Global existence of a nonlinear wave equation arising from Nordstr\"om's theory of gravitation

TL;DR

This paper proves the global existence of solutions for a scalar wave equation from Nordstr"om's gravity theory on a three-dimensional torus, using Fourier and hyperbolic system methods, with results on asymptotics and blow-up conditions.

Contribution

It introduces two novel methods for establishing global solutions to a semi-linear wave equation in Nordstr"om's gravity, including a Fourier series approach and a symmetric hyperbolic system formulation.

Findings

Global solutions exist under small source term conditions.

Solutions exhibit specific asymptotic behaviors.

Blow-up occurs if conditions for global existence are violated.

Abstract

We show global existence of classical solutions for the nonlinear Nordstr\"om theory with a source term and a cosmological constant under the assumption that the source term is small in an appropriate norm, while in some cases no smallness assumption on the initial data is required. In this theory, the gravitational field is described by a single scalar function that satisfies a certain semi-linear wave equation. We consider spatial periodic deviation from the background metric, that is why we study the semi-linear wave equation on the three-dimensional torus $\setT^3$ in the Sobolev spaces $H^m(\setT^3)$. We apply two methods to achieve the existence of global solutions, the first one is by Fourier series, and in the second one, we write the semi-linear wave equation in a non-conventional way as a symmetric hyperbolic system. We also provide results concerning the asymptotic behavior of these solutions and, finally, a blow-up result if the conditions of our global existence theorems are not met.