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

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^λ-\mathrm{div}(d_λ(x)(|\nabla u^λ|^{p(x)-2}+η) \nabla u^λ)+ |u^λ|^{p(x)-2}u^λ=B(u^λ), & \mbox{ in } Ω\\ u^λ= 0, & \mbox{ on } \partialΩ\\ u^λ(0)=u^λ_0 \in L^2(Ω),& \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.

21 pages