Global existence of small data solutions for wave models with super-exponential propagation speed

We consider semi-linear classical damped wave models with time-dependent speed of propagation and time-dependent dissipation. The right-hand side is a power nonlinearity which can be interpreted only as a source term. We are interested in the interplay of time-dependent coefficients on the global existence of small data solutions. The considerations are divided into two cases depending on the behavior of the propagation speed: the super-exponential and the sub-exponential case. In this paper we study the super-exponential case. One main tool of our approach is the use of Matsumura type estimates for a family of Cauchy problems depending on a parameter.