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: u t t − λ ( t ) 2 Δ u + b ( t ) u t = 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.