Decay estimates for the critical semilinear wave equation

Hajer Bahouri; Jalal Shatah

Annales de l'I.H.P. Analyse non linéaire (1998)

  • Volume: 15, Issue: 6, page 783-789
  • ISSN: 0294-1449

How to cite


Bahouri, Hajer, and Shatah, Jalal. "Decay estimates for the critical semilinear wave equation." Annales de l'I.H.P. Analyse non linéaire 15.6 (1998): 783-789. <>.

author = {Bahouri, Hajer, Shatah, Jalal},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {global space-time estimate; Cauchy problem; finite energy solution; dilation identity},
language = {eng},
number = {6},
pages = {783-789},
publisher = {Gauthier-Villars},
title = {Decay estimates for the critical semilinear wave equation},
url = {},
volume = {15},
year = {1998},

AU - Bahouri, Hajer
AU - Shatah, Jalal
TI - Decay estimates for the critical semilinear wave equation
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 1998
PB - Gauthier-Villars
VL - 15
IS - 6
SP - 783
EP - 789
LA - eng
KW - global space-time estimate; Cauchy problem; finite energy solution; dilation identity
UR -
ER -


  1. [1] H. Bahouri, P. Gerard, Private communications. 
  2. [2] J. Ginibre, A. Soffer, G. Velo, The global Cauchy problem for the critical nonlinear wave equation, J. Func. Analysis, Vol. 110, 1992, pp. 96-130. Zbl0813.35054MR1190421
  3. [3] J. Ginibre, G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Zeit., Vol. 189, 1985, pp. 487-505. Zbl0549.35108MR786279
  4. [4] M. Grillakis, Regularity and asymptotic behavior of the wave equation with a critical nonlinearity, Ann. of Math., Vol. 132, 1990, pp. 485-509. Zbl0736.35067MR1078267
  5. [5] K. Jörgen, Das Anfangswertproblem in Grossen für eine Klasse nichtlinearer wellengleichunger, Math. Zeit., Vol. 77, 1961, pp. 295-308. Zbl0111.09105MR130462
  6. [6] J. Rauch, The u5 Klein-Gordon equation, Nonlinear PDE's and Applications. Pitman Research Notes in Math., Vol. 53, pp. 335-364. Zbl0473.35055MR631403
  7. [7] I.E. Segal, The global Cauchy problem for a relativistic scalar field with power interaction, Bull. Soc. Math. France, Vol. 91, 1963, pp. 129-135. Zbl0178.45403MR153967
  8. [8] J. Shatah, M. Struwe, Regularity results for nonlinear wave equation, Ann. of Math., Vol. 138, 1993, pp. 503-518. Zbl0836.35096MR1247991
  9. [9] J. Shatah, M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth, Internat. Math. Res. Notices, Vol. 7, 1994, pp. 303-309. Zbl0830.35086MR1283026
  10. [10] W. Strauss, Decay and asymptotics for ⊑u = F(u), J. Funct. Anal., Vol. 2, 1968, pp. 405-457. Zbl0182.13602MR233062
  11. [11] W. Strauss, Nonlinear invariant wave equations, Lecture Notes in Physics, Vol. 23, 1978, 197-249, Springer, New York. MR498955
  12. [12] W.A. Strauss, Nonlinear scattering theory of low energy, J. Funct. Anal., Vol. 41, 1981, pp. 110-133. Zbl0466.47006MR614228
  13. [13] R.S. Strichartz, Convolutions with kernels having singularities on a sphere, Trans. A.M.S., Vol. 148, 1970, pp. 461-471. Zbl0199.17502MR256219
  14. [14] R.S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., Vol. 44, 1977, pp. 705-714. Zbl0372.35001MR512086
  15. [15] M. Struwe, Globally regular solutions to the u5 Klein-Gordon equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4, Vol. 15, 1988, pp. 495-513. Zbl0728.35072MR1015805
  16. [16] C. Zuily, Solutions en grand temps d' equations d' ondes non linéaresSéminaire BouBaki, 46ème année, 1993–94, n° 779. Zbl0830.35088

Citations in EuDML Documents

  1. Matthew D. Blair, Hart F. Smith, Christopher D. Sogge, Strichartz estimates for the wave equation on manifolds with boundary
  2. Gilles Lebeau, Optique non linéaire et ondes sur critiques
  3. Slim Ibrahim, Mohamed Majdoub, Solutions globales de l’équation des ondes semi-linéaire critique à coefficients variables
  4. Carlos E. Kenig, Global Well-posedness, scattering and blow up for the energy-critical, focusing, non-linear Schrödinger and wave equations
  5. Mohamed Madjoub, Existence globale de solutions pour une équation des ondes semi-linéaire en deux dimensions d’espace
  6. Pierre Germain, Solutions globales d’énergie infinie pour l’équation des ondes critique

NotesEmbed ?


You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.


Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.