Bi-spatial Pullback Attractors of Fractional Nonclassical Diffusion Equations on Unbounded Domains with (p, q)-Growth Nonlinearities

Abstract

This article is concerned with the existence as well as regularity of pullback attractors for a wide class of non-autonomous, fractional, nonclassical diffusion equations with (p, q)-growth nonlinearities defined on unbounded domains. We first establish the well-posedness of solutions as well as the existence of an energy equation, and then prove the existence of a unique pullback attractor in the fractional Sobolev space $H^s({\mathbb {R}}^N)$ for all $s\in (0,1]$. Finally, we show that this attractor is a bi-spatial $(H^s({\mathbb {R}}^N),L^r({\mathbb {R}}^N))$-attractor in two cases:

$$\begin{aligned} 2\le r\le \max (p,q)\text { for }s=1;\text { and }2\le r<\max (p,q) \text { for }s\in (0,1). \end{aligned}$$

The idea of energy equations and the method of asymptotic a priori estimates are employed to establish the pullback asymptotic compactness of the solutions in $H^s({\mathbb {R}}^N)\cap L^r({\mathbb {R}}^N)$ in order to overcome the non-compactness of the Sobolev embedding on the unbounded domain as well as the weak dissipation of the equations. This is the first time to study the bi-spatial attractor of the equation when the initial space is $H^s({\mathbb {R}}^N)$, and the result of this article is new even in $H^1({\mathbb {R}}^N)\cap L^r({\mathbb {R}}^N)$ when the fractional Laplace operator reduces to the standard Laplace operator.