Visco-elastic damped wave models with time-dependent coefficient

TL;DR

This paper investigates the effects of a time-dependent damping coefficient on the behavior of solutions to a visco-elastic damped wave equation, using WKB-analysis to derive decay estimates and analyze the parabolic effect.

Contribution

It introduces a novel analysis of a wave model with a general time-dependent damping coefficient using combined elliptic and hyperbolic WKB-analysis techniques.

Findings

Decay estimates for higher order energies

Insights into the parabolic effect in visco-elastic models

Application of WKB-analysis to time-dependent coefficients

Abstract

In this paper, we study the following Cauchy problem for linear visco-elastic damped wave models with a general time-dependent coefficient $g=g(t)$: \begin{equation} \label{EqAbstract} \tag{$\star$} \begin{cases} u_{tt}- \Delta u + 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} We are interested to study the influence of the damping term $g(t)(-\Delta)u_t$ on qualitative properties of solutions to \eqref{EqAbstract} as decay estimates for energies of higher order and the parabolic effect. The main tools are related to WKB-analysis. We apply elliptic as well as hyperbolic WKB-analysis in different parts of the extended phase space.