Figure 1.
Comparison of energy-mass
-
Previous Article
$ L^p $ Neumann problems in homogenization of general elliptic operators
- DCDS Home
- This Issue
-
Next Article
Lifespan of solutions to the Strauss type wave system on asymptotically flat space-times
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 |
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 and Continuous Dynamical Systems,
2020, 40
(8)
: 5001-5017.
doi: 10.3934/dcds.2020209
![]()
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [12] |
P. L. Kelley,
Self focusing of optical beams, Phys. Rev. Lett, 15 (1965), 1005-1008.
doi: 10.1109/IQEC.2005.1561150. |
| [13] |
E. Lieb, Analysis, in Graduate Studies in Mathematics, American Mathematical Society, 2001. |
| [14] |
P. L. Lions,
The Choquard equation and related questions, Nonlinear. Anal, 4 (1980), 1063-1072.
doi: 10.1016/0362-546X(80)90016-4. |
| [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. |
| [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. |
| [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. |
| [18] |
S. Pekar, Untersuchung über Die Elektronentheorie Der Kristalle, Akademie-Verlag, Berlin, 1954. |
| [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. |
| [20] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Commun. Math. Phys, 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
| [21] |
Y. Tsutsumi,
Scattering problem for nonlinear Schrödinger equation, Nonlinear Ann. Inst. Henri. Poincaré Physique Théorique, 43 (1985), 321-347.
|
| [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. |
| [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. |
| [24] |
M. I. Weinstein,
Nonlinear Schrödinger equations and sharp interpolations estimates, Commun. Math. Phys, 87 (1982/83), 567-576.
|
| [25] |
V. E. Zakharov,
Collapse of Langmuir waves, Sov. Phys. JETP, 35 (1972), 909-912.
|
| [26] |
S. H. Zhu, On the Davey-Stewartson system with competing nonlinearities, J. Math. Phys, 57 (2016), 031501, 13pp.
doi: 10.1063/1.4942633. |
| [27] |
S. H. Zhu, Dynamical Properties of Blow-Up Solutions to Nonlinear Schrödinger Equations, Ph.D. thesis (in Chinese), Sichuan University, 2011. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [12] |
P. L. Kelley,
Self focusing of optical beams, Phys. Rev. Lett, 15 (1965), 1005-1008.
doi: 10.1109/IQEC.2005.1561150. |
| [13] |
E. Lieb, Analysis, in Graduate Studies in Mathematics, American Mathematical Society, 2001. |
| [14] |
P. L. Lions,
The Choquard equation and related questions, Nonlinear. Anal, 4 (1980), 1063-1072.
doi: 10.1016/0362-546X(80)90016-4. |
| [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. |
| [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. |
| [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. |
| [18] |
S. Pekar, Untersuchung über Die Elektronentheorie Der Kristalle, Akademie-Verlag, Berlin, 1954. |
| [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. |
| [20] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Commun. Math. Phys, 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
| [21] |
Y. Tsutsumi,
Scattering problem for nonlinear Schrödinger equation, Nonlinear Ann. Inst. Henri. Poincaré Physique Théorique, 43 (1985), 321-347.
|
| [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. |
| [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. |
| [24] |
M. I. Weinstein,
Nonlinear Schrödinger equations and sharp interpolations estimates, Commun. Math. Phys, 87 (1982/83), 567-576.
|
| [25] |
V. E. Zakharov,
Collapse of Langmuir waves, Sov. Phys. JETP, 35 (1972), 909-912.
|
| [26] |
S. H. Zhu, On the Davey-Stewartson system with competing nonlinearities, J. Math. Phys, 57 (2016), 031501, 13pp.
doi: 10.1063/1.4942633. |
| [27] |
S. H. Zhu, Dynamical Properties of Blow-Up Solutions to Nonlinear Schrödinger Equations, Ph.D. thesis (in Chinese), Sichuan University, 2011. |
Figure 2.
Comparison of $ H^1 $ norm
| [1] |
Binhua Feng, Xiangxia Yuan. On the Cauchy problem for the Schrödinger-Hartree equation. Evolution Equations and Control Theory, 2015, 4 (4) : 431-445. doi: 10.3934/eect.2015.4.431 |
| [2] |
Carlos Banquet, Élder J. Villamizar-Roa. On the management fourth-order Schrödinger-Hartree equation. Evolution Equations and Control Theory, 2020, 9 (3) : 865-889. doi: 10.3934/eect.2020037 |
| [3] |
Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903 |
| [4] |
Jianqing Chen. A variational argument to finding global solutions of a quasilinear Schrödinger equation. Communications on Pure and Applied Analysis, 2008, 7 (1) : 83-88. doi: 10.3934/cpaa.2008.7.83 |
| [5] |
Jianqing Chen. Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1613-1628. doi: 10.3934/dcdss.2016066 |
| [6] |
Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107 |
| [7] |
Hans Zwart, Yann Le Gorrec, Bernhard Maschke. Relating systems properties of the wave and the Schrödinger equation. Evolution Equations and Control Theory, 2015, 4 (2) : 233-240. doi: 10.3934/eect.2015.4.233 |
| [8] |
Maria J. Esteban. Gagliardo-Nirenberg-Sobolev inequalities on planar graphs. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2101-2114. doi: 10.3934/cpaa.2022051 |
| [9] |
Felipe Hernandez. A decomposition for the Schrödinger equation with applications to bilinear and multilinear estimates. Communications on Pure and Applied Analysis, 2018, 17 (2) : 627-646. doi: 10.3934/cpaa.2018034 |
| [10] |
Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control and Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119 |
| [11] |
César E. Torres Ledesma. Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well. Communications on Pure and Applied Analysis, 2016, 15 (2) : 535-547. doi: 10.3934/cpaa.2016.15.535 |
| [12] |
Vincenzo Ambrosio. The nonlinear fractional relativistic Schrödinger equation: Existence, multiplicity, decay and concentration results. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5659-5705. doi: 10.3934/dcds.2021092 |
| [13] |
Vincenzo Ambrosio. Concentration phenomena for critical fractional Schrödinger systems. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2085-2123. doi: 10.3934/cpaa.2018099 |
| [14] |
Guoyuan Chen, Youquan Zheng. Concentration phenomenon for fractional nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2359-2376. doi: 10.3934/cpaa.2014.13.2359 |
| [15] |
Zhongyuan Liu. Concentration of solutions for the fractional Nirenberg problem. Communications on Pure and Applied Analysis, 2016, 15 (2) : 563-576. doi: 10.3934/cpaa.2016.15.563 |
| [16] |
Qingxuan Wang, Binhua Feng, Yuan Li, Qihong Shi. On Asymptotic Properties of Semi-relativistic Hartree Equation with combined Hartree-type nonlinearities. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1225-1247. doi: 10.3934/cpaa.2022017 |
| [17] |
Yong Luo, Shu Zhang. Concentration behavior of ground states for $ L^2 $-critical Schrödinger Equation with a spatially decaying nonlinearity. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1481-1504. doi: 10.3934/cpaa.2022026 |
| [18] |
Benjamin Dodson. Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ via a linear-nonlinear decomposition. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1905-1926. doi: 10.3934/dcds.2013.33.1905 |
| [19] |
Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120 |
| [20] |
Daniele Cassani, João Marcos do Ó, Abbas Moameni. Existence and concentration of solitary waves for a class of quasilinear Schrödinger equations. Communications on Pure and Applied Analysis, 2010, 9 (2) : 281-306. doi: 10.3934/cpaa.2010.9.281 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]