Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications

By Carlos E. Kenig, F. Merle. 2008

135 citations recorded by the source corpus.

In this paper we establish optimal pointwise decay estimates for non-dispersive (compact) radial solutions to non-linear wave equations in 3 dimensions, in the energy supercritical range. As an application, we show for the full energy supercritical range, in the defocusing case, that if the scale invariant Sobolev norm of a radial solution remains bounded in its maximal interval of existence, then the solution must exist for all times and scatter.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8f35fba3b25eb14ae25ce807a268cd3eca0ac95d

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In Mathematical Physics

1. B. Dodson
2. Pierre Raphael
3. F. Merle
4. Jérémie Szeftel
5. A. Lawrie
6. Aynur Bulut
7. Oana Pocovnicu
8. Tadahiro Oh

Citations in the corpus

• Global Well-Posedness and Scattering for the Defocusing Energy-Supercritical Cubic Nonlinear Wave Equation (2010, methodology): “We note that when the cubic nonlinearity F (u) = |u|(2)u is replaced by the ddimensional energy-supercritical nonlinearity |u|u, p > 4 d−2 , the above theorem was proved by Kenig and Merle [20] in d = 3 for radial initial data, and by Killip and Visan [26] for general data in d = 3 with even values ”
• On collapsing ring blow up solutions to the mass supercritical NLS (2012, background): “9) This relates to the regularity results for the 3D Navier Stokes [8] and the regularity result [14], and shows a major dynamical difference between critical and super critical blow up.”
• Scattering for the radial 3D cubic wave equation (2014, background): “MR 2011b:35486 Zbl 1188.35180 [Kenig and Merle 2011] C. E. Kenig and F. Merle, “Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications”, Amer.”
• On the Probabilistic Cauchy Theory of the Cubic Nonlinear Schrödinger Equation on Rd, d≥3 (2014, result): “ existence, then u exists globally in time and scatters. For d≥ 5, the cubic NLS is supercritical with respect to any known conservation law. Nonetheless, motivated by a similar result of Kenig-Merle [36] on radial solutions to the energy-supercritical nonlinear wave equation (NLW) on R3, Killip-Vi¸”