Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity

Abstract

In this article, we study the asymptotic behaviour of solutions of a first-order stochastic lattice dynamical system with an additive noise. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth and dissipative conditions so that uniqueness of the Cauchy problem fails to be true. Using the theory of multi-valued random dynamical systems, we prove the existence of a random compact global attractor.

Keywords:

• stochastic lattice dynamical systems
• set-valued dynamical system
• random attractor

AMS Subject Classification::

• 35B40
• 35B41
• 35K40
• 35K57
• 37B25
• 37L55
• 60H10

Acknowledgements

This work was partially supported by Spanish Ministerio de Ciencia e Innovación, Projects MTM2008-00088 and MTM2009-11820, the Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía), Proyecto de Excelencia P07-FQM-02468 and the Consejería de Cultura y Educación (Comunidad Autónoma de Murcia), grant 08667/PI/08.