Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension

TL;DR

This paper proves that the set of characteristic blow-up points for solutions of a one-dimensional semilinear wave equation is locally finite, providing insight into the structure of singularities.

Contribution

It establishes the local finiteness of the set of characteristic points in blow-up solutions of a semilinear wave equation in one dimension.

Findings

The set of characteristic points is locally finite.

Characteristic points are isolated in the blow-up set.

Provides a structural understanding of blow-up behavior.

Abstract

We consider the semilinear wave equation with power nonlinearity in one space dimension. We consider an arbitrary blow-up solution $u(x,t)$, the graph $x\mapsto T(x)$ of its blow-up points and ${\cal S}\subset {\mathbb R}$ the set of all characteristic points. We show that ${\ca S}$ is locally finite.