Skip to main content
SpringerLink
Log in

Determination of the blow-up rate for a critical semilinear wave equation

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract.

In this paper, we determine the blow-up rate for the semilinear wave equation with critical power nonlinearity related to the conformal invariance.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alinhac, S.: Blowup for nonlinear hyperbolic equations, vol. 17 Progress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston Inc., Boston, MA, 1995

  2. Antonini, C., Merle, F.: Optimal bounds on positive blow-up solutions for a semilinear wave equation. Internat. Math. Res. Notices 21, 1141–1167 (2001)

    Article  MATH  Google Scholar 

  3. Caffarelli, L.A., Friedman, A.: The blow-up boundary for nonlinear wave equations. Trans. Am. Math. Soc. 297(1), 223–241 (1986)

    MATH  Google Scholar 

  4. Filippas, S., Herrero, M.A., Velázquez, J.J.L.: Fast blow-up mechanisms for sign-changing solutions of a semilinear parabolic equation with critical nonlinearity. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456(2004), 2957–2982 (2000)

    MATH  Google Scholar 

  5. Giga, Y., Kohn, R.V.: Nondegeneracy of blowup for semilinear heat equations. Comm. Pure Appl. Math. 42(6), 845–884 (1989)

    MATH  Google Scholar 

  6. Herrero, M.A., Velázquez, J.J.L.: Blow-up behaviour of one-dimensional semilinear parabolic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 10(2), 131–189 (1993)

    MATH  Google Scholar 

  7. John, F.: Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscripta Math. 28(1–3), 235–268 (1979)

    Google Scholar 

  8. Kichenassamy, S., Littman, W.: Blow-up surfaces for nonlinear wave equations. I. Comm. Partial Diff. Eqs. 18(3–4), 431–452 (1993)

    Google Scholar 

  9. Kichenassamy, S., Littman, W.: Blow-up surfaces for nonlinear wave equations. II. Comm. Partial Diff. Eqs. 18(11), 1869–1899 (1993)

    MATH  Google Scholar 

  10. Lindblad, H., Sogge, C.D.: On existence and scattering with minimal regularity for semilinear wave equations. J. Funct. Anal. 130(2), 357–426 (1995)

    Article  MATH  Google Scholar 

  11. Merle, F., Raphaël, P.: On universality of blow-up profile for L2 critical nonlinear Schrödinger equation. Invent. Math. 156, 565–672 (2004)

    Article  MathSciNet  Google Scholar 

  12. Merle, F., Raphaël, P.: Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation. Geom. Funct. Anal. 13(3), 591–642 (2003)

    Article  MATH  Google Scholar 

  13. Merle, F., Raphaël, P.: Blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Anal. Math. To appear 2004

  14. Merle, F., Zaag, H.: Blow-up rate near the blow-up curve for semilinear wave equations. In preparation

  15. Merle, F., Zaag, H.: A Liouville theorem for vector-valued nonlinear heat equations and applications. Math. Annalen 316(1), 103–137 (2000)

    Article  MATH  Google Scholar 

  16. Merle, F., Zaag, H.: Determination of the blow-up rate for the semilinear wave equation. Amer. J. Math. 125, 1147–1164 (2003)

    MATH  Google Scholar 

  17. Shatah, J., Struwe, M.: Geometric wave equations. New York University Courant Institute of Mathematical Sciences, New York, 1998

Download references

Author information

Authors and Affiliations

  1. Départment de mathématiques, Université de Cergy Pontoise, 2 avenue Adolphe Chauvin, BP 222, 95302, Cergy Pontoise cedex, France

    Frank Merle

  2. Département de mathématiques et applications, CNRS UMR 8553, École Normal Supérieure, 45 rue d’Ulm, 75005, Paris, France

    Hatem Zaag

Authors
  1. Frank Merle
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hatem Zaag
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hatem Zaag.

Additional information

Mathematics Subject classification (2000): 35L05, 35L67

Membre de l’Institut Universitaire de France

Rights and permissions

Reprints and permissions

About this article

Cite this article

Merle, F., Zaag, H. Determination of the blow-up rate for a critical semilinear wave equation. Math. Ann. 331, 395–416 (2005). https://doi.org/10.1007/s00208-004-0587-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-004-0587-1

Keywords

Search

Navigation