Almost global existence for some semilinear wave equations

By Markus Keel, Hart F. Smith, Christopher D. Sogge. 2001

143 citations recorded by the source corpus.

We prove almost global existence for semilinear wave equations outside of nontrapping obstacles. We use the vector field method, but only use the generators of translations and Euclidean rotations. Our method exploits 1/r decay of wave equations, as opposed to the much harder to prove 1/t decay.

Subfields: Control and Systems Engineering, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3af28199e3251ea5998fc9ea4cf0759f2e1c143f

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In Control and Systems Engineering

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Local decay of waves on asymptotically flat stationary space-times (2009, background): “ed in the work of Morawetz [35], [36], [37]. There is by now an extensive literature devoted to this topic and its applications; without being exhaustive we mention Strauss [43], Keel-SmithSogge [22],[23], Burq-Planchon-Stalker-Tahvildar-Zadeh [6], Metcalfe-Sogge [31],[32]. The sharp form of the est”
Global existence of null-form wave equations in exterior domains (2006, methodology): “See [13]. The reader is also encouraged to see [ 11 ] for the first example of how this decay can be paired with KSS estimates to yield long time existence for nonlinear equations.”
Price's Law on Nonstationary Space-Times (2011, unknown): “applications; without being exhaustive we mention [59], [36], [56], [34], [35], [9], [46], [47], [58], [31].”
Strichartz Estimates on Schwarzschild Black Hole Backgrounds (2008, unknown): “2) See for instance [20,22,40–42].”
Hyperbolic trapped rays and global existence of quasilinear wave equations (2003, methodology): “As in [ 10 ] and [11] we shall control the local L2 norms, such as the last term in (2.19) by using weighted L2t L2x estimates.”
Global parametrices and dispersive estimates for variable coefficient wave equations (2007, unknown): “In the simplest form (see e.g. [1], [ 10 ], [13], [15],”
Angular Regularity and Strichartz Estimates for the Wave Equation (2004, background): “Estimate (95), as well as as its version with η = 0 and a logarithmic loss in time, was proved in the work of Keel-Smith-Sogge [ 5 ] in dimension three with the help of the sharp Huygen’s principle.”