On the energy critical focusing non-linear wave equation

Carlos E. Kenig; Frank Merle

Séminaire Équations aux dérivées partielles (2006-2007)

  • Volume: 166, Issue: 3, page 1-12

How to cite


Kenig, Carlos E., and Merle, Frank. "On the energy critical focusing non-linear wave equation." Séminaire Équations aux dérivées partielles 166.3 (2006-2007): 1-12. <http://eudml.org/doc/11161>.

author = {Kenig, Carlos E., Merle, Frank},
journal = {Séminaire Équations aux dérivées partielles},
keywords = {global well-posedness; blow-up; radial case; nonlinear Schrödinger equation; defocusing; focusing},
language = {eng},
number = {3},
pages = {1-12},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {On the energy critical focusing non-linear wave equation},
url = {http://eudml.org/doc/11161},
volume = {166},
year = {2006-2007},

AU - Kenig, Carlos E.
AU - Merle, Frank
TI - On the energy critical focusing non-linear wave equation
JO - Séminaire Équations aux dérivées partielles
PY - 2006-2007
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 166
IS - 3
SP - 1
EP - 12
LA - eng
KW - global well-posedness; blow-up; radial case; nonlinear Schrödinger equation; defocusing; focusing
UR - http://eudml.org/doc/11161
ER -


  1. C. Antonini and F. Merle, Optimal bounds on positive blow-up solutions for a semilinear wave equation, Internat. Math. Res. Notices 21 (2001), 1141–1167. Zbl0989.35090MR1861514
  2. N. Aronszajn, A. Krzywicki and J. Szarski, A unique continuation theorem for exterior differential forms on Riemannian manifolds, Ark. Mat. 4 (1962), 417–453. Zbl0107.07803MR140031
  3. T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant le courbure scalaire, J. Math. Pures Appl. (9), 55, 1976, 3, 269–296. Zbl0336.53033MR431287
  4. H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math 121 (1999), 131–175. Zbl0919.35089MR1705001
  5. H. Brézis and M. Marcus, Hardy’s inequalities revisited, Ann. Scuola Norm. Piza 25 (1997), 217–237. Zbl1011.46027
  6. M. Christ and M. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg–de Vries equation, J. Funct. Anal. 100 (1991), 87–109. Zbl0743.35067MR1124294
  7. P. Gérard, Description du défaut de compacité de l’injection de Sobolev, ESAIM Control Optim. Calc. Var. 3 (1998), 213–233. 
  8. Y. Giga and R. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math. 42 (1989), 223–241. Zbl0703.35020MR1003437
  9. J. Ginibre, A. Soffer and G. Velo, The global Cauchy problem for the critical nonlinear wave equation, J. Funct. Anal. 110 (1992), 96–130. Zbl0813.35054MR1190421
  10. J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133 (1995), 50–68. Zbl0849.35064MR1351643
  11. M. Grillakis, Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity, Ann. of Math. 132 (1990), 485–509. Zbl0736.35067MR1078267
  12. M. Grillakis, Regularity for the wave equation with a critical nonlinearity, Comm. Pure Appl. Math. 45 (1992), 749–774. Zbl0785.35065MR1162370
  13. L. Hörmander, “The analysis of linear partial differential operators III”, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1984. Zbl0601.35001
  14. D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math. 121 (1985), 463–494. Zbl0593.35119MR794370
  15. L. Kapitanski, Global and unique weak solutions of nonlinear wave equations, Math. Res. Lett., 1 (1994), no. 2, 211–223. Zbl0841.35067MR1266760
  16. C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy critical, focusing, non-linear Schrödinger equation, Preprint. Zbl1183.35202
  17. C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy critical, focusing, non-linear Schrödinger equation in the radical case, Invent. Math. 166 (2006), no. 3, 645–675. Zbl1115.35125MR2257393
  18. C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), 527–620. Zbl0808.35128MR1211741
  19. S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations 175 (2001), 353–392. Zbl1038.35119MR1855973
  20. J. Krieger and W. Schlag, On the focusing critical semi-linear wave equation, to appear, Amer. J. of Math. Zbl1219.35144MR2325106
  21. H. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form P u t t = - A u + ( u ) , Trans. Amer. Math. Soc. 192 (1974), 1–21. Zbl0288.35003MR344697
  22. H. Lindblad and C. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal. 130 (1995), 357–426. Zbl0846.35085MR1335386
  23. F. Merle, Existence of blow-up solutions in the energy space for the critical generalized KdV equation, J. Amer. Math. Soc. 14 (2001), 555–578. Zbl0970.35128MR1824989
  24. F. Merle and H. Zaag, Determination of the blow-up rate for the semilinear wave equation, Amer. J. of Math. 125 (2003), 1147–1164. Zbl1052.35043MR2004432
  25. F. Merle and H. Zaag, A Liouville theorem for vector-valued nonlinear heat equations and applications, Math. Ann. 316 (2000), no. 1, 103–137. Zbl0939.35086MR1735081
  26. L.E. Payne and D.H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22, (1975), 273–303. Zbl0317.35059MR402291
  27. H. Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z. 185 (1984), 261–270. Zbl0538.35063MR731347
  28. D.H. Sattinger, On global solutions of nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 30, (1968), 148–172. Zbl0159.39102MR227616
  29. J. Shatah and M. Struwe, Regularity results for nonlinear wave equations, Ann. of Math. 138 (1993), 503–518. Zbl0836.35096MR1247991
  30. J. Shatah and M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth, Internat. Math. Res. Notices 7 (1994), 303–309. Zbl0830.35086MR1283026
  31. J. Shatah and M. Struwe, “Geometric wave equations,” Courant Lecture Notes in Mathematics, 2 (1998). Zbl0993.35001
  32. C. Sogge, “Lectures on nonlinear wave equations,” Monographs in Analysis II, International Press, 1995. Zbl1089.35500
  33. G. Staffilani, On the generalized Korteweg-de Vries-type equations, Differential Integral Equations 10 (1997), 777–796. Zbl0891.35135MR1741772
  34. W. Strauss, “Nonlinear wave equations,” CBMS Regional Conference Series in Mathematics, 73, American Math. Soc., Providence, RI, 1989. Zbl0714.35003
  35. M. Struwe, Globally regular solutions to the u 5 Klein-Gordon equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 15 (1988), 495–513. Zbl0728.35072MR1015805
  36. G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372. Zbl0353.46018MR463908
  37. T. Tao, Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions, preprint, http://arxiv.org/abs/math.AP/0601164. MR2227039
  38. T. Tao and M. Visan, Stability of energy-critical nonlinear Schrödinger equations in high dimensions, Electron. J. Differential Equations 118 (2005), 28 pp. (electronic). Zbl1245.35122MR2174550
  39. M. Taylor, “Tools for PDE. Pseudodifferential operators, paradifferential operators and layer potentials,” Math. Surveys and Monographs 81, AMS, Providence RI 2000. Zbl0963.35211
  40. N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa 22 (1968), 265–274. Zbl0159.23801MR240748
  41. T. Wolff, Recent work on sharp estimates in second-order elliptic unique continuation problems, J. Geom. Anal. 3 (1993), 621–650. Zbl0787.35017MR1248088

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