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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 $$x\equiv 0$$. We show in addition that this stochastic equation has a finite-dimensional random attractor, and from our results conjecture a possible bifurcation scenario.

MSC:
 37H20 Bifurcation theory for random and stochastic dynamical systems 35B35 Stability in context of PDEs 35K57 Reaction-diffusion equations 35R60 PDEs with randomness, stochastic partial differential equations 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems 60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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