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

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M5">\begin{document}$ {\mathcal{O}} \subset {\mathbb{R}}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ N \leqslant 3 $\end{document}</tex-math></inline-formula>, with Dirichlet boundary condition:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M7">\begin{document}$ \beta&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ g\in H $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M9">\begin{document}$ W $\end{document}</tex-math></inline-formula> a scalar and two-sided Wiener process with a regular perturbation intensity <inline-formula><tex-math id="M10">\begin{document}$ h $\end{document}</tex-math></inline-formula>. We first construct an <inline-formula><tex-math id="M11">\begin{document}$ H^2 $\end{document}</tex-math></inline-formula> tempered random absorbing set of the system, and then prove an <inline-formula><tex-math id="M12">\begin{document}$ (H,H^2) $\end{document}</tex-math></inline-formula>-smoothing property and conclude that the random attractor of the system is in fact a finite-dimensional tempered random set in <inline-formula><tex-math id="M13">\begin{document}$ H^2 $\end{document}</tex-math></inline-formula> and pullback attracts tempered random sets in <inline-formula><tex-math id="M14">\begin{document}$ H $\end{document}</tex-math></inline-formula> under the topology of <inline-formula><tex-math id="M15">\begin{document}$ H^2 $\end{document}</tex-math></inline-formula>. The main technique we shall employ is comparing the regularity of the stochastic equation to that of the corresponding deterministic equation for which the asymptotic <inline-formula><tex-math id="M16">\begin{document}$ H^2 $\end{document}</tex-math></inline-formula> regularity is already known.</p>