Time-dependent global attractors for strongly damped wave equations with time-dependent memory kernels

Abstract

In this paper, we consider the longtime behaviour for the strongly damped wave equation with time-dependent memory kernels on a bounded domain . The main novelty of our result is that the memory kernel depends on time, allowing to describe the dynamics of aging materials. We first investigate the existence and uniqueness of weak solutions and then, we obtain the existence of the time-dependent global attractors in . We also prove the regularity of the time-dependent global attractor , i.e. is bounded in , with a bound independent of t. Finally, when approaches a multiple of the Dirac mass at zero as , we prove that the asymptotic dynamics of our problem is close to the one of its formal limit describing viscoelastic solids of Kelvin–Voigt type.