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Blowup for nonlinear wave equations describing boson stars
Author(s) -
Fröhlich Jürg,
Lenzmann Enno
Publication year - 2007
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20186
Subject(s) - physics , stars , boson , nonlinear system , gravitational collapse , instability , star (game theory) , mathematical physics , negative energy , norm (philosophy) , gravitational wave , equation of state , type (biology) , classical mechanics , astrophysics , quantum mechanics , ecology , political science , law , biology
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, u 0 ( x ) ∈ C 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. © 2007 Wiley Periodicals, Inc.