A class of dissipative wave equations with time-dependent speed and damping

We study the long time behavior of the energy for wave-type equations with time-dependent speed and damping: utt−λ(t)2Δu+b(t)ut=0. We investigate the interaction between the speed of propagation λ(t) and the damping coefficient b(t), showing how to describe the dissipative effect on the energy. We study a class of dissipations for which the equation keeps its hyperbolic structure and properties.