## Analysis & PDE

• Anal. PDE
• Volume 9, Number 8 (2016), 1999-2030.

### Finite-time blowup for a supercritical defocusing nonlinear wave system

Terence Tao

#### Abstract

We consider the global regularity problem for defocusing nonlinear wave systems

$□u = (∇ℝmF)(u)$

on Minkowski spacetime $ℝ1+d$ with d’Alembertian $□ := −∂t2 + ∑ i=1d∂xi2$, where the field $u : ℝ1+d → ℝm$ is vector-valued, and $F : ℝm → ℝ$ 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 $F(v) = 1∕(p + 1)|v|p+1$. It is well known that in the energy-subcritical and energy-critical cases when $d ≤ 2$ or $d ≥ 3$ and $p ≤ 1 + 4∕(d − 2)$, one has global existence of smooth solutions from arbitrary smooth initial data $u(0),∂tu(0)$, at least for dimensions $d ≤ 7$. 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 $u(0),∂tu(0)$ 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.

#### Article information

Source
Anal. PDE, Volume 9, Number 8 (2016), 1999-2030.

Dates
Revised: 6 August 2016
Accepted: 29 September 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.apde/1510843381

Digital Object Identifier
doi:10.2140/apde.2016.9.1999

Mathematical Reviews number (MathSciNet)
MR3599524

Zentralblatt MATH identifier
1365.35111

#### Citation

Tao, Terence. Finite-time blowup for a supercritical defocusing nonlinear wave system. Anal. PDE 9 (2016), no. 8, 1999--2030. doi:10.2140/apde.2016.9.1999. https://projecteuclid.org/euclid.apde/1510843381

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