Upper semi-continuity of attractors for non-autonomous fractional stochastic parabolic equations with delay

Abstract

This paper is concerned with the asymptotic behavior of the solutions to a class of non-autonomous nonlocal fractional stochastic parabolic equations with delay defined on bounded domain. We first prove the existence of a continuous non-autonomous random dynamical system for the equations as well as the uniform estimates of solutions with respect to the delay time and noise intensity. We then show pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractors by utilizing the Arzela-Ascoli theorem and the uniform estimates of solutions in fractional Sobolev space H α (R n) with α ∈ (0, 1) as well as their time derivatives in L 2 (R n). Finally, we establish the upper semi-continuity of the random attractors when noise intensity and time delay approaches zero, respectively.

Existence and asymptotic behavior of square-mean S-asymptotically periodic solutions of fractional stochastic evolution equations

“…In recent years, more and more mathematicians have paid attention to the qualitative study of solutions of semilinear stochastic evolution equations(see [3,16,17,36] and the references therein). Very recently, in [7,10,11], Chen and Zhang investigated the non-autonomous stochastic evolution equations with nonlinear noise and nonlocal initial conditions, and obtained the existence results of mild solutions under some weaker growth conditions on nonlinear functions. It is also worth noting that in [8], Chen et al established a sufficient condition to judge the relative compactness of abstract continuous function families on infinite intervals.…”

Existence and upper semicontinuity of random pullback attractors for 2D and 3D non-autonomous stochastic convective Brinkman-Forchheimer equations on whole domain

“…Using this theory, the periodic pullback random attractor for stochastic NSE with time dependent deterministic term on Poincaré domains is established in [48]. For more applications of this theory, the interested readers are referred to see [12,19,21,46,50,53], etc.…”

“…In [49], the author extended the results on upper semicontinuity of the works [8,46] to nonautonomous RDS (compact as well as non-compact), and it has been applied in the works [12,50,53], etc. We fix r ∈ [1, ∞) with any µ, β > 0 in 2D, and r ∈ (3, ∞) with any µ, β > 0 and r = 3 with 2µβ ≥ 1 in 3D.…”

Upper semi-continuity of non-autonomous fractional stochastic $ p $-Laplacian equation driven by additive noise on $ \mathbb{R}^n $

“…The key is to establish the convergence of solutions in L 2 (R n ) as intensity of noise goes to zero. For more details about the convergence of attractors, we refer the readers to [4,5,14,22,27,24] for deterministic and stochastic differential equations. This paper is organized as follows.…”