Blow-up behavior outside the origin for a semilinear wave equation in the radial case

TL;DR

This paper studies the blow-up behavior of solutions to a radial semilinear wave equation with subcritical nonlinearity, extending one-dimensional results to higher dimensions and characterizing the blow-up set structure.

Contribution

It introduces a new Lyapunov functional for the radial case and extends previous one-dimensional results to describe the blow-up set in higher dimensions.

Findings

Blow-up set near non-zero non-characteristic points is of class C^1.

Characteristic points form a finite set of concentric spheres.

Extended the analysis of blow-up behavior to higher-dimensional radial cases.

Abstract

We consider the semilinear wave equation in the radial case with conformal subcritical power nonlinearity. If we consider a blow-up point different from the origin, then we exhibit a new Lyapunov functional which is a perturbation of the one dimensional case and extend all our previous results known in the one-dimensional case. In particular, we show that the blow-up set near non-zero non-characteristic points is of class $C^1$, and that the set of characteristic points is made of concentric spheres in finite number in $\{\frac 1R \le |x|\le R\}$ for any $R>1$.