Qualitative behavior of a class of stochastic parabolic  PDEs with dynamical boundary conditions

Igor Chueshov; Björn Schmalfuss

doi:10.3934/dcds.2007.18.315

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2007, Volume 18, Issue 2&3: 315-338. Doi: 10.3934/dcds.2007.18.315

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Qualitative behavior of a class of stochastic parabolic PDEs with dynamical boundary conditions

Igor Chueshov^1, and
Björn Schmalfuss^2,

1.
Department of Mechanics and Mathematics, Kharkov National University, Kharkov, 61077
2.
Mathematical Institute, University of Paderborn, Paderborn, 33098

Received: January 31, 2006

Revised: August 31, 2006

Published: May 2007

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Abstract

We consider non-linear parabolic stochastic partial differential equations with dynamical boundary conditions and with a noise which acts in the domain but also on the boundary and is presented by the temporal generalized derivative of an infinite dimensional Wiener process. We prove that solutions to this stochastic partial differential equation generate a random dynamical system. Under additional conditions we show that this system is monotone. Our main result states the existence of a compact global (pullback) attractor.

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Mathematics Subject Classification: primary: 37L55; secondary: 37H10, 37L30, 60H15.

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