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September/October 2014 Almost global existence for 4-dimensional quasilinear wave equations in exterior domains
John Helms, Jason Metcalfe
Differential Integral Equations 27(9/10): 837-878 (September/October 2014).

Abstract

This article focuses on almost global existence for quasilinear wave equations with small initial data in 4-dimensional exterior domains. The nonlinearity is allowed to depend on the solution at the quadratic level as well as its first and second derivatives. In the boundaryless setting, Hörmander proved that the lifespan $T_\epsilon \gtrsim \exp(c/\epsilon)$, where $\epsilon\gt 0$ denotes the size of the Cauchy data. Later Du, the second author, Sogge, and Zhou showed that this inequality also holds for star-shaped obstacles. Following up on the authors' work in the 3-dimensional case, we only require that the obstacle allow for a sufficiently rapid decay of local energy for the linear homogeneous wave equation. The key innovation of this paper is the combination of the boundary term estimates of the second author and Sogge with a variant of an estimate of Klainerman and Sideris, which is obtained via a Sobolev inequality of Du and Zhou.

Citation

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John Helms. Jason Metcalfe. "Almost global existence for 4-dimensional quasilinear wave equations in exterior domains." Differential Integral Equations 27 (9/10) 837 - 878, September/October 2014.

Information

Published: September/October 2014
First available in Project Euclid: 1 July 2014

zbMATH: 1340.35223
MathSciNet: MR3229094

Subjects:
Primary: 35B30, 35L05, 35L53, 35L72

Rights: Copyright © 2014 Khayyam Publishing, Inc.

JOURNAL ARTICLE
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Vol.27 • No. 9/10 • September/October 2014
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