On the Exterior Stability of Nonlinear Wave Equations

Abstract

We consider a general class of nonlinear wave equations, which admit trivial solutions and not necessarily verify any form of null conditions. For data prescribed on \({{\mathbb {R}}}^3\setminus B_R\) with small weighted energy, without some form of null conditions on the nonlinearity, the exterior stability is not expected to hold in the full domain of dependence, due to the known results of formation of shocks with data on annuli. The classical method can only give the well-posedness upto a finite time. In this paper, we prove that, there exists a constant \(R_0\ge 2\), if the weighted energy of the data is sufficiently small on \({\mathbb R}^3\setminus B_R\) with the fixed number \(R\ge R_0\), then the solution exists and is unique in the entire exterior of a Schwarzschild cone initiating from \(\{|x|=R\}\) (including the boundary) with a small negative mass \(-M_0\). Such \(M_0\) is determined according to the size of the initial data. The result is achieved by obtaining in the exterior region a series of sharp decay properties for the solution, which are stronger than the general known behavior of free wave. For semi-linear equations, the stability region can be any close to \(\{|x|-t>R\}\) if the weighted energy of the data is sufficiently small on \(\{|x|\ge R\}\). As a quick application, for Einstein (massive and massless) scalar fields, we show the solution converges to a small static solution, stable in the entire exterior of a Schwarzschild cone with positive mass, and hence patchable to the interior results.