Research Article

On the long-time behaviour of a class of parabolic SPDE's: monotonicity methods and exchange of stability

Bergé, Benjamin^a1 and Saussereau, Bruno^a2

^a1 Institut de Mathématiques, Université de Neuchâtel, Rue Émile Argand, 11, 2007 Neuchâtel, Switzerland; benjamin.berge@unine.ch

^a2 Département de Mathématiques, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France; bruno.saussereau@univ-fcomte.fr

Abstract

In this article we prove new results concerning the structure and the stability properties of the global attractor associated with a class of nonlinear parabolic stochastic partial differential equations driven by a standard multidimensional Brownian motion. We first use monotonicity methods to prove that the random fields either stabilize exponentially rapidly with probability one around one of the two equilibrium states, or that they set out to oscillate between them. In the first case we can also compute exactly the corresponding Lyapunov exponents. The last case of our analysis reveals a phenomenon of exchange of stability between the two components of the global attractor. In order to prove this asymptotic property, we show an exponential decay estimate between the random field and its spatial average under an additional uniform ellipticity hypothesis.

(Received October 12 2004)

(Revised February 15 2005)

(Online publication November 15 2005)

Key Words:

• Parabolic stochastic partial differential equations;
• asymptotic behaviour;
• monotonicity methods.

Mathematics Subject Classification:

• 60H10;
• 60H15