Existence of a Local Classical Solution to the Einstein-Scalar System in Higher Dimensions

We study the existence of a local classical solution to the Einstein-Scalar equations in higher dimensions. We reduce the problem to a single first-order integro-differential equation. Then, we employ the contraction mapping in the appropriate Banach space. Using the Banach fixed theorem, we show that there exists a unique fixed point, which is the solution to the main problem. Finally, for given initial data, we prove the existence of a local classical solution.