Dynamics for subcritical fractional nonclassical diffusion equations with nonlinear Wong-Zakai noise and delays

Qiangheng Zhang; Lianbing She

doi:10.3934/dcdsb.2022234

Article Contents

2023, Volume 28, Issue 6: 3629-3661. Doi: 10.3934/dcdsb.2022234

This issue Previous Article Pullback attractors and upper semicontinuity for non-autonomous extensible two-beams Next Article Monotonicity results for fractional parabolic equations in the whole space

Dynamics for subcritical fractional nonclassical diffusion equations with nonlinear Wong-Zakai noise and delays

Qiangheng Zhang^1, and
Lianbing She^2, ,

1.
School of Mathematics and Statistics, Heze University, Heze 274015, China
2.
School of Mathematics and Statistics, Liupanshui normal college, Liupanshui 553004, China

^*Corresponding author: Lianbing She
^*Corresponding author: Lianbing She

Received: July 2022

Revised: November 2022

Early access: December 7, 2022

Published online: December 7, 2022

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Abstract

The multi-scale stability of random attractors in the space of continuous functions from the delay interval into the fractional Sobolev space for a class of nonclassical diffusion equations on $ \mathbb{R}^n $ with time-delay and nonlinear Wong-Zakai noise is studied. First, we prove the existence of a unique pullback random attractor, which is a family of compact sets depending on two parameters: the step size of noise and the current time. Secondly, we study the semicontinuity of the pullback random attractor as the current time goes to positive infinity. Thirdly, we consider the semicontinuity of the pullback random attractor as the delay time approach zero. Finally, we prove the semicontinuity of the pullback random attractor as the step size of noise tends to positive infinity. In order to overcome the lack of compact Sobolev embeddings on $ \mathbb{R}^n $ and the weak dissipation of equations, we employ the spectral method and idea of uniform tail-estimates to prove that the solution operator is pullback asymptotic compactness over a time-uniformly tempered attracting universe.

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Mathematics Subject Classification: Primary: 37L55, 35B40; Secondary: 35B41.

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