Scattering for radial, bounded solutions of focusing supercritical wave equations

Thomas Duyckaerts; Carlos Kenig; Frank Merle

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arXiv · 1208.2158

Scattering for radial, bounded solutions of focusing supercritical wave equations

Thomas Duyckaerts·Carlos Kenig·Frank Merle

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Abstract

In this paper, we consider the wave equation in space dimension 3 with an energy-supercritical, focusing nonlinearity. We show that any radial solution of the equation which is bounded in the critical Sobolev space is globally defined and scatters to a linear solution. As a consequence, finite time blow-up solutions have critical Sobolev norm converging to infinity (along some sequence of times). The proof relies on the compactness/rigidity method, pointwise estimates on compact solutions obtained by the two last authors, and channels of energy arguments used by the authors in previous works on the energy-critical equation.

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Thomas Duyckaerts, Carlos Kenig, Frank Merle. 2012-08-10. Scattering for radial, bounded solutions of focusing supercritical wave equations. https://arxiv.org/abs/1208.2158

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