Spatial homogeneity in parabolic problems with nonlinear boundary conditions
Alexandre Nolasco de Carvalho Marcos Roberto Teixeira Primo
In this work we prove that global attractors of systems of weakly coupled parabolic equations with nonlinear boundary conditions and large diffusivity are close to attractors of an ordinary differential equation. The limiting ordinary differential equation is given explicitly in terms of the reaction, boundary flux, the $n$-dimensional Lebesgue measure of the domain and the $(n-1)-$Hausdorff measure of its boundary. The tools are invariant manifold theory and comparison results.
keywords: comparison and positivity results spatial homogeneity Invariant manifold theory global attractors
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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