Analysis & PDE

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

Finite-time blowup for a supercritical defocusing nonlinear wave system

Terence Tao

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

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

Received: 24 February 2016
Revised: 6 August 2016
Accepted: 29 September 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10] 35L71: Semilinear second-order hyperbolic equations
Secondary: 35L67: Shocks and singularities [See also 58Kxx, 76L05]

nonlinear wave equation Nash embedding theorem


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.

Export citation


  • P. Brenner and P. Kumlin, “On wave equations with supercritical nonlinearities”, Arch. Math. $($Basel$)$ 74:2 (2000), 129–147.
  • P. Brenner and W. von Wahl, “Global classical solutions of nonlinear wave equations”, Math. Z. 176:1 (1981), 87–121.
  • N. Burq, S. Ibrahim, and P. Gérard, “Instability results for nonlinear Schrödinger and wave equations”, preprint, 2007.
  • T. Cazenave, J. Shatah, and A. S. Tahvildar-Zadeh, “Harmonic maps of the hyperbolic space and development of singularities in wave maps and Yang–Mills fields”, Ann. Inst. H. Poincaré Phys. Théor. 68:3 (1998), 315–349.
  • M. Christ, J. Colliander, and T. Tao, “Ill-posedness for nonlinear Schrodinger and wave equations”, preprint, 2003.
  • M. G. Grillakis, “Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity”, Ann. of Math. $(2)$ 132:3 (1990), 485–509.
  • M. G. Grillakis, “Regularity for the wave equation with a critical nonlinearity”, Comm. Pure Appl. Math. 45:6 (1992), 749–774.
  • M. Günther, “Isometric embeddings of Riemannian manifolds”, pp. 1137–1143 in Proceedings of the International Congress of Mathematicians, Vol. II (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991.
  • S. Ibrahim, M. Majdoub, and N. Masmoudi, “Well- and ill-posedness issues for energy supercritical waves”, Anal. PDE 4:2 (2011), 341–367.
  • K. J örgens, “Das Anfangswertproblem im Grossen für eine Klasse nichtlinearer Wellengleichungen”, Math. Z. 77 (1961), 295–308.
  • C. E. Kenig and F. Merle, “Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation”, Acta Math. 201:2 (2008), 147–212.
  • R. Killip and M. Visan, “The defocusing energy-supercritical nonlinear wave equation in three space dimensions”, Trans. Amer. Math. Soc. 363:7 (2011), 3893–3934.
  • R. Killip and M. Visan, “The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions”, Proc. Amer. Math. Soc. 139:5 (2011), 1805–1817.
  • G. Lebeau, “Non linear optic and supercritical wave equation”, Bull. Soc. Roy. Sci. Liège 70:4–6 (2001), 267–306.
  • G. Lebeau, “Perte de régularité pour les équations d'ondes sur-critiques”, Bull. Soc. Math. France 133:1 (2005), 145–157.
  • H. Lindblad and C. D. Sogge, “Long-time existence for small amplitude semilinear wave equations”, Amer. J. Math. 118:5 (1996), 1047–1135.
  • F. Planchon, “Self-similar solutions and semi-linear wave equations in Besov spaces”, J. Math. Pures Appl. $(9)$ 79:8 (2000), 809–820.
  • F. Ribaud and A. Youssfi, “Global solutions and self-similar solutions of semilinear wave equation”, Math. Z. 239:2 (2002), 231–262.
  • T. Roy, “Global existence of smooth solutions of a 3D log-log energy-supercritical wave equation”, Anal. PDE 2:3 (2009), 261–280.
  • R. T. Seeley, “Extension of $C\sp{\infty }$ functions defined in a half space”, Proc. Amer. Math. Soc. 15:4 (1964), 625–626.
  • I. E. Segal, “The global Cauchy problem for a relativistic scalar field with power interaction”, Bull. Soc. Math. France 91 (1963), 129–135.
  • J. Shatah and M. Struwe, Geometric wave equations, Courant Lecture Notes in Mathematics 2, New York University, Courant Institute of Mathematical Sciences, New York, 1998.
  • W. A. Strauss, Nonlinear wave equations, CBMS Regional Conference Series in Mathematics 73, American Mathematical Society, Providence, RI, 1989.
  • M. Struwe, “Globally regular solutions to the $\smash{u\sp 5}$ Klein–Gordon equation”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. $(4)$ 15:3 (1988), 495–513.
  • T. Tao, Nonlinear dispersive equations: local and global analysis, CBMS Regional Conference Series in Mathematics 106, American Mathematical Society, Providence, RI, 2006.
  • T. Tao, “Global regularity for a logarithmically supercritical defocusing nonlinear wave equation for spherically symmetric data”, J. Hyperbolic Differ. Equ. 4:2 (2007), 259–265.
  • Y. X. Zheng, “Concentration in sequences of solutions to the nonlinear Klein–Gordon equation”, Indiana Univ. Math. J. 40:1 (1991), 201–235.