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# Communications in Mathematical Sciences

## Volume 21 (2023)

### Number 3

### Well-posedness and long-time dynamics of fractional nonclassical diffusion equations with locally Lipschitz noise

Pages: 731 – 774

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n3.a6

#### Authors

#### Abstract

The global well-posedness and long-time dynamics are investigated for a class of nonautonomous, stochastic, *fractional*, nonclassical diffusion equations on $\mathbb{R}^N$ with polynomial growth drifts of *arbitrary* order $p \gt 2$ and *locally* Lipschitz diffusions. By a regularization method, we establish the well-posedness of the equations in $L^2 (\Omega, \mathcal{F}, H^s (\mathbb{R}^N))$ for any $s \in (0,1]$ and $N \in \mathbb{N}$ in four instances: regular additive noise, general additive noise, globally Lipschitz noise and locally Lipschitz noise. Then we prove that the mean random dynamical systems generated by the solution operators have a unique weak pullback mean random attractor in $L^2 (\Omega, \mathcal{F}, H^s (\mathbb{R}^N))$. Our results do not depend on any restrictions on the triple $(N,s,p)$, and even original in $L^2 (\Omega, \mathcal{F} ,H^1 (\mathbb{R}^N))$ when the fractional Laplacian reduces to the standard one.

#### Keywords

nonclassical diffusion equation, regional fractional Laplacian, locally Lipschitz noise, weak pullback mean random attractors, mean random dynamical system

#### 2010 Mathematics Subject Classification

35A15, 35B41, 37L55, 60H15

Renhai Wang was supported by the China Postdoctoral Science Foundation under grant numbers 2020TQ0053 and 2020M680456.

Boling Guo was supported by the NSFC under grant numbers 11731014 and 11571254.

Chunxiao Guo was supported by the NSFC under grant number 11771444.

Received 2 August 2021

Accepted 25 July 2022

Published 27 February 2023