Modified scattering for the boson star equation (Q462888)

The author considers 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, \[ i\partial_tu-\sqrt{m^2-\Delta u}=\lambda\left(|x|^{-1}\ast|u|^2\right)u, \tag{eqn:euler} \] with \(u :(t,x)\in\mathbb{R}\times\mathbb{R}^3\longrightarrow \mathbb{C}\), \(m>0\) and \(\lambda\in\mathbb{R}\). The operator \(\sqrt{m^2-\Delta}\) is defined as usual by its symbol \(\sqrt{m^2+|\xi|^2}\) in Fourier space, and \(\ast\) denotes the convolution on \(\mathbb{R}^3\). The main result of the paper is the following : For any given \(u_0(x)=u(t=0,x)\) which is small enough in a suitable weighted Sobolev space, there exists a unique global solution of (1) which decays pointwise over time like a solution of the linear equation, but, as time goes to infinity, scatters in a nonlinear fashion. An additional result contained in the present paper concerns some generalizations of (1) with potentials decaying faster than the Coulomb potential \(|x|^{-1}\). The author proves regular scattering for those models, closing some gaps in the existing literature. In summary, the author combines 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. He describes the behavior of such solutions at infinity by identifying a suitable nonlinear asymptotic correction to scattering. As a byproduct of the weighted energy estimates alone, he obtains global existence and linear scattering for solutions of semi-relativistic Hartree equations with potentials decaying faster than Coulomb.