Stability of stochastic reaction-diffusion equation under random influences in high regular spaces

Abstract

In this paper, we systematically study the high-order stability of the stochastic reaction-diffusion equation driven by additive noise as the noise intensity vanishes. First, with a general assumption on the nonlinear term, we obtain the convergence of solutions and upper semi-continuity of random attractors in L2(RN). Second, by using the nonlinear decomposition method, we technically establish the convergence of solutions in Lp(RN)∩H1(RN)(p>2), and therefore, the upper semi-continuity of random attractors is proved, where p is the growth exponent of the nonlinearity. Finally, by induction argument, we prove that the solution is uniformly bounded near the initial time in Lδ(RN) for arbitrary δ > p, in which space the convergence of solutions and the upper semi-continuity of random attractors are also established.