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On blowup for the supercritical quadratic wave equation

Elek Csobo, Irfan Glogić and Birgit Schörkhuber

Vol. 17 (2024), No. 2, 617–680
Abstract

We study singularity formation for the quadratic wave equation in the energy supercritical case, i.e., for d 7. We find in closed form a new, nontrivial, radial, self-similar blow-up solution u which exists for all d 7. For d = 9, we study the stability of u without any symmetry assumptions on the initial data and show that there is a family of perturbations which lead to blowup via u . In similarity coordinates, this family represents a codimension-1 Lipschitz manifold modulo translation symmetries. The stability analysis relies on delicate spectral analysis for a non-self-adjoint operator. In addition, in d = 7 and d = 9, we prove nonradial stability of the well-known ODE blow-up solution. Also, for the first time we establish persistence of regularity for the wave equation in similarity coordinates.

Keywords
nonlinear wave equation, supercritical, blowup, singularity formation, self-similar solution, stability
Mathematical Subject Classification
Primary: 35B44
Milestones
Received: 8 October 2021
Accepted: 11 July 2022
Published: 6 March 2024
Authors
Elek Csobo
Universität Innsbruck
Institut für Mathematik
Technikerstraße 13
6020 Innsbruck
Austria
Irfan Glogić
Faculty of Mathematics
University of Vienna
Vienna
Austria
Birgit Schörkhuber
Universität Innsbruck
Institut für Mathematik
Technikerstraße 13
6020 Innsbruck
Austria

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