2022/01/10 by Yanan Li, Zhijian Yang, Li, Yanan +1
Engineering · Mathematics · #Stability and Controllability of Differential Equations #Advanced Mathematical Physics Problems #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.2201.03241
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"older 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: \eρ(t) utt+α ut -Δ u+f(u)=g, with perturbation parameter \e∈(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].