Advertisement

Communications in Mathematical Physics

, Volume 327, Issue 1, pp 309–332 | Cite as

Threshold Phenomenon for the Quintic Wave Equation in Three Dimensions

  • Joachim Krieger
  • Kenji Nakanishi
  • Wilhelm SchlagEmail author
Article

Abstract

For the critical focusing wave equation \({\square u = u^5 \, {\rm on} \, \mathbb{R}^{3+1}}\) in the radial case, we establish the role of the “center stable” manifold \({\Sigma}\) constructed in Krieger and Schlag (Am J Math 129(3):843–913, 2007) near the ground state (W, 0) as a threshold between blowup and scattering to zero, establishing a conjecture going back to numerical work by Bizoń et al. (Nonlinearity 17(6):2187–2201, 2004). The underlying topology is stronger than the energy norm.

Keywords

Wave Equation Invariant Manifold Nonlinear Wave Equation Limit Absorption Principle Bootstrap Assumption 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bahouri H., Gérard P.: High frequency approximation of solutions to critical nonlinear wave equations. Am. J. Math. 121(1), 131–175 (1999)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bates, P.W., Jones, C.K.R.T.: Invariant manifolds for semilinear partial differential equations. In: Dynamics reported, Vol. 2, Dynam. Report. Ser. Dynam. Systems Appl., Vol. 2, Chichester: Wiley, 1989, pp. 1–38Google Scholar
  3. 3.
    Bizoń P., Chmaj T., Tabor Z.: On blowup for semilinear wave equations with a focusing nonlinearity. Nonlinearity 17(6), 2187–2201 (2004)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Donninger R., Krieger J.: Nonscattering solutions and blow up at infinity for the critical wave equation. Mathematische Annaten 357(1), 89–163 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Duyckaerts T., Kenig C., Merle F.: Universality of blow-up profile for small radial type II blow-up solutions of energy-critical wave equation. J. Eur. Math. Soc. 13(3), 533–599 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Duyckaerts T., Kenig C., Merle F.: Universality of the blow-up profile for small type II blow-up solutions of energy-critical wave equation: the non-radial case. J. Eur. Math. Soc. 14(5), 1389–1454 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Duyckaerts T., Kenig C., Merle F.: Profiles of bounded radial solutions of the focusing, energy-critical wave equation. Geom. Funct. Anal. 22(3), 639–698 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Duyckaerts, T., Kenig, C., Merle, F.: Classification of radial solutions of the focusing, energy-critical wave equation, preprint, arXiv:1204.0031v1 (2012)
  9. 9.
    Duyckaerts T., Merle F.: Dynamic of threshold solutions for energy-critical NLS. Geom. Funct. Anal. 18(6), 1787–1840 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Duyckaerts, T., Merle, F.: Dynamic of threshold solutions for energy-critical wave equation. Int. Math. Res. Pap. IMRP (2008)Google Scholar
  11. 11.
    Hillairet M., Raphaël P.: Smooth type II blow up solutions to the four dimensional energy critical wave equation. Anal. PDE 5(4), 777–829 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Ibrahim S., Masmoudi N., Nakanishi K.: Scattering threshold for the focusing nonlinear Klein–Gordon equation. Anal. PDE 4(3), 405–460 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Karageorgis P., Strauss W.: Instability of steady states for nonlinear wave and heat equations. J. Differ. Equ. 241(1), 184–205 (2007)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Kenig C., Merle F.: Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear Schrödinger equation in the radial case. Invent. Math. 166(3), 645–675 (2006)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Kenig C., Merle F.: Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation. Acta Math. 201(2), 147–212 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Krieger, J., Nakanishi, K., Schlag, W.: Global dynamics away from the ground state for the energy-critical nonlinear wave equation. Am. J. Math. (2014)Google Scholar
  17. 17.
    Krieger, J., Nakanishi, K., Schlag, W.: Global dynamics of the nonradial energy-critical wave equation above the ground state energy. Disc. Cont. Dyn. Syst. A (2014)Google Scholar
  18. 18.
    Krieger, J., Nakanishi, K., Schlag, W.: Center-Stable manifold of the ground state in the energy space for the critical wave equation, preprint (2013)Google Scholar
  19. 19.
    Krieger J., Schlag W.: On the focusing critical semi-linear wave equation. Am. J. Math. 129(3), 843–913 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Krieger J., Schlag W., Tataru D.: Slow blow-up solutions for the \({H^1(\mathbb{R}^3)}\) critical focusing semilinear wave equation. Duke Math. J. 147(1), 1–53 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Nakanishi K., Schlag W.: Global dynamics above the ground state energy for the focusing nonlinear Klein–Gordon equation. J. Differ. Equ. 250, 2299–2233 (2011)Google Scholar
  22. 22.
    Nakanishi K., Schlag W.: Global dynamics above the ground state energy for the cubic NLS equation in 3D. Calc. Var. PDE 44(1–2), 1–45 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Nakanishi K., Schlag W.: Global dynamics above the ground state for the nonlinear Klein–Gordon equation without a radial assumption. Arch. Rational Mech. Anal. 203(3), 809–851 (2012)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Nakanishi, K., Schlag, W.: Invariant manifolds and dispersive Hamiltonian evolution equations. In: Zürich Lectures in Advanced Mathematics, EMS, 2011Google Scholar
  25. 25.
    Palmer K.: Linearization near an integral manifold. J. Math. Anal. Appl. 51, 243–255 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Payne L.E., Sattinger D.H.: Saddle points and instability of nonlinear hyperbolic equations. Israel J. Math. 22(3–4), 273–303 (1975)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Shoshitaishvili A.N.: Bifurcations of topological type of singular points of vector fields that depend on parameters. Funkcional. Anal. i Prilozen. 6(2), 97–98 (1972)MathSciNetGoogle Scholar
  28. 28.
    Shoshitaishvili A.N.: The bifurcation of the topological type of the singular points of vector fields that depend on parameters. Trudy Sem. Petrovsk. Vyp. 1, 279–309 (1975)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Joachim Krieger
    • 1
  • Kenji Nakanishi
    • 2
  • Wilhelm Schlag
    • 3
    Email author
  1. 1.Bâtiment des Mathématiques, EPFLLausanneSwitzerland
  2. 2.Department of MathematicsKyoto UniversityKyotoJapan
  3. 3.Department of MathematicsThe University of ChicagoChicagoUSA

Personalised recommendations