Pullback attractor in $H^{1}$ for nonautonomous stochastic reaction-diffusion equations on $\mathbb{R}^n$

Abstract

In this paper we study the asymptotic dynamics of the weak solutions of nonautonomous stochastic reaction-diffusion equations driven by a time-dependent forcing term and the multiplicative noise. By conducting the uniform estimates we show that the cocycle generated by this SRDE has a pullback \begin{document}$(L^2, H^1)$\end{document} absorbing set and it is pullback asymptotically compact through the pullback flattening approach. The existence of a pullback \begin{document}$(L^2, H^1)$\end{document} random attractor for this random dynamical system in space \begin{document}$H^{1}(\mathbb{R}^{n})$\end{document} is proved.