Appearance of Strauss-type exponent in semilinear wave equations with time-dependent speed of propagation

In this paper, blowup phenomenon for the semilinear wave equation with time-dependent speed of propagation and scattering damping is considered under the smallness of initial data. Our result contains small data blowup for sub-Strauss exponent for the simplest semilinear wave equation and also the one for semilinear generalized Tricomi equation. Key ingredient is so-called test function method (developed in Ikeda–Sobajima–Wakasa [10]) with a certain conservative quantity via a special solution with the Liouville–Green (or WKB) approximation.