Openness of the Set of Non-characteristic Points and Regularity of the Blow-up Curve for the 1 D Semilinear Wave Equation

  • Frank Merle
  • Hatem ZaagEmail author


We consider here the 1 D semilinear wave equation with a power nonlinearity and with no restriction on initial data. We first prove a Liouville Theorem for that equation. Then, we consider a blow-up solution, its blow-up curve \({x\mapsto T(x)}\) and \({I_0\subset \mathbb{R}}\) the set of non-characteristic points. We show that I 0 is open and that T(x) is C 1 on I 0. All these results fundamentally use our previous result in [19] showing the convergence in selfsimilar variables for \({x\in I_0}\) .


Characteristic Point Nonlinear Wave Equation Liouville Theorem Nonlinear Heat Equation Energy Argument 
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  1. 1.
    Alinhac, S.: Blowup for nonlinear hyperbolic equations. Volume 17 of Progress in Nonlinear Differential Equations and their Applications. Boston, MA: Birkhäuser Boston Inc. 1995Google Scholar
  2. 2.
    Alinhac, S.: A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations. In: Journées “Équations aux Dérivées Partielles” (Forges-les-Eaux, 2002), Nantes:Univ. Nantes, 2002 pp. Exp. No. I, 33Google Scholar
  3. 3.
    Antonini C., Merle F.: Optimal bounds on positive blow-up solutions for a semilinear wave equation. Internat. Math. Res. Notices. 21, 1141–1167 (2001)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Caffarelli L.A., Friedman A.: Differentiability of the blow-up curve for one-dimensional nonlinear wave equations. Arch. Rati. Mech. Anal. 91(1), 83–98 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Caffarelli L.A., Friedman. A.: The blow-up boundary for nonlinear wave equations. Trans. Amer. Math. Soc. 297(1), 223–241 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Ginibre J., Soffer A., Velo G.: The global Cauchy problem for the critical nonlinear wave equation. J. Funct. Anal. 110(1), 96–130 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kichenassamy S., Littman W.: Blow-up surfaces for nonlinear wave equations. I. Comm. Partial Differe. Eq. 18(3-4), 431–452 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Kichenassamy S., Littman W.: Blow-up surfaces for nonlinear wave equations. II. Comm. Partial Differe. Eq. 18(11), 1869–1899 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Levine H.A.: Instability and nonexistence of global solutions to nonlinear wave equations of the form \({Pu\sb{tt}=-Au+{\cal F}(u)}\). Trans. Amer.Math. Soc. 192, 1–21 (1974) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Martel Y., Merle F.: A Liouville theorem for the critical generalized Korteweg-de Vries equation. J. Math. Pures Appl. (9). 79(4), 339–425 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Martel Y., Merle F.: Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation. Ann. of Math. (2). 155(1), 235–280 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Merle F., Raphael P.: On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation. Invent. Math. 156(3), 565–672 (2004)zbMATHCrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Merle F., Raphael P.: The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Annals of Math (2). 161((1), 157–222 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Merle F., Zaag H.: Optimal estimates for blowup rate and behavior for nonlinear heat equations. Comm. Pure Appl. Math. 51(2), 139–196 (1998)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Merle F., Zaag H.: A Liouville theorem for vector-valued nonlinear heat equations and applications. Math. Ann. 316(1), 103–137 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Merle F., Zaag H.: Determination of the blow-up rate for the semilinear wave equation. Amer. J. Math. 125, 1147–1164 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Merle F., Zaag H.: Blow-up rate near the blow-up surface for semilinear wave equations. Internat. Math. Res. Notices. 19, 1127–1156 (2005)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Merle F., Zaag H.: Determination of the blow-up rate for a critical semilinear wave equation. Math. Annalen. 331(2), 395–416 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Merle F., Zaag H.: Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension. J. Funct. Anal. 253(1), 43–121 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Merle, F., Zaag, H.: Existence and characterization of characteristic points for a semilinear wave equation in one space dimension. In preparation (2008)Google Scholar
  21. 21.
    Nouaili, N.: C 1, α regularity of the blow-up curve at non-characteristic points for the one dimensional semilinear wave equation. Comm. Partial Diff. Eq. (2008) (to appear)Google Scholar
  22. 22.
    Zaag H.: On the regularity of the blow-up set for semilinear heat equations. Ann. Inst. H. Poincaré Anal. Non Linéaire. 19(5), 505–542 (2002)zbMATHCrossRefMathSciNetADSGoogle Scholar
  23. 23.
    Zaag H.: One dimensional behavior of singular N dimensional solutions of semilinear heat equations. Comm. Math. Phys. 225(3), 523–549 (2002)zbMATHCrossRefADSMathSciNetGoogle Scholar
  24. 24.
    Zaag, H.: Regularity of the blow-up set and singular behavior for semilinear heat equations. In: Mathematics & mathematics education (Bethlehem, 2000), River Edge, NJ: World Sci. Publishing, (2002) pp 337–347Google Scholar
  25. 25.
    Zaag H.: Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation. Duke Math. J. 133(3), 499–525 (2006)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Département de mathématiquesUniversité de Cergy Pontoise, IHES and CNRSCergy Pontoise cedexFrance
  2. 2.Université Paris 13, Institut Galilée, Laboratoire Analyse, Géométrie et Applications, CNRS UMR 7539VilletaneuseFrance

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