Scattering for radial, bounded solutions of focusing supercritical wave equations

By Thomas Duyckaerts, Carlos E. Kenig, F. Merle. 2012

114 citations recorded by the source corpus.

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.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ff38254c45da43ea77d638603d5a4567220efd84

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In Mathematical Physics

1. B. Dodson
2. A. Lawrie
3. Hao Jia
4. Carlos E. Kenig
5. Aynur Bulut
6. Raphael Cote
7. Gunther Uhlmann
8. F. Merle

Citations in the corpus

• Scattering for the radial 3D cubic wave equation (2014, methodology): “MR 3272053 Zbl 06266536 [Duyckaerts et al. 2014] T. Duyckaerts, C. E. Kenig, and F. Merle, “Scattering for radial, bounded solutions of focusing supercritical wave equations”, Int.”
• Asymptotic decomposition for semilinear Wave and equivariant wave map equations (2015, background): “ for the associated linear wave equation, and have since been extended to general dimensions [12,25] and applied to many other situations, see for example [10,11,23–25] for applications to wave maps, [7,16,22,32] for application to semilinear focusing wave equations. It turns out the outer energy 1 ”