Continuity of the attractors in time-dependent spaces and applications

In this paper, we investigate the continuity of the attractors in time-dependent phase spaces. (i) We establish two abstract criteria on the upper semicontinuity and the residual continuity of the pullback D-attractor with respect to the perturbations, and an equivalence criterion between their continuity and the pullback equiattraction, which generalize the continuity theory of attractors developed recently in [27,28] to that in time-dependent spaces. (ii) We propose the notion of pullback D-exponential attractor, which includes the notion of time-dependent exponential attractor [33] as its spacial case, and establish its existence and Holder continuity criterion via quasi-stability method introduced originally by Chueshov and Lasiecka [12,13]. (iii) We apply above-mentioned criteria to the semilinear damped wave equations with perturbed time-dependent speed of propagation: Ep(t)utt + alpha ut - Delta u + f (u) = g, with perturbation parameter E is an element of (0, 1], to realize above mentioned continuity of pullback D and D-exponential attractors in time-dependent phase spaces, and the method developed here allows to overcome the difficulty of the hyperbolicity of the model. These results deepen and extend recent theory of attractors in time-dependent spaces in literatures [15,20,29]. (c) 2023 Elsevier Inc. All rights reserved.