On blowup for the supercritical quadratic wave equation

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.