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Abstract 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 ≥ 2 $p\geq2$ 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 ) $H_{0}^{1}(U)$ is first obtained. As an application, we show the existence of ( L 2 ( U ) , L p ( U ) ) $(L^{2}(U),L^{p}(U))$ and ( L 2 ( U ) , H 0 1 ( U ) ) $(L^{2}(U),H_{0}^{1}(U))$ -pullback random attractors, respectively.