Modified scattering for the Boson Star Equation

TL;DR

This paper proves global existence and describes the long-term behavior of solutions to the boson star equation in three dimensions, using weighted estimates and Fourier analysis to identify nonlinear scattering corrections.

Contribution

It introduces a novel combination of weighted estimates and Fourier analysis to establish scattering and global solutions for the boson star equation with small initial data.

Findings

Global solutions exist for small, localized initial data.

Solutions exhibit a nonlinear asymptotic correction at infinity.

Results extend to semi-relativistic Hartree equations with fast-decaying potentials.

Abstract

We consider the question of scattering for the boson star equation in three space dimensions. This is a semi-relativistic Klein-Gordon equation with a cubic nonlinearity of Hartree type. We combine weighted estimates, obtained by exploiting a special null structure present in the equation, and a refined asymptotic analysis performed in Fourier space, to obtain global solutions evolving from small and localized Cauchy data. We describe the behavior at infinity of such solutions by identifying a suitable nonlinear asymptotic correction to scattering. As a byproduct of the weighted energy estimates alone, we also obtain global existence and (linear) scattering for solutions of semi-relativistic Hartree equations with potentials decaying faster than Coulomb.