Radial blow-up standing solutions for the semilinear wave equation

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.