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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The defocusing energy-supercritical nonlinear wave equation in three space dimensions


Authors: Rowan Killip and Monica Visan
Journal: Trans. Amer. Math. Soc. 363 (2011), 3893-3934
MSC (2010): Primary 35L71
Published electronically: January 28, 2011
MathSciNet review: 2775831
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Abstract: We consider the defocusing nonlinear wave equation $ u_{tt}-\Delta u + \vert u\vert^p u=0$ in the energy-supercritical regime $ p>4$. For even values of the power $ p$, we show that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that solutions with bounded critical Sobolev norm are global and scatter. The impetus to consider this problem comes from recent work of Kenig and Merle who treated the case of spherically-symmetric solutions.


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Additional Information

Rowan Killip
Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095-1555

Monica Visan
Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095-1555

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05400-0
PII: S 0002-9947(2011)05400-0
Received by editor(s): January 22, 2010
Received by editor(s) in revised form: June 14, 2010
Published electronically: January 28, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.