  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.
