# Propagation of singularities for generalized solutions to nonlinear wave equations

## TL;DR

This paper investigates how initial singularities in generalized solutions to semilinear wave equations propagate along characteristic lines in one dimension, using Colombeau algebras and fixed point methods.

## Contribution

It extends the classical singularity propagation results to the Colombeau algebra setting for semilinear wave equations in one dimension.

## Key findings

- Singularities propagate along characteristic lines in the Colombeau framework.
- The proof employs a fixed point theorem in ultra-metric topology.
- Results generalize classical propagation phenomena to generalized solutions.

## Abstract

The paper is devoted to regularity theory of generalized solutions to semilinear wave equations with a small nonlinearity. The setting is the one of Colombeau algebras of generalized functions. It is shown that in one space dimension, an initial singularity at the origin propagates along the characteristic lines emanating from the origin, as in the linear case. The proof relies on a fixed point theorem in the ultra-metric topology on the algebras involved. The paper takes up the initiating research of the 1970s on anomalous singularities in classical solutions to semilinear hyperbolic equations and transplants the methods into the Colombeau setting.