Existence of Positive Solutions and Its Asymptotic Behavior of (p(x), q(x))-Laplacian Parabolic System

Abstract

This paper deals with the existence of positively solution and its asymptotic behavior for parabolic system of $\left(p\right(x),q(x\left)\right)$-Laplacian system of partial differential equations using a sub and super solution according to some given boundary conditions, Our result is an extension of Boulaaras’s works which studied the stationary case, this idea is new for evolutionary case of this kind of problem.

1. Introduction

In this paper, we consider the following evolutionary problem: find $u\in L^2(0,T,H_0^1\left(\mathsf{\Omega }\right))$ solution of

$$\left\{\begin{array}{c}\frac{\partial u}{\partial t}-{\Delta }_{p\left(x\right)}u={\lambda }^{p\left(x\right)}\left[{\lambda }_{1}a\left(x\right)f\left(v\right)+{\mu }_{1}c\left(x\right)h\left(u\right)\right]\mathrm{in}{Q}_T=(0,T)\times \mathsf{\Omega },\hfill \\ \frac{\partial v}{\partial t}-{\Delta }_{q\left(x\right)}v={\lambda }^{q\left(x\right)}\left[{\lambda }_{2}b\left(x\right)g\left(u\right)+{\mu }_{2}d\left(x\right)\tau \left(v\right)\right]\mathrm{in}{Q}_T=(0,T)\times \mathsf{\Omega },\hfill \\ u=v=0\mathrm{on}\partial Q_T=(0,T)\times \partial \mathsf{\Omega },\hfill \\ u(x,0)={\kappa }_1\left(x\right),u(x,0)={\kappa }_2\left(x\right),\hfill \end{array}\right.$$

(1)

where $\mathsf{\Omega }$⊂${\mathbb{R}}^N$ is a bounded domain and the functions $p\left(x\right),q\left(x\right)$ belong to $C^1\left(\overline{\mathsf{\Omega }}\right)$ and satisfying the following conditions:

$$1<p^{-}:=\underset{x\in \mathsf{\Omega }}{\inf }p\left(x\right)\le p^{+}:=\underset{\mathsf{\Omega }}{\sup }p\left(x\right)<\infty ,1<q^{-}:=\underset{x\in \mathsf{\Omega }}{\inf }q\left(x\right)\le q^{+}:=\underset{x\in \mathsf{\Omega }}{\sup }q\left(x\right)<\infty$$

(2)

and satisfy some natural growth condition at $u=\infty .$

${\Delta }_{p\left(x\right)}$ is given by ${\Delta }_{p\left(x\right)}u=\mathrm{div}({|\nabla u|}^{p\left(x\right)-2}\nabla u)$ is called $p\left(x\right)$-Laplacian, the parameters $\lambda ,{\lambda }_1,{\lambda }_2,{\mu }_1$ and ${\mu }_2$ are positive with $a,b,c,d$ are regular functions. In addition we did not consider any sign condition on $f\left(0\right),g\left(0\right),h\left(0\right),\tau \left(0\right).$

The linear and nonlinear stationary equations with operators of quasilinear homogeneous type as p-Laplace operator can be carried out according to the standard Sobolev spaces theory of $W^{m,p}$, and thus we can find the weak solutions. The last spaces consist of functions having weak derivatives which verify some conditions of integrability. Thus, we can have the nonhomogeneous case of $p(.)$-Laplace operators in this last condition. We will use Sobolev spaces of the exponential variable in our standard framework, so that $L^{p\left(.\right)}\left(\mathsf{\Omega }\right)$ will be used instead of Lebesgue spaces $L^p\left(\mathsf{\Omega }\right).$

We denote new Sobolev space by $W^{m,p}\left(\mathsf{\Omega }\right)$, if we replace $L^p\left(\mathsf{\Omega }\right)$ by $L^{p\left(.\right)}\left(\mathsf{\Omega }\right)$, the Sobolev spaces becomes $W^{m,p\left(.\right)}\left(\mathsf{\Omega }\right)$. Several Sobolev spaces properties have been extended to spaces of Orlicz-Sobolev, particularly by O’Neill in the reference ([1]). The spaces $W^{m,p\left(.\right)}\left(\mathsf{\Omega }\right)$ and $L^{p\left(.\right)}\left(\mathsf{\Omega }\right)$ have been carefully studied by many researchers team (see the references ([2] and [3,4,5]).

Here, in our study we consider the boundedness condition in domain $\mathsf{\Omega }$, because many results under p-Laplacian theory are not usually verified for the $p\left(x\right)$-Laplacian theory; for that in ([6]) the quotient

$${\lambda }_{p\left(x\right)}=\underset{u\in W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)\setminus \left\{0\right\}}{\inf }\frac{{\int }_{\mathsf{\Omega }}\frac{1}{p\left(x\right)}{\left|\nabla u\right|}^{p\left(x\right)}dx}{{\int }_{\mathsf{\Omega }}\frac{1}{p\left(x\right)}{\left|u\right|}^{p\left(x\right)}dx}$$

(3)

becomes 0 generally. Then ${\lambda }_{p\left(x\right)}$ can be positive only for some given conditions. In fact, the first eigenvalue of $p\left(x\right)$-Laplacian and its associated eigenfunction cannot exist, the existence of the positive first eigenvalue ${\lambda }_p$ and getting its eigenfunction are very important in the p-Laplacian problem study. Therefore, the study of existence of solutions of our problems have more meaning. Many studies of the experimental side have been studied on various materials that rely on this advanced theory, as they are important in electrical fluids, which states that viscosity relates to the electric field in a certain liquid.

Recently, in ([2,6,7,8]), we have proved the existence of positive solutions of many classes of $\left(p\left(x\right),q\left(x\right)\right)$-Laplacian stationary problems by using the sub -super solution concept. The current results are an extension of our previous stationary study to the parabolic case, where we follow-up the same procedures mathematical proofs similar to that in ([2,7]) by using difference time scheme taking into consideration the stability analysis of the used scheme and the same conditions which have given in references mentioned earlier. Our result is an extension for our previous study in ( [2,7,9]) which studied the stationary case, this idea is new for evolutionary case of this kind of problem.

The outline of paper consists as follow: In first section we give some definitions, basic theorems and necessarily propositions in the functional analysis which will be used in our study. Then in Section 3, we prove our main result.

2. Preliminaries Results and Assumptions

In order to discuss problem (1), we need some theories on $W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)$ which we call variable exponent Sobolev space. Firstly we state some basic properties of spaces $W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)$ which will be used later (for details, see [10]).

Let us define

$$L^{p\left(x\right)}\left(\mathsf{\Omega }\right)=\left\{u:u\mathrm{is}\mathrm{a}\mathrm{measurable}\mathrm{real}-\mathrm{valued}\mathrm{function}\mathrm{such}\mathrm{that}\underset{\mathsf{\Omega }}{\int }{\left|u\left(x\right)\right|}^{p\left(x\right)}dx<\infty \right\}.$$

We introduce the norm on $L^{p\left(x\right)}\left(\mathsf{\Omega }\right)$ by

$${\left|u\left(x\right)\right|}_{p\left(x\right)}=\inf \left\{\lambda >0:\underset{\mathsf{\Omega }}{\int }{\left|\frac{u\left(x\right)}{\lambda }\right|}^{p\left(x\right)}dx\le 1\right\}$$

and

$$W^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)=\left\{u\in L^{p\left(x\right)}\left(\mathsf{\Omega }\right);\left|\nabla u\right|\in L^{p\left(x\right)}\left(\mathsf{\Omega }\right)\right\},$$

with the norm

$$∥u∥={\left|u\right|}_{p\left(x\right)}+{\left|\nabla u\right|}_{p\left(x\right)},\forall u\in W^{1,p\left(x\right)}\left(\mathsf{\Omega }\right).$$

We denote by $W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)$ the closure of $C_0^{\infty }\left(\mathsf{\Omega }\right)$ in $W^{1,p\left(x\right)}\left(\mathsf{\Omega }\right).$

We introduce in this applying for problem (2), we will assume that:

• $\left(H_1\right)$$p,q\in C^1\left(\overline{\mathsf{\Omega }}\right)$ and $1<p_{-}<p_{+},1<q_{-}<q_{+};$
• $\left(H_2\right)$$f,g,h$ and $\tau :\left[0,+\infty \right[\to \mathbb{R}$ are $C^1,$ monotone functions, such that

  $$\underset{u\to +\infty }{\lim }f\left(u_k\right)=+\infty \underset{u\to +\infty }{\lim }g\left(u_k\right)=+\infty ,\underset{u\to +\infty }{\lim }h\left(u_k\right)=+\infty ,\underset{u\to +\infty }{\lim }\tau \left(u_k\right)=+\infty ,$$
• $\left(H_3\right)$$\underset{u\to +\infty }{\lim }\frac{f(M{\left(g\left(u_k\right)\right)}^{\frac{1}{q^{-}-1}})}{u_k^{p^{-}-1}}=0,$ for all $M>0;$
• $\left(H_4\right) \underset{u\to +\infty }{\lim }\frac{h\left(u_k\right)}{u_k^{p^{-}-1}}=0,$ and $\underset{u\to +\infty }{\lim }\frac{\tau \left(u_k\right)}{u_k^{p^{-}-1}}=0;$
• $\left(H_5\right)$$a,b,c,d:$$\overline{\mathsf{\Omega }}\to (0,+\infty )$ are contionous functions, such that

  $$a_1=\underset{x\in \overline{\mathsf{\Omega }}}{\min }a\left(x\right),b_1=\underset{x\in \overline{\mathsf{\Omega }}}{\min }b\left(x\right),c_1=\underset{x\in \overline{\mathsf{\Omega }}}{\min }c\left(x\right),d_1=\underset{x\in \overline{\mathsf{\Omega }}}{\min }d\left(x\right),$$

  $$a_2=\underset{x\in \overline{\mathsf{\Omega }}}{\max }a\left(x\right),b_2=\underset{x\in \overline{\mathsf{\Omega }}}{\max }b\left(x\right),c_2=\underset{x\in \overline{\mathsf{\Omega }}}{\max }c\left(x\right),d_2=\underset{x\in \overline{\mathsf{\Omega }}}{\max }d\left(x\right).$$

The Semi-Discrete Problem

We discrete the problem (1) by difference time scheme, we obtain the following problems

$$\left\{\begin{array}{c}{u}_k-{\tau }^{\prime }{\Delta }_{p\left(x\right)}{u}_k={\tau }^{\prime }{\lambda }^{p\left(x\right)}\left[{\lambda }_{1}a\left(x\right)f\left(v\right)+{\mu }_{1}c\left(x\right)h\left(u_k\right)\right]+u_{k-1}\mathrm{in}\mathsf{\Omega },\hfill \\ v_k-{\tau }^{\prime }{\Delta }_{q\left(x\right)}v={\tau }^{\prime }{\lambda }^{q\left(x\right)}\left[{\lambda }_{2}b\left(x\right)g\left(u_k\right)+{\mu }_{2}d\left(x\right)\tau \left(v\right)\right]+v_{k-1}\mathrm{in}\mathsf{\Omega },\hfill \\ u_k=v=0\mathrm{on}\partial \mathsf{\Omega },\hfill \\ u_0={\phi }_0,\hfill \end{array}\right.$$

(4)

where $N{\tau }^{\prime }=T,$ $0<{\tau }^{\prime }<1,$ and for $1\le k\le N.$

We define

$$⟨L\left(u_k\right),v⟩=\underset{\mathsf{\Omega }}{\int }{\left|\nabla u_k\right|}^{p\left(x\right)-2}\nabla u_k\nabla vdx,\forall u_k,v\in W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right).$$

According to ([11] in Theorem 3.1), the bounded operator $L:W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)\to {\left(W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)\right)}^{\ast }$ is a continuous and strictly monotone, and it is a homeomorphism.

We considere mapping $A:W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)\to {\left(W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)\right)}^{\ast }$ as

$$⟨A\left(u_k\right),\phi ⟩=\underset{\mathsf{\Omega }}{\int }\left({\left|\nabla u_k\right|}^{p\left(x\right)-2}\nabla u_k\nabla \phi +h(x,u_k)\phi \right)dx,\mathrm{for}\mathrm{all}{u}_k,v\in W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right),$$

where $h(x,u_k)$ is continuous on $\overline{\mathsf{\Omega }}\times \mathbb{R},$ and $h(x,.)$ is increasing function.It is easy to verify that A is a continuous bounded mapping. By the proof ([12]).

Definition 1.

An weak solution to discretized problems $\left(P_k\right)$ is a sequence ${(u_k,v)}_{0\le k\le N}$ such that $u_0={\phi }_0$ and $(u_k,v)$ is defined by

$$\left\{\begin{array}{c}{u}_k-{\tau }^{\prime }{\Delta }_{p\left(x\right)}{u}_k={\tau }^{\prime }{\lambda }^{p\left(x\right)}\left[{\lambda }_{1}a\left(x\right)f\left(v\right)+{\mu }_{1}c\left(x\right)h\left(u_k\right)\right]+u_{k-1}in\mathsf{\Omega },\hfill \\ v_k-{\tau }^{\prime }{\Delta }_{q\left(x\right)}v={\tau }^{\prime }{\lambda }^{q\left(x\right)}\left[{\lambda }_{2}b\left(x\right)g\left(u_k\right)+{\mu }_{2}d\left(x\right)\tau \left(v\right)\right]+v_{k-1}in\mathsf{\Omega },\hfill \\ u_k=v=0on\partial \mathsf{\Omega },\hfill \end{array}\right.$$

such that

$$\left\{\begin{array}{c}-{\Delta }_{p\left(x\right)}{u}_k={\lambda }^{p\left(x\right)}\left[{\lambda }_{1}a\left(x\right)f\left(v\right)+{\mu }_{1}c\left(x\right)h\left(u_k\right)\right]-\frac{u_k-u_{k-1}}{{\tau }^{\prime }}in\mathsf{\Omega },\hfill \\ -{\Delta }_{q\left(x\right)}v={\lambda }^{q\left(x\right)}\left[{\lambda }_{2}b\left(x\right)g\left(u_k\right)+{\mu }_{2}d\left(x\right)\tau \left(v\right)\right]-\frac{v_k-v_{k-1}}{{\tau }^{\prime }}in\mathsf{\Omega },\hfill \\ u_k=v=0on\partial \mathsf{\Omega }.\hfill \end{array}\right.$$

(5)

We have the following:

$\left(1\right)$ If $(u_k,v)\in \left(W_0^{1.p\left(x\right)}\left(\mathsf{\Omega }\right)\times W_0^{1.q\left(x\right)}\left(\mathsf{\Omega }\right)\right),(u_k,v)$ is called a weak solution of (5) if it satisfies

$$\underset{\mathsf{\Omega }}{\int }{\left|\nabla u_k\right|}^{p\left(x\right)-2}\nabla u_k.\nabla \phi dx=\underset{\mathsf{\Omega }}{\int }\left[{\lambda }^{p\left(x\right)}\left[{\lambda }_{1}a\left(x\right)f\left(v\right)+{\mu }_{1}c\left(x\right)h\left(u_k\right)\right]-\frac{u_k-u_{k-1}}{{\tau }^{\prime }}\right]\phi dx,$$

$$\underset{\mathsf{\Omega }}{\int }{\left|\nabla v\right|}^{q\left(x\right)-2}\nabla v.\nabla \psi dx=\underset{\mathsf{\Omega }}{\int }\left[{\lambda }^{q\left(x\right)}\left[{\lambda }_{2}b\left(x\right)g\left(u_k\right)+{\mu }_{2}d\left(x\right)\tau \left(v\right)\right]-\frac{v_k-v_{k-1}}{{\tau }^{\prime }}\right]\psi dx.$$

(6)

for all

$$(\phi ,\psi )\in \left(W_0^{1.p(.)}\left(\mathsf{\Omega }\right)\times W_0^{1.q(.)}\left(\mathsf{\Omega }\right)\right)$$

with $(\phi ,\psi )⩾0.$

$\left(2\right)$ We say called a sub solution (respectively a super solution) of (1) if

$$\underset{\mathsf{\Omega }}{\int }{\left|\nabla u_k\right|}^{p\left(x\right)-2}\nabla u_k.\nabla \phi dx\le (\mathrm{respectively}⩾)\underset{\mathsf{\Omega }}{\int }\left[{\lambda }^{p\left(x\right)}\left[{\lambda }_{1}a\left(x\right)f\left(v\right)+{\mu }_{1}c\left(x\right)h\left(u_k\right)\right]-\frac{u_k-u_{k-1}}{{\tau }^{\prime }}\right]\phi dx,$$

$$\underset{\mathsf{\Omega }}{\int }{\left|\nabla v\right|}^{q\left(x\right)-2}\nabla v.\nabla \psi dx\le (\mathrm{respectively}⩾)\underset{\mathsf{\Omega }}{\int }\left[{\lambda }^{q\left(x\right)}\left[{\lambda }_{2}b\left(x\right)g\left(u_k\right)+{\mu }_{2}d\left(x\right)\tau \left(v\right)\right]-\frac{v_k-v_{k-1}}{{\tau }^{\prime }}\right]\psi dx.$$

Lemma 1.

(Comparison principle) Let $u_k,v\in W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)$ verify $A{u}_k-Av⩾0$ in ${\left(W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)\right)}^{\ast },\phi \left(x\right)=\min \left\{u_k\left(x\right)-v\left(x\right),0\right\}.$ If $\phi \left(x\right)\in W_0^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)$ (i.e., $u_k⩾v$ on $\partial \mathsf{\Omega }$), then $u_k⩾v$ a.e in $\mathsf{\Omega }.$

Here, we will use the notation $d(x,\partial \mathsf{\Omega })$ to denote the distance of $x\in \mathsf{\Omega }$ to denote the distance of $\mathsf{\Omega }.$

Denote $d\left(x\right)=d(x,\partial \mathsf{\Omega })$ and $\partial {\mathsf{\Omega }}_{\epsilon }=\left\{x\in \mathsf{\Omega }:d(x,\partial \mathsf{\Omega })<\epsilon \right\}.$

Since $\partial \mathsf{\Omega }$ is $C^2$ regularly, there exists a constant $\delta$$\in (0,1)$ such that $d\left(x\right)\in C^2\left({\overline{\partial \mathsf{\Omega }}}_{3\delta }\right)$ and $|\nabla d(x\left)\right|=1.$

Denote also

$$v_1\left(x\right)=\left\{\begin{array}{c}\gamma d\left(x\right),d\left(x\right)<\delta ,\hfill \\ \gamma \delta +{\int }_{\delta }^{d\left(x\right)}\gamma {\left(\frac{2\delta -t}{\delta }\right)}^{\frac{2}{p^{-}-1}}{({\lambda }_1{a}_1+{\mu }_1{c}_1)}^{\frac{2}{p^{-}-1}}dt,\delta \le d\left(x\right)\le 2\delta ,\hfill \\ \gamma \delta +{\int }_{\delta }^{2\delta }\gamma {\left(\frac{2\delta -t}{\delta }\right)}^{\frac{2}{p^{-}-1}}{({\lambda }_1{b}_1+{\mu }_1{d}_1)}^{\frac{2}{p^{-}-1}}dt,2\delta \le d\left(x\right)\hfill \end{array}\right.$$

and

$$v_2\left(x\right)=\left\{\begin{array}{c}\gamma d\left(x\right),d\left(x\right)<\delta ,\hfill \\ \gamma \delta +{\int }_{\delta }^{d\left(x\right)}\gamma {\left(\frac{2\delta -t}{\delta }\right)}^{\frac{2}{p^{-}-1}}{({\lambda }_2{a}_2+{\mu }_2{c}_2)}^{\frac{2}{q^{-}-1}}dt,\delta \le d\left(x\right)\le 2\delta ,\hfill \\ \gamma \delta +{\int }_{\delta }^{2\delta }\gamma {\left(\frac{2\delta -t}{\delta }\right)}^{\frac{2}{p^{-}-1}}{({\lambda }_2{b}_2+{\mu }_2{d}_2)}^{\frac{2}{q^{-}-1}}dt,2\delta \le d\left(x\right).\hfill \end{array}\right.$$

Obviously,

$$0\le v_1\left(x\right),v_2\left(x\right)\in C^1\left(\overline{\mathsf{\Omega }}\right).$$

Considering

$$\left\{\begin{array}{c}-{\Delta }_{p\left(x\right)}w\left(x\right)=\eta \mathrm{in}\mathsf{\Omega }\hfill \\ w=0\mathrm{on}\partial \mathsf{\Omega }.\hfill \end{array}\right.$$

(7)

Lemma 2.

([13]), If positive parameter η is large enough and w is the unique solution of (7), then we have

(i) 

For any $\theta \in (0,1)$ there exists a positive constant $C_1,$ such that

$$C_1{\eta }^{\frac{1}{p^{+}-1+\theta }}\le \underset{x\in \overline{\mathsf{\Omega }}}{\max }w\left(x\right).$$

(ii) 

There exists a positive constant $C_2,$ such that

$$\underset{x\in \overline{\mathsf{\Omega }}}{\max }w\left(x\right)\le C_2{\eta }^{\frac{1}{p^{-}-1}}$$

3. Main Result

In the following, once we have no misunderstanding, we always use $C_i$ to denote the positive constants.

Theorem 1.

Assume that the conditions $\left(H_1\right)$–$\left(H_5\right)$ are statisfied.Then, problem (1) has a positive solution when λ is large enough.

Proof. 

We establish Theorem 1 by constructing a positive subsolution $({\varphi }_{k_1},{\varphi }_{k_2})$ and supersolution $(z_{k_1},z_{k_2})$ of (1) such that ${\varphi }_{k_1}$$\le z_{k_1}$ and ${\varphi }_{k_2}$≤$z_{k_2}$, that is $({\varphi }_{k_1},{\varphi }_{k_2})$ and $(z_{k_1},z_{k_2})$ satisfies

$$\begin{array}{ccc}\hfill \underset{\mathsf{\Omega }}{\int }{\left|\nabla {\varphi }_{k_1}\right|}^{p\left(x\right)-2}\nabla {\varphi }_{k_1}.\nabla \phi dx& \le & \underset{\mathsf{\Omega }}{\int }\left[{\lambda }^{p\left(x\right)}\left[{\lambda }_{1}a\left(x\right)f\left({\varphi }_{k_2}\right)+{\mu }_{1}c\left(x\right)h\left({\varphi }_{k_1}\right)\right]-\frac{{\varphi }_{k_1}-{\varphi }_{k_1-1}}{{\tau }^{\prime }}\right]\phi dx,\hfill \\ \hfill \underset{\mathsf{\Omega }}{\int }{\left|\nabla {\varphi }_{k_2}\right|}^{q\left(x\right)-2}\nabla {\varphi }_{k_2}.\nabla \psi dx& \le & \underset{\mathsf{\Omega }}{\int }\left[{\lambda }^{q\left(x\right)}\left[{\lambda }_{2}b\left(x\right)g\left({\varphi }_{k_1}\right)+{\mu }_{2}d\left(x\right)\tau \left({\varphi }_{k_2}\right)\right]-\frac{{\varphi }_{k_1}-{\varphi }_{k_1-1}}{{\tau }^{\prime }}\right]\psi dx,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \underset{\mathsf{\Omega }}{\int }{\left|\nabla z_{k_1}\right|}^{p\left(x\right)-2}\nabla z_{k_1}.\nabla \phi dx& \ge & \underset{\mathsf{\Omega }}{\int }\left[{\lambda }^{p\left(x\right)}\left[{\lambda }_{1}a\left(x\right)f\left(z_{k_2}\right)+{\mu }_{1}c\left(x\right)h\left(z_{k_1}\right)\right]-\frac{z_{k_1}-z_{k_1-1}}{{\tau }^{\prime }}\right]\phi dx,\hfill \\ \hfill \underset{\mathsf{\Omega }}{\int }{\left|\nabla z_{k_2}\right|}^{q\left(x\right)-2}\nabla z_{k_2}.\nabla \psi dx& \ge & \underset{\mathsf{\Omega }}{\int }\left[{\lambda }^{q\left(x\right)}\left[{\lambda }_{2}b\left(x\right)g\left(z_{k_1}\right)+{\mu }_{2}d\left(x\right)\tau \left(z_{k_2}\right)\right]-\frac{z_{k_1}-z_{k_1-1}}{{\tau }^{\prime }}\right]\psi dx,\hfill \end{array}$$

for all $(\phi ,\psi )\in \left(W_0^{1.p\left(x\right)}\left(\mathsf{\Omega }\right)\times W_0^{1.q\left(x\right)}\left(\mathsf{\Omega }\right)\right)$ with $(\phi ,\psi )⩾0.$ According to the sub-super solution method for $\left(\left(p\right(x),q(x\left)\right)\right)—$Laplacian systems see ([9,13]), the problem (1) has a positive solution.

Step 1. We will construct a subsolution of (1). Let $\sigma \in (0,\delta )$ is small enough. Denote

$${\varphi }_{k_1}\left(x\right)=\left\{\begin{array}{c}{e}^{kd\left(x\right)}-1,d\left(x\right)<\sigma ,\hfill \\ e^{kd\left(x\right)}-1+{\int }_{\delta }^{d\left(x\right)}k{e}^{k\sigma }{(\frac{2\delta -t}{2\delta -\sigma })}^{\frac{2}{p^{-}-1}}dt,\sigma \le d\left(x\right)<2\delta ,\hfill \\ e^{kd\left(x\right)}-1+{\int }_{\sigma }^{2\delta }k{e}^{k\sigma }{(\frac{2\delta -t}{2\delta -\sigma })}^{\frac{2}{p^{-}-1}}dt,2\delta \le d\left(x\right)\hfill \end{array}\right.$$

and

$${\varphi }_{k_2}\left(x\right)=\left\{\begin{array}{c}{e}^{kd\left(x\right)}-1,d\left(x\right)<\sigma ,\hfill \\ e^{kd\left(x\right)}-1+{\int }_{\delta }^{d\left(x\right)}k{e}^{k\sigma }{(\frac{2\delta -t}{2\delta -\sigma })}^{\frac{2}{q^{-}-1}}dt,\sigma \le d\left(x\right)<2\delta ,\hfill \\ e^{kd\left(x\right)}-1+{\int }_{\sigma }^{2\delta }k{e}^{k\sigma }{(\frac{2\delta -t}{2\delta -\sigma })}^{\frac{2}{q^{-}-1}}dt,2\delta \le d\left(x\right).\hfill \end{array}\right.$$

It easy to see that ${\varphi }_{k_1},{\varphi }_{k_2}\in C^1\left(\overline{\mathsf{\Omega }}\right).$

Denote

$$\alpha =\min \left\{\frac{\inf p\left(x\right)-1}{4\left(\sup \left|\nabla p\left(x\right)+1\right|\right)},\frac{\inf q\left(x\right)-1}{4\left(\sup \left|\nabla q\left(x\right)+1\right|\right)},1\right\}$$

and

$$\xi =\min \left\{{\lambda }_1{a}_{1}f\left(0\right)+{\mu }_1{c}_{1}h\left(0\right),{\lambda }_2{b}_{1}g\left(0\right)+{\mu }_2{d}_1\sigma \left(0\right),-1\right\}.$$

By some simple computations we obtain

$$-{\Delta }_{p\left(x\right)}{\varphi }_{k_1}=\left\{\begin{array}{c}-k{\left(e^{kd\left(x\right)}\right)}^{p\left(x\right)-1}\left[(p\left(x\right)-1)+(d\left(x\right)+\frac{\ln k}{k})\nabla p\nabla d+\frac{\Delta d}{k}\right],d\left(x\right)<\sigma \hfill \\ \left\{\frac{1}{2\delta -\sigma }\frac{2\left(p\right(x)-1)}{p^{-}-1}-\left(\frac{2\delta -d}{2\delta -\sigma }\right)\left[\left(\ln k{e}^{k\sigma }\right){\left(\frac{2\delta -d}{2\delta -\sigma }\right)}^{\frac{2}{p^{-}-1}}\nabla p\nabla d+\Delta d\right]\right\}\hfill \\ \times {\left(K{e}^{k\sigma }\right)}^{p\left(x\right)-1}{\left(\frac{2\delta -d}{2\delta -\sigma }\right)}^{\frac{2\left(p\right(x)-1)}{p^{-}-1}-1},\sigma \le d\left(x\right)<2\delta ,\hfill \\ 0,2\delta \le d\left(x\right)\hfill \end{array}\right.$$

and

$$-{\Delta }_{p\left(x\right)}{\varphi }_{k_2}=\left\{\begin{array}{c}-k{\left(e^{kd\left(x\right)}\right)}^{q\left(x\right)-1}\left[(q\left(x\right)-1)+(d\left(x\right)+\frac{\ln k}{k})\nabla q\nabla d+\frac{\Delta d}{k}\right],d\left(x\right)<\sigma ,\hfill \\ \left\{\frac{1}{2\delta -\sigma }\frac{2\left(q\right(x)-1)}{q^{-}-1}-\left(\frac{2\delta -d}{2\delta -\sigma }\right)\left[\left(\ln k{e}^{k\sigma }\right){\left(\frac{2\delta -d}{2\delta -\sigma }\right)}^{\frac{2}{q^{-}-1}}\nabla q\nabla d+\Delta d\right]\right\}\hfill \\ \times {\left(K{e}^{k\sigma }\right)}^{q\left(x\right)-1}{\left(\frac{2\delta -d}{2\delta -\sigma }\right)}^{{\displaystyle \frac{2\left(q\right(x)-1)}{q^{-}-1}}-1},\sigma \le d\left(x\right)<2\delta ,\hfill \\ 0,2\delta \le d\left(x\right).\hfill \end{array}\right.$$

From $\left(H_3\right)$ there exists a positive constant $M>1$ such that

$$\begin{array}{ccc}\hfill f(M-1)& ⩾& 1,g(M-1)⩾1,\hfill \\ \hfill h(M-1)& ⩾& 1,\sigma (M-1)⩾1.\hfill \end{array}$$

Let $\sigma =\frac{1}{k}\ln M,$ then

$$\sigma k=\ln M.$$

(8)

If k is sufficiently large, from (8), we have

$$-{\Delta }_{p\left(x\right)}{\varphi }_{k_1}\le -k^{p\left(x\right)}\alpha ,d\left(x\right)<\sigma .$$

(9)

Let $\lambda \xi =k\alpha ,$ then

$$k^{p\left(x\right)}\alpha ⩾-{\lambda }^{p\left(x\right)}\xi .$$

From (9), we have

$$\left\{\begin{array}{c}-{\Delta }_{p\left(x\right)}{\varphi }_{k_1}\le {\lambda }^{p\left(x\right)}\xi \le {\lambda }^{p\left(x\right)}({\lambda }_1{a}_{1}f\left(0\right)+{\mu }_1{c}_{1}h\left(0\right))\hfill \\ \le {\lambda }^{p\left(x\right)}({\lambda }_{1}a\left(x\right)f\left({\varphi }_{k_2}\right)+{\mu }_{1}c\left(x\right)h\left({\varphi }_{k_1}\right)),d\left(x\right)<\sigma .\hfill \end{array}\right.$$

(10)

Since $d\left(x\right)$$\in C^2\left(\overline{\partial {\mathsf{\Omega }}_{3\delta }}\right)$, there exists a positive constant $C_3,$ such that

$$\begin{array}{ccc}\hfill -{\Delta }_{p\left(x\right)}{\varphi }_{k_1}& \le & {\left(K{e}^{k\sigma }\right)}^{p\left(x\right)-1}{\left(\frac{2\delta -d}{2\delta -\sigma }\right)}^{\frac{2\left(p\right(x)-1)}{p^{-}-1}-1}({\lambda }_1{a}_1+{\mu }_1{c}_1)\hfill \\ & & \times \left|\left\{\frac{1}{2\delta -\sigma }\frac{2\left(p\right(x)-1)}{p^{-}-1}-\left(\frac{2\delta -d}{2\delta -\sigma }\right)\right.\right.\hfill \\ & & \left.\left.\times \left[\left(\ln k{e}^{k\sigma }\right){\left(\frac{2\delta -d}{2\delta -\sigma }\right)}^{\frac{2}{p^{-}-1}}\nabla p\nabla d+\Delta d\right]\right\}\right|\hfill \\ & \le & C_3{\left(K{e}^{k\sigma }\right)}^{p\left(x\right)-1}({\lambda }_1{a}_1+{\mu }_1{c}_1)\ln k,\sigma \le d\left(x\right)<2\delta .\hfill \end{array}$$

If k is sufficiently large, let $\lambda \xi$$=k\alpha$, then we have

$$\begin{array}{ccc}\hfill C_3{\left(K{e}^{k\sigma }\right)}^{p\left(x\right)-1}({\lambda }_1{a}_1+{\mu }_1{c}_1)\ln k& =& C_3{\left(kM\right)}^{p\left(x\right)-1}({\lambda }_1{a}_1+{\mu }_1{c}_1)\ln k\hfill \\ & \le & {\lambda }^{p\left(x\right)}({\lambda }_1{a}_1+{\mu }_1{c}_1),\hfill \end{array}$$

then

$$-{\Delta }_{p\left(x\right)}{\varphi }_{k_1}\le {\lambda }^{p\left(x\right)}({\lambda }_1{a}_1+{\mu }_1{c}_1),\sigma \le d\left(x\right)<2\delta$$

(11)

Since ${\varphi }_{k_1}\left(x\right)$, ${\varphi }_{k_2}\left(x\right)$ and $f,h$ are monotone, when $\lambda$ is large enough, we have

$$-{\Delta }_{p\left(x\right)}{\varphi }_{k_1}\le {\lambda }^{p\left(x\right)}({\lambda }_{1}a\left(x\right)f\left({\varphi }_{k_2}\right)+{\mu }_{1}c\left(x\right)h\left({\varphi }_{k_1}\right)),\sigma \le d\left(x\right)<2\delta$$

and

$$\begin{array}{c}-{\Delta }_{p\left(x\right)}{\varphi }_{k_1}=0\le {\lambda }^{p\left(x\right)}({\lambda }_1{a}_1+{\mu }_1{c}_1)\le {\lambda }^{p\left(x\right)}({\lambda }_{1}a\left(x\right)f\left({\varphi }_{k_2}\right)\\ +{\mu }_{1}c\left(x\right)h\left({\varphi }_{k_1}\right)),2\delta \le d\left(x\right).\end{array}$$

(12)

Combining (10), (12) and (13), we can deduce that

$$-{\Delta }_{p\left(x\right)}{\varphi }_{k_1}\le {\lambda }^{p\left(x\right)}({\lambda }_{1}a\left(x\right)f\left({\varphi }_{k_2}\right)+{\mu }_{1}c\left(x\right)h\left({\varphi }_{k_1}\right)),a.e.\mathrm{on}\mathsf{\Omega }.$$

(13)

Similarly

$$-{\Delta }_{q\left(x\right)}{\varphi }_{k_2}\le {\lambda }^{q\left(x\right)}({\lambda }_{2}b\left(x\right)g\left({\varphi }_{k_1}\right)+{\mu }_{2}d\left(x\right)\tau \left({\varphi }_{k_2}\right)),a.e.\mathrm{on}\mathsf{\Omega }$$

(14)

From (13) and (14), we can see that $({\varphi }_{k_1},{\varphi }_{k_2})$ is a subsolution of problem (1).

Step 2. We will construct a supersolution of problem (1), we consider

$$\left\{\begin{array}{c}-{\Delta }_{p\left(x\right)}{z}_{k_1}={\lambda }^{p+}({\lambda }_1{a}_2+{\mu }_1{c}_2)\mu \mathrm{in}\mathsf{\Omega },\hfill \\ -{\Delta }_{q\left(x\right)}{z}_{k_2}={\lambda }^{q+}({\lambda }_1{b}_2+{\mu }_1{d}_2)g\left(\beta \left({\lambda }^{p+}({\lambda }_1{a}_2+{\mu }_1{c}_2)\mu \right)\right)\mathrm{in}\mathsf{\Omega },\hfill \\ z_{k_1}=z_{k_2}=0\mathrm{on}\partial \mathsf{\Omega },\hfill \end{array}\right.$$

where

$$\beta =\beta \left({\lambda }^{p+}({\lambda }_1{a}_2+{\mu }_1{c}_2)\mu \right)=\underset{x\in \overline{\mathsf{\Omega }}}{\max }{z}_{k_1}\left(x\right).$$

We shall prove that $(z_{k_1},z_{k_2})$ is a supersolution of problem (1).

From Lemma 2, we have

$$\underset{x\in \overline{\mathsf{\Omega }}}{\max }{z}_{k_1}\left(x\right)\le C_2{\left[{\lambda }^{p+}({\lambda }_1{a}_2+{\mu }_1{c}_2)\mu \right]}^{\frac{1}{p^{-}-1}}$$

and

$$\underset{x\in \overline{\mathsf{\Omega }}}{\max }{z}_{k_2}\left(x\right)\le C_2{\left[{\lambda }^{q+}({\lambda }_2{b}_2+{\mu }_2{d}_2)g\left(\beta \left({\lambda }^{p+}({\lambda }_1{a}_2+{\mu }_1{c}_2)\mu \right)\right)\right]}^{\frac{1}{q^{-}-1}}.$$

For $\psi$$\in W_0^{1,q\left(x\right)}\left(\mathsf{\Omega }\right)$ with $\psi ⩾0$, it is easy to see that

$$\begin{array}{c}\underset{\mathsf{\Omega }}{\int }{\left|\nabla z_{k_2}\right|}^{q\left(x\right)-2}\nabla z_{k_2}.\nabla \psi dx=\underset{\mathsf{\Omega }}{\int }{\lambda }^{q+}({\lambda }_2{b}_2+{\mu }_2{d}_2)g\left(\beta \left({\lambda }^{p+}({\lambda }_1{a}_2+{\mu }_1{c}_2)\mu \right)\right)\psi dx⩾\hfill \\ \underset{\mathsf{\Omega }}{\int }{\lambda }^{q+}{\lambda }_{2}b\left(x\right)g\left(z_{k_1}\right)\psi dx+\underset{\mathsf{\Omega }}{\int }{\lambda }^{q+}{\mu }_{2}d\left(x\right)g\left(\beta \left({\lambda }^{p+}({\lambda }_1{a}_2+{\mu }_1{c}_2)\mu \right)\right)\psi dx.\hfill \end{array}$$

By $\left(H_4\right)$, for $\mu$ a large enough, using Lemma 2, we have

$$\begin{array}{c}g\left(\beta \left({\lambda }^{p+}({\lambda }_1{a}_2+{\mu }_1{c}_2)\mu \right)\right)\hfill \\ ⩾\tau (C_2{\left[{\lambda }^{q+}({\lambda }_2{b}_2+{\mu }_2{d}_2)g\left(\beta \left({\lambda }^{p+}({\lambda }_1{a}_2+{\mu }_1{c}_2)\mu \right)\right)\right]}^{\frac{1}{q^{-}-1}})\hfill \\ ⩾\tau \left(z_{k_2}\right).\hfill \end{array}$$

(15)

Hence

$$\underset{\mathsf{\Omega }}{\int }{\left|\nabla z_{k_2}\right|}^{q\left(x\right)-2}\nabla z_{k_2}.\nabla \psi dx⩾\underset{\mathsf{\Omega }}{\int }{\lambda }^{q+}{\lambda }_{2}b\left(x\right)g\left(z_{k_1}\right)\psi dx+\underset{\mathsf{\Omega }}{\int }{\lambda }^{q+}{\mu }_{2}d\left(x\right)\tau \left(z_{k_2}\right)\psi dx.$$

(16)

Also, for $\phi$$\in W^{1,p\left(x\right)}\left(\mathsf{\Omega }\right)$ with $\phi \ge 0$, it is easy to see that

$$\underset{\mathsf{\Omega }}{\int }{\left|\nabla z_{k_1}\right|}^{p\left(x\right)-2}\nabla z_{k_1}.\nabla \phi dx=\underset{\mathsf{\Omega }}{\int }{\lambda }^{p+}({\lambda }_1{a}_2+{\mu }_1{c}_2)\mu \phi dx.$$

By $\left(H_3\right),\left(H_4\right)$ and Lemma 2, when $\mu$ is sufficiently large, we have

$$\begin{array}{ccc}\hfill ({\lambda }_1{a}_2+{\mu }_1{c}_2)\mu & ⩾& \frac{1}{{\lambda }^{p+}}{\left[\frac{1}{C_2}\beta \left({\lambda }^{p+}({\lambda }_1{a}_2+{\mu }_1{c}_2)\mu \right)\right]}^{p^{-}-1}\hfill \\ & ⩾& {\mu }_{1}h\left(\beta \left({\lambda }^{p+}({\lambda }_1{a}_2+{\mu }_1{c}_2)\mu \right)\right)\hfill \\ & & +{\lambda }_{1}f\left(C_2{\left[{\lambda }^{q+}({\lambda }_2{b}_2+{\mu }_2{d}_2)g\left(\beta \left({\lambda }^{p+}({\lambda }_1{a}_2+{\mu }_1{c}_2)\mu \right)\right)\right]}^{\frac{1}{q^{-}-1}}\right).\hfill \end{array}$$

Then

$$\underset{\mathsf{\Omega }}{\int }{\left|\nabla z_{k_1}\right|}^{p\left(x\right)-2}\nabla z_{k_1}.\nabla \phi dx⩾\underset{\mathsf{\Omega }}{\int }{\lambda }^{p+}{\lambda }_{1}a\left(x\right)f\left(z_{k_2}\right)\phi dx+\underset{\mathsf{\Omega }}{\int }{\lambda }^{p+}{\mu }_{1}c\left(x\right)h\left(z_{k_1}\right)\phi dx.$$

(17)

According to (16) and (17), we can conclude that $(z_{k_1},z_{k_2})$ is a supersolution of problem (1). It only remains to prove that ${\varphi }_{k_1}$$\le z_{k_1}$ and ${\varphi }_{k_2}$≤$z_{k_2}$.

In the definition of $v_1\left(x\right)$, let

$$\gamma =\frac{2}{\delta }\left(\underset{\overline{\mathsf{\Omega }}}{\max }{\varphi }_{k_1}\left(x\right)+\underset{\overline{\mathsf{\Omega }}}{\max }\left|\nabla {\varphi }_{k_1}\right|\left(x\right)\right).$$

We claim that

$${\varphi }_{k_1}\left(x\right)\le v_1\left(x\right),\forall x\in \mathsf{\Omega }.$$

(18)

From the definition of $v_1,$ it is easy to see that

$${\varphi }_{k_1}\left(x\right)\le 2\underset{\overline{\mathsf{\Omega }}}{\max }{\varphi }_{k_1}\left(x\right)\le v_1\left(x\right),\mathrm{when}d\left(x\right)=\delta ,$$

$${\varphi }_{k_1}\left(x\right)\le 2\underset{\overline{\mathsf{\Omega }}}{\max }{\varphi }_{k_1}\left(x\right)\le v_1\left(x\right),\mathrm{when}d\left(x\right)⩾\delta$$

and

$${\varphi }_{k_1}\left(x\right)\le v_1\left(x\right)\mathrm{when}d\left(x\right)<\delta .$$

Since $v_1-{\varphi }_{k_1}\in C^1\left(\overline{\partial {\mathsf{\Omega }}_{\delta }}\right),$ there exists a point $x_0\in \overline{\partial {\mathsf{\Omega }}_{\delta }},$ such that

$$v_1\left(x_0\right)-{\varphi }_{k_1}\left(x_0\right)=\underset{x_0\in \overline{\partial {\mathsf{\Omega }}_{\delta }}}{\min }(v_1\left(x_0\right)-{\varphi }_{k_1}\left(x_0\right)).$$

If $v_1\left(x_0\right)-{\varphi }_{k_1}\left(x_0\right)<0,$ It is easy to see that $0<d\left(x\right)<\delta$ and then

$$\nabla v_1\left(x_0\right)-\nabla {\varphi }_{k_1}\left(x_0\right)=0.$$

From the definition of $v_1,$ we have

$$\left|\nabla v_1\left(x_0\right)\right|=\gamma =\frac{2}{\delta }\left(\underset{\overline{\mathsf{\Omega }}}{\max }{\varphi }_{k_1}\left(x_0\right)+\underset{\overline{\mathsf{\Omega }}}{\max }\left|\nabla {\varphi }_{k_1}\right|\left(x_0\right)\right)>\left|\nabla {\varphi }_{k_1}\right|\left(x_0\right).$$

It is a contradiction to

$$\nabla v_1\left(x_0\right)-\nabla {\varphi }_{k_1}\left(x_0\right)=0.$$

Thus, (18) is valid.

Obviously, there exists a positive constants $C_3,$ such that $\gamma \le C_3\lambda .$

Since $d\left(x\right)\in C^2\left(\overline{\partial {\mathsf{\Omega }}_{3\delta }}\right),$ according to the proof of Lemma 2, there exists a positive constant $C_4$, such that

$$-{\Delta }_{p\left(x\right)}{v}_1\left(x\right)\le C_{\ast }{\gamma }^{p\left(x\right)-1+\theta }\le C_4{\lambda }^{p\left(x\right)-1+\theta }\mathrm{a}.\mathrm{e}\mathsf{\Omega },\mathrm{where}\theta \in (0,1).$$

Since $\eta ⩾{\lambda }^{p+}$ is large enough, we have $-{\Delta }_{p\left(x\right)}{v}_1\left(x\right)\le \eta .$

Under the comparaison principle, we have

$$v_1\left(x\right)\le w\left(x\right),\mathrm{for}\mathrm{all}x\in \mathsf{\Omega }.$$

(19)

From (18) and (19), when $\eta ⩾{\lambda }^{p+}$ and $\lambda ⩾1$ is sufficiently large, we have

$${\varphi }_{k_1}\left(x\right)\le v_1\left(x\right)\le w\left(x\right),\mathrm{for}\mathrm{all}x\in \mathsf{\Omega }.$$

(20)

According to the comparaison principle, when $\mu$ is large enough, we have

$$v_1\left(x\right)\le w\left(x\right)\le z_{k_1}\left(x\right),\mathrm{for}\mathrm{all}x\in \mathsf{\Omega }.$$

Combining the definition of $v_1\left(x\right)$ and (20), it is easy to see that

$${\varphi }_{k_1}\left(x\right)\le v_1\left(x\right)\le w\left(x\right)\le z_{k_1}\left(x\right),\mathrm{for}\mathrm{all}x\in \mathsf{\Omega }.$$

When $\mu ⩾1$ and $\lambda$ is a large enough, from Lemma 2, we can note that

$\beta ({\lambda }^{p+}({\lambda }_1{a}_2+{\mu }_1{c}_2)\mu$ is large enough, then

$${\lambda }^{q+}({\lambda }_2{b}_2+{\mu }_2{d}_2)g\left(\beta \left({\lambda }^{p+}({\lambda }_1{a}_2+{\mu }_1{c}_2)\mu \right)\right)$$

is a large enough. Similarly, we have ${\varphi }_{k_2}\left(x\right)\le z_{k_2}\left(x\right)$. This completes the proof. □

Asymptotic Behavior of Solutions

Definition 2.

A measurable funtion $u:{\mathsf{\Omega }}_T\to \mathbb{R}$ is an weak solution to hyperbolic systems involving of $\left(p\right(x),q(x\left)\right)—$Laplacien (1) in ${\mathsf{\Omega }}_T$ if $u(.,0)=u_0$ in $\mathsf{\Omega },$

$$u\in C(0,T;L^2\left(\mathsf{\Omega }\right))\cap L^p(0,T;H_0^1\left(\mathsf{\Omega }\right)),$$

$$\frac{\partial u}{\partial t}\in L^2\left({\mathsf{\Omega }}_T\right),\nabla u\in {\left(L^2\left({\mathsf{\Omega }}_T\right)\right)}^N$$

and for all $\phi \in C^1\left({\mathsf{\Omega }}_T\right)$ and $\psi \in C^1\left({\mathsf{\Omega }}_T\right),$ we have

$$\begin{array}{c}\underset{0}{\overset{T}{\int }}{\int }_{\mathsf{\Omega }}\frac{\partial u}{\partial t}\phi dxdt+\underset{0}{\overset{T}{\int }}{\int }_{\mathsf{\Omega }}{\left|\nabla u\right|}^{p\left(x\right)-2}\nabla u\nabla \phi dxdt+\underset{0}{\overset{T}{\int }}{\int }_{\mathsf{\Omega }}(-{\lambda }^{p\left(x\right)}{\mu }_{1}c\left(x\right)h\left(u\right))\phi dxdt\\ =\underset{0}{\overset{T}{\int }}{\int }_{\mathsf{\Omega }}{\lambda }^{p\left(x\right)}{\lambda }_{1}a\left(x\right)f\left(v\right)\phi dxdt\end{array}$$

(21)

Lemma 3.

$$\begin{array}{c}\underset{0}{\overset{T}{\int }}{\int }_{\mathsf{\Omega }}\frac{\partial v}{\partial t}\psi dxdt+\underset{0}{\overset{T}{\int }}{\int }_{\mathsf{\Omega }}{\left|\nabla v\right|}^{q\left(x\right)-2}\nabla v\nabla \psi dxdt+\underset{0}{\overset{T}{\int }}{\int }_{\mathsf{\Omega }}(-{\lambda }^{q\left(x\right)}{\lambda }_{2}b\left(x\right)g\left(u\right))\psi dxdt\\ =\underset{0}{\overset{T}{\int }}{\int }_{\mathsf{\Omega }}{\lambda }^{q\left(x\right)}{\mu }_{2}d\left(x\right)\sigma \left(v\right)\psi dxdt\end{array}$$

Lemma 4.

Let $\underline{u},\overline{u}$ be the solutions of (1) with $\underline{u}\left(x,0\right)={\phi }_1,\overline{u}\left(x,0\right)={\phi }_2$ Than $\underline{u}\left(x,t\right)$ is nondercreasing in t, $\overline{u}\left(x,t\right)$ is nonincreasing and $\overline{u}>\underline{u}$ for all $t\ge 0,x\in \mathsf{\Omega }$

Theorem 2.

Let hypotheses $\left(H_1\right),\left(H_2\right)$ and $\left(H_3\right)$ be satisfied. and let $u\left(x,t\right)$ the solution of a new class of hyperbolic systems (1) with $\mathsf{\Psi }\in S^{\ast }$ than

$$\underset{t\to \infty }{lim}u\left(x,t\right)=\left\{\begin{array}{c}{\underline{u}}_s\left(x\right)if{\widehat{u}}_s\le \mathsf{\Psi }\le {\underline{u}}_s\\ {\overline{u}}_s\left(x\right)if{\overline{u}}_s\le \mathsf{\Psi }\le {\tilde{u}}_s\end{array}\right.$$

.

Proof. 

The pair $\left({\underline{u}}_s,{\widehat{u}}_s\right)$ and the pair $\left({\tilde{u}}_s,{\overline{u}}_s\right)$ are both sub-super solutions of (4), the maximale and minimale property of ${\overline{u}}_s$ and ${\underline{u}}_s$ in $S^{\ast }$ ensures that:

${\underline{u}}_s$ is the unique solution in $\left[{\widehat{u}}_s,{\underline{u}}_s\right]$ and ${\overline{u}}_s$ is the unique solution in $\left[{\overline{u}}_s,{\tilde{u}}_s\right]$. □

4. Conclusions

Our result is an extension for our previous study in ( [2,7,8]) which studied the stationary case, this idea is new for evolutionary case of this kind of problem, This paper deals with the existence of positively solution and its asymptotic behavior for parabolic system of $\left(p\right(x),q(x\left)\right)$-Laplacian system of partial differential equations using a sub and super solution according to some given boundary conditions, which is familiar in physics, since it appears clearly natural in inflation cosmology and supersymmetric filed theories, quantum mechanics, and nuclear physics (see [3,14]). This sort of problem has many applications in several branches of physics such as nuclear physics, optics, and geophysics (see [7,15]). In future work, we will try to extend this study for the hyperbolic case of the presented problem, but by using the semigroup theory.