Blowup for nonlinear wave equations describing boson stars

We consider the nonlinear wave equation modeling the dynamics of (pseudorelativistic) boson stars. For spherically symmetric initial data, u 0 ( x ) ∈ C (ℝ 3 ), with negative energy, we prove blowup of u ( t, x ) in the H 1/2 ‐norm within a finite time. Physically this phenomenon describes the onset of “gravitational collapse” of a boson star. We also study blowup in external, spherically symmetric potentials, and we consider more general Hartree‐type nonlinearities. As an application, we exhibit instability of ground state solitary waves at rest if m = 0.