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In this paper, we consider the longtime behaviour for the strongly damped wave equation with time-dependent memory kernelson a bounded domain. The main novelty of our result is that the memory kerneldepends 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 attractorsin. We also prove the regularity of the time-dependent global attractor, i.e.is bounded in, with a bound independent of  t . Finally, whenapproaches a multipleof the Dirac mass at zero as, we prove that the asymptotic dynamics of our problem is close to the one of its formal limitdescribing viscoelastic solids of Kelvin–Voigt type.

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