Global attractors for $p$-Laplacian differential inclusions in unbounded domains
Jacson Simsen José Valero
In this work we consider a differential inclusion governed by a p-Laplacian operator with a diffusion coefficient depending on a parameter in which the space variable belongs to an unbounded domain. We prove the existence of a global attractor and show that the family of attractors behaves upper semicontinuously with respect to the diffusion parameter. Both autonomous and nonautonomous cases are studied.
keywords: upper semicontinuity. differential inclusions Unbounded domains $p$-Laplacian operator attractors
Pullback attractors for non-autonomous evolution equations with spatially variable exponents
Peter E. Kloeden Jacson Simsen
Dissipative problems in electrorheological fluids, porous media and image processing often involve spatially dependent exponents. They also include time-dependent terms as in equation \begin{eqnarray} \frac{\partial u_\lambda}{\partial t}(t)-\textrm{div}(D_\lambda(t)|\nabla u_\lambda(t)|^{p(x)-2}\nabla u_\lambda(t))+|u_\lambda(t)|^{p(x)-2}u_\lambda(t) = B(t,u_\lambda(t)) \end{eqnarray} on a bounded smooth domain $\Omega$ in $\mathbb{R}^n$, $n\geq 1$, with a homogeneous Neumann boundary condition, where the exponent $p(\cdot)\in C(\bar{\Omega}, \mathbb{R}^+)$ satisfying $p^-$ $:=$ $\min p(x)$ $>$ $2$, and $\lambda$ $\in$ $[0,\infty)$ is a parameter.
The existence and upper semicontinuity of pullback attractors are established for this equation under the assumptions, amongst others, that $B$ is globally Lipschitz in its second variable and $D_\lambda$ $ \in $ $L^\infty([\tau,T] \times \Omega, \mathbb{R}^+)$ is bounded from above and below, is monotonically nonincreasing in time and continuous in the parameter $\lambda$. The global existence and uniqueness of strong solutions is obtained through results of Yotsutani.
keywords: upper semicontinuity. variable exponents pullback attractors Non-autonomous parabolic problems
Reaction-Diffusion equations with spatially variable exponents and large diffusion
Jacson Simsen Mariza Stefanello Simsen Marcos Roberto Teixeira Primo
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.
keywords: parabolic problems variable exponents Reaction-Diffusion equations attractors upper semicontinuity.

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