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

In the paper (Bui and Reissig, 2015) we explained that it is reasonable to divide the study of global existence of small data solutions to semi-linear classical damped wave models with time-dependent speed of propagation, time-dependent dissipation and power nonlinearity into two cases. The super-exponential case was treated in Bui and Reissig (2015). The present paper is devoted to the sub-exponential case. Both cases arise from a different influence of the interplay of time-dependent coefficients on the critical exponent. Here we only sketch differences to the approach for the super-exponential case. These differences appear in handling the nonlinearity. The corresponding Matsumura type estimates for a family of linear Cauchy problems depending on a parameter coincide in both cases (see Bui and Reissig (2014, 2015)).