A note on the stability of self-similar blow-up solutions for superconformal semilinear wave equations

Abstract

In this note, we investigate the stability of self-similar blow-up solutions for superconformal semilinear wave equations in all dimensions. A central aspect of our analysis is the spectral equivalence of the linearized operators under Lorentz transformations in self-similar variables. This observation serves as a useful tool in proving mode stability and provides insights that may aid the study of self-similar solutions in related problems. As a direct consequence, we establish the asymptotic stability of the ODE blow-up family, extending the classical results of Merle and Zaag [Merle-Zaag, 2007, 2016] to the superconformal case and generalizing the recent findings of Ostermann [Ostermann, 2024] to include the entire ODE blow-up family.

17 pages