Finite fractal dimension of random attractors for non-autonomous fractional stochastic reaction–diffusion equations in ℝ

Abstract

This paper investigates the asymptotic behavior of solutions for non-autonomous fractional stochastic reaction–diffusion equations with multiplicative noise in . We prove the existence and uniqueness of the tempered pullback random attractor for the equations in and obtain the finite fractal dimension for the pullback random attractor. Two main difficulties here are that the fractional Laplacian operator is non-local and the Sobolev embedding is not compact on unbounded domains. To solve this, we derive the tail-estimates of solutions of the equation and decompose the solutions into a sum of three parts, which one part is finite-dimensional and other two parts are quickly decay in mean sense.