Uniform attractors for a strongly damped wave equation with linear memory

Abstract

A strongly damped semilinear wave equation with linear memory is considered in a history space setting. Namely, the evolution of the past history of the displacement vector is contained in the dynamical system. Existence, uniqueness and continuous dependence results are discussed. Under proper assumptions on the memory kernel, the existence of uniform absorbing sets is achieved. Moreover, when the source term is translation compact in a suitable functional space, the system is shown to possess a uniform attractor.