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On the Cauchy problem for the Hartree type equation in the Wiener algebra


Authors: Rémi Carles and Lounès Mouzaoui
Journal: Proc. Amer. Math. Soc. 142 (2014), 2469-2482
MSC (2010): Primary 35Q55; Secondary 35A01, 35B30, 35B45, 35B65
DOI: https://doi.org/10.1090/S0002-9939-2014-12072-7
Published electronically: March 17, 2014
MathSciNet review: 3195768
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Abstract: We consider the mass-subcritical Hartree equation with a homogeneous kernel in the space of square integrable functions whose Fourier transform is integrable. We prove a global well-posedness result in this space. On the other hand, we show that the Cauchy problem is not even locally well-posed if we simply work in the space of functions whose Fourier transform is integrable. Similar results are proven when the kernel is not homogeneous and is such that its Fourier transform belongs to some Lebesgue space.


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Rémi Carles
Affiliation: Mathématiques, CC 051, CNRS and Université Montpellier 2, 34095 Montpellier, France
Email: Remi.Carles@math.cnrs.fr

Lounès Mouzaoui
Affiliation: Mathématiques, CC 051, CNRS and Université Montpellier 2, 34095 Montpellier, France
Email: lounes.mouzaoui@univ-montp2.fr

DOI: https://doi.org/10.1090/S0002-9939-2014-12072-7
Keywords: Hartree equation, well-posedness, Wiener algebra
Received by editor(s): May 16, 2012
Received by editor(s) in revised form: July 17, 2012, and August 1, 2012
Published electronically: March 17, 2014
Additional Notes: This work was supported by the French ANR project R.A.S. (ANR-08-JCJC-0124-01)
Communicated by: James E. Colliander
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.