Co-dimension one stable blowup for the supercritical cubic wave equation

Abstract

For the focusing cubic wave equation, we find an explicit, non-trivial self-similar blowup solution $u_T^{\ast }$, which is defined on the whole space and exists in all supercritical dimensions $d\ge 5$. For $d=7$, we analyze its stability properties without any symmetry assumptions and prove the existence of a set of perturbations which lead to blowup via $u_T^{\ast }$ in a backward light cone. Moreover, this set corresponds to a co-dimension one Lipschitz manifold modulo translation symmetries in similarity coordinates.

61 pages; v3 introduction expanded, several sections added for completion and clarification of the methods used, will appear in Advances in Mathematics