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

Renhai Wang (School of Mathematical Sciences, Guizhou Normal University, Guiyang, China)

Boling Guo (Institute of Applied Physics and Computational Mathematics, Beijing, China)

Chunxiao Guo (Department of Mathematics, China University of Mining and Technology Beijing, China)

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

Received 2 August 2021

Accepted 25 July 2022

Published 27 February 2023