Asymptotic Expansions Close to the Singularity in Gowdy Spacetimes

We consider Gowdy spacetimes under the assumption that the spatial hypersurfaces are diffeomorphic to the torus. The relevant equations are then wave map equations with the hyperbolic space as a target. In an article by Grubǐsić and Moncrief, a formal expansion of solutions in the direction toward the singularity was proposed. Later, Kichenassamy and Rendall constructed a family of real analytic solutions with the maximum number of free functions and the desired asymptotics at the singularity. The condition of real analyticity was subsequently removed by Rendall. In an article by the author, it was shown that one can put a condition on initial data that leads to asymptotic expansions. In this article, we show the following. By fixing a point in hyperbolic space, we can consider the hyperbolic distance from this point to the solution at a given spacetime point. If we fix a spatial point for the solution, it is enough to put conditions on the rate at which the hyperbolic distance tends to infinity as time tends to the singularity in order to conclude that there are smooth expansions in a neighbourhood of the given spatial point.