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

TL;DR

This paper analyzes the decay behavior of solutions to a wave equation with time-dependent damping terms, focusing on how these coefficients influence energy decay and solution properties.

Contribution

It introduces a classification of damping mechanisms and employs the WKB-method to derive decay estimates for higher order energy norms in such wave equations.

Findings

Derived decay estimates for higher order energy norms.

Classified damping mechanisms based on their influence.

Analyzed the interplay between frictional and viscoelastic damping.

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.