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

TL;DR

This paper investigates the existence and structure of characteristic blow-up points in solutions to a one-dimensional semilinear wave equation, revealing their geometric and solitonic decomposition properties.

Contribution

It establishes the existence of characteristic blow-up points, analyzes their geometric structure, and describes the soliton decomposition near these points in self-similar variables.

Findings

Characteristic points form a set with empty interior.

Near characteristic points, solutions decompose into multiple solitons with alternating signs.

The blow-up graph forms a right-angled corner at characteristic points.

Abstract

We consider the semilinear wave equation with power nonlinearity in one space dimension. We first show the existence of a blow-up solution with a characteristic point. Then, we consider an arbitrary blow-up solution $u(x,t)$, the graph $x\mapsto T(x)$ of its blow-up points and $\SS\subset $ the set of all characteristic points, and show that the $\SS$ has an empty interior. Finally, given $x_0\in \SS$, we show that in selfsimilar variables, the solution decomposes into a decoupled sum of (at least two) solitons with alternate signs and that $T(x)$ forms a corner of angle $\frac \pi 2$ at $x_0$.