Random attractors for semilinear reaction-diffusion equation with distribution derivatives and multiplicative noise on R n

In this paper, we investigate the existence of random attractors for a semilinear reaction-diffusion equation with a nonlinearity having a polynomial growth of arbitrary order p − 1 ( p ≥ 2 ) , and with distribution derivatives and multiplicative noise defined on unbounded domains. The random attractors are obtained in L 2 ( R n ) and L p ( R n ) respectively. The semilinear reaction-diffusion equation is recast as a continuous random dynamical system and asymptotic compactness for this demonstrated by using uniform a priori estimates for far-field values of solutions as well as the cut-off technique.