Vol. 9, No. 8, 2016

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Finite-time blowup for a supercritical defocusing nonlinear wave system

Terence Tao

Vol. 9 (2016), No. 8, 1999–2030

We consider the global regularity problem for defocusing nonlinear wave systems

u = (mF)(u)

on Minkowski spacetime 1+d with d’Alembertian := t2 + i=1dxi2, where the field u : 1+d m is vector-valued, and F : m 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.

nonlinear wave equation, Nash embedding theorem
Mathematical Subject Classification 2010
Primary: 35Q30, 35L71
Secondary: 35L67
Received: 24 February 2016
Revised: 6 August 2016
Accepted: 29 September 2016
Published: 11 December 2016
Terence Tao
Department of Mathematics
Los Angeles, CA 90095-1555
United States