Existence of Viscosity Solutions for Weakly Coupled Cooperative Parabolic Systems with Fully Nonlinear Principle Part

Abstract

In this paper, the existence of viscosity solutions for weakly coupled, degenerate, and cooperative parabolic systems is studied in a bounded domain. In particular, we consider the viscosity solutions of parabolic systems with fully nonlinear degenerated principal symbol and linear coupling part. The maximum principle theorem is given as well.

1. Introduction

Parabolic systems with a fully nonlinear degenerated principal symbol model many problems arising in optimal control, physics, etc. For instance, such systems emerge in study of the Ricci flow and some other geometric evolution equations like the mean curvature flow. Pseudo-parabolic equations model the fluid flow in fissure porous media (see [1]), two-phase flow in porous media with dynamical capillary pressure (see [2,3]), heat conduction in two temperature systems, see [4], etc.

In this paper, we study a particular class of fully nonlinear parabolic systems, namely the one of weakly coupled, degenerate, and cooperative systems with fully nonlinear degenerated principal symbol and linear coupling part. For such systems, the class of viscosity solutions is suitable.

Let $\mathsf{\Omega }$ be a bounded domain in $R^n$. We study nonlinear quasi-monotone degenerate parabolic systems with null boundary data

$$u_t^i+E^i(t,x,u^i,D{u}^i,D^2{u}^i)+\sum _{j=1}^N{c}_{ij}(t,x)u^j=f^i(t,x)$$

(1)

where $i=1,\dots ,N$ and $(t,x)\in G=(0,T)\times \mathsf{\Omega }$. The parabolic boundary of G is denoted by $\mathsf{\Gamma }=\{[0,T]\times \partial \mathsf{\Omega }\}\cup \{0\times \overline{\mathsf{\Omega }}\}$.

Functions $E^i$ and $c_{ij}$ are supposed to be continuous, i.e., for every $i=1,\dots ,N$

$$E^i(t,x,z^i,p^i,X^i)\in C(R\times \mathsf{\Omega }\times R\times R^n\times S^n)$$

(2)

where $S^n$ is the set of all real symmetric $n\times n$ matrices, and $c_{ij}(t,x)\in C\left(\overline{G}\right)$ for $i=1,\dots ,N$. The right-hand side $f^i(x,t)\in L^2\left(G\right)$ is a bounded function

$$|f^i(x,t)|\le C$$

(3)

in $\overline{G}$ for every $i=1,\dots ,N$, where C is a positive constant.

System (1) is degenerate parabolic, where

$$E^i(t,x,z^i,p^i,X^i)\ge E^i(t,x,z^i,p^i,Y^i)\mathrm{whenever}{X}^i\le Y^i;$$

(4)

and monotone increasing w.r.t. z, i.e.,

$$E^i(t,x,z^i,p^i,X^i)\ge E^i(t,x,y^i,p^i,X^i)\mathrm{whenever}{z}^i\ge y^i,$$

(5)

for $i=1,\dots ,N$, $0\le t\le T$, $x\in \mathsf{\Omega }$, $p^i\in R^n$ and $X^i,Y^i\in S^n$.

Furthermore, system (1) is a quasi-monotone one, i.e., for any $(t,x)\in \overline{G}\backslash \mathsf{\Gamma }$ holds

$$\sum _{j=1}^N{c}_{ij}(t,x)\ge 0\mathrm{and}{c}_{ij}(t,x)\le 0\mathrm{for}i\ne j$$

(6)

Condition (6) means that $c_{ii}$ is “dominant” over $c_{ij}$, $i\ne j$.

In this paper, the existence of viscosity solutions to system (1) by Perron’s method is proven. For the sake of completeness, the validity of maximum principle (MP) is also considered for system (1). The main difficulty is that the principle part of system (1) is nonlinear. On the other hand, system (1) can be considered to be degenerated elliptic and the results for the comparison and maximum principles for elliptic nonlinear systems are applicable. The main obstacle is that the parabolic boundary differs from the elliptic one and the novelty in this paper is that the corresponding proofs and inequalities are given on the piece of the parabolic boundary $\{T\times \overline{\mathsf{\Omega }}\}$.

There is a number of recent results that have been obtained which demonstrate the existence of an MP for nonlinear parabolic systems. For instance, in [5], the authors investigate the existence of solutions to a nonlinear parabolic system that couples a non-homogeneous reaction–diffusion-type equation and a non-homogeneous viscous Hamilton–Jacobi one with non-negative initial data and right-hand, satisfying suitable integrability conditions. The existence, uniqueness, and asymptotic behavior of a solution are studied in [6] for a class of coupled nonlinear parabolic equations in a general unbounded domain as the whole or a half space in $R^n$ and the exterior of a bounded domain. The asymptotic behavior of the solution is given with respect to a pair of quasi-solutions of the corresponding elliptic system, and when these two quasi-solutions coincide, the solution of the parabolic system converges to a unique solution of the elliptic system. Perron’s method is applied to general nonlinear elliptic equations in [7,8], to elliptic nonlinear systems in [9], and to the systems of Hamilton–Jacobi equations in [10,11]. In [12], the existence of periodic solutions in an autonomous parabolic system of differential equations with weak diffusion on the circle is proven. The existence and stability of an arbitrarily large finite number of cycles is studied for a parabolic system with weak diffusion.

The global existence of a class of strongly coupled parabolic systems is given in [13]. The necessary a priori estimates are obtained via new approach to the regularity theory of parabolic scalar equations with integrable data and new $W^{1,p}$ estimates of their solutions. The key assumption here is that the $L^p$ norms of solutions are uniformly bounded for some sufficiently large $p\in (1,\infty )$, an assumption which can be easily affirmed for systems with polynomial growth data. This replaces the usual condition that the solutions are uniformly bounded, which is very difficult to verify because the maximum principles for systems are generally unavailable.

The global existence and blow-up of solutions in a finite time is studied in [14] for a class of pseudo p-Laplacian evolution equations with logarithmic nonlinearity, and in [15], for a class of parabolic systems with fractional p-Laplacian operator with logarithmic nonlinearity. In [15], the existence and decay exponential estimates of the global weak solutions with critical initial energy $I(u_0,v_0)<d$ are proven by the classical Galerkin method and some remarks on the finite time blow-up of weak solutions are given as well.

As for the maximum principle (MP), it is well studied for weakly coupled linear and semi-linear parabolic systems. For instance, in [16], the classical MP for weakly coupled quasi-linear parabolic systems is proven, while the strong MP is proven in [17] for cooperative periodic–parabolic linear systems. In [18], R. Martin proved the validity of MP for parabolic systems that are in fact cooperative ones. In [19], the MP for parabolic systems is studied with coupling terms of the kind $u^{\alpha }{v}^{\beta }$, where $\alpha ,\beta \ge 0$ are constants. In [20], strongly coupled parabolic systems and cooperativeness are considered, which are replaced by a considerably restrictive structural condition, namely that of an inward unit normal $\nu$ being a left-eigenvector of $\left\{b_i^k\right\}$ at any point of $\partial \mathsf{\Omega }$ and the scalar product $(\nu .f)\ge 0$.

One interesting result on the generalized maximum principle of Alexandrov—Bakelman—Pucci type is obtained for linear elliptic systems in [21] and in [22] for viscosity solutions of fully nonlinear cooperative elliptic systems, as well as a Harnack inequality for such systems.

One fundamental comparison theorem is established in [23] for general semi-linear parabolic systems via the notions of sectorial operators, analytic semi-groups, and the application of the Tychonoff fixed-point theorem. Based on this result, we obtain a maximum principle for systems of general parabolic operators and general comparison theorems for parabolic systems with quasi-monotone or mixed quasi-monotone nonlinearities. These results cover and extend most currently used forms of maximum principles and comparison theorems. A global existence theorem for parabolic systems is derived as an application which, in particular, gives rise to some global existence results for Fujita-type systems and certain generalizations.

The author in [24] considers strongly coupled nonlinear parabolic systems of the type

$$\frac{\partial u}{\partial t}-A(x,t,u)\sum _{i,j=1}^n{a}_{ij}(x,t,u)\frac{{\partial }^{2}u}{\partial x_i\partial x_j}+\sum _{i=1}^n{B}_i(x,t,u)\frac{\partial u}{\partial x_i}=f(x,t,u)$$

in some domain $D(0,T)$. The strong maximum principle for these systems is proved under the following considerably restrictive condition: there exists a convex $C^2$ domain $S\in R^m$ such that for any u in $\partial S$, the inward unit normal $v\left(u\right)$ at u is a left eigenvector of A and $B_i$ for $i=1,2,\dots ,n$, and $v\left(x\right). f(x,t,u)\ge 0$ for all $(x,t)\in D(0,T)$. Two extensions of Hamilton’s MP for parabolic systems on manifolds are given in [25]. The one named by Bennett Chow and Peng Lu “the maximum principle with time-dependent convex sets” is applicable to the study of the Ricci flow and some other geometric evolution equations like the mean curvature flow.

2. Viscosity Solutions

For the sake of completeness, we recall the definition of viscosity solution of system (1). Let $v(t,x):\overline{G}\backslash \mathsf{\Gamma }\to R$ be a locally bounded function. Parabolic semi-jets of second-order $P^{2,+}$ and $P^{2,-}$ of v at point $(s,y)\in \overline{G}\backslash \mathsf{\Gamma }$, are defined as

$$P^{2,+}v(s,y)=\{(a,p,X)\in R\times R^n\times S^n:\mathrm{for}\mathrm{any}(t,x)\to (s,y)$$

(7)

$$v(t,x)\le v(s,y)+a·(t-s)+⟨p,x-y⟩+\frac{1}{2}{⟨X(x-y),x-y⟩+o\left(\right|t-s|+|x-y|}^2\left)\right\}.$$

Analogously $P^{2,-}v(s,y)=-P^{2,+}(-v(s,y))$. The closure of $P^{2,+}$ is defined as

$${\overline{P}}^{2,+}u(t,x)=\left\{(a,p,X)\in R\times R^n\times S^n:\mathrm{for}\mathrm{some}\mathrm{sequence}\right.$$

$$(t^k,x^k,p^k,X^k)\in [0,T]\times \mathsf{\Omega }\times R^n\times S^n,(p^k,X^k)\in P^{2,+}v(t,x)$$

$$\left.\mathrm{holds}(t^k,x^k,v\left(x^k\right),p^k,X^k)\to (t,x,v\left(x\right),p,X)\mathrm{as}k\to \infty \right\}.$$

The definition of viscosity sub- and super-solution to (1) is similar to the one for elliptic systems, as can be seen in Definition 2,1, page 1997, [9]. In the following definition 1 set $USC(\overline{G}\backslash \mathsf{\Gamma })$ contains all upper semicontinuous functions in $\overline{G}\backslash \mathsf{\Gamma }$, while set $LSC(\overline{G}\backslash \mathsf{\Gamma })$ contains all lower semicontinuous functions in $\overline{G}\backslash \mathsf{\Gamma }$.

Definition 1.

Let $u=(u^1,\dots ,u^N):\overline{G}\to R^N$ be a locally bounded function. Then:

(i) $u(t,x)\in USC(\overline{G}\backslash \mathsf{\Gamma })$ is a viscosity subsolution to (1) if

$$a^i+E^i(t,x,u^i(t,x),p^i,X^i)+\sum _{j=1}^N{c}_{ij}(t,x)u^j(t,x)\le 0$$

for every $(a^i,p^i,X^i)\in P^{2,+}u(t,x)$, $i=1,\dots ,N$.

(ii) $u(t,x)\in LSC(\overline{G}\backslash \mathsf{\Gamma })$ is a viscosity supersolution to (1) if

$$b^i+E^i(t,x,u^i(t,x),q^i,Y^i)+\sum _{j=1}^N{c}_{ij}(t,x)u^j(t,x)\ge 0$$

for every $(b^i,q^i,Y^i)\in P^{2,-}u(t,x)$, $i=1,\dots ,N$.

(iii) Finally, $u(t,x)$ is a viscosity solution to (1) if it is both viscosity sub-and supersolution of (1).

3. Existence of Viscosity Solutions

The existence of viscosity solutions to system (1) is proven by the generalized Peron method. The key argument is the validity of the comparison principle (CP) for the viscosity sub- and super- solutions to (1). It is well studied in the elliptic case. Following [7], let us recall that CP holds for nonlinear elliptic system if there is a number $\gamma$ such that

$$\gamma (r-s)\le E^i(t,x,r,p,X)-E^i(t,x,s,p,X)$$

(8)

for $r\ge s$, $(t,x,p,X)\in G\times R^N\times S^n$ and there is a function $\omega :R^{+}\to R^{+}$ such that $\omega \left(\sigma \right)\to 0$ for $\sigma \to 0^{+}$ and

$$E^i(t,x,r,\alpha (x-y),Y)-E^i(t,x,r,\alpha (x-y),X)\le {\omega \left(\alpha \right|x-y|}^2+|x-y|)$$

(9)

for $x,y\in \mathsf{\Omega }$, $r\in R$ and $X,Y\in S^n$ satisfying

$$-3\alpha \left(\begin{array}{cc}I& 0\\ 0& I\end{array}\right)\le \left(\begin{array}{cc}X& 0\\ 0& Y\end{array}\right)\le 3\alpha \left(\begin{array}{cc}I& -I\\ -I& I\end{array}\right)$$

(10)

for a number $\alpha >1$.

System (1) is in fact a degenerated elliptic one with no second derivative on variable t. It can be rewritten as

$$P^k{v}^k=v_t^k+F^k(t,x,v,D{v}^k,D^2{v}^k)-f^k(t,x)=0$$

(11)

in $G=(0,T)\times \mathsf{\Omega }$, where $v^k=e^{-\lambda t}{u}^k$ and

$$F^k(t,x,r,p^k,X^k)=e^{-\lambda t}{E}^i(t,x,e^{\lambda t}{r}^k,e^{\lambda t}{p}^k,e^{\lambda t}{X}^k)+\sum _{j=1}^N{c}_{kj}(t,x)r^j+\lambda r^k.$$

(12)

Let us check whether CP holds for system (11). By (6) and the monotonicity condition (5), one obtains the inequality

$$F^k(t,x,r,p^k,X^k)-F^k(t,x,s,p^k,X^k)\ge \lambda (r^k-s^k)$$

(13)

for every $r,s\in R^N$ such that

$$\underset{1\le i\le N}{max}(r^i-s^i)=r^k-s^k$$

and every $(t,x)\in G$, $p^k\in R^n$, $X^k\in S^n$.

Indeed, from (5) and (6), inequality (13) becomes

$$F^k(t,x,r,p^k,X^k)-F^k(t,x,s,p^k,X^k)\ge \sum _{j=1}^N{c}_{kj}(t,x)(r^j-s^j)+\lambda (r^k-s^k)\ge$$

$$\ge \sum _{j=1,j\ne k}^N{c}_{kj}(t,x)[r^j-s^j-(r^k-s^k)]+\lambda (r^k-s^k)\ge \lambda (r^k-s^k).$$

In order to employ the CP to system (11), the following condition should hold

There is a continuous function $\omega :R^{+}\to R^{+}$, $\omega \left(0\right)=0$, such that, if $X^i,Y^i\in S^n$, $\alpha >1$ and

$$-3\alpha \left(\begin{array}{cc}I& 0\\ 0& I\end{array}\right)\le \left(\begin{array}{cc}{X}^i& 0\\ 0& Y^i\end{array}\right)\le 3\alpha \left(\begin{array}{cc}I& -I\\ -I& I\end{array}\right).$$

(14)

then

$$F^i(t,y,r,\alpha (x^i-y^i),Y^i)-F^i(t,x,r,\alpha (x^i-y^i),X^i)\le {\omega \left(\alpha \right|x-y|}^2+\frac{1}{\alpha })$$

(15)

for all $i=1,\dots ,N$, $t\in (0,T)$, $x,y\in \mathsf{\Omega }$, $r\in R^N$ and $\alpha >1$.

Then, the following theorem holds:

Theorem 1.

Suppose that conditions (2)–(6) and (15) hold and that $w\in USC(\overline{G}\backslash \mathsf{\Gamma })$, $W\in LSC(\overline{G}\backslash \mathsf{\Gamma })$ are viscosity sub- and super-solutions to (11) satisfying

$$w^i(t,x)\le W^i(t,x)$$

(16)

for $(t,x)\in \mathsf{\Gamma }$, $i=1,\dots ,N$. Then, $w^i(t,x)\le W^i(t,x)$ for $(t,x)\in \overline{G}$ and $i=1,\dots ,N$.

Proof of Theorem 1.

For $ϵ>0$, the function $w^k(t,x)-\frac{ϵ}{T-t}$ is also a sub-solution of (11), i.e., in G

$$P^i\left(w^i-\frac{ϵ}{T-t}\right)=P^i\left(w^i\right)-\frac{ϵ}{{(T-t)}^2}<0,$$

(17)

$w^k(t,x)-\frac{ϵ}{T-t}\le W^i$ on $\mathsf{\Gamma }$ for $i=1,\dots ,N$. Without a loss of generality, we suppose that $w^i$ satisfies

$$P^i{w}^i<0\mathrm{in}\mathrm{G},\underset{t\to T}{lim}{w}^i(t,x)=-\infty$$

(18)

for $i=1,2,\dots ,N$.

We suppose by contradiction that

$$\underset{i}{max}\underset{\overline{G}}{sup}\left(w^i(t,x)-W^i(t,x)\right)=\delta >0.$$

(19)

The function

$$\Phi (t,x,y,i)=w^i(t,x)-W^i(t,x)-\frac{\alpha }{2}{|x-y|}^2$$

(20)

has a positive maximum $H_{\alpha }\ge \delta$ at the point $(t_{\alpha },x_{\alpha },y_{\alpha },i)\in (0,T)\times \mathsf{\Omega }\times \mathsf{\Omega }\times [1,N]$.

Following the proof of Th.4.1 in [9], we obtain

$$\underset{\alpha \to \infty }{lim}|x_{\alpha }-y_{\alpha }|=0$$

(21)

and repeating the proof of Lemma 4.6 in [9] and Th. 8.3 in [7], we obtain that there exist $X^k,Y^k\in S^n$ and $a^k\in R$ such that $(a^k,\alpha (x_{\alpha }-y_{\alpha }),X^k)\in {\overline{P}}^{2,+}{w}^k(t_{\alpha },x_{\alpha })$, $(a^k,\alpha (x_{\alpha }-y_{\alpha }),Y^k)\in {\overline{P}}^{2,-}{W}^k(t_{\alpha },x_{\alpha })$, and $X^k,Y^k$ satisfy (14).

By Definition 1, in (13) and (21), we obtain the following impossible chain of inequalities for $\alpha \to \infty ,ϵ\to 0$

$$\delta \alpha \le \alpha (w^k(t_{\alpha },x_{\alpha })-W^k(t_{\alpha },x_{\alpha }))\le$$

$$\le a^k+F^k(t_{\alpha },x_{\alpha },w(t_{\alpha },x_{\alpha }),\alpha (x_{\alpha }^k-y_{\alpha }^k),X^k)-a^k-F^k(t_{\alpha },x_{\alpha },W(t_{\alpha },x_{\alpha }),\alpha (x_{\alpha }^k-y_{\alpha }^k),Y^k)\le$$

$$\le w\left(\alpha \right|x_{\alpha }-y_{\alpha }{|}^2+\frac{1}{\alpha }).$$

□

Theorem 2.

Suppose that conditions (2)–(6) and (15) hold and $g^i\in USC(\overline{G}\backslash \mathsf{\Gamma })$, $h^i\in LSC(\overline{G}\backslash \mathsf{\Gamma })$ are the viscosity sub- and super-solutions of (1), respectively, satisfying $g^i(t,x)\le h^i(t,x)$ on Γ, $i=1,\dots ,N$. Then, there exists a solution $u(t,x)$ of (1), such that $g^i(t,x)\le u^i(t,x)\le h^i(t,x)$ for $(t,x)\in G$ and $i=1,\dots ,N$.

Proof of Theorem 2.

The existence result follows from the generalized Perron method, which is applied to system (1).

First step: Let U be the set of all subsolutions $z(t,x)$ of (11) such that $z^i(t,x)\le h^i(t,x)$ in G for $i=1,\dots ,N$ and define $u^i(t,x)={sup}_{\overline{G}}\left\{z^i(t,x),z\in U\right\}$ for $i=1,\dots ,N$ and $(t,x)\in \overline{G}$. Moreover, following the proof of Lemma 3.1 in [9], one can prove that $u(t,x)$ is a subsolution of (11).

Second step: We prove by contradiction that $u(t,x)$ is a supersolution. Suppose the opposite, i.e., that $u(t,x)$ is not a supersolution; therefore, there exists $s\in (0,T],y\in \mathsf{\Omega },k\in [1,N],\psi (t,x)\in C^2\left(\overline{G}\right)$ and $\mu >0$ such that $u^k(t,x)-\psi (t,x)$ attains its local minimum at $(s,y)$. Without loss of generality, we have

$$(u^k-\psi )(s,y)=0,(u^k-\psi )(t,x)\ge {|x-y|}^4$$

(22)

for all $(t,x)\in \overline{G}$, but

$${\psi }_t(s,y)+F^k(s,y,u^k(s,y),D\psi (s,y),D^2\psi (s,y))\le -\mu <0.$$

(23)

Third step: We prove that $\psi (s,y)\le u^k(s,y)$. If not, from (13), (22), (23), and $u^k(s,y)=h^k(s,y)$, it follows that $h^k(t,x)-\psi (t,x)$ attains a minimum at $(s,y)$ if $h^k(s,y)=\psi (s,y)$. Hence,

$$0\le {\psi }_t(s,y)+F^k(s,y,h^k(s,y),D\psi (s,y),D^2\psi (s,y))\le$$

$$\le {\psi }_t(s,y)+F^k(s,y,u^k(s,y),D\psi (s,y),D^2\psi (s,y))\le -\mu <0$$

that is impossible, i.e., $\psi (s,y)<u^k(s,y)$.

By (23), there is $\delta >0$ such that $\left\{\right|y|<2\delta \}\subset \mathsf{\Omega }$ and

$${\psi }_t(t,x)+F^k(t,x,u^k(t,x),D\psi (t,x),D^2\psi (t,x))\le -\frac{\mu }{2}$$

(24)

for $(t,x)\in G_{2\delta }$, $G_{\delta }=\{0<s-\delta \le min(T,s+\delta ),|x-y|<\delta \}$.

It is clear that $w(t,x)$, $w^i(t,x)=\psi (t,x)$ for $i\ne k$, $w^k(t,x)=u^k(t,x)$ is a subsolution of (11) in $G_{2\delta }$. By (22), we have

$$u^k(t,x)\ge \psi (t,x)+{\delta }^4$$

(25)

for $(t,x)\in G\backslash G_{\delta }$. If $\delta$ is small enough, the function $z(t,x),z^k=\psi +{\delta }^4,z^i=u^i$ is a subsolution of (11) by (25). We construct the function $Z^k=max\{u^k,\psi +{\delta }^4\}$ in $G_{2\delta }$ and $Z^k=u^k$ in $G\backslash G_{2\delta }$, $Z^i=u^i$ for $i\ne k$, which is a subsolution of (11) in the interior of $G_{2\delta }$ and in $G\backslash G_{\delta }$, i.e., it is a subsolution of (11) in G. Moreover, $Z(t,x)\le h(t,x)$, i.e., $Z\in U$. Since $Z^k(s,y)<\psi (s,y)+{\delta }^4$ there exists a point $(a,b)$ near $(s,y)$ such that $Z^k(a,b)>u^k(a,b)$, which contradicts the maximality of $u^k$. □

4. Strong Maximum Principle

The existence of sub- and super solutions $V(t,x)$ and $W(t,x)$ follows from the strong maximum principle for parabolic system (1).

Theorem 3.

(Strong interior maximum principle) Suppose that conditions (2)–(6) hold for system (1) and

$$E^k(t,x,0,0,0)\ge 0.$$

(26)

If $u=(u^1,\dots ,u^N)$, $u^k(t,x)\in USC(\overline{G}\backslash \mathsf{\Gamma })$ is a viscosity subsolution to (1) in $\overline{G}\backslash \mathsf{\Gamma }$, then $u(t,x)$ attains a positive absolute maximum on the parabolic boundary Γ of the cylinder G.

Furthermore, we also use the notion “absolute maximum” of $u=(u^1,u^2,\dots ,u^N)$:

Definition 2.

If $su{p}_{\overline{G}}{u}^k(t,x)=M_k$ then $M=ma{x}_{1\le k\le N}\left\{M_k\right\}$ is the absolute maximum of $u(t,x)$.

Proof of Theorem 3.

Suppose by contradiction that $u(t,x)$ attains its positive absolute maximum M at some point of $\overline{G}\backslash \mathsf{\Gamma }$ but

$$\underset{1\le k\le N}{max}\left\{\underset{\mathsf{\Gamma }}{sup}{u}^k(t,x)\right\}=m<M.$$

(27)

For sufficiently small positive constant $\lambda$, satisfying

$$m.e^{\lambda T}<M.$$

(28)

we construct the function $w=(w^1,w^2,\dots ,w^N)$, $w^k(t,x)=e^{-\lambda t}{u}^k(t,x)$. It is obvious that $w(t,x)$ is a viscosity subsolution of the system

$$w_t^k+\lambda w^k+e^{-\lambda t}{F}^k(t,x,e^{\lambda t}{w}^k,e^{\lambda t}D{w}^k,e^{\lambda t}{D}^2{w}^k)+\sum _{j=1}^N{c}_{kj}(t,x)w^j=0$$

(29)

for $(t,x)\in \overline{G}\backslash \mathsf{\Gamma }$ and $k=1,\dots ,N$.

Since

$$\underset{1\le k\le N}{max}\left\{\underset{\mathsf{\Gamma }}{sup}{w}^k(t,x)\right\}\le \underset{1\le k\le N}{max}\left\{\underset{\mathsf{\Gamma }}{sup}{u}^k(t,x)\right\}=m$$

and

$$\underset{1\le k\le N}{max}\left\{\underset{\overline{G}\backslash \mathsf{\Gamma }}{sup}{w}^k(t,x)\right\}\ge e^{-\lambda T}\underset{1\le k\le N}{max}\left\{\underset{\overline{G}\backslash \mathsf{\Gamma }}{sup}{u}^k(t,x)\right\}=M.e^{-\lambda T}>m$$

from (28), it follows that $w(t,x)$ attains its positive absolute maximum at a point $(t_1,x_1)\in \overline{G}\backslash \mathsf{\Gamma }$, i.e., $x_1\in \mathsf{\Omega }$ and $0<t_1\le T$.

System (29) is a degenerate elliptic one of the variables $(t,x)$ in G, i.e.

$$L^{k}w=e^{-\lambda t}{E}^k(t,x,e^{\lambda t}{w}^k,e^{\lambda t}D{w}^k,e^{\lambda t}{D}^2{w}^k)+w_t^k+\sum _{j=1}^N\left(c_{kj}(t,x)+\lambda {\delta }_k^j\right)w^j=0$$

(30)

for $(t,x)\in G$ and $k=1,\dots ,N$, where ${\delta }_k^j$ is a Kroneker symbol, ${\delta }_k^j=0$ for $j\ne k$, ${\delta }_k^k=1$, and one can apply Theorems 1 and 3 in [26] to system (29). Indeed, all conditions (2), (4), (5), and (26) are trivially satisfied for (30). Furthermore, by (6), we have

$$\sum _{j=1}^N\left(c_{kj}(t,x)+\lambda .{\delta }_k^j\right)=\lambda +\sum _{j=1}^N{c}_{kj}(t,x)\ge \lambda >0.$$

(31)

If $t_1<T$, then $w(t,x)$ does not attain positive absolute maximum at an interior point of G by the strong interior maximum principle for elliptic systems, Theorem 1 in [26]. This fact contradicts the assumption $(t_1,x_1)\in G$.

Let us consider the case that $t_1=T$, $x_1\in \mathsf{\Omega }$. Without loss of generality we suppose that

$$\underset{1\le k\le N}{max}\left\{\underset{\overline{G}}{sup}{w}^k(t,x)\right\}=w^1(T_1,x_1)>0.$$

(32)

Applying the strong boundary maximum principle to the elliptic system (30) in G, Theorem 3 in [26], it follows for $k=1$, $\rho =(-1,0)$ that

$$\underset{s\to +0}{lim}\frac{u^1(T-s,x_1)-u^1(T,x_1)}{s}=-a<0.$$

(33)

Since $u^1(T,x)$ has a maximum at $x_1\in \mathsf{\Omega }$,

$$(0,0)\in J^{2,+}{u}^1(T,x_1),$$

(34)

($J^{2,+}$ is the elliptic semijet, as can be seen in [26]), and (33), (34) give us that $(a,0,0)\in P^{2,+}{u}^1(T,x_1)$ for some $a>0$.

By Definition 1 (1), (2), (5), (6), (7), and (32), we obtain the following impossible chain of inequalities:

$$0\ge a+e^{-\lambda T}{F}^1(T,x_1,e^{\lambda .T}w(T,x_1),0,0)+\sum _{j=1}^N\left(c_{kj}(T,x_1)+{\delta }_1^j\right)w^j(T,x_1)\ge$$

$$\ge a+e^{-\lambda T}{F}^1(T,x_1,0,0,0)+\sum _{j=1}^N{c}_{kj}(T,x_1)M+\left(\lambda +c_{11}(T,x_1)\right)w^1(T,x_1)+$$

$$\sum _{j=2}^N{c}_{kj}(T,x_1)\left(w^j(T,x_1)-M\right)\ge a+\lambda w^1(T,x_1)>0$$

because $c_{11}(T,x_1)\ge -{\sum }_{j=2}^N{c}_{1j}(T,x_1)\ge 0$.

Theorem 3 is completed. □

A more precise result for the strong interior maximum principle could be derived as a consequence of the proof of Theorem 3, since (31) holds.

Corollary 1.

Suppose that conditions (2), (4), (5), and (26) hold. If $u=(u^1,\dots ,u^N)$, $u(t,x)\in USC(\overline{G}\backslash \mathsf{\Gamma })$ is a viscosity subsolution to (1) in $\overline{G}\backslash \mathsf{\Gamma }$ then $u(t,x)$ does not attain a positive absolute maximum in $\overline{G}\backslash \mathsf{\Gamma }$.

The proof of this Corollary replicates the proof of Theorem 3 for $\lambda =0$.

Finally, we formulate the strong boundary maximum principle for parabolic systems.

Theorem 4.

(Strong boundary maximum principle) Suppose that conditions (2), (4), (5), (6), and (26) hold, then Ω satisfies an interior sphere condition. Let $u(t,x)=(u^1(t,x),\dots ,u^N(t,x))$, $u^k(t,x)\in USC(\overline{G}\backslash \mathsf{\Gamma })$, be a viscosity subsolution to (1). If $u(t,x)$ attains a positive absolute maximum at some boundary point $(t_0,x_0)$, $t_0\in (0,T)$, $x_0\in \partial \mathsf{\Omega }$, i.e., $u^k(t_0,x_0)=M>0$ for some $1\le k\le N$, then the following inequality holds:

$$li{m}_{s\to 0}\frac{u^k(t_0+s\tau ,x_0+s\rho )-u^k(t_0,x_0)}{s}<0$$

(35)

for every non-tangential to $\partial \mathsf{\Omega }$ direction $(\tau ,\rho )$ pointing into G.

The proof of Theorem 4 follows from Theorem 3 in [26] being applied to the degenerate elliptic system (1) of the variables $(t,x_1,\dots ,x_N)\in G\subset R^{N+1}$, taking into account that (31) holds for system (30).

5. Discussion

The results in this paper can easily be extended to parabolic systems with a fully nonlinear principal part and a quasi-monotone coupling part. On the other hand, for non-cooperative fully nonlinear systems, there are no results on the validity of the comparison and maximum principles.

An open problem is the one of spectral properties of system (1).