Time-dependent asymptotic behavior of the solution for wave equations with linear memory

In this article, we consider the long-time behavior of solutions for the wave equation with linear memory. Within the theory of process on time-dependent spaces, we investigate the existence of the time-dependent attractor by using the operator decomposition technique and compactness of translation theorem and more detailed estimates. Furthermore, the asymptotic structure of time-dependent attractor, which converges to the attractor of parabolic equation with memory, is proved. Besides, we obtain a further regular result about ut.