Blow-Up for Nonlinear Wave Equations describing Boson Stars

TL;DR

This paper proves finite-time blow-up for spherically symmetric solutions of a nonlinear wave equation modeling boson stars, indicating gravitational collapse, and explores stability and blow-up under various conditions.

Contribution

It establishes blow-up results for the pseudo-relativistic boson star model and analyzes stability of ground states, extending understanding of collapse phenomena.

Findings

Finite-time blow-up for negative energy initial data.

Instability of ground state solitary waves when mass parameter m=0.

Blow-up persists under external potentials and generalized nonlinearities.

Abstract

We consider the nonlinear wave equation $i \partial_t u= \sqrt{-\Delta + m^2} u - (|x|^{-1} \ast |u|^2) u$ on $\RR^3$ modelling the dynamics of (pseudo-relativistic) boson stars. For spherically symmetric initial data, $u_0(x) \in C^\infty_{\mathrm{c}}(\RR^3)$, with negative energy, we prove blow-up of $u(t,x)$ in $H^{1/2}$-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 at rest if $m=0$.