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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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