Blowup results and concentration in focusing Schrödinger-Hartree equation

Yingying Xie; Jian Su; Liquan Mei

doi:10.3934/dcds.2020209

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

Blowup results and concentration in focusing Schrödinger-Hartree equation

Yingying Xie , Jian Su and Liquan Mei ^,

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.

Keywords: Schrödinger-Hartree equation, profile decomposition, Gagliardo-Nirenberg inequality, bootstrap argument, concentration properties.

Mathematics Subject Classification: Primary: 35Q55; Secondary: 74H35.

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

References:

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[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

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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

Figure 1. Comparison of energy-mass

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Figure 2. Comparison of $ H^1 $ norm

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