Asymptotic behavior of non-autonomous fractional p-Laplacian equations driven by additive noise on unbounded domains

Abstract

This paper deals with the asymptotic behavior of solutions to non-autonomous, fractional, stochastic p-Laplacian equations driven by additive white noise and random terms defined on the unbounded domain ℝN. We first prove the existence and uniqueness of solutions for polynomial drift terms of arbitrary order. We then establish the existence and uniqueness of pullback random attractors for the system in L2(ℝN). This attractor is further proved to be a bi-spatial (L2(ℝN),Lr(ℝN))-attractor for any r ∈ [2,∞), which is compact, measurable in Lr(ℝN) and attracts all random subsets of L2(ℝN) with respect to the norm of Lr(ℝN). Finally, we show the robustness of these attractors as the intensity of noise and the random coefficients approach zero. The idea of uniform tail-estimates as well as the method of higher-order estimates on difference of solutions are employed to derive the pullback asymptotic compactness of solutions in Lr(ℝN) for r ∈ [2,∞) in order to overcome the non-compactness of Sobolev embeddings on ℝN and the nonlinearity of the fractional p-Laplace operator.

Keywords

• fractional p-laplacian equation
• bi-spatial random attractor
• additive noise
• robustness
• unbounded domain