Dh. Chae, Global existence of spherically symmetric solutions to the coupled Einstein and nonlinear Klein-Gordon system, CLASS QUANT, 18(21), 2001, pp. 4589-4605
We prove the global unique existence of classical solutions to the Einstein
equations coupled with the nonlinear Klein-Gordon equations for small init
ial data under spherical symmetry. We also obtain the decay estimates of th
e solutions. For its proof we reduce the system to a single first-order int
egro-differential equation, and use the contraction mapping theorem in the
appropriate function spaces.