# On blowup for the supercritical quadratic wave equation

Csobo, Elek; Glogić, Irfan; Schörkhuber, Birgit

## Abstract:

We study singularity formation for the focusing quadratic wave equation in the energy supercritical case, i.e., for $d \ge 7$. We find in closed form a new, non-trivial, radial, self-similar blowup solution $u^∗$ which exists for all d $d \ge 7$. 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 co-dimension one Lipschitz manifold modulo translation symmetries. In addition, in $d = 7$ and $d = 9$, we prove non-radial stability of the well-known ODE blowup solution. Also, for the first time we establish persistence of regularity for the wave equation in similarity coordinates.

 Zugehörige Institution(en) am KIT Institut für Analysis (IANA)Sonderforschungsbereich 1173 (SFB 1173) Publikationstyp Forschungsbericht/Preprint Publikationsmonat/-jahr 09.2021 Sprache Englisch Identifikator ISSN: 2365-662X KITopen-ID: 1000138775 Verlag Karlsruher Institut für Technologie (KIT) Umfang 63 S. Serie CRC 1173 Preprint ; 2021/40 Projektinformation SFB 1173/2 (DFG, DFG KOORD, SFB 1173/2 2019) Externe Relationen Siehe auch Schlagwörter nonlinear wave equation, singularity formation, self-similar solution
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