Asymptotic profiles and singular limits for the viscoelastic damped wave equation with memory of type I

Abstract

In this paper, we are interested in the Cauchy problem for the viscoelastic damped wave equation with memory of type I. By applying the WKB analysis and the Fourier analysis, we explain the memory’s influence on dissipative structures and asymptotic profiles of solutions to the model with additionally weighted L1 initial data. Furthermore, concerning standard energy and the solution itself, we establish the singular limit between the dissipative Moore–Gibson–Thompson equation with memory and the viscoelastic damped wave equation with memory, where a crucial compatibility condition of the initial datum is found.