Finite-time blow-up for the real-valued defocusing energy-supercritical NLW

Abstract

We construct real-valued solutions of the defocusing energy-supercritical nonlinear wave equation ${\partial }_t^{2}u-\Delta u+u^p=0$ that blow up in finite time from smooth, compactly supported initial data. We consider three cases: for $p=3$ we construct blow-up solutions in all dimensions $d\ge 9$, for $p=5$ we treat all dimensions $d\ge 7$, while for $p=7$ we treat all dimensions $d\ge 6$. The solutions we construct here appear to be the first blow-up solutions of defocusing energy-supercritical nonlinear wave equations with smooth real-valued data. The main step is the construction of nonzero \emph{discretely self-similar profiles} that are periodic in self-similar time and real-analytic across the light cone. We find approximate profiles by numerical continuation from rotating self-similar solutions of the complex-valued equation and upgrade them to exact solutions by a computer-assisted Newton--Kantorovich argument in a Banach algebra of Fourier--Chebyshev coefficients. The main analytic ingredients are an exact description of the linear mode operators and uniform bounds for their inverses. The codes for the computer-assisted parts of the proofs, together with the approximate solutions, are available on GitHub \cite{code}.