Asymptotically autonomous dynamics for fractional subcritical nonclassical diffusion equations driven by nonlinear colored noise

Abstract

The asymptotically autonomous dynamics in $$H^s(\mathbb R^n)$$ for any $$s\in (0,1)$$ and $$n\in \mathbb {N}$$ are discussed for a class of highly nonlinear nonclassical diffusion equations perturbed by colored noise. The main feature of this model is that the time-dependent drift and diffusion terms have subcritical and superlinear polynomial growth of orders p and q, respectively, satisfying $$\begin{aligned} 2\le 2q<p<\infty \text { if }n=1\text { and }s\in [\frac{1}{2},1);\text { otherwise, }\, 2\le 2q<p<\frac{2n}{n-2s}. \end{aligned}$$ The number $$\frac{2n}{n-2s}$$ is called the fractional critical Sobolev embedding exponent. Under this setting, we prove the existence, time-dependent uniform compactness and asymptotically autonomous convergence of pathwise random attractors of the equations in $$H^s(\mathbb R^n)$$ when the time-dependent nonlinearities satisfy some new conditions. The time-dependent uniform pullback asymptotical compactness of the solution operators in $$H^s(\mathbb R^n)$$ is proved by virtue of a cut-off technique [38], a spectral decomposition approach and uniform estimates in a time-uniformly tempered attracting universe in order to overcome several difficulties caused by the lack of compact Sobolev embeddings on unbounded domains, the weak dissipativeness of the systems and the unknown measurability of attractors.