Revista Matemática Iberoamericana

On global solutions to a defocusing semi-linear wave equation

Isabelle Gallagher and Fabrice Planchon

Full-text: Open access


We prove that the 3D cubic defocusing semi-linear wave equation is globally well-posed for data in the Sobolev space $\dot{H}^{s}$ where $s>3/4$. This result was obtained in [Kenig-Ponce-Vega, 2000] following Bourgain's method ([Bourgain, 1998]). We present here a different and somewhat simpler argument, inspired by previous work on the Navier-Stokes equations ([Calderon, 1990], [Gallagher-Planchon, 2002])

Article information

Rev. Mat. Iberoamericana Volume 19, Number 1 (2003), 161-177.

First available in Project Euclid: 31 March 2003

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35L70: Nonlinear second-order hyperbolic equations 35L05: Wave equation

Wave equation Global solution


Gallagher, Isabelle; Planchon, Fabrice. On global solutions to a defocusing semi-linear wave equation. Rev. Mat. Iberoamericana 19 (2003), no. 1, 161--177.

Export citation


  • Bahouri, Hajer and Gérard, Patrick: High frequency approximation of solutions to critical nonlinear wave equations. Amer. J. Math. 121 (1999), no. 1, 131--175.
  • Bony, Jean-Michel: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 2, 209--246.
  • Bourgain, Jean: Refinements of Strichartz' inequality and applications to $2$D-NLS with critical nonlinearity. Internat. Math. Res. Notices 5 (1998), 253--283.
  • Calderón, Calixto P.: Existence of weak solutions for the Navier-Stokes equations with initial data in $L\sp p$. Trans. Amer. Math. Soc. 318 (1990), no. 1, 179--200.
  • Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao, T.: Global well-posedness for the schrodinger equations with derivative. SIAM J. Math. Anal. 33 (2001), no. 3, 649--669.
  • Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao, T.: Global well-posedness for KdV in Sobolev spaces of negative index. Electron. J. Differential Equations 2001, no. 26, 7 pp. (electronic).
  • Gallagher, Isabelle and Planchon, Fabrice: On infinite energy solutions to the Navier-Stokes equations: global 2D existence and 3D weak-strong uniqueness. Arch. Rat. Mech. An. 161 (2002), no. 4, 307--337.
  • Ginibre, J. and Velo, G.: The global Cauchy problem for the nonlinear Klein-Gordon equation. Math. Z. 189 (1985), no. 4, 487--505.
  • Ginibre, Jean and Velo, Giorgio: Generalized Strichartz inequalities for the wave equation. J. Funct. Anal. 133 (1995), no. 1, 50--68.
  • Jörgens, Konrad: Das Anfangswertproblem im Grossen für eine Klasse nichtlinearer Wellengleichungen. Math. Z. 77 (1961), 295--308.
  • Kenig, Carlos E., Ponce, Gustavo and Vega, Luis: Global well-posedness for semi-linear wave equations. Comm. Partial Differential Equations 25 (2000), no. 9-10, 1741--1752.
  • Kenig, Carlos E., Ponce, Gustavo and Vega, Luis: On the ill-posedness of some canonical dispersive equations. Duke Math. J. 106 (2001), no. 3, 617--633.
  • Klainerman, Sergiu and Tataru, Daniel: On the optimal local regularity for Yang-Mills equations in $\mathbbR\sp 4+1$. J. Amer. Math. Soc. 12 (1999), no. 1, 93--116.
  • Lindblad, Hans and Sogge, Christopher D.: On existence and scattering with minimal regularity for semilinear wave equations. J. Funct. Anal. 130 (1995), no. 2, 357--426.
  • Planchon, Fabrice: Self-similar solutions and semi-linear wave equations in Besov spaces. J. Math. Pures Appl. (9) 79 (2000), no. 8, 809--820.
  • Planchon, Fabrice: On self-similar solutions, well-posedness and the conformal wave equation. Commun. Contemp. Math. 4 (2002), 211--222.