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 $\mathscr D$-attractor with respect to the perturbations, and an equivalence criterion between their continuity and the pullback equi-attraction, 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 $\mathscr D$-exponential attractor, which includes the notion of time-dependent exponential attractor [33] as its spacial case, and establish its existence and H\"{o}lder 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: $\e\rho(t) u_{tt}+\alpha u_t -\Delta u+f(u)=g$, with perturbation parameter $\e\in(0, 1]$, to realize above mentioned continuity of pullback $\mathscr D$ and $\mathscr D$-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 [15,20,19].