Global Future Existence for a Nonlinear Wave Equation in a (1+3+n)-dimensional Cosmological Supersymmetric Spacetime

Abstract

This paper is concerned with the Cauchy problem for a semilinear wave equation with arbitrary quadratic nonlinearities, posed on a cosmological supersymmetric spacetime of the product form (Open Milne) $\times K$, where $K$ is a compact, Ricci-flat $n$-dimensional Riemannian manifold. We establish the global future existence of solutions for small initial data. Our proof, based on a novel modified energy, avoids spectral decompositions of the Laplace operator on the internal manifold and does not require any special nonlinear coupling structure. This robust framework is expected to be readily extendable to tensorial wave equations, including the Einstein equations.