*Mathematical Methods in the Applied Sciences*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.

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