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Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Itô’s case. (English) Zbl 0968.60058
Long-time behavior is investigated for generalized solutions of \[ \begin{split} du(x,t,\omega)= (\text{div}(k(x,t)\nabla u(x,t,\omega))+s^{\sharp} g(u(x,t,\omega))) dt+ g(u(x,t,\omega)) dB(t,\omega),\\ (x,t,\omega)\in D\times R^+\times \Omega,\end{split} \] \[ u(x,0,\omega)= \varphi(x,\omega)\in(u_0,u_1),(x,\omega)\in D\times \Omega, \qquad \frac{\partial u(x,t,\omega)}{\partial n(k)} = 0, (x,t,\omega)\in \partial D\times R^+\times \Omega. \] \(B\) denotes a one-dimensional Brownian motion. The existence of a global attractor, stabilization and stability results are proved.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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