Existence and upper semicontinuity of global attractors for a $p$-Laplacian inclusion

Abstract

In this work we study the asymptotic behavior of a $p$-Laplacian
inclusion of the form $\displaystyle\frac{\partial
u_\lambda}{\partial t} - div(D^\lambda|\nabla
u_\lambda|^{p-2}\nabla u_\lambda) + |u_\lambda|^{p-2}u_\lambda$ $\in F(u_\lambda) + h,$ where $p>2$, $h\in L^2(\Omega),$ with
$\Omega\subset\mathbb{R}^n,\; n\geq 1,$ a bounded smooth domain,
$D^\lambda \in L^\infty(\Omega)$, $\infty > M\geq D^\lambda(x)
\geq \sigma >0$ a.e. in $\Omega$, $\lambda \in [0,\infty)$ and
$D^\lambda\rightarrow D^{\lambda_1}$ in $L^\infty(\Omega)$ as
$\lambda \to \lambda_1$, $F:\mathcal{D}(F)\subset
L^{2}(\Omega)\rightarrow\mathcal{P}(L^{2}(\Omega))$, given by
$F(y(\cdot))=\{\xi(\cdot)\in L^{2}(\Omega):\xi(x)\in
f(y(x))\;x\mbox{-a.e. in}\; \Omega\}$ with
$f:\mathbb{R}\rightarrow\mathcal{C}_{v}(\mathbb{R})$ Lipschitz
($\mathcal{C}_{v}(\mathbb{R})$ is the set of all nonempty,
bounded, closed, convex subsets of $\mathbb{R}$) be a multivalued
map. We prove the existence of a global attractor in $L^2(\Omega)$
for each positive finite diffusion coefficient and we show that
the family of attractors behaves upper semicontinuously on
positive finite diffusion parameters.