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

Killip, Rowan; Visan, Monica

doi:10.1090/S0002-9939-2010-10615-9

The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions
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by Rowan Killip and Monica Visan

Proc. Amer. Math. Soc. 139 (2011), 1805-1817

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

Published electronically: November 1, 2010

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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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