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

TL;DR

This paper investigates blowup phenomena in semilinear wave equations with time-dependent speeds, revealing the appearance of Strauss-type exponents through a test function method and Liouville-Green approximation.

Contribution

It extends blowup results to equations with variable speeds, including the generalized Tricomi equation, using a novel test function approach.

Findings

Small data blowup for sub-Strauss exponent in wave equations.

Identification of Strauss-type exponent in variable speed wave equations.

Application of Liouville-Green approximation in blowup analysis.

Abstract

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.