Blowup for nonlinear wave equations describing boson stars

We consider the nonlinear wave equation $$i \partial_{t}u = \sqrt{-\Delta + m^{2}} \; u - (|{x}|^{-1} \ast |{u}|^{2})u \;\;\; {\rm on}\;\; {\tt R}^{3}$$ modeling the dynamics of (pseudorelativistic) boson stars. For spherically symmetric initial data, u0(x) ∈ C c∞ (ℝ3), with negative energy, we prov...

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Bibliographic Details
Published in:Communications on pure and applied mathematics 2007-11, Vol.60 (11), p.1691-1705
Main Authors: Fröhlich, Jürg, Lenzmann, Enno
Format: Article
Language:English
Subjects:
Exact sciences and technology
Mathematical analysis
Mathematical functions
Mathematical problems
Mathematics
Nonlinear equations
Partial differential equations
Sciences and techniques of general use
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Summary:We consider the nonlinear wave equation $$i \partial_{t}u = \sqrt{-\Delta + m^{2}} \; u - (|{x}|^{-1} \ast |{u}|^{2})u \;\;\; {\rm on}\;\; {\tt R}^{3}$$ modeling the dynamics of (pseudorelativistic) boson stars. For spherically symmetric initial data, u0(x) ∈ C c∞ (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. © 2007 Wiley Periodicals, Inc.
ISSN:0010-3640
1097-0312
DOI:10.1002/cpa.20186