Pathwise solutions and attractors for retarded SPDEs with time smooth diffusion coefficients

Hakima Bessaih; María J. Garrido–Atienza; Björn Schmalfuss

doi:10.3934/dcds.2014.34.3945

Article Contents

2014, Volume 34, Issue 10: 3945-3968. Doi: 10.3934/dcds.2014.34.3945

This issue Previous Article Estimates on the distance of inertial manifolds Next Article On global in time dynamics of a planar Bingham flow subject to a subdifferential boundary condition

Pathwise solutions and attractors for retarded SPDEs with time smooth diffusion coefficients

1.
University of Wyoming, Department of Mathematics, Dept. 3036, 1000 East University Avenue, Laramie, WY 82071
2.
Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla
3.
Institut für Stochastik, Friedrich Schiller Universität Jena, Ernst Abbe Platz 2, 07737 Jena

Received: February 2013

Revised: May 2013

Published: October 2014

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Abstract

In this paper we study the long--time dynamics of mild solutions to retarded stochastic evolution systems driven by a Hilbert-valued Brownian motion. For this purpose, we begin by showing the existence and uniqueness of a cocycle solution of such an equation. We do not assume that the noise is given in additive form or that it is a very simple multiplicative noise. However, we need some smoothing property for the coefficient in front of the noise. The main idea of this paper consists of expressing the stochastic integral in terms of non-stochastic integrals and the noisy path by using an integration by parts. This latter term causes that at first, only a local mild solution can be obtained, since in order to apply the Banach fixed point theorem it is crucial to have the Hölder norm of the noisy path to be sufficiently small. Subsequently, by using appropriate stopping times, we shall derive the existence and uniqueness of a global mild solution. Furthermore, the asymptotic behavior is investigated by using the Random Dynamical Systems theory. In particular, we shall show that the global mild solution generates a random dynamical system that, under an appropriate smallness condition for the time lag, has an associated random attractor.

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Mathematics Subject Classification: Primary: 37L55; Secondary: 60H15, 35R60, 60J65, 34D45.

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