Continuity of non-autonomous attractors for hyperbolic perturbation of parabolic equations

Abstract

In this work we prove the lower and upper semicontinuity of pullback, uniform, and cocycle attractors for the non-autonomous dynamical system given by hyperbolic equation on a bounded domain Ω⊂R3ϵutt+ut−Δu=fϵ(t,u). For each ϵ>0, this equation has uniform, pullback, and cocycle attractors in H01(Ω)×L2(Ω) and for ϵ=0 the limit parabolic equationut−Δu=f0(u) has a global attractor A0 in H01(Ω) which can be naturally embedded into a compact set A0 in H01(Ω)×L2(Ω). We prove that all three types of non-autonomous attractors converge, both upper and lower-semicontinuously to A0. The study of the detailed structure of the non-autonomous attractors under perturbation plays the crucial role in the arguments.