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Threshold Phenomenon for the Quintic Wave Equation in Three Dimensions

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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.

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Authors and Affiliations

  1. Bâtiment des Mathématiques, EPFL, Station 8, 1015, Lausanne, Switzerland

    Joachim Krieger

  2. Department of Mathematics, Kyoto University, 606-8502, Kyoto, Japan

    Kenji Nakanishi

  3. Department of Mathematics, The University of Chicago, 5734 South University Avenue, Chicago, IL, 60615, USA

    Wilhelm Schlag

Authors
  1. Joachim Krieger
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  2. Kenji Nakanishi
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  3. Wilhelm Schlag
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Corresponding author

Correspondence to Wilhelm Schlag.

Additional information

Communicated by P. Constantin

Support of the National Science Foundation DMS-0617854, DMS-1160817 for the third author, and the Swiss National Fund for the first author are gratefully acknowledged. The latter would like to thank the University of Chicago for its hospitality in August 2012.

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Krieger, J., Nakanishi, K. & Schlag, W. Threshold Phenomenon for the Quintic Wave Equation in Three Dimensions. Commun. Math. Phys. 327, 309–332 (2014). https://doi.org/10.1007/s00220-014-1900-9

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  • DOI: https://doi.org/10.1007/s00220-014-1900-9

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