A modulation technique for the blow-up profile of the vector-valued semilinear wave equation

TL;DR

This paper studies the blow-up behavior of vector-valued solutions to a semilinear wave equation in one dimension, characterizing stationary solutions and showing convergence in self-similar variables with a novel solution structure.

Contribution

It introduces a new structure of stationary solutions for the vector-valued semilinear wave equation and analyzes the blow-up profile in the energy norm.

Findings

Characterization of all stationary solutions as an m-parameter family

Convergence of solutions to a particular stationary solution in self-similar variables

Identification of a new structure of stationary solutions

Abstract

We consider a vector-valued blow-up solution with values in $\mathbb{R}^m$ for the semilinear wave equation with power nonlinearity in one space dimension (this is a system of PDEs). We first characterize all the solutions of the associated stationary problem as an m-parameter family. Then, we show that the solution in self-similar variables approaches some particular stationary one in the energy norm, in the non-characteristic cases. Our analysis is not just a simple adaptation of the already handled real or complex case. In particular, there is a new structure of the set a stationary solutions.