩽\underset{k\to \infty }{lim\; inf}{\int }_0^T{\int }_0^{\tau }{\int }_{\Omega }{|\nabla u_k|}^{2}dxdtd\tau ,$$

which together with (23) implies that there exists a subsequence $\left\{u_k\right\}$, which we denote again by $\left\{u_k\right\}$ such that for a.e. $\tau \in (0,T)$

$$\underset{k\to \infty }{lim}{\int }_0^{\tau }{\int }_{\Omega }{|\nabla u_k|}^{2}dxdt={\int }_0^{\tau }{\int }_{\Omega }{|\nabla u|}^{2}dxdt.$$

That means

$$\underset{k\to \infty }{lim}{\int }_0^T{\int }_{\Omega }{|\nabla u_k|}^{2}dxdt={\int }_0^T{\int }_{\Omega }{|\nabla u|}^{2}dxdt,$$

and the conclusion follows. □

Finally, to establish the interior Hölder regularity of weak solutions to system (16) and thus complete the proof of Theorem 1, we need the regularity theorem for weak solutions of quasilinear parabolic equations in the paper [37].

Theorem 4

([37]). Consider the quasilinear parabolic equation

$$u_t-\mathrm{div}a(x,t,u,\nabla u)=b(x,t,u,\nabla u)\mathrm{in}{Q}_T,$$

where $a:Q_T\times {\mathbb{R}}^{N+1}\to {\mathbb{R}}^n$ and $b:Q_T\times {\mathbb{R}}^{N+1}\to \mathbb{R}$ are measurable functions satisfying the following structure conditions:

(A₁) 

$a(x,t,u,\nabla u)·\nabla u\ge C_0{|\nabla u|}^2-C_0^{\prime }$,

(A₂) 

$\left|a(x,t,u,\nabla u)\right|\le C_1|\nabla u|+C_1^{\prime }$,

(A₃) 

$\left|b(x,t,u,\nabla u)\right|\le C_2{|\nabla u|}^2+C_2^{\prime }$,

where $C_i>0$ and $C_i^{\prime }\ge 0,i=0,1,2$ are given constants. Let $u\in L^2(0,T;H^1(\Omega ))$ be a bounded weak solution of the equation in $Q_T$. Then u is locally $C^{\lambda }$ continuous in $Q_T$, for every $\lambda \in (0,1)$.

Remark 2.

In Theorem 4, we only state the local regularity for $L^2(0,T;H^1(\Omega ))$ weak solutions, for the corresponding results concerning $L^p(0,T;W^{1,p}(\Omega ))$ weak solutions under more general structure conditions, we refer the reader to [37] (Theorem 1.1).

Lemma 5.

$y\in C^{\lambda ,\frac{\lambda }{2}}(\Omega \times (0,T)),w\in C^{2+\lambda ,\frac{\lambda }{2}}(\Omega \times (0,T))$ for all $\lambda \in (0,1)$.

Proof. 

We shall complete the proof by applying Theorem 4. Please note that here

$$a(x,t,y,\nabla y)=\nabla y-h\left(y\right)\nabla w,b(x,t,y,\nabla y)={\mu }_1{y}^{+}(1-y^{+})-{\mu }_2{y}^{-}(1-b{y}^{-}).$$

From (14) and Lemma 4, $y\in L^{\infty }\left(Q_T\right)\cap L^2(0,T;H^1(\Omega ))$. Hence, ${\parallel h\left(y\right)\parallel }_{L^{\infty }}$ and ${\mu }_1{y}^{+}(1-y^{+})-{\mu }_2{y}^{-}(1-b{y}^{-})$ are uniformly bounded, from which condition (A3) in Theorem 4 holds. Moreover, by the Cauchy–Schwarz inequality, we have

$$a(x,t,y,\nabla y)·\nabla y={|\nabla y|}^2-h\left(y\right)\nabla w·\nabla y\ge {\displaystyle \frac{1}{2}}{|\nabla y|}^2-{\displaystyle \frac{1}{2}}{h}^2\left(y\right){|\nabla w|}^2$$

and

$$\left|a\right(x,t,y,\nabla y\left)\right|\le |\nabla y|+\left|h\right(y\left)\right||\nabla w|,$$

which together with Remark 1 imply that conditions (A1) and (A2) in Theorem 4 hold. Now form Theorem 4, we have $y\in C^{\lambda ,\frac{\lambda }{2}}(\Omega \times (0,T))$, and hence $w\in C^{2+\lambda ,\frac{\lambda }{2}}(\Omega \times (0,T))$ for all $\lambda \in (0,1)$. □

Remark 3.

From Theorem 1.3 in [37], the Hölder continuity of y can be claimed up to the boundary. Also, if both $u_0$ and $v_0$ are Hölder continuous in $\overline{\Omega }$ then y is Hölder continuous in ${\overline{Q}}_T$.

4. Uniform Hölder Bounds with Respect to $\mathit{k}$

This section is devoted to the proof of Theorem 2. In order to obtain a localized version of the uniform Hölder regularity for solutions to (4), we aim to show that the solutions are uniformly $C^{\alpha }$-bounded in K, uniformly in k, for any compact set $K\subset \subset K^{\prime }\subset \subset Q$. Take a standard cut-off function $\eta$ such that $0\le \eta \le 1$, $\eta \equiv 1$ in K. We will prove that there exists a constant $C>0$ independent of k such that

$$max\left\{\underset{K^{\prime }}{sup}{\displaystyle \frac{|\left(\eta u_k\right)(x,t)-\left(\eta u_k\right)(y,s)|}{d^{\alpha }\left((x,t),(y,s)\right)}},\underset{K^{\prime }}{sup}{\displaystyle \frac{|\left(\eta v_k\right)(x,t)-\left(\eta v_k\right)(y,s)|}{d^{\alpha }\left((x,t),(y,s)\right)}}\right\}\le C,$$

from which the desired result follows.

The proof of Theorem 2 follows the mainstream of [24,29,30], based upon a blow-up analysis and the validity of some Liouville-type theorems. Precisely, assuming the uniform Hölder estimate fails, we construct sequences of points where the Hölder seminorm is unbounded. After appropriate scaling, we obtain rescaled functions that converge to a nontrivial global solution to some semilinear systems, but the corresponding Liouville theorem implies that the solution is trivial, a contradiction. To begin with, we assume by contradiction that, up to a subsequence, it holds

$$L_k:=max\left\{\underset{X,Y\in K^{\prime }}{sup}{\displaystyle \frac{|\left(\eta u_k\right)\left(X\right)-\left(\eta u_k\right)\left(Y\right)|}{d^{\alpha }(X,Y)}},\underset{X,Y\in K^{\prime }}{sup}{\displaystyle \frac{|\left(\eta v_k\right)\left(X\right)-\left(\eta v_k\right)\left(Y\right)|}{d^{\alpha }(X,Y)}}\right\}\to +\infty .$$

We can take $X_k=(x_k,t_k)$, $Y_k=(y_k,s_k)\in K^{\prime }$ such that

$$L_k={\displaystyle \frac{|\left(\eta u_k\right)\left(X_k\right)-\left(\eta u_k\right)\left(Y_k\right)|}{d^{\alpha }(X_k,Y_k)}}.$$

Without loss of generality we can assume $s_k\le t_k$. Since $u_k$ is uniformly bounded, we must have

$$\begin{array}{ccc}\hfill d^{\alpha }(X_k,Y_k)& =& |\left(\eta u_k\right)\left(X_k\right)-\left(\eta u_k\right)\left(Y_k\right)|/L_k\hfill \\ & \le & {\displaystyle \frac{\parallel u_k{\parallel }_{L^{\infty }}}{L_k}}\left(\eta \left(X_k\right)+\eta \left(Y_k\right)\right)\hfill \\ & \to & 0,\hfill \end{array}$$

as $k\to +\infty$.

Our analysis is based on two different blow-up sequences, take $r_k\to 0$, which will be chosen later. Define

$${\tilde{u}}_k\left(X\right)=\eta \left(X_k\right){\displaystyle \frac{u_k(X_k+r_{k}X)}{L_k{r}_k^{\alpha }}},{\tilde{v}}_k\left(X\right)=\eta \left(X_k\right){\displaystyle \frac{v_k(X_k+r_{k}X)}{L_k{r}_k^{\alpha }}},$$

and

$${\overline{u}}_k\left(X\right)={\displaystyle \frac{\left(\eta u_k\right)(X_k+r_{k}X)}{L_k{r}_k^{\alpha }}},{\overline{v}}_k\left(X\right)={\displaystyle \frac{\left(\eta v_k\right)(X_k+r_{k}X)}{L_k{r}_k^{\alpha }}},$$

for $X\in {\tilde{Q}}_k:={\displaystyle \frac{1}{r_k}}(K^{\prime }-X_k)={\Omega }_k\times (-T_k,+\infty )$ (Here $T_k>0$). Please note that since $K^{\prime }\subset \subset Q$, we have ${\tilde{Q}}_k\to {\tilde{Q}}_{\infty }={\mathbb{R}}^{N+1}$ as $k\to +\infty$.

Then, denote ${\tilde{Y}}_k:=(Y_k-X_k)/r_k$, we observe that

$$\begin{array}{c}\hfill max\left\{\underset{X,Y\in \overline{{\tilde{Q}}_k}}{max}{\displaystyle \frac{|{\overline{u}}_k\left(X\right)-{\overline{u}}_k\left(Y\right)|}{d^{\alpha }(X,Y)}},\underset{X,Y\in \overline{{\tilde{Q}}_k}}{max}{\displaystyle \frac{|{\overline{v}}_k\left(X\right)-{\overline{v}}_k\left(Y\right)|}{d^{\alpha }(X,Y)}}\right\}={\displaystyle \frac{|{\overline{u}}_k\left(0\right)-{\overline{u}}_k\left({\tilde{Y}}_k\right)|}{d^{\alpha }({\tilde{Y}}_k,0)}}=1,\end{array}$$

(24)

and hence the functions ${\overline{u}}_k,{\overline{v}}_k$ have a uniform bound on Hölder seminorm and at least ${\overline{u}}_k$ cannot be a constant. Moreover, define

$${\tilde{w}}_k\left(X\right)=w_k(X_k+r_{k}X),\mathrm{for}X\in {\tilde{Q}}_k,$$

then $({\tilde{u}}_k,{\tilde{v}}_k,{\tilde{w}}_k)$ solves

$$\left\{\begin{array}{cc}{\displaystyle \frac{\partial {\tilde{u}}_k}{\partial t}}-▵{\tilde{u}}_k=-{\chi }_1\nabla {\tilde{u}}_k·\nabla {\tilde{w}}_k-{\chi }_1(r_k^2{\tilde{w}}_k-M_k{\tilde{v}}_k-M_k{\tilde{u}}_k){\tilde{u}}_k\hfill & \\ +{\mu }_1(r_k^2-M_k{\tilde{u}}_k){\tilde{u}}_k-k{M}_k{\tilde{u}}_k{\tilde{v}}_k\hfill & \mathrm{in}{\tilde{Q}}_k,\hfill \\ {\displaystyle \frac{\partial {\tilde{v}}_k}{\partial t}}-▵{\tilde{v}}_k=-{\chi }_2\nabla {\tilde{v}}_k·\nabla {\tilde{w}}_k-{\chi }_2(r_k^2{\tilde{w}}_k-M_k{\tilde{u}}_k-M_k{\tilde{v}}_k){\tilde{v}}_k\hfill & \\ +{\mu }_2(r_k^2-M_k{\tilde{v}}_k){\tilde{v}}_k-kb{M}_k{\tilde{u}}_k{\tilde{v}}_k\hfill & \mathrm{in}{\tilde{Q}}_k,\hfill \\ 0=▵{\tilde{w}}_k-r_k^2{\tilde{w}}_k+M_k{\tilde{u}}_k+M_k{\tilde{v}}_k\hfill & \mathrm{in}{\tilde{Q}}_k,\hfill \end{array}\right.$$

(25)

with $M_k:=L_k{r}_k^{2+\alpha }/\eta \left(X_k\right)$.

Remark 4.

The uniform bound of $\parallel (u_k,v_k){\parallel }_{L^{\infty }\left(Q\right)}$ and Remark 1 imply that

$$r_k^2{\tilde{u}}_k{\tilde{w}}_k,r_k^2{\tilde{v}}_k{\tilde{w}}_k,r_k^2{\tilde{u}}_k,r_k^2{\tilde{v}}_k\to 0\mathit{in}{L}^{\infty }\left({\tilde{Q}}_k\right),$$

$$M_k{\tilde{u}}_k,M_k{\tilde{v}}_k,M_k{\tilde{u}}_k^2,M_k{\tilde{v}}_k^2\to 0\mathit{in}{L}^{\infty }\left({\tilde{Q}}_k\right).$$

Lemma 6.

Let $K\subset {\mathbb{R}}^{N+1}$ be compact. Then

(i) we have

$$\begin{array}{c}\hfill \underset{X\in K\cap {\overline{\tilde{Q}}}_k}{max}|{\tilde{u}}_k\left(X\right)-{\overline{u}}_k\left(X\right)|\to 0,\underset{X\in K\cap {\overline{\tilde{Q}}}_k}{max}|{\tilde{v}}_k\left(X\right)-{\overline{v}}_k\left(X\right)|\to 0.\end{array}$$

(ii) there exists C, only depending on K, such that

$$|{\tilde{u}}_k\left(X\right)-{\tilde{u}}_k\left(0\right)|\le C,|{\tilde{v}}_k\left(X\right)-{\tilde{v}}_k\left(0\right)|\le C,$$

for every $X\in K$.

Proof. 

We will prove the estimates of ${\tilde{u}}_k$ and ${\overline{u}}_k$, that of ${\tilde{v}}_k$ and ${\overline{v}}_k$ are similar. This is a consequence of the Lipschitz continuity of $\eta$ and the uniform boundedness of $u_k$. Indeed, we have, for every k,

$$\begin{array}{ccc}\hfill |{\tilde{u}}_k\left(X\right)-{\overline{u}}_k\left(X\right)|& =& {\displaystyle \frac{u_k(X_k+r_{k}X)}{L_k{r}_k^{\alpha }}}|\eta \left(X_k\right)-\eta (X_k+r_{k}X)|\hfill \\ & \le & \tilde{m}{r}_k^{-\alpha }{L}_k^{-1}|\eta \left(X_k\right)-\eta (X_k+r_{k}X)|\hfill \\ & \le & l\tilde{m}{r}_k^{1-\alpha }{L}_k^{-1}\left|X\right|,\hfill \end{array}$$

where l denotes the Lipschitz constant of $\eta$, $\tilde{m}$ is the uniform boundedness of $\left\{u_k\right\}$. Since $L_k\to +\infty$, K is compact, which implies the first part. Moreover, by definition, ${\tilde{u}}_k\left(0\right)={\overline{u}}_k\left(0\right)$, and $|{\overline{u}}_k\left(X\right)-{\overline{u}}_k\left(0\right)|\le C{\left|X\right|}^{\alpha }$ for every $X\in {\tilde{Q}}_k$. Then we can conclude by noticing that

$$|{\tilde{u}}_k\left(X\right)-{\tilde{u}}_k\left(0\right)|\le |{\tilde{u}}_k\left(X\right)-{\overline{u}}_k\left(X\right)|+|{\overline{u}}_k\left(X\right)-{\overline{u}}_k\left(0\right)|$$

by applying the first part. □

In the blow-up procedure, we need to establish some Liouville-type theorems. Fortunately, due to Remark 1, ${\tilde{w}}_k$ (minus a constant) will converge to 0 locally uniformly (see Lemma 10 below), in such a way that the Liouville-type theorems given in [30] are still valid for our problem. For convenient, let us recall some of them, which will be used in the sequel.

Lemma 7

([30], Corollary 2.2). Assume in ${\mathbb{R}}^N\times (-\infty ,0]$, u and v are continuous non-negative functions satisfying

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{\partial u}{\partial t}}-▵u\le 0,\hfill \\ {\displaystyle \frac{\partial v}{\partial t}}-▵v\le 0,\hfill \\ ({\displaystyle \frac{\partial }{\partial t}}-▵)u-({\displaystyle \frac{\partial }{\partial t}}-▵)v\ge 0,\hfill \\ uv=0,\hfill \\ u(0,0)=v(0,0)=0,\hfill \end{array}\right.\end{array}$$

Moveover, if $u,v$ have strong sublinear growth in the sense that there exist $\alpha \in (0,1)$ and $C>0$ such that

$$max\left\{u(x,t),v(x,t)\right\}\le C{\left(1+d(x,t)\right)}^{\alpha }.$$

(26)

Then $u=v\equiv 0$.

Lemma 8

([30], Corollary 2.5). Assume in ${\mathbb{R}}^N\times (-\infty ,0]$, u and v are smooth non-negative functions satisfying

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{\partial u}{\partial t}}-▵u=-uv,\hfill \\ {\displaystyle \frac{\partial v}{\partial t}}-▵v=-uv.\hfill \end{array}\right.\end{array}$$

Assume moreover that (26) holds for some $\alpha \in (0,1)$. Then $u=v\equiv 0$.

We also need the following technical lemma, which generalizes the estimate in [30] (Lemma 3.2); its proof can be found in [38].

Lemma 9

([38], Lemma 4.4). If in the parabolic cylinder $Q_{2R}$, u satisfies the following

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t-▵u\le \lambda \nabla m·\nabla u-Hu\hfill & \mathit{in}{Q}_{2R},\hfill \\ u\ge 0\hfill & \mathit{in}{Q}_{2R},\hfill \\ u\le A\hfill & \mathit{in}{Q}_{2R},\hfill \end{array}\right.\end{array}$$

where $m\in C^{2,1}\left({\overline{B}}_{2R}\right)$ and $\lambda ,H,A$ are positive constant. If H is sufficiently large compared to R, then we have

$$\underset{Q_{R/2}}{sup}u\le C_{1}A{e}^{-C_{2}R\sqrt{H}},$$

where $C_1,C_2$ are two positive constants not depending on $A,H$ and R.

In order to manage the different parts of the proof, we need to make different choices of the sequence $r_k$. Once $r_k$ is chosen, we wish to pass to the limit (on compact sets), and to this aim we will use Ascoli–Arzelà’s theorem. Now, since ${\overline{u}}_k,{\overline{v}}_k$ are uniformly Hölder continuous, it suffices to show that $\left\{{\overline{u}}_k\left(0\right)\right\}$, $\left\{{\overline{v}}_k\left(0\right)\right\}$ are bounded in k. The following lemma provides a sufficient condition on $r_k$ for such a bound to hold.

Lemma 10.

Under the previous notations. Let $r_k\to 0$ as $k\to \infty$ be such that

$\left(i\right)$ ${\displaystyle \frac{d(X_k,Y_k)}{r_k}}⩽R^{\prime }$ for some $R^{\prime }>0$.

$\left(ii\right)$ $k{M}_k\nrightarrow 0$.

Then $\left\{{\overline{u}}_k\left(0\right)\right\}$, $\left\{{\overline{v}}_k\left(0\right)\right\}$ are uniformly bounded in k.

Proof. 

We prove the estimate for ${\overline{u}}_k\left(0\right)$, and that for ${\overline{v}}_k\left(0\right)$ follows similarly. Assume by contradiction that $\left\{{\overline{u}}_k\left(0\right)\right\}$ is unbounded, and let $R⩾R^{\prime }$, since ${\overline{u}}_k$ is made of continuous functions which share the same $C^{\alpha }$-seminorm, we have ${inf}_{Q_{2R}\cap {\tilde{Q}}_k}{\overline{u}}_k\to +\infty$. Furthermore, ${inf}_{Q_{2R}\cap {\tilde{Q}}_k}{\tilde{u}}_k\to +\infty$ as well. Therefore

$$N_k:=\underset{Q_{2R}\left(0\right)\cap {\tilde{Q}}_k}{inf}k{M}_k{\tilde{u}}_k\to +\infty .$$

In particular, on $Q_{2R}\cap ({\Omega }_k\times \{-T_k\})$, ${\tilde{u}}_k>0$ and ${\tilde{v}}_k=0$.

We claim that:

$${\parallel k{M}_k{\tilde{u}}_k{\tilde{v}}_k\parallel }_{L^{\infty }(Q_R\left(0\right)\cap {\tilde{Q}}_k)}\to 0\mathrm{as}k\to \infty .$$

Combine with Remark 4 we have

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial {\tilde{v}}_k}{\partial t}}-▵{\tilde{v}}_k& =& -{\chi }_2(\nabla {\tilde{v}}_k·\nabla {\tilde{w}}_k+r_k^2{\tilde{v}}_k{\tilde{w}}_k-M_k{\tilde{u}}_k{\tilde{v}}_k-M_k{\tilde{v}}_k^2)+{\mu }_2{r}_k^2{\tilde{v}}_k-{\mu }_2{M}_k{\tilde{v}}_k^2-kb{M}_k{\tilde{u}}_k{\tilde{v}}_k\hfill \\ & \le & -{\chi }_2(\nabla {\tilde{v}}_k·\nabla {\tilde{w}}_k+r_k^2{\tilde{v}}_k{\tilde{w}}_k-M_k{\tilde{u}}_k{\tilde{v}}_k-M_k{\tilde{v}}_k^2)+{\mu }_2{r}_k^2{\tilde{v}}_k-{\mu }_2{M}_k{\tilde{v}}_k^2-b{N}_k{\tilde{v}}_k\hfill \\ & \le & -{\chi }_2\nabla {\tilde{v}}_k·\nabla {\tilde{w}}_k-{\displaystyle \frac{b{N}_k}{2}}{\tilde{v}}_k.\hfill \end{array}$$

(27)

Let us choose a cut-off function $\eta \in C_0^{\infty }\left(Q_{2R}\left(0\right)\right)$ such that $\eta \equiv 0$ on $R^{N+1}\setminus Q_{2R}\left(0\right)$. Then multiplying Equation (27) by ${\eta }^2{\tilde{v}}_k$ and integrating on $Q_{2R}\left(0\right)\cap {\tilde{Q}}_k$. We obtain

$$\int {\int }_{Q_{2R}\left(0\right)\cap {\tilde{Q}}_k}{\displaystyle \frac{\partial {\tilde{v}}_k}{\partial t}}{\eta }^2{\tilde{v}}_k-▵{\tilde{v}}_k{\eta }^2{\tilde{v}}_k\le \int {\int }_{Q_{2R}\left(0\right)\cap {\tilde{Q}}_k}-{\chi }_2\nabla {\tilde{v}}_k·\nabla {\tilde{w}}_k{\eta }^2{\tilde{v}}_k-{\displaystyle \frac{b{N}_k}{2}}{\tilde{v}}_k^2{\eta }^2.$$

Then

$$\begin{array}{ccc}& & \int {\int }_{Q_{2R}\left(0\right)\cap {\tilde{Q}}_k}{\displaystyle \frac{b{N}_k}{2}}{\tilde{v}}_k^2{\eta }^2+{\eta }^2{|\nabla {\tilde{v}}_k|}^2\hfill \\ & \le & \int {\int }_{Q_{2R}\left(0\right)\cap {\tilde{Q}}_k}\eta \left|{\displaystyle \frac{\partial \eta }{\partial t}}\right|{\tilde{v}}_k^2-2\eta {\tilde{v}}_k\nabla \eta ·\nabla {\tilde{v}}_k-{\chi }_2\nabla {\tilde{v}}_k·\nabla {\tilde{w}}_k{\eta }^2{\tilde{v}}_k\hfill \\ & \le & \int {\int }_{Q_{2R}\left(0\right)\cap {\tilde{Q}}_k}\eta \left|{\displaystyle \frac{\partial \eta }{\partial t}}\right|{\tilde{v}}_k^2+{\eta }^2|\nabla {\tilde{v}}_k{|}^2+{2|\nabla \eta |}^2{\tilde{v}}_k^2+{\displaystyle \frac{1}{2}}{\chi }_2^2{\eta }^2{\tilde{v}}_k^2{|\nabla {\tilde{w}}_k|}^2.\hfill \end{array}$$

Thus

$$\begin{array}{ccc}\hfill \int {\int }_{Q_{2R}\left(0\right)\cap {\tilde{Q}}_k}{\displaystyle \frac{b{N}_k}{2}}{\tilde{v}}_k^2{\eta }^2& \le & \int {\int }_{Q_{2R}\left(0\right)\cap {\tilde{Q}}_k}\eta \left|{\displaystyle \frac{\partial \eta }{\partial t}}\right|{\tilde{v}}_k^2+{2|\nabla \eta |}^2{\tilde{v}}_k^2+{\displaystyle \frac{1}{2}}{\chi }_2^2{\eta }^2{|\nabla {\tilde{w}}_k|}^2{\tilde{v}}_k^2\hfill \\ & \le & C\left(R\right)sup\int {\int }_{Q_{2R}\left(0\right)\cap {\tilde{Q}}_k}{\tilde{v}}_k^2.\hfill \end{array}$$

(28)

On the other hand, using again the uniform Hölder bounds of the sequence $\left\{{\overline{v}}_k\right\}$ and the uniformly control given by Lemma 6, we infer

$$\begin{array}{ccc}\hfill {\displaystyle \frac{b{N}_k}{2}}\int {\int }_{Q_{2R}\left(0\right)\cap {\tilde{Q}}_k}{\eta }^2{\tilde{v}}_k^2& \ge & {\displaystyle \frac{b{N}_k}{2}}\underset{Q_{2R}\left(0\right)\cap {\tilde{Q}}_k}{inf}{\tilde{v}}_k^2\int {\int }_{Q_{2R}\left(0\right)\cap {\tilde{Q}}_k}{\eta }^2\hfill \\ & \ge & {\displaystyle \frac{b{N}_k}{2}}\left({\displaystyle \frac{1}{2}}\underset{Q_{2R}\left(0\right)\cap {\tilde{Q}}_k}{sup}{\tilde{v}}_k^2-{\left(2R\right)}^{2\alpha }\right)\int {\int }_{Q_{2R}\left(0\right)\cap {\tilde{Q}}_k}{\eta }^2\hfill \\ & \ge & {\displaystyle \frac{b{N}_k}{2}}{C}^{\prime }\left(R\right)\underset{Q_{2R}\left(0\right)\cap {\tilde{Q}}_k}{sup}{\tilde{v}}_k^2-C^{″}\left(R\right)N_k.\hfill \end{array}$$

(29)

Combining (28) with (29) gives

$${\displaystyle \frac{b{N}_k}{2}}{C}^{\prime }\left(R\right)\underset{Q_{2R}\left(0\right)\cap {\tilde{Q}}_k}{sup}{\tilde{v}}_k^2\le C\left(R\right)\underset{Q_{2R}\left(0\right)\cap {\tilde{Q}}_k}{sup}{\tilde{v}}_k^2+C^{″}\left(R\right)N_k,$$

which implies the boundedness of ${\tilde{v}}_k$ in $Q_{2R}\left(0\right)\cap {\tilde{Q}}_k$. Now Lemma 9 gives that

$${\parallel {\tilde{v}}_k\parallel }_{L^{\infty }(Q_{R/2}\left(0\right)\cap {\tilde{Q}}_k)}\le C{e}^{-C^{\prime }R\sqrt{N_k}}.$$

The claim follows. Therefore ${\tilde{u}}_k$ satisfies

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial {\tilde{u}}_k}{\partial t}}-▵{\tilde{u}}_k+{\chi }_1\nabla {\tilde{u}}_k·\nabla {\tilde{w}}_k={\epsilon }_k,\end{array}$$

where ${\epsilon }_k\to 0$ uniformly on $Q_{\frac{R}{2}}\left(0\right)\cap {\tilde{Q}}_k$.

Consider now

$${\widehat{u}}_k\left(X\right):={\overline{u}}_k\left(X\right)-{\overline{u}}_k\left(0\right),{\check{u}}_k\left(X\right):={\tilde{u}}_k\left(X\right)-{\tilde{u}}_k\left(0\right).$$

We have ${\widehat{u}}_k$ is Hölder continuous and ${\widehat{u}}_k\left(0\right)=0$. Since (24),

$${\displaystyle \frac{|{\widehat{u}}_k\left({\tilde{Y}}_k\right)-{\widehat{u}}_k\left(0\right)|}{d^{\alpha }({\tilde{Y}}_k,0)}}=1.$$

(30)

By Ascoli–Arzelà’s theorem we know that, up to a subsequence, ${\widehat{u}}_k\to u_{\infty }$ on compact sets. Combining this fact with the assumption $d(X_k,Y_k)\le R^{\prime }{r}_k$, we deduce from (30) that there exist $\tilde{Y}\in R^{N+1}\setminus \left\{0\right\}$ such that ${lim}_{k\to +\infty }{\tilde{Y}}_k=\tilde{Y}$. Moreover, (30) can be passed to the limit, which is

$${\displaystyle \frac{|u_{\infty }\left(\tilde{Y}\right)-u_{\infty }\left(0\right)|}{d^{\alpha }(\tilde{Y},0)}}=1.$$

(31)

On the other hand, ${\check{u}}_k$ satisfies

$${\displaystyle \frac{\partial {\check{u}}_k}{\partial t}}-▵{\check{u}}_k+{\chi }_1\nabla {\check{u}}_k·\nabla {\tilde{w}}_k={\epsilon }_k,$$

from this equation, we can infer from the standard parabolic estimate that ${\check{u}}_k$ is uniformly Lipschitz continuous. Since ${\tilde{w}}_k\in C^{1+\alpha }$, then for every $X\in \overline{{\tilde{Q}}_k}$,

$$|\nabla {\tilde{w}}_k\left(X\right)|=r_k|\nabla w_k(X_k+r_{k}X)|\to 0\mathrm{as}k\to +\infty .$$

Thus it holds after passing to a subsequence,

$${\displaystyle \frac{\partial u_{\infty }}{\partial t}}-▵u_{\infty }=0\mathrm{in}{\tilde{Q}}_{\infty }.$$

Since ${\tilde{Q}}_{\infty }={\mathbb{R}}^{N+1}$, $u_{\infty }$ must be a constant function by the Liouville theorem. This contradicts (31). □

Lemma 11.

Under the previous notations, we have (up to a subsequence)

$$k{L}_k{d}^{2+\alpha }(X_k,Y_k)/\eta \left(X_k\right)\to +\infty .$$

Proof. 

Assume by contraction that $k{L}_k{d}^{2+\alpha }(X_k,Y_k)\le R^{\prime }$ for some $R^{\prime }>0$ by contradiction. We now choose

$$r_k={\left({\displaystyle \frac{k{L}_k}{\eta \left(X_k\right)}}\right)}^{-\frac{1}{2+\alpha }}.$$

So that $k{M}_k=k{L}_k{r}_k^{2+\alpha }/\eta \left(X_k\right)=1$, and the assumptions of Lemma 10 are satisfied, and thus ${\overline{u}}_k\left(0\right)$ and ${\overline{v}}_k\left(0\right)$ are uniformly bounded. By the uniform Hölder continuity and Ascoli–Arzelà’s theorem, we have that, up to a subsequence, there exist ${\overline{u}}_{\infty }$ and ${\overline{v}}_{\infty }$ such that ${\overline{u}}_k\to {\overline{u}}_{\infty }$, ${\overline{v}}_k\to {\overline{v}}_{\infty }$ uniformly on the compact sets of $R^{N+1}$. Moreover, by Lemma 6, we also find that ${\tilde{u}}_k\to {\overline{u}}_{\infty }$, ${\tilde{v}}_k\to {\overline{v}}_{\infty }$.

Moreover, the choice of $r_k$ gives that the equation of ${\tilde{u}}_k$ and ${\tilde{v}}_k$ are

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{\partial {\tilde{u}}_k}{\partial t}}-▵{\tilde{u}}_k=-{\chi }_1(\nabla {\tilde{u}}_k·\nabla {\tilde{w}}_k+r_k^2{\tilde{u}}_k{\tilde{w}}_k-M_k{\tilde{u}}_k{\tilde{v}}_k-M_k{\tilde{u}}_k^2)+{\mu }_1{r}_k^2{\tilde{u}}_k-{\mu }_1{M}_k{\tilde{u}}_k^2-{\tilde{u}}_k{\tilde{v}}_k,\hfill \\ {\displaystyle \frac{\partial {\tilde{v}}_k}{\partial t}}-▵{\tilde{v}}_k=-{\chi }_2(\nabla {\tilde{v}}_k·\nabla {\tilde{w}}_k+r_k^2{\tilde{v}}_k{\tilde{w}}_k-M_k{\tilde{u}}_k{\tilde{v}}_k-M_k{\tilde{v}}_k^2)+{\mu }_2{r}_k^2{\tilde{v}}_k-{\mu }_2{M}_k{\tilde{v}}_k^2-b{\tilde{u}}_k{\tilde{v}}_k.\hfill \end{array}\right.\end{array}$$

From the first equation, we can also obtain a uniform Lipschitz estimate of ${\tilde{u}}_k$, and we can deduce from (24) that

$${\displaystyle \frac{|{\overline{u}}_{\infty }\left(\tilde{Y}\right)-{\overline{u}}_{\infty }\left(0\right)|}{d^{\alpha }(\tilde{Y},0)}}=1,$$

(32)

In particular, ${\overline{u}}_{\infty }$ is not a constant function. On the other hand, since $|\nabla {\tilde{w}}_k|\to 0$ as $k\to +\infty$, at the limit, we have

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \frac{\partial {\overline{u}}_{\infty }}{\partial t}}-▵{\overline{u}}_{\infty }=-{\overline{u}}_{\infty }{\overline{v}}_{\infty }\hfill & \mathrm{in}{\tilde{Q}}_{\infty },\hfill \\ {\displaystyle \frac{\partial {\overline{v}}_{\infty }}{\partial t}}-▵{\overline{v}}_{\infty }=-b{\overline{u}}_{\infty }{\overline{v}}_{\infty }\hfill & \mathrm{in}{\tilde{Q}}_{\infty }.\hfill \end{array}\right.\end{array}$$

Therefore, by Lemma 8, ${\overline{u}}_{\infty }$ must be a constant function, which is in contradiction with (32). □

Now we come to the proof of the Theorems.

Proof of Theorem 2. 

From Lemma 11, we have $k{L}_k{d}^{2+\alpha }(X_k,Y_k)/\eta \left(X_k\right)\to +\infty$, letting $r_k=d(X_k,Y_k)$. With this choice, we have that all the assumptions of Lemma 10 are satisfied and hence ${\overline{u}}_k\left(0\right)$ and ${\overline{v}}_k\left(0\right)$ are uniformly bounded. Again, by the uniform Hölder continuity and Ascoli–Arzelà’s theorem we have that, up to a subsequence, there exist ${\overline{u}}_{\infty },{\overline{v}}_{\infty }$ such that ${\overline{u}}_k\to {\overline{u}}_{\infty }$, ${\overline{v}}_k\to {\overline{v}}_{\infty }$ uniformly on the compact subsets of $R^{N+1}$. By Lemma 6, we deduce that ${\tilde{u}}_k\to {\overline{u}}_{\infty }$, ${\tilde{v}}_k\to {\overline{v}}_{\infty }$. Please note that $d(\tilde{Y},0)=1$, (24) implies that

$$\begin{array}{c}\hfill |{\overline{u}}_{\infty }\left(\tilde{Y}\right)-{\overline{u}}_{\infty }\left(0\right)|=1.\end{array}$$

(33)

Moreover ${\tilde{u}}_k,{\tilde{v}}_k$ satisfy the following inequalities from (25)

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial {\tilde{u}}_k}{\partial t}}-▵{\tilde{u}}_k+{\chi }_1\nabla {\tilde{u}}_k·\nabla {\tilde{w}}_k\le C_1{r}_k^2{\tilde{u}}_k,\hfill \\ {\displaystyle \frac{\partial {\tilde{v}}_k}{\partial t}}-▵{\tilde{v}}_k+{\chi }_2\nabla {\tilde{v}}_k·\nabla {\tilde{w}}_k\le C_2{r}_k^2{\tilde{v}}_k.\hfill \end{array}\right.$$

(34)

We also note that

$$\begin{array}{ccc}\hfill b({\displaystyle \frac{\partial {\tilde{u}}_k}{\partial t}}-▵{\tilde{u}}_k)-({\displaystyle \frac{\partial {\tilde{v}}_k}{\partial t}}-▵{\tilde{v}}_k)& =& -b{\chi }_1(\nabla {\tilde{u}}_k·\nabla {\tilde{w}}_k+r_k^2{\tilde{u}}_k{\tilde{w}}_k-M_k{\tilde{u}}_k{\tilde{v}}_k-M_k{\tilde{u}}_k^2)\hfill \\ & & +{\chi }_2(\nabla {\tilde{v}}_k·\nabla {\tilde{w}}_k+r_k^2{\tilde{v}}_k{\tilde{w}}_k-M_k{\tilde{u}}_k{\tilde{v}}_k-M_k{\tilde{v}}_k^2)\hfill \\ & & +b{\mu }_1{r}_k^2{\tilde{u}}_k-b{\mu }_1{M}_k{\tilde{u}}_k^2-{\mu }_2{r}_k^2{\tilde{v}}_k+{\mu }_2{M}_k{\tilde{v}}_k^2.\hfill \end{array}$$

(35)

Then (34) and (35) can be passed to the limit (up to a subsequence), so in the distributional sense ${\overline{u}}_{\infty }$ and ${\overline{v}}_{\infty }$ satisfy

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial {\overline{u}}_{\infty }}{\partial t}}-▵{\overline{u}}_{\infty }\le 0,\hfill \\ {\displaystyle \frac{\partial {\overline{v}}_{\infty }}{\partial t}}-▵{\overline{v}}_{\infty }\le 0,\hfill \\ b({\displaystyle \frac{\partial }{\partial t}}-▵){\overline{u}}_{\infty }-({\displaystyle \frac{\partial }{\partial t}}-▵){\overline{v}}_{\infty }=0.\hfill \end{array}\right.$$

(36)

In order to apply Lemma 7, we must show that ${\overline{u}}_{\infty }{\overline{v}}_{\infty }=0$ in ${\tilde{Q}}_{\infty }$.

$\forall K⋐{\tilde{Q}}_{\infty }$, we may let k sufficient large such that $K⋐{\tilde{Q}}_k$. Let us consider a smooth cut-off function $\eta \in C_0^{\infty }\left({\tilde{Q}}_k\right)$ such that $0\le \eta \le 1$ and $\eta \equiv 1$ in K. Multiplying the first inequality in (25) by $\eta$ and integrating by parts, we obtain

$$\begin{array}{ccc}\hfill k{M}_k\int {\int }_K{\tilde{u}}_k{\tilde{v}}_k& \le & \int {\int }_{{\tilde{Q}}_k}▵{\tilde{u}}_k\eta -{\chi }_1(\nabla {\tilde{u}}_k·\nabla {\tilde{w}}_k+r_k^2{\tilde{u}}_k{\tilde{w}}_k-M_k{\tilde{u}}_k{\tilde{v}}_k-M_k{\tilde{v}}_k^2)\eta \hfill \\ & & +{\mu }_1{r}_k^2{\tilde{u}}_k\eta -{\mu }_1{M}_k{\tilde{u}}_k^2\eta -\int {\int }_{{\tilde{Q}}_k}{\displaystyle \frac{\partial {\tilde{u}}_k}{\partial t}}\eta \hfill \\ & \le & \int {\int }_{{\tilde{Q}}_k}\left|▵\eta +{\chi }_1\nabla {\tilde{w}}_k·\nabla \eta +{\mu }_1{r}_k^2\eta -{\mu }_1{M}_k{\tilde{u}}_k\eta +{\displaystyle \frac{\partial \eta }{\partial t}}\right|{\tilde{u}}_k.\hfill \end{array}$$

(37)

Since ${\overline{u}}_k$ is uniformly Hölder continuous and ${\overline{u}}_k\left(0\right)$ is uniformly bounded, ${\overline{u}}_k$ is uniformly bounded on compact sets. Again, by Lemma 6, we can also conclude that ${\tilde{u}}_k$ is uniformly bounded on compact sets. Since $k{M}_k\to +\infty$, we can deduce from (37) that

$$\underset{k\to +\infty }{lim}\int {\int }_K{\tilde{u}}_k{\tilde{v}}_k=0,$$

which implies that ${\overline{u}}_{\infty }{\overline{v}}_{\infty }=0$ in ${\tilde{Q}}_{\infty }$. Finally, we can apply Lemma 7, which gives ${\overline{u}}_{\infty }$ must be a constant function. This contradicts (33), and we complete the proof of Theorem 2 □