Asymptotics for quasilinear wave equations in exterior domains

Abstract

The main concern of this paper is the asymptotic behavior of global classical solution to exterior domain problem for three-dimensional quasilinear wave equations satisfying null condition, in the small data setting. For this purpose, we first provide an alternative proof for the global existence result via purely energy approach, in which only the general derivatives and spatial rotation operators are employed as commuting vector fields. Then based on this new proof, we show that the global solution will scatter, that is, it will converge to some solution of homogeneous linear wave equations, in the energy sense, as time tends to infinity. We also show that the global solution can be determined by the scattering data uniquely, i.e., the inverse scattering property holds.