The random dynamics in $ H^s(\mathbb{R}^n) with s\in (0,1) is investigated for the fractional nonclassical diffusion equations driven by colored noise. Both existence and uniqueness of pullback attractors are established a wide class nonlinear terms. In case additive noise, upper semi-continuity these proved as correlation time noise approaches zero. methods uniform tail-estimate spectral decomposition employed to obtain asymptotic compactness solutions order overcome non-compactness Sobolev...

A bi-spatial pullback attractor is obtained for non-autonomous and stochastic FitzHugh-Nagumo equations when the initial space $L^2(\mathbb{R}^n)^2$ terminate $H^1(\mathbb{R}^n)\times L^2(\mathbb{R}^n)$. Some new techniques of positive negative truncations are used to investigate regularity attractors coupling correct essential mistake in [T. Q. Bao, Discrete Cont. Dyn. Syst. 35(2015), 441-466]. counterexample given an important lemma $H^1$-attractor several literatures included above.

In this article, we study the asymptotic behavior of a nonlocal semilinear degenerate heat equation with past history in bounded domain. We approximate obtained after Dafermos transformation non-degenerate to obtain existence, uniqueness, and regularity solutions equation. Then, existence global attractor original

Existence and connection of numerical attractors for discrete-time p-Laplace lattice systems via the implicit Euler scheme are proved.The shown to have an optimized bound, which leads continuous convergence when graph nonlinearity closes vertical axis or external force vanishes.A new type Taylor expansion without Fréchet derivatives is established applied show discretization error order two, crucial prove that converge upper semicontinuously global attractor original continuous-time system...

This paper is concerned with the regular random dynamics for reaction-diffusion equation defined on a thin domain and perturbed by rough noise, where usual Winner process replaced general stochastic satisfied basic convergence. A bi-spatial attractor obtained when non-initial space $p$-times Lebesgue or Sobolev space. The measurability of solution operator proved, which leads to in both state spaces. Finally, upper semi-continuity attractors under $p$-norm established narrow degenerates onto...

In this paper, we consider the dynamic behavior of stochastic [Formula: see text]-Laplacian-type lattice equations perturbed by a multiplicative noise. Under weaker dissipative conditions compared to cases in bounded and unbounded domains, first obtain existence unique random attractor. We also establish approximation attractors from finite infinite lattice, which indicates that family is upper lower semi-continuous when number nodes tends infinity.

Backward compact dynamics is deduced for a non-autonomous Benjamin-Bona-Mahony (BBM) equation on an unbounded 3D-channel. A backward attractor defined by time-dependent family of compact, invariant and pullback attracting sets. The theoretical existence result such derived from the flattening property, this property proved to be equivalent asymptotic compactness in uniformly convex Banach space. Finally, it shown that BBM has Sobolev space under some suitable assumptions, as, translation...

This paper is concerned with the robustness of a pullback attractor as time tends to infinity. A called forward (resp. backward) compact if union over future past) pre-compact. We prove that compactness necessary and sufficient condition such upper semi-continuous set at positive negative) infinity, also obtain minimal limit-set. further lower semi-continuity get maximal limit-set Some criteria for are established when evolution process or backward omega-limit compact. Those theoretical...