The paper considers the asymptotic behaviour of the solution of a free boundary value problem of one-dimensional model system associated with the motion of a viscous compressible fluid. The governing equations are the conservation of mass, momentum and energy along with a state equation for a barotropic gas or an ideal polytropic gas. These equations are subject to initial data and boundary conditions. The author has obtained some existence theorems of unique solution, globally in time, for this free boundary value problem. Thus, he has shown that the solution is approached at a decay rate of \(t^{-\alpha}\) as \(t\to \infty\), where \(\alpha\) is a small positive constant, on the smallness conditions of the initial data and \(\gamma\geq 1\), \(\gamma\) being a specific heat constant. The analysis in this paper is essentially theoretical, albeit very clear. The paper proofs to be of interest mainly to pure mathematicians.