The Interplay Between Time-dependent Speed of Propagation and Dissipation in Wave Models

Abstract

The aim of this paper is to understand the influence of the interplay between a time-dependent increasing speed of propagation and a timedependent coefficient in the dissipation on qualitative properties of solutions to the wave model

$$u_{tt}-a^2(t)\bigtriangleup\mathrm{u}+ b(t)u_t=0,\,u(0,x)=u_1(x),u_{t}(0,x)=u_{2}(x).$$

Our considerations are focused to energy estimates. The main difficulty is to find a good description of non-effective and effective dissipations depending on a given speed of propagation. The obtained energy estimates are optimal as special examples will show. At the end we will sketch very briefly how to get scattering and over-damping results. So, we propose a classification of different damping terms which is motivated by the thesis of J. Wirth for the case $a\equiv1$

Mathematics Subject Classification (2010). Primary 35L15; Secondary 35L05, 35B40, 35B45.