Random dynamics of non-autonomous fractional stochastic p-Laplacian equations on $${\mathbb {R}}^N$$

Abstract

This article is concerned with the random dynamics of a wide class of non-autonomous, non-local, fractional, stochastic p-Laplacian equations driven by multiplicative white noise on the entire space $${\mathbb {R}}^N$$ . We first establish the well-posedness of the equations when the time-dependent non-linear drift terms have polynomial growth of arbitrary orders $$p,q\ge 2$$ . We then prove that the equation has a unique bi-spatial pullback random attractor that is measurable, compact in $$L^2({\mathbb {R}}^N)\cap L^p({\mathbb {R}}^N)\cap L^q({\mathbb {R}}^N)$$ and attracts all random subsets of $$L^2({\mathbb {R}}^N)$$ under the topology of $$L^2({\mathbb {R}}^N)\cap L^p({\mathbb {R}}^N)\cap L^q({\mathbb {R}}^N)$$ . In addition, we establish the upper semi-continuity of these attractors in $$L^2({\mathbb {R}}^N)\cap L^p({\mathbb {R}}^N)\cap L^q({\mathbb {R}}^N)$$ when the density of noise shrinks to zero. The idea of uniform tail estimates and the method of asymptotic a priori estimates are applied to prove the pullback asymptotic compactness of the solutions in $$L^2({\mathbb {R}}^N)\cap L^p({\mathbb {R}}^N)\cap L^q({\mathbb {R}}^N)$$ to overcome the non-compactness of Sobolev embeddings on $${\mathbb {R}}^N$$ as well as the almost sure nondifferentiability of the sample paths of the Wiener process.