Semi-classical analysis of the Hartree equation around and before the caustic

Abstract

We consider the asymptotic behavior of the solution to the semi-classical Hartree equation in the limit of short wave length, with an initial data which causes focusing at a point. It is known that there exists a critical index which indicates whether or not the asymptotic behavior reflects the effect of the nonlinearity in the neighborhood of the caustic. Investigating time range where the nonlinear effect appears, we improve previous convergence results. In particular, we show that the solution behaves as a free solution before the caustic in some super critical cases.