High-order stability for a stochastic reaction-diffusion equation under random fluctuation on ℝ N

Abstract

In this article, we systematically study the high-order stability for a stochastic reaction-diffusion equation under random fluctuation. We obtain the convergence of solutions and upper semi-continuity of random attractors for the fluctuation equations in the initial space L 2(ℝ N ), without an additional assumption on the nonlinearity. By means of a splitting method, we technically prove that the L p ∩H 1-difference of solutions is controlled by the L 2-difference of initial values near the initial time, where p , p > 2 , is the dissipation exponent of the nonlinearity. This along with the L 2-stability is sufficient for us to establish the L p ∩H 1-stability of random attractors. By utilizing some iteration arguments, we show that the solution is uniformly bounded on a bounded interval near the initial time in L δ (ℝ N ) for arbitrary δ > p, in which space the convergence of solutions and the upper semi-continuity of random attractors are also established.