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Profiles for the Radial Focusing 4d Energy-Critical Wave Equation

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Abstract

Consider a finite energy radial solution to the focusing energy critical semilinear wave equation in 1 + 4 dimensions. Assume that this solution exhibits type-II behavior, by which we mean that the critical Sobolev norm of the evolution stays bounded on the maximal interval of existence. We prove that along a sequence of times tending to the maximal forward time of existence, the solution decomposes into a sum of dynamically rescaled solitons, a free radiation term, and an error tending to zero in the energy space. If, in addition, we assume that the critical norm of the evolution localized to the light cone (the forward light cone in the case of global solutions and the backwards cone in the case of finite time blow-up) is less than 2 times the critical norm of the ground state solution W, then the decomposition holds without a restriction to a subsequence.

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Authors and Affiliations

  1. Institut de Recherche Mathématique Avancée UMR 7501, Université de Strasbourg, 7 rue René-Descartes, 67084, Strasbourg Cedex, France

    R. Côte

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

    C. E. Kenig & W. Schlag

  3. Department of Mathematics, MIT, 77 Massachusetts Ave 2-267, Cambridge, MA, 02139, USA

    A. Lawrie

Authors
  1. R. Côte
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  2. C. E. Kenig
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  3. A. Lawrie
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  4. W. Schlag
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Corresponding author

Correspondence to W. Schlag.

Additional information

Communicated by H. T. Yau

The first author gratefully acknowledges Support of the European Research Council through the ERC Advanced Grant No. 291214, BLOWDISOL. Support of the National Science Foundation DMS-0968472 and DMS-1265249 for the second author, DMS-1302782 for the third author, and DMS-0617854, DMS-1160817 for the fourth author is gratefully acknowledged.

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Côte, R., Kenig, C.E., Lawrie, A. et al. Profiles for the Radial Focusing 4d Energy-Critical Wave Equation. Commun. Math. Phys. 357, 943–1008 (2018). https://doi.org/10.1007/s00220-017-3043-2

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