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Lifespan of solutions to the Strauss type wave system on asymptotically flat space-times
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.
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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[27] |
S. H. Zhu, Dynamical Properties of Blow-Up Solutions to Nonlinear Schrödinger Equations, Ph.D. thesis (in Chinese), Sichuan University, 2011. Google Scholar |
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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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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