On the quasilinear wave equations in time dependent inhomogeneous media

Abstract

We consider the problem of small data global existence for quasilinear wave equations with null condition on a class of Lorentzian manifolds $({\mathbb{R}}^{3+1},g)$ with \textbf{time dependent} inhomogeneous metric. We show that sufficiently small data give rise to a unique global solution for metric which is merely $C^1$ close to the Minkowski metric inside some large cylinder $\{(t,x)||x|\le R\}$ and approaches the Minkowski metric weakly as $|x|\to \infty$. Based on this result, we give weak but sufficient conditions on a given large solution of quasilinear wave equations such that the solution is globally stable under perturbations of initial data.

arXiv comment46pages

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• Evanescent ergosurface instability
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• On the geodesic hypothesis in general relativity
• Global solutions of nonlinear wave equations with large energy
• Decay estimates for the massless Vlasov equation on Schwarzschild spacetimes
• A geometric approach for sharp Local well-posedness of quasilinear wave equations
• Global existence for semilinear damped wave equations in relation with the Strauss conjecture
• On global behavior of solutions of the Maxwell-Klein-Gordon equations
• Nonlinear Wave Equations With Null Condition On Extremal Reissner-Nordström Spacetimes I: Spherical Symmetry
• Uniform boundedness of highest norm for 2D quasilinear wave
• The weak null condition on Kerr backgrounds
• Global stability dynamics of the timelike extremal hypersurfaces in Minkowski space