Analysis of an equation describing large-scale structure formation

The subject of this talk is a linear hyperbolic equation which arises in cosmological perturbation theory. I will describe work of Paul Allen and myself in which we determine the asymptotics of solutions of this equation. The equation is defined for positive times with a singularity at the origin. We have studied the asymptotics both in the approach to the origin and at late times. The coefficients of the equation depend on a function which represents the equation of state of a fluid. In some cases it is possible to parametrize all solutions by the coefficient functions in the asymptotic expansions in either of the asymptotic regimes.