Document ESAIM: P&S, October 2005, Vol. 9, pp. 254-276
DOI: 10.1051/ps:2005015
On the long-time behaviour of a class of parabolic SPDE's: monotonicity methods and exchange of stability
Benjamin Bergé1 and Bruno Saussereau21 Institut de Mathématiques, Université de Neuchâtel, Rue Émile Argand, 11, 2007 Neuchâtel, Switzerland; benjamin.berge@unine.ch
2 Département de Mathématiques, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France; bruno.saussereau@univ-fcomte.fr
(Received October 12, 2004. Revised February 15, 2005.)
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.
Mathematics Subject Classification. 60H10, 60H15.
Key words: Parabolic stochastic partial differential equations, asymptotic behaviour, monotonicity methods.
© EDP Sciences, SMAI 2005