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 $\left({\partial }^{2}u/\partial t^2\right)-\mathrm{\Delta }u-\mathrm{\Delta }\left(\partial u/\partial t\right)-\mathrm{\Delta }\left({\partial }^{2}u/\partial t^2\right)=f\left(t,u\left(x,t-\rho \left(t\right)\right)\right)+g\left(t,x\right),\text{in}\left(\tau ,\infty \right)\times \mathrm{\Omega }$ with some hereditary characteristics, where $\mathrm{\Omega }$ is an open-bounded domain of ${ℝ}^N\left(N\ge 3\right)$ with smooth boundary $\partial \mathrm{\Omega }$ . 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 $\left\{U\left(t,\tau \right)\right\}$ on $C_{H_0^1\left(\mathrm{\Omega }\right),H_0^1\left(\mathrm{\Omega }\right)}$ , we obtain the existence of pullback attractors in the Banach spaces $C_{H_0^1\left(\mathrm{\Omega }\right),H_0^1\left(\mathrm{\Omega }\right)}$ for the multivalued process generated by a class of second-order nonautonomous evolution equations with hereditary characteristics and ill-posedness.