Local Existence of Classical Solutions to Scalar Field Equation on Spatially Compact Spacetime as a Background

Abstract

The goal of this paper is to prove the wellposedness of scalar field equation on spatially compact spacetime of Riemannian manifold. We construct the equation of motion from the Lagrangian of scalar field with non-minimal coupling, where the coupling interaction of the scalar field ϕ is proportional to the scalar curvature of the spacetime. The equation of motion has the form like non-linear wave equation. The next step is to prove local existence of solutions. We have show that both the k th linear energy and energy norm are bounded for the finite time with the initial data in H k+1 × H k. Finally, we prove the uniqueness and smoothness properties of the solution.