Pullback attractors for stochastic heat equations in materials with memory

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May 2008, 9(3&4, May): 525-539. doi: 10.3934/dcdsb.2008.9.525

Pullback attractors for stochastic heat equations in materials with memory

Tomás Caraballo ^1, , José Real ^2, and I. D. Chueshov ^3,

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, Ukraine

Received September 2006 Revised May 2007 Published February 2008

Abstract
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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.

Keywords: stochastic heat equation with memory., Random pullback attractor, asymptotic compactness, Ornstein-Uhlenbeck process.

Mathematics Subject Classification: Primary: 60H15, 35B25, 35B40, 37L3.

Citation: Tomás Caraballo, José Real, I. D. Chueshov. Pullback attractors for stochastic heat equations in materials with memory. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 525-539. doi: 10.3934/dcdsb.2008.9.525

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