PDE and ODE limit problems for p(x)-Laplacian parabolic equations

AbstractIn this work we prove continuity of solutions with respect to initial conditions and parameters and we prove upper semicontinuity of a family of global attractors for problems of the formut−div(Dλ|∇uλ|p(x)−2∇uλ)=B(uλ) in a bounded smooth domain Ω in RN

Coupled nonautonomous inclusion systems with spatially variable exponents

A family of nonautonomous coupled inclusions governed by p(x)-Laplacian operators with large diffusion is investigated. The existence of solutions and pullback attractors as well as the generation of a generalized process are established. It is shown that the asymptotic dynamics is determined by a two dimensional ordinary nonautonomous coupled inclusion when the exponents converge to constants provided the absorption coefficients are independent of the spatial variable. The pullback attractor and forward attracting set of this limiting system is investigated

Weak upper semicontinuity of pullback attractors for nonautonomous reaction-diffusion equations

We consider nonautonomous reaction-diffusion equations with variable exponents and large diffusion and we prove continuity of the flow and weak upper semicontinuity of a family of pullback attractors when the exponents go to 2 in L

Weak upper semicontinuity of pullback attractors for nonautonomous reaction-diffusion equations

We consider nonautonomous reaction-diffusion equations with variable exponents and large diffusion and we prove continuity of the flow and weak upper semicontinuity of a family of pullback attractors when the exponents go to 2 in L

Systems of p-Laplacian differential inclusions with large diffusion

AbstractIn this paper we consider coupled systems of p-Laplacian differential inclusions and we prove, under suitable conditions, that a homogenization process occurs when diffusion parameters become arbitrarily large. In fact we obtain that the attractors are continuous at infinity on L2(Ω)×L2(Ω) topology, with respect to the diffusion coefficients, and the limit set is the attractor of an ordinary differential problem

Robustness with respect to exponents for nonautonomous reaction–diffusion equations

In this work we consider a family of nonautonomous problems with homogeneous Neumann boundary conditions and spatially variable exponents with equation of the form \begin{equation*} \frac{\partial u_{\lambda}}{\partial_t}(t)-\operatorname{div}\left(D(t)|\nabla u_{\lambda}(t)|^{p_{\lambda}(x)-2}\nabla u_{\lambda}(t)\right)+|u_{\lambda}(t)|^{p_{\lambda}(x)-2}u_{\lambda}(t)=B(t,u_{\lambda}(t)). \end{equation*} We study the continuity of the flow and we study the behavior of attractors when $p_{\lambda}(\cdot)\to p(\cdot)$ in $L^{\infty}(\Omega)$ as $\lambda\to\infty$ where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$