  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.
