On blowup for the supercritical quadratic wave equation

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 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 D 7 and d 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. 1. Introduction 617 2. The stability problem in similarity coordinates 625 3. The free wave evolution in similarity variables 629 4. Linearization around a self-similar solution 635 5. Spectral analysis for perturbations around U a 639 6. Perturbations around U a 651 7. Nonlinear theory 654 8. Proof of Theorem 1.6 670 Appendix. Proof of Lemma 3.4 674 References 678 Glogić is supported by the Austrian Science Fund FWF Projects P 34378 and P 30076.