Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 40, pp. 1-16.

Title: Random attractors and their stability for nonclassical diffusion equations driven by additive white noise with delay and intensity

Authors: Wenhui Ma (Northwest Normal Univ., Lanzhou, China)
         Qiaozhen Ma (Northwest Normal Univ., Lanzhou, China)

Abstract:
In this article, we study the asymptotic behavior of solutions of 
nonclassical diffusion equation driven by an additive noise with delay 
and intensity $\epsilon\in(0,1]$ on $\mathbb{R}^n$. We first establish 
the existence and uniqueness of tempered pullback random attractors for 
the equations in $C([-\rho,0],H^{1}(\mathbb{R}^n))$, and then the upper 
semicontinuity of random attractors is also obtained when the intensity of 
noise approaches zero. It's worth mentioning that the Arzela-Ascoli theorem, 
spectral decomposition, and uniform tail-estimates have been utilized to 
demonstrate the asymptotic compactness of the solutions.


Submitted October 2, 2024. Published April 11, 2025.
Math Subject Classifications: 35B40, 35B41, 35R60, 37L55.
Key Words: Pullback random attractors; nonclassical diffusion equation; nonlinear delay; additive white noise