Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE

Abstract

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

Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients

“…We can prove an abstract result that a periodic random attractor is backward compact if and only if it is local compact. A random attractor may not be locally compact (see [10]), although a deterministic attractor is automatically locally compact (see [27]).…”

Convergences of asymptotically autonomous pullback attractors towards semigroup attractors

“…We first recall a locally uniform compactness. For the pullback attractor A of a process U , the locally uniform compactness of A is often trivial since the mapping t → A(t) is often continuous in the full Hausdorff metric sense, provided that s → U (s, τ, x) is continuous, see [7, p31], also [11]. In the framework of random attractors Cui et al [4] studied the case without the continuity in s.…”

Dynamics for stochastic Fitzhugh–Nagumo systems with general multiplicative noise on thin domains

“…Notably, problems on thin domains for random dynamical systems are contrary to those in expanding domains 30 and different from those in time-varying domains. 31,32 Moreover, the upper semi-continuity of thin domains is also different from that of the time variable [33][34][35][36][37] and delay. [38][39][40][41] This paper is organized as follows.…”