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

Abstract

In this work we study the asymptotic behavior of a $p$-Laplacianinclusion of the form $\displaystyle\frac{\partialu_\lambda}{\partial t} - div(D^\lambda|\nablau_\lambda|^{p-2}\nabla u_\lambda) + |u_\lambda|^{p-2}u_\lambda$ $\in F(u_\lambda) + h,$ 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$\lambda \to \lambda_1$, $F:\mathcal{D}(F)\subsetL^{2}(\Omega)\rightarrow\mathcal{P}(L^{2}(\Omega))$, given by$F(y(\cdot))=\{\xi(\cdot)\in L^{2}(\Omega):\xi(x)\inf(y(x))\;x\mbox{-a.e. in}\; \Omega\}$ 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 multivaluedmap. We prove the existence of a global attractor in $L^2(\Omega)$for each positive finite diffusion coefficient and we show thatthe family of attractors behaves upper semicontinuously onpositive finite diffusion parameters.