Asymptotical behavior of non-autonomous stochastic reaction-diffusion equations with variable delay on RN

In this paper, we study the asymptotical behavior of solutions of stochastic reaction- diffusion equations with a super non-linearity and a Lipschizt continuous variable delayed term. The existence and uniqueness of tempered measurable pullback attractors are established in C([-h, 0]; H-1(R-N)). On account of the unobtainable bound of solution in H-2(R-N) caused by the non-differentiability of Brownian motion, the compact embedding on bounded domains is unavailable. To surmount this obstacle, the time point-wise pullback asymptotical compactness of the solutions in H-1(R-N) is proved by employing jointly uniform L-2 (P-2)-truncation estimates, spectral decomposition technique, and uniform tail estimates. In addition, the uniform equi-continuity of solutions in C([-h, 0]; H-1(R-N)) is checked mainly by transforming the infinite dimension problem into a finite dimension problem plus an infinite small principle part.