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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions


Authors: Rowan Killip and Monica Visan
Journal: Proc. Amer. Math. Soc. 139 (2011), 1805-1817
MSC (2010): Primary 35L71
Published electronically: November 1, 2010
MathSciNet review: 2763767
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Abstract: We consider the defocusing nonlinear wave equation $ u_{tt}-\Delta u + \vert u\vert^p u=0$ with spherically-symmetric initial data in the regime $ \frac4{d-2}<p<\frac4{d-3}$ (which is energy-supercritical) and dimensions $ 3\leq d\leq 6$; we also consider $ d\geq 7$, but for a smaller range of $ p>\frac4{d-2}$. The principal result is that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that maximal-lifespan solutions with bounded critical Sobolev norm are global and scatter.


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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/S0002-9939-2010-10615-9
Received by editor(s): February 8, 2010
Received by editor(s) in revised form: May 26, 2010
Published electronically: November 1, 2010
Additional Notes: The first author was supported by NSF grant DMS-0701085
The second author was supported by NSF grant DMS-0901166
Communicated by: Hart F. Smith
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.