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March  2007, 18(2&3): 271-293. doi: 10.3934/dcds.2007.18.271

Existence of exponentially attracting stationary solutions for delay evolution equations

Tomás Caraballo 1, , M. J. Garrido-Atienza 2, and  B. Schmalfuss 3,

1. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla
2. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080–Sevilla, Spain
3. 

Institut für Mathematik Fakultät EIM, Universität Paderborn, Warburger Strasse 100, 33098 Paderborn

Received  March 2006 Revised  September 2006 Published  March 2007

We consider the exponential stability of semilinear stochastic evolution equations with delays when zero is not a solution for these equations. We prove the existence of a non-trivial stationary solution exponentially stable, for which we use a general random fixed point theorem for general cocycles. We also construct stationary solutions with the stronger property of attracting bounded sets uniformly, by means of the theory of random dynamical systems and their conjugation properties.
Keywords: stationary solutions, exponential stability., random dynamical systems, delay equations, cocycles.
Mathematics Subject Classification: Primary: 60H15, 35K4.
Citation: Tomás Caraballo, M. J. Garrido-Atienza, B. Schmalfuss. Existence of exponentially attracting stationary solutions for delay evolution equations. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 271-293. doi: 10.3934/dcds.2007.18.271
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