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Asymptotic $ H^2$ regularity of a stochastic reaction-diffusion equation

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  • In this paper we study the asymptotic dynamics for the weak solutions of the following stochastic reaction-diffusion equation defined on a bounded smooth domain $ {\mathcal{O}} \subset {\mathbb{R}}^N $, $ N \leqslant 3 $, with Dirichlet boundary condition:

    $ \begin{equation} \nonumber\begin{aligned} { {{\rm{d}}} u } +(-\Delta u + u ^3- \beta u ) {{\rm{d}}} t = g(x) {{\rm{d}}} t+h(x) {{\rm{d}}} W , \quad u|_{t = 0} = u_0\in H: = L^2( {\mathcal{O}}), \end{aligned} \end{equation} $

    where $ \beta>0 $, $ g\in H $, and $ W $ a scalar and two-sided Wiener process with a regular perturbation intensity $ h $. We first construct an $ H^2 $ tempered random absorbing set of the system, and then prove an $ (H,H^2) $-smoothing property and conclude that the random attractor of the system is in fact a finite-dimensional tempered random set in $ H^2 $ and pullback attracts tempered random sets in $ H $ under the topology of $ H^2 $. The main technique we shall employ is comparing the regularity of the stochastic equation to that of the corresponding deterministic equation for which the asymptotic $ H^2 $ regularity is already known.

    Mathematics Subject Classification: Primary: 35B40; Secondary: 35B41, 37L30.


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