Time dependent perturbation in a non-autonomous non-classical parabolic equation

Felipe Rivero

doi:10.3934/dcdsb.2013.18.209

Article Contents

2013, Volume 18, Issue 1: 209-221. Doi: 10.3934/dcdsb.2013.18.209

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Time dependent perturbation in a non-autonomous non-classical parabolic equation

Felipe Rivero^1,

1.
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080, Sevilla

Received: December 31, 2011

Revised: May 31, 2012

Published: December 2012

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Abstract

n this paper we study the existence and characterization of a pullback attractor for a non-autonomous non-classical parabolic equation of the form \begin{equation}\label{EQnoncla} \left\{ \begin{split} &u_t-\gamma(t)\Delta u_t-\Delta u=f(u) \mbox{ in }\Omega,\\ &u=0 \mbox{ on }\partial\Omega \end{split} \right. (1) \end{equation} in a sufficiently smooth bounded domain $\Omega\subset\mathbb R^n$ with $f$ and $\gamma$ satisfying some suitable natural conditions. We prove the well posedness of this model and the existence of a pullback attractor. We show that this pullback attractor is characterized as the union of unstable sets of the associated equilibria and that this characterization is stable under time dependent perturbation of the nonlinearity.

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Mathematics Subject Classification: 35B20, 35B50, 35K55, 37B55, 37C70.

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