On the blow-up phenomenon for the mass-critical focusing Hartree equation in R4
نویسندگان
چکیده
منابع مشابه
On the blow up phenomenon for the mass critical focusing Hartree equation in R
Here f(u) = λ ( V ∗|u|2 ) u, V (x) = |x|−γ , 0 < γ < d, and ∗ denotes the convolution in Rd. If λ > 0, we call the equation (1.1) defocusing; if λ < 0, we call it focusing. This equation describes the mean-field limit of many-body quantum systems; see, e.g., [6], [7] and [36]. An essential feature of Hartree equation is that the convolution kernel V (x) still retains the fine structure of micro...
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For the defocusing with 2 < γ < min(4, d), J. Ginibre and G. Velo [6] proved the global well-posedness and scattering results in the energy space. Later, K. Nakanishi [26] made use of a new Morawetz estimate to obtain the similar results for the more general functions V (x). Recently, the authors proved the global wellposedness and scattering for the defocusing, energy critical Hartree equation...
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Consider the focusing mass-critical nonlinear Hartree equation iut + u=−(| · |−2 ∗ |u|2)u for spherically symmetric H 1 x initial data with ground state mass M(Q) in dimension d 5. We show that any global solution u which does not scatter must be the solitary wave eitQ up to phase rotation and scaling. © 2008 Elsevier Inc. All rights reserved. MSC: 35Q55
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with u0 ∈ H1 = {u,∇u ∈ L2(RN )} in dimension N ≥ 1. This equation for N = 2 appears in physics as a universal model to describe self trapping of waves propagating in nonlinear media. The physical expectation for large smooth data is the concentration of part of the L2 mass in finite time corresponding to the focusing of the laser beam. If some explicit examples of this phenomenon are known, and...
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ژورنال
عنوان ژورنال: Colloquium Mathematicum
سال: 2010
ISSN: 0010-1354,1730-6302
DOI: 10.4064/cm119-1-2