On the exterior stability of nonlinear wave equations

Abstract

We consider a very general class of nonlinear wave equations, which admit trivial solutions and not necessarily verify any form of null conditions. For compactly supported small data, one can only have a semi-global result which states that the solutions are well-posed upto a finite time-span depending on the size of the Cauchy data. For some of the equations of the class, the solutions blow up within a finite time for the compactly supported data of any size. 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. In this paper, we prove that, there exists a constant $R(\ga_0)\ge 2$, depending on the fixed weight exponent $\ga_0>1$ in the weighted energy norm, if the norm of the data are sufficiently small on ${\mathbb R}^3\setminus B_R$ with the fixed number $R\ge R(\ga_0)$, the solution exists and is unique in the entire exterior of a schwarzschild cone initiating from $\{|x|=R\}$ (including the boundary) with small negative mass $-M_0$. $M_0$ is determined according to the size of the initial data. In this exterior region, by constructing the schwarzschild cone foliation, we can improve the linear behavior of wave equations in particular on the transversal derivative $\Lb \phi$. Such improvement enables us to control the nonlinearity violating the null condition without loss, and thus show the solutions converge to the trivial solution. As an application, we give the exterior stability result for Einstein (massive and massless) scalar fields. We prove the solution converges to a small static solution, stable in the entire exterior of a schwarzschild cone with positive mass, which then is patchable to the interior results.