Refined regularity for the blow-up set at non characteristic points for the complex semilinear wave equation

TL;DR

This paper studies the regularity of the blow-up set for complex semilinear wave equations, proving that the set of non characteristic points is open and that the blow-up curve and phase are smoothly regular.

Contribution

It introduces a Liouville theorem and establishes refined regularity properties of the blow-up set at non characteristic points.

Findings

The set of non characteristic points is open.

The blow-up curve is of class C^{1,μ_0}.

The phase θ is C^{μ_0} on this set.

Abstract

In this paper, we consider a blow-up solution for the complex-valued semilinear wave equation with power non-linearity in one space dimension. We show that the set of non characteristic points $I_0$ is open and that the blow-up curve is of class $C^{1,\mu_0}$ and the phase $\theta$ is $C^{\mu_0}$ on this set. In order to prove this result, we introduce a Liouville Theorem for that equation.