Existence of exponential attractor to a family of problems dominated by a perturbation of $p(x)$-laplacian with localized large diffusion via the $l$-trajectories method

TL;DR

This paper proves the existence of an exponential attractor for a family of PDEs with localized large diffusion, using the $l$-trajectory method, and analyzes their limit problem.

Contribution

It introduces a novel application of the $l$-trajectory method to PDEs with localized large diffusion and establishes the existence of exponential attractors in this context.

Findings

Existence of exponential attractor for the family of problems.

Convergence analysis of the problems to a limit problem.

Application of the $l$-trajectory method to localized diffusion issues.

Abstract

This article is devoted to the study of the existence of an exponential attractor for a family of problems, in which diffusion $d_{\lambda}$ blows up in localized regions inside the domain, \begin{equation*} \begin{cases} \displaystyle u_t^\lambda-\mathrm{div}(d_\lambda(x)(|\nabla u^\lambda|^{p(x)-2}+\eta ) \nabla u^\lambda)+ |u^\lambda|^{p(x)-2}u^\lambda=B(u^\lambda), & \mbox{ in } \Omega \\ u^\lambda = 0, & \mbox{ on } \partial\Omega\\ u^\lambda(0)=u^\lambda_0 \in L^2(\Omega),& \end{cases} \end{equation*} and their limit problem via the $l$-trajectory method. \vskip .1 in \noindent {\it Mathematical Subject Classification 2010:} 35K55, 37L30, 35B40, 35B41. \newline {\it Key words and phrases:} global attractor, exponential attractor, $l-$trajectory, limit problem, localized large diffusion.