Global solutions of nonlinear wave equations in time dependent inhomogeneous media

Details Global solutions of nonlinear wave equations in time dependent inhomogeneous media Global solutions of nonlinear wave equations in time dependent inhomogeneous media

2010-10-21

arxiv.org/abs/1010.4341v3

Mathematics

Analysis of PDEs

Typos corrected

Scientific paper

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 \textbf{time dependent metric $g$} coinciding with 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 these estimates 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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