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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions
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by Rowan Killip and Monica Visan PDF
Proc. Amer. Math. Soc. 139 (2011), 1805-1817 Request permission

Abstract:

We consider the defocusing nonlinear wave equation $u_{tt}-\Delta u + |u|^p u=0$ with spherically-symmetric initial data in the regime $\frac 4{d-2}<p<\frac 4{d-3}$ (which is energy-supercritical) and dimensions $3\leq d\leq 6$; we also consider $d\geq 7$, but for a smaller range of $p>\frac 4{d-2}$. 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
  • 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
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 139 (2011), 1805-1817
  • MSC (2010): Primary 35L71
  • DOI: https://doi.org/10.1090/S0002-9939-2010-10615-9
  • MathSciNet review: 2763767