Pullback attractors for  non-autonomous evolution equations with spatially variable exponents

Peter E. Kloeden; Jacson Simsen

doi:10.3934/cpaa.2014.13.2543

Article Contents

2014, Volume 13, Issue 6: 2543-2557. Doi: 10.3934/cpaa.2014.13.2543

This issue Previous Article A quasi-linear heat transmission problem in a periodic two-phase dilute composite. A functional analytic approach Next Article Global gradient estimates for $p(x)$-Laplace equation in non-smooth domains

Pullback attractors for non-autonomous evolution equations with spatially variable exponents

Peter E. Kloeden^1, and
Jacson Simsen^2,

1.
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074
2.
Instituto de Matemática e Computaçã, Universidade Federal de Itajubá, 37500-903 Itajubá MG

Received: January 31, 2014

Revised: May 31, 2014

Published: October 2014

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K55, 35K92; Secondary: 35A16, 35B40, 35B41, 37B55.

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