Global Existence and Completeness of Classical Solutions in Higher Dimensional Einstein-Klein-Gordon System

In this paper we study the global existence and completeness of classical solutions of gravity coupled a scalar field system called Einstein-Klein-Gordon system in higher dimensions. We introduce a new ansatz function to reduce the problem into a single first-order integro-differential equation. Then, we employ the contraction mapping in the appropriate Banach space. Using Banach fixed theorem, we show that there exists a unique fixed point, which is the solution of the theory. For a given initial data, we prove the existence of both local and global classical solutions. We also study the completeness properties of the spacetime. Here, we introduce a mass-like function for $D\geq 4$ in Bondi coordinates. The completeness of spacetime along the future directed timelike lines outward to a region which resembles the event horizon of the black hole.