Regularity of pullback attractors for nonautonomous nonclassical diffusion equations

Random dynamics of fractional nonclassical diffusion equations driven by colored noise

“…If s = 1, then the fractional operator (−∆) s becomes the standard Laplace operator. In this special case, the attractor of the nonclassical diffusion equation has been studied in [4,5,52,54,56,61] for the deterministic systems, and in [7,59,60] for the stochastic systems driven by linear white noise. However, even for s = 1, the attractors of the nonclassical diffusion equations driven by colored noise have not been discussed in the literature.…”

Strong pullback attractors for a nonclassical diffusion equation

“…During the last years, the existence and asymptotic behavior of the solutions to problem (1) have been investigated by many authors under the different assumption conditions. There have been some results [38,31,47,32,22,1,42,46,50,35,41,45,18] for problem (1) when the perturbation parameter µ > 0 is a fixed constant. Authors in [38] investigated the existence of strong global attractors for a nonclassical diffusion equation with subcritical nonlinearity, and proved the global attractors A µ → A 0 + in the sense of Hausdorff semidistance in H 1 0 (Ω) as µ → 0 + .…”

“…Recently, authors in [41] considered the upper semi-continuity of uniform attractors in H 1 0 (Ω) for a non-autonomous nonclassical diffusion equation with critical nonlinearity. Wang, Zhu and Li [45] proved the regularity of pullback attractors for a three dimensional non-autonomous nonclassical diffusion equation with critical nonlinearity. Under a Sobolev type growth condition of the nonlinearity g, Lee and Toi [18] investigated the existence and upper semi-continuity of global attractors of nonclassical diffusion equation in a smooth bounded domain Ω with dynamic boundary condition.…”

“…Existence and uniqueness of solutions. We know that there have been some results [1,41,42,45] on the existences and properties of the pullback attractors for nonclassical diffusion equations ( 1) in H 1 0 (Ω). We state the known result for the existence and uniqueness of weak solution to problem (1) as the following lemma.…”

Dynamics of fractional nonclassical diffusion equations with delay driven by additive noise on $ \mathbb{R}^n $

“…When α = 1, the fractional Laplacian (−∆) α becomes the standard Laplace operator −∆. In this special case, the deterministic attractors of the equation have been investigated in [4,5,7,18,19,54,57,63,69,70,73], and the random attractors have been studied in [8,27,41,76] and [65,66] for white noise and colored noise, respectively. Recently, the existence of random attractors of the stochastic nonclassical diffusion equations with time-varying delay has been reported in [22].…”