Long-time behavior of stochastic reaction–diffusion equation with multiplicative noise

In this paper, we study the dynamical behavior of the solution for the stochastic reaction–diffusion equation with the nonlinearity satisfying the polynomial growth of arbitrary order $p\geq2$ and any space dimension N. Based on the inductive principle, the higher-order integrability of the difference of the solutions near the initial data is established, and then the (norm-to-norm) continuity of solutions with respect to the initial data in $H_{0}^{1}(U)$ is first obtained. As an application, we show the existence of $(L^{2}(U),L^{p}(U))$ and $(L^{2}(U),H_{0}^{1}(U))$-pullback random attractors, respectively.