On blowup for the supercritical quadratic wave equation

We study singularity formation for the quadratic wave equation in the energy supercritical case, i.e., for $d\ge 7$. We find in closed form a new, nontrivial, radial, self-similar blow-up solution $u^{\ast }$ which exists for all $d\ge 7$. For $d=9$, we study the stability of $u^{\ast }$ without any symmetry assumptions on the initial data and show that there is a family of perturbations which lead to blowup via $u^{\ast }$. 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.