Radial blow-up standing solutions for the semilinear wave equation

TL;DR

This paper constructs radial blow-up solutions for the semilinear wave equation that converge exponentially to a soliton near a non-characteristic point, using self-similar variables and energy estimates.

Contribution

It introduces a method to handle the additional gradient term in radial blow-up solutions, extending previous one-dimensional energy estimates to the radial case.

Findings

Existence of blow-up solutions converging to a soliton

Identification of non-characteristic points in radial solutions

Extension of energy estimate techniques to radial setting

Abstract

We consider the semilinear wave equation with a power nonlinearity in the radial case. Given $r_0>0$, we construct a blow-up solution such that the solution near $(r_0,T(r_0))$ converges exponentially to a soliton. Moreover, we show that $r_0$ is a non-characteristic point. For that, we translate the question in self-similar variables and use a modulation technique. We will also use energy estimates from the one dimensional case treated by Merle and Zaag in 2007. Of course because of the radial setting, we have an additional gradient term which is delicate to handle. That's precisely the purpose of our paper.