Blow-Up for Nonlinear Wave Equations describing Boson Stars

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 $m=0$.

final version; to appear in Comm. Pure Appl. Math; 14 pages

References in corpus (3)

• Boson Stars as Solitary Waves
• Well-Posedness for Semi-Relativistic Hartree Equations of Critical Type
• Effective Dynamics for Boson Stars

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• Nondispersive solutions to the L2-critical half-wave equation
• On Singularity formation for the L^2-critical Boson star equation
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• Effective Dynamics for Boson Stars
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• On Blowup for time-dependent generalized Hartree-Fock equations
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• On Stability of Pseudo-Conformal Blowup for L^2-critical Hartree NLS
• On the transport of Gaussian measures under the one-dimensional fractional nonlinear Schrödinger equations
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• On finite time blowup for mass-critical Hartree equations
• Rate of Convergence Towards Semi-Relativistic Hartree Dynamics
• Stable ground states for the relativistic gravitational Vlasov-Poisson system
• The Hartree limit of Born's ensemble for the ground state of a bosonic atom or ion
• Discrete Solitary Waves in Systems with Nonlocal Interactions and the Peierls-Nabarro Barrier
• An Improvement on the Brézis-Gallouët technique for 2D NLS and 1D half-wave equation
• Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities
• The Lieb-Yau Conjecture for Ground States of Pseudo-Relativistic Boson Stars
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• Optimal convergence rate of nonrelativistic limit for the nonlinear pseudo-relativistic equations
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