Nonlinear stability of self-similar solutions for semilinear wave equations

TL;DR

This paper proves the nonlinear stability of a fundamental self-similar solution for certain focusing semilinear wave equations in the radial case, using a semigroup approach in similarity coordinates.

Contribution

It introduces a novel stability proof for self-similar solutions of wave equations with focusing power nonlinearities for odd integer powers.

Findings

Stability established for p=3,5,7,... in the radial case.

Uses semigroup formulation in similarity coordinates.

Provides a framework for analyzing nonlinear stability of self-similar solutions.

Abstract

We prove nonlinear stability of the fundamental self--similar solution of the wave equation with a focusing power nonlinearity $\psi_{tt}-\Delta \psi=\psi^p$ for $p=3,5,7,...$ in the radial case. The proof is based on a semigroup formulation of the wave equation in similarity coordinates.