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Global Solutions of Nonlinear Wave Equations in Time Dependent Inhomogeneous Media

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Abstract

We consider the problem of small data global existence for a class of semilinear wave equations with null condition on a Lorentzian background \({(\mathbb{R}^{3 + 1}, g)}\) with a time dependent metric g coinciding with the Minkowski metric outside the cylinder \({\{(t, x) || x | \leq R\}}\). We show that the small data global existence result can be reduced to two integrated local energy estimates and demonstrate that these estimates work in the particular case when g is merely C 1 close to the Minkowski metric. One of the novel aspects of this work is that it applies to equations on backgrounds which do not settle to any particular stationary metric.

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Authors and Affiliations

  1. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Shiwu Yang

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  1. Shiwu Yang
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Correspondence to Shiwu Yang.

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Communicated by C. Dafermos

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Yang, S. Global Solutions of Nonlinear Wave Equations in Time Dependent Inhomogeneous Media. Arch Rational Mech Anal 209, 683–728 (2013). https://doi.org/10.1007/s00205-013-0631-y

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  • DOI: https://doi.org/10.1007/s00205-013-0631-y

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