Self-similar blow up for energy supercritical semilinear wave equation

TL;DR

This paper investigates self-similar blow-up solutions for the energy supercritical semilinear wave equation, establishing existence and stability of these profiles in a specific parameter regime.

Contribution

It introduces a family of self-similar solutions bifurcating from solitons and proves their finite codimensional stability in the supercritical setting.

Findings

Existence of countably many self-similar profiles.

Stability of these profiles under perturbations.

Extension of stability analysis to non-radial solutions.

Abstract

We analyse the energy supercritical semilinear wave equation $$\Phi_{tt}-\Delta\Phi-|\Phi|^{p-1}\Phi=0$$ in $\mathbb R^d$ space. We first prove in a suitable regime of parameters the existence of a countable family of self similar profiles which bifurcate from the soliton solution. We then prove the non radial finite codimensional stability of these profiles by adapting the functional setting of arXiv:1912.11005.