  Motivated by the initial value problem in semiclassical gravity, we study the
initial value problem of a system consisting of a quantum scalar field weakly
interacting with a classical one. The quantum field obeys a Klein-Gordon
equation with a potential proportional to the classical field. The classical
field obeys an inhomogeneous Klein-Gordon equation sourced by the renormalised
expectation value of the squared quantum field in a Hadamard state, $\langle
\Psi| \Phi^2 \Psi \rangle$. Thus, the system of equations for the scalar fields
reminisces of the semi-classical Einstein field equations with a Klein-Gordon
field, where classical geometry is sourced by the renormalised stress-energy
tensor of the quantum field, and the Klein-Gordon equation depends on the
metric explicitly. We show that a unique asymptotic solution for the system can
be obtained perturbatively at any fixed finite order in the weak coupling from
initial data provided that the interaction is switched on and off smoothly in a
spacetime region to the future of the initial data surface. This allows one to
provide "free" initial data for the decoupled system that guarantees that the
Wightman function of the quantum field be of Hadamard form, and hence that the
renormalised $\langle \Psi| \Phi^2 \Psi \rangle$ exist (in a perturbative
sense) and be smooth. We comment on how to relax the switching of the
interaction, which might be relevant for the corresponding problem in
semiclassical gravity.
