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Reaction-Diffusion equations with spatially variable exponents and large diffusion

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  • In this work we prove continuity of solutions with respect to initial conditions and couple parameters and we prove joint upper semicontinuity of a family of global attractors for the problem \begin{eqnarray} &\frac{\partial u_{s}}{\partial t}(t)-\textrm{div}(D_s|\nabla u_{s}|^{p_s(x)-2}\nabla u_{s})+|u_s|^{p_s(x)-2}u_s=B(u_{s}(t)),\;\; t>0,\\ &u_{s}(0)=u_{0s}, \end{eqnarray} under homogeneous Neumann boundary conditions, $u_{0s}\in H:=L^2(\Omega),$ $\Omega\subset\mathbb{R}^n$ ($n\geq 1$) is a smooth bounded domain, $B:H\rightarrow H$ is a globally Lipschitz map with Lipschitz constant $L\geq 0$, $D_s\in[1,\infty)$, $p_s(\cdot)\in C(\bar{\Omega})$, $p_s^-:=\textrm{ess inf}\;p_s\geq p,$ $p_s^+:=\textrm{ess sup}\;p_s\leq a,$ for all $s\in \mathbb{N},$ when $p_s(\cdot)\rightarrow p$ in $L^\infty(\Omega)$ and $D_s\rightarrow\infty$ as $s\rightarrow\infty,$ with $a,p>2$ positive constants.
    Mathematics Subject Classification: Primary: 35K55, 35K92; Secondary: 35A16, 35B40, 35B41.


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