Pullback attractors for stochastic heat equations in materials with memory

Tomás Caraballo; José Real; I. D. Chueshov

doi:10.3934/dcdsb.2008.9.525

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2008, Volume 9, Issue 3&4: 525-539. Doi: 10.3934/dcdsb.2008.9.525

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Pullback attractors for stochastic heat equations in materials with memory

1.
Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla
2.
Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla
3.
Department of Mechanics and Mathematics, Kharkov National University, 4 Svobody sq., 61077, Kharkov

Received: August 31, 2006

Revised: April 30, 2007

Published: April 2008

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Abstract

We study the asymptotic behaviour of a non-autonomous stochastic reaction-diffusion equation with memory. In fact, we prove the existence of a random pullback attractor for our stochastic parabolic PDE with memory. The randomness enters in our model as an additive Hilbert valued noise. We first prove that the equation generates a random dynamical system (RDS) in an appropriate phase space. Due to the fact that the memory term takes into account the whole past history of the phenomenon, we are not able to prove compactness of the generated RDS, but its asymptotic compactness, ensuring thus the existence of the random pullback attractor.

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Mathematics Subject Classification: Primary: 60H15, 35B25, 35B40, 37L30.

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