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 [Formula: see text] with time dependent inhomogeneous metric. We show that sufficiently small data give rise to a unique global solution for metric which is merely [Formula: see text] close to the Minkowski metric inside some large cylinder [Formula: see text] and approaches the Minkowski metric slowly as [Formula: see text]. 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.