Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation

We study the energy-critical focusing non-linear wave equation, with data in the energy space, in dimensions 3, 4 and 5. We prove that for Cauchy data of energy smaller than the one of the static solution W which gives the best constant in the Sobolev embedding, the following alternative holds. If t...

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Bibliographic Details
Published in:Acta mathematica 2008-12, Vol.201 (2), p.147-212
Main Authors: Kenig, Carlos E., Merle, Frank
Format: Article
Language:English
Subjects:
Cauchy problems
Energy
Estimates
Exact sciences and technology
General mathematics
General, history and biography
Linear equations
Mathematics
Mathematics and Statistics
Sciences and techniques of general use
Studies
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Summary:We study the energy-critical focusing non-linear wave equation, with data in the energy space, in dimensions 3, 4 and 5. We prove that for Cauchy data of energy smaller than the one of the static solution W which gives the best constant in the Sobolev embedding, the following alternative holds. If the initial data has smaller norm in the homogeneous Sobolev space H 1 than the one of W , then we have global well-posedness and scattering. If the norm is larger than the one of W , then we have break-down in finite time.
ISSN:0001-5962
1871-2509
DOI:10.1007/s11511-008-0031-6