Finite-time blowup for a supercritical defocusing nonlinear wave system

Abstract

We consider the global regularity problem for defocusing nonlinear wave systems

$$\square u=\left({\nabla }_{{ℝ}^m}F\right)\left(u\right)$$

on Minkowski spacetime ${ℝ}^{1+d}$ with d’Alembertian $\square :=-{\partial }_t^2+{\sum }_{i=1}^d{\partial }_{x_i}^2$, where the field $u:{ℝ}^{1+d}\to {ℝ}^m$ is vector-valued, and $F:{ℝ}^m\to ℝ$ is a smooth potential which is positive and homogeneous of order $p+1$ outside of the unit ball for some $p>1$. This generalises the scalar defocusing nonlinear wave (NLW) equation, in which $m=1$ and $F\left(v\right)=1∕\left(p+1\right)|v{|}^{p+1}$. It is well known that in the energy-subcritical and energy-critical cases when $d\le 2$ or $d\ge 3$ and $p\le 1+4∕\left(d-2\right)$, one has global existence of smooth solutions from arbitrary smooth initial data $u\left(0\right),{\partial }_{t}u\left(0\right)$, at least for dimensions $d\le 7$. We study the supercritical case where $d=3$ and $p>5$. We show that in this case, there exists a smooth potential $F$ for some sufficiently large $m$ (in fact we can take $m=40$), positive and homogeneous of order $p+1$ outside of the unit ball, and a smooth choice of initial data $u\left(0\right),{\partial }_{t}u\left(0\right)$ for which the solution develops a finite-time singularity. In fact the solution is discretely self-similar in a backwards light cone. The basic strategy is to first select the mass and energy densities of $u$, then $u$ itself, and then finally design the potential $F$ in order to solve the required equation. The Nash embedding theorem is used in the second step, explaining the need to take $m$ relatively large.