Blow-Up for Nonlinear Wave Equations describing Boson Stars

We consider the nonlinear wave equation i∂tu = √ −∆+m2u− (|x| ∗ |u|)u on R modelling the dynamics of (pseudo-relativistic) boson stars. For spherically symmetric initial data, u0(x) ∈ C∞ c (R), with negative energy, we prove blow-up of u(t, x) inH-norm within a finite time. Physically, this phenomenon describes the onset of “gravitational collapse” of a boson star. We also study blow-up in external, spherically symmetric potentials and we consider more general Hartree-type nonlinearities. As an application, we exhibit instability for ground state solitary waves of the equation with m = 0.