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Licensed Unlicensed Requires Authentication Published by De Gruyter July 20, 2012

The lifespan for 3-dimensional quasilinear wave equations in exterior domains

  • John Helms EMAIL logo and Jason Metcalfe
From the journal Forum Mathematicum

Abstract

This article focuses on long-time existence for quasilinear wave equations with small initial data in exterior domains. The nonlinearity is permitted to fully depend on the solution at the quadratic level, rather than just the first and second derivatives of the solution. The corresponding lifespan bound in the boundaryless case is due to Lindblad, and Du and Zhou first proved such long-time existence exterior to star-shaped obstacles. Here we relax the hypothesis on the geometry and only require that there is a sufficiently rapid decay of local energy for the linear homogeneous wave equation, which permits some domains that contain trapped rays. The key step is to prove useful energy estimates involving the scaling vector field for which the approach of the second author and Sogge provides guidance.

MSC: 42B99; 35L72

Funding source: NSF

Award Identifier / Grant number: DMS-1054289

Funding source: NSF

Award Identifier / Grant number: DMS-0800678

Received: 2012-4-20
Published Online: 2012-7-20
Published in Print: 2014-11-1

© 2015 by De Gruyter

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