Existence of exponential attractor to $p(x)$-laplacian via the $l$-trajectories method.

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} 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.