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 utt-a2(t)Δu + b(t)ut = 0, u(0,x) = u1(x), ut(0,x) = u2(x). (0.1) 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 ≡ 1.