# American Institute of Mathematical Sciences

August  2020, 40(8): 5001-5017. doi: 10.3934/dcds.2020209

## Blowup results and concentration in focusing Schrödinger-Hartree equation

 School of Mathematics and Statistics, Jiaotong University, Xi'an, Shanxi 710049, China

* Corresponding author: Liquan Mei

Received  October 2019 Published  May 2020

Fund Project: The work is partially supported by Science Challenge Project, No. TZ2016002

This paper is concerned with the Cauchy problem of the Schrödinger-Hartree equation. Applying the profile decomposition of bounded sequence in $\dot{H}^1(\mathbb{R}^N)\cap\dot{H}^{S_c}(\mathbb{R}^N)$ and corresponding variational structure, a refined Gagliardo-Nirenberg inequality is established and the sharp constant for this inequality is deduced. Secondly, via construction and analysis of some invariant manifolds, we derive a different criterion of global existence and blowup results. Under the discussion of Bootstrap argument, we additionally obtain other sufficient condition for global existence. Finally, A compactness result is applied to show that the blowup solutions with bounded $\dot{H}^{S_c}$ norm definitely have concentration properties related to a fixed $\dot{H}^{S_c}$ norm of certain standing waves.

Citation: Yingying Xie, Jian Su, Liquan Mei. Blowup results and concentration in focusing Schrödinger-Hartree equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (8) : 5001-5017. doi: 10.3934/dcds.2020209
##### References:
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show all references

##### References:
 [1] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar [2] T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys, 85 (1982), 549-561.  doi: 10.1007/BF01403504.  Google Scholar [3] J. Chen and B. Guo, Strong instability of standing waves for a nonlocal Schrödinger equation, Physica. D, 227 (2007), 142-148.  doi: 10.1016/j.physd.2007.01.004.  Google Scholar [4] B. Feng, Sharp threshold of global existence and instability of standing wave for the Schrödinger-Hartree equation with a harmonic potential, Nonlinear. Anal. Real. World. Appl, 31 (2016), 132-145.  doi: 10.1016/j.nonrwa.2016.01.012.  Google Scholar [5] B. Feng and Y. Cai, Concentration for blow-up solutions of the Davey-Stewartson system in $\mathbb{R}^3$, Nonlinear. Anal. Real. World. Appl, 26 (2015), 330-342.  doi: 10.1016/j.nonrwa.2015.06.003.  Google Scholar [6] B. H. Feng and X. Yuan, The global well-posedness and blow-up solutions for the generalized Choquard equation, Evol. Equ. Control. Theory, 4 (2015), 281-296.   Google Scholar [7] B. H. Feng and H. H. Zhang, Stability of standing waves for the fractional Schrödinger-Choquard equation, Comput. Math. Appl, 75 (2018), 2499-2507.  doi: 10.1016/j.camwa.2017.12.025.  Google Scholar [8] J. Fröehlich and E. Lenzmann, Mean-field limit of quantum bose gases and nonlinear Hartree equation, in Seminaire EDP, Ecole Polytechnique, (2004), 2003–2004.  Google Scholar [9] H. Genev and G. Venkov, Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 903-923.  doi: 10.3934/dcdss.2012.5.903.  Google Scholar [10] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations.I. The Cauchy problem, general case, J. Funct. Anal, 32 (1979), 1-32.  doi: 10.1016/0022-1236(79)90076-4.  Google Scholar [11] R. T. Glassey, On the blowing up of solution to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys, 18 (1977), 1794-1797.  doi: 10.1063/1.523491.  Google Scholar [12] P. L. Kelley, Self focusing of optical beams, Phys. Rev. Lett, 15 (1965), 1005-1008.  doi: 10.1109/IQEC.2005.1561150.  Google Scholar [13] E. Lieb, Analysis, in Graduate Studies in Mathematics, American Mathematical Society, 2001. Google Scholar [14] P. L. Lions, The Choquard equation and related questions, Nonlinear. Anal, 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4.  Google Scholar [15] C. Miao, G. Xu and L. Zhao, On the blow up phenomenon for the $L^2$-critical focusing Hartree equation in $R^3$, Colloq. Math, 119 (2010), 23-50.  doi: 10.4064/cm119-1-2.  Google Scholar [16] V. Moroz and J. V. Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal, 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007.  Google Scholar [17] T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differ. Equations, 92 (1991), 317-330.  doi: 10.1016/0022-0396(91)90052-B.  Google Scholar [18] S. Pekar, Untersuchung über Die Elektronentheorie Der Kristalle, Akademie-Verlag, Berlin, 1954. Google Scholar [19] W. A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal, 41 (1981), 110-133.  doi: 10.1016/0022-1236(81)90063-X.  Google Scholar [20] W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys, 55 (1977), 149-162.  doi: 10.1007/BF01626517.  Google Scholar [21] Y. Tsutsumi, Scattering problem for nonlinear Schrödinger equation, Nonlinear Ann. Inst. Henri. Poincaré Physique Théorique, 43 (1985), 321-347.   Google Scholar [22] Y. J. Wang, Strong instability of standing waves for Hartree equation with harmonic potential, Physica. D, 237 (2008), 998-1005.  doi: 10.1016/j.physd.2007.11.018.  Google Scholar [23] X. Wang, X. M. Sun and W. H. Lv, Orbital stability of generalized Choquard equation, Bound. Value Probl, 2016 (2016), Paper No. 190, 8 pp. doi: 10.1186/s13661-016-0697-1.  Google Scholar [24] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolations estimates, Commun. Math. Phys, 87 (1982/83), 567-576.   Google Scholar [25] V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP, 35 (1972), 909-912.   Google Scholar [26] S. H. Zhu, On the Davey-Stewartson system with competing nonlinearities, J. Math. Phys, 57 (2016), 031501, 13pp. doi: 10.1063/1.4942633.  Google Scholar [27] S. H. Zhu, Dynamical Properties of Blow-Up Solutions to Nonlinear Schrödinger Equations, Ph.D. thesis (in Chinese), Sichuan University, 2011. Google Scholar
Comparison of energy-mass
Comparison of $H^1$ norm
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