This paper deals with the existence of positively solution and its asymptotic behavior for parabolic system of ( p ( x ) , q ( x ) ) -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.

Abstract In this article, we discuss the existence of positive solutions by using sub-super solutions concepts of the following {p(x)} -Kirchhoff system: \left\{\begin{aligned} &\displaystyle{-}M(I_{0}(u))\triangle_{p(x)}u=\lambda^{% p(x)}[\lambda_{1}f(v)+\mu_{1}h(u)]&&\displaystyle\text{in }\Omega,\\ &\displaystyle{-}M(I_{0}(v))\triangle_{p(x)}v=\lambda^{p(x)}[\lambda_{2}g(u)+% \mu_{2}\tau(v)]&&\displaystyle\text{in }\Omega,\\ &\displaystyle u=v=0&&\displaystyle\text{on }\partial\Omega,\end{aligned}\right. where {\Omega\subset\mathbb{R}^{N}} is a bounded smooth domain with {C^{2}} boundary {\partial\Omega} , {\triangle_{p(x)}u=\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u)} , {p(x)\in C^{1}(\overline{\Omega})} , with {1<p(x)} , is a function satisfying {1<p^{-}=\inf_{\Omega}p(x)\leq p^{+}=\sup_{\Omega}p(x)<\infty} , λ, {\lambda_{1}} , {\lambda_{2}} , {\mu_{1}} and {\mu_{2}} are positive parameters, {I_{0}(u)=\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}\,dx} , and {M(t)} is a continuous function.

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