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Stability and random attractors for a reaction-diffusion equation with multiplicative noise. (English) Zbl 1011.37031
Summary: We study the asymptotic behaviour of a reaction-diffusion equation, and prove that the addition of multiplicative white noise (in the sense of Itô) stabilizes the stationary solution x0. We show in addition that this stochastic equation has a finite-dimensional random attractor, and from our results conjecture a possible bifurcation scenario.
MSC:
37H20Bifurcation theory
35B35Stability of solutions of PDE
35K57Reaction-diffusion equations
35R60PDEs with randomness, stochastic PDE
37L30Attractors and their dimensions, Lyapunov exponents
60H15Stochastic partial differential equations