Pullback Attractors for a Class of Semilinear Second-Order Nonautonomous Evolution Equations with Hereditary Characteristics

Abstract

In this paper, we investigate the long-time behavior for the nonautonomous semilinear second-order evolution equation ∂2u/∂t2−Δu−Δ∂u/∂t−Δ∂2u/∂t2=ft,ux,t−ρt+gt,x,inτ,∞×Ω with some hereditary characteristics, where Ω is an open-bounded domain of ℝNN≥3 with smooth boundary ∂Ω. Firstly, we establish the existence of solutions for the second-order nonautonomous evolution equation by the standard Faedo–Galerkin method, but without the uniqueness of solutions. Then by proving the pullback asymptotic compactness for the multivalued process Ut,τ on CH01Ω,H01Ω, we obtain the existence of pullback attractors in the Banach spaces CH01Ω,H01Ω for the multivalued process generated by a class of second-order nonautonomous evolution equations with hereditary characteristics and ill-posedness.