Evolution models with time-dependent coefficients in friction and viscoelastic damping terms

Abstract

We study the following Cauchy problem for the linear wave equation with both time-dependent friction and time-dependent viscoelastic damping: \begin{equation} \label{EqAbstract}\tag{$\ast$} \begin{cases} u_{tt}- \Delta u + b(t)u_t - g(t)\Delta u_t=0, &(t,x) \in (0,\infty) \times \mathbb{R}^n, \\ u(0,x)= u_0(x),\quad u_t(0,x)= u_1(x), &x \in \mathbb{R}^n. \end{cases} \end{equation} Our goal is to derive decay estimates for higher order energy norms of solutions to this problem. We focus on the interplay between the time-dependent coefficients in both damping terms and their influence on the qualitative behavior of solutions. The analysis is based on a classification of the damping mechanisms, frictional damping $b(t)u_t$ and viscoelastic damping $-g(t)\Delta u_t$ as well, and employs the WKB-method in the extended phase space.