A global existence result for a semilinear wave equation with scale-invariant damping and mass in even space dimension

TL;DR

This paper proves the global existence of radial solutions for a semilinear wave equation with scale-invariant damping and mass in even dimensions, identifying the critical exponent and confirming part of a conjecture in the massless case.

Contribution

It establishes global existence results under specific conditions and confirms a conjecture related to the wave equation with damping and mass in even dimensions.

Findings

Global existence of solutions in even dimensions

Identification of the critical Strauss exponent

Validation of a conjecture in the massless case

Abstract

In the present article a semilinear wave equation with scale-invariant damping and mass is considered. The global (in time) existence of radial symmetric solutions in even spatial dimension $n$ is proved using weighted $L^\infty-L^\infty$ estimates, under the assumption that the multiplicative constants, which appear in the coefficients of damping and of mass terms, fulfill an interplay condition which yields somehow a "wave-like" model. In particular, combining this existence result with a recently proved blow-up result, a suitable shift of Strauss exponent is proved to be the critical exponent for the considered model. Moreover, the still open part of a conjecture done by D'Abbicco - Lucente - Reissig is proved to be true in the massless case.