  The Fourier-transformed version of the BGK model in one-dimension is solved
in order to determine the general solution's asymptotics. The ultimate goal of
this paper is to demonstrate that the solution to the model Boltzmann possesses
a special property that was conjectured by Truesdell and Muncaster: that
solutions decay to a subclass of the solution set uniquely determined by the
initial first moment of the gas. First we determine the spectrum and
eigendistributions of the associated homogeneous equation. Then, using Case's
method of elementary solutions, we find analytic time-dependent solutions to
the original problem. In doing so, we show that the spectrum separates the
solutions into two distinct parts; one that behaves as a set of transient
solutions and the other limiting to a stable subclass of solutions. This
demonstrates that in time all gas flows for the one-dimensional BGK model
Boltzmann act as grossly determined solutions.
