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

DOI:
https://doi.org/10.1090/S0002-9939-2010-10615-9

Published electronically:
November 1, 2010

MathSciNet review:
2763767

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.