Blowup for nonlinear wave equations describing boson stars

We consider the nonlinear wave equation [image] modeling the dynamics of (pseudorelativistic) boson stars. For spherically symmetric initial data, u0(x) ∈ Cmath image (ℝ3), with negative energy, we prove blowup of u(t, x) in the H1/2-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.

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