This paper considers the asymptotic and numerical solution of a simple model for the expansion of a collisionless plasma into a plasma of lower density. The dependence on the density ratio of qualitative and quantitative features of solutions of the well-known cold-ion model is explored. In the cold-ion limit, we find that a singularity develops in the ion density in finite time unless the density ratio is zero or close to unity. The classical cold-ion model may cease to be valid when such a singularity occurs and we then regularize the model by the finite ion-temperature Vlasov-Poisson system. Numerical evidence suggests the emergence of a multi-modal velocity distribution.