-
Previous Article
Remarks on global solutions of dissipative wave equations with exponential nonlinear terms
- CPAA Home
- This Issue
-
Next Article
On the pointwise decay estimate for the wave equation with compactly supported forcing term
A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation
1. | Laboratory of Mathematics, Institute of Engineering, Hiroshima university, Higashihiroshima Hirhosima, 739-8527, Japan |
References:
[1] |
T. Akahori and H. Nawa, Blowup and scattering problems for the nonlinear Schrödinger equations, Kyoto J. Math., 53 (2013), 629-672.
doi: 10.1215/21562261-2265914. |
[2] |
H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175. |
[3] |
J. E. Barab, Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation, J. Math. Phys., 25 (1984), 3270-3273.
doi: 10.1063/1.526074. |
[4] |
J. Bergh and J. Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223. |
[5] |
T. Cazenave, Semilinear Schrödinger equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University Courant Institute of Mathematical Sciences, New York, 2003. |
[6] |
T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100. |
[7] |
F. M. Christ and M. I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109.
doi: 10.1016/0022-1236(91)90103-C. |
[8] |
P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc., 1 (1988), 413-439.
doi: 10.2307/1990923. |
[9] |
B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, archived as arXiv1104:1114., 2011. |
[10] |
T. Duyckaerts, J. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250.
doi: 10.4310/MRL.2008.v15.n6.a13. |
[11] |
D. Fang, J. Xie and T. Cazenave, Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062.
doi: 10.1007/s11425-011-4283-9. |
[12] |
G. Fibich, Singular solution of the subcritical nonlinear Schrödinger equation, Phys. D, 240 (2011), 1119-1122.
doi: 10.1016/j.physd.2011.04.004. |
[13] |
D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24.
doi: 10.1142/S0219891605000361. |
[14] |
M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J., 34 (1985), 777-799.
doi: 10.1512/iumj.1985.34.34041. |
[15] |
P. Gerard, Y. Meyer and F. Oru, Inégalités de Sobolev précisées, in Séminaire sur les Équations aux Dérivées Partielles, 1996-1997, École Polytech., Palaiseau, 1997, Exp. No. IV, 11. |
[16] |
J. Ginibre, T. Ozawa and G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994), 211-239. |
[17] |
K. Hidano, Nonlinear Schrödinger equations with radially symmetric data of critical regularity, Funkcial. Ekvac., 51 (2008), 135-147.
doi: 10.1619/fesi.51.135. |
[18] |
J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467.
doi: 10.1007/s00220-008-0529-y. |
[19] |
T. Kato, An $L^{q,r}$-theory for nonlinear Schrödinger equations, in Spectral and scattering theory and applications, vol. 23 of Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, 1994, 223-238. |
[20] |
C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.
doi: 10.1007/s00222-006-0011-4. |
[21] |
C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.
doi: 10.1002/cpa.3160460405. |
[22] |
S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations, 175 (2001), 353-392.
doi: 10.1006/jdeq.2000.3951. |
[23] |
R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., 132 (2010), 361-424.
doi: 10.1353/ajm.0.0107. |
[24] |
Y. Koh, Improved inhomogeneous Strichartz estimates for the Schrödinger equation, J. Math. Anal. Appl., 373 (2011), 147-160.
doi: 10.1016/j.jmaa.2010.06.019. |
[25] |
S. Masaki, On minimal non-scattering solution to focusing mass-subcritical nonlinear Schrödinger equation, archived as arXiv:1301.1742., 2013. |
[26] |
K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the cubic NLS equation in 3D, Calc. Var. Partial Differential Equations, 44 (2012), 1-45.
doi: 10.1007/s00526-011-0424-9. |
[27] |
K. Nakanishi, Asymptotically-free solutions for the short-range nonlinear Schrödinger equation, SIAM J. Math. Anal., 32 (2001), 1265-1271 (electronic).
doi: 10.1137/S0036141000369083. |
[28] |
K. Nakanishi and T. Ozawa, Remarks on scattering for nonlinear Schrödinger equations, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 45-68.
doi: 10.1007/s00030-002-8118-9. |
[29] |
P. Sjölin, Regularity of solutions to the Schrödinger equation, Duke Math. J., 55 (1987), 699-715.
doi: 10.1215/S0012-7094-87-05535-9. |
[30] |
W. A. Strauss, Nonlinear scattering theory,, \emph{Scattering Theory in Mathematical Physics, (): 53.
|
[31] |
T. Tao and M. Visan, Stability of energy-critical nonlinear Schrödinger equations in high dimensions,, \emph{Electron. J. Differential Equations}, ().
|
[32] |
Y. Tsutsumi, Scattering problem for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 43 (1985), 321-347. |
[33] |
Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125. |
[34] |
Y. Tsutsumi and K. Yajima, The asymptotic behavior of nonlinear Schrödinger equations, Bull. Amer. Math. Soc. (N.S.), 11 (1984), 186-188.
doi: 10.1090/S0273-0979-1984-15263-7. |
[35] |
L. Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc., 102 (1988), 874-878.
doi: 10.2307/2047326. |
[36] |
M. C. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 2123-2136 (electronic).
doi: 10.1090/S0002-9947-06-04099-2. |
[37] |
M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., 138 (2007), 281-374.
doi: 10.1215/S0012-7094-07-13825-0. |
[38] |
M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, \emph{Comm. Math. Phys.}, 87 (): 567.
|
show all references
References:
[1] |
T. Akahori and H. Nawa, Blowup and scattering problems for the nonlinear Schrödinger equations, Kyoto J. Math., 53 (2013), 629-672.
doi: 10.1215/21562261-2265914. |
[2] |
H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175. |
[3] |
J. E. Barab, Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation, J. Math. Phys., 25 (1984), 3270-3273.
doi: 10.1063/1.526074. |
[4] |
J. Bergh and J. Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223. |
[5] |
T. Cazenave, Semilinear Schrödinger equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University Courant Institute of Mathematical Sciences, New York, 2003. |
[6] |
T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100. |
[7] |
F. M. Christ and M. I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109.
doi: 10.1016/0022-1236(91)90103-C. |
[8] |
P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc., 1 (1988), 413-439.
doi: 10.2307/1990923. |
[9] |
B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, archived as arXiv1104:1114., 2011. |
[10] |
T. Duyckaerts, J. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250.
doi: 10.4310/MRL.2008.v15.n6.a13. |
[11] |
D. Fang, J. Xie and T. Cazenave, Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062.
doi: 10.1007/s11425-011-4283-9. |
[12] |
G. Fibich, Singular solution of the subcritical nonlinear Schrödinger equation, Phys. D, 240 (2011), 1119-1122.
doi: 10.1016/j.physd.2011.04.004. |
[13] |
D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24.
doi: 10.1142/S0219891605000361. |
[14] |
M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J., 34 (1985), 777-799.
doi: 10.1512/iumj.1985.34.34041. |
[15] |
P. Gerard, Y. Meyer and F. Oru, Inégalités de Sobolev précisées, in Séminaire sur les Équations aux Dérivées Partielles, 1996-1997, École Polytech., Palaiseau, 1997, Exp. No. IV, 11. |
[16] |
J. Ginibre, T. Ozawa and G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994), 211-239. |
[17] |
K. Hidano, Nonlinear Schrödinger equations with radially symmetric data of critical regularity, Funkcial. Ekvac., 51 (2008), 135-147.
doi: 10.1619/fesi.51.135. |
[18] |
J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467.
doi: 10.1007/s00220-008-0529-y. |
[19] |
T. Kato, An $L^{q,r}$-theory for nonlinear Schrödinger equations, in Spectral and scattering theory and applications, vol. 23 of Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, 1994, 223-238. |
[20] |
C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.
doi: 10.1007/s00222-006-0011-4. |
[21] |
C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.
doi: 10.1002/cpa.3160460405. |
[22] |
S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations, 175 (2001), 353-392.
doi: 10.1006/jdeq.2000.3951. |
[23] |
R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., 132 (2010), 361-424.
doi: 10.1353/ajm.0.0107. |
[24] |
Y. Koh, Improved inhomogeneous Strichartz estimates for the Schrödinger equation, J. Math. Anal. Appl., 373 (2011), 147-160.
doi: 10.1016/j.jmaa.2010.06.019. |
[25] |
S. Masaki, On minimal non-scattering solution to focusing mass-subcritical nonlinear Schrödinger equation, archived as arXiv:1301.1742., 2013. |
[26] |
K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the cubic NLS equation in 3D, Calc. Var. Partial Differential Equations, 44 (2012), 1-45.
doi: 10.1007/s00526-011-0424-9. |
[27] |
K. Nakanishi, Asymptotically-free solutions for the short-range nonlinear Schrödinger equation, SIAM J. Math. Anal., 32 (2001), 1265-1271 (electronic).
doi: 10.1137/S0036141000369083. |
[28] |
K. Nakanishi and T. Ozawa, Remarks on scattering for nonlinear Schrödinger equations, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 45-68.
doi: 10.1007/s00030-002-8118-9. |
[29] |
P. Sjölin, Regularity of solutions to the Schrödinger equation, Duke Math. J., 55 (1987), 699-715.
doi: 10.1215/S0012-7094-87-05535-9. |
[30] |
W. A. Strauss, Nonlinear scattering theory,, \emph{Scattering Theory in Mathematical Physics, (): 53.
|
[31] |
T. Tao and M. Visan, Stability of energy-critical nonlinear Schrödinger equations in high dimensions,, \emph{Electron. J. Differential Equations}, ().
|
[32] |
Y. Tsutsumi, Scattering problem for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 43 (1985), 321-347. |
[33] |
Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125. |
[34] |
Y. Tsutsumi and K. Yajima, The asymptotic behavior of nonlinear Schrödinger equations, Bull. Amer. Math. Soc. (N.S.), 11 (1984), 186-188.
doi: 10.1090/S0273-0979-1984-15263-7. |
[35] |
L. Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc., 102 (1988), 874-878.
doi: 10.2307/2047326. |
[36] |
M. C. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 2123-2136 (electronic).
doi: 10.1090/S0002-9947-06-04099-2. |
[37] |
M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., 138 (2007), 281-374.
doi: 10.1215/S0012-7094-07-13825-0. |
[38] |
M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, \emph{Comm. Math. Phys.}, 87 (): 567.
|
[1] |
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 |
[2] |
Jianbo Cui, Jialin Hong, Liying Sun. On global existence and blow-up for damped stochastic nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6837-6854. doi: 10.3934/dcdsb.2019169 |
[3] |
Dapeng Du, Yifei Wu, Kaijun Zhang. On blow-up criterion for the nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3639-3650. doi: 10.3934/dcds.2016.36.3639 |
[4] |
Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure and Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264 |
[5] |
Zaihui Gan, Jian Zhang. Blow-up, global existence and standing waves for the magnetic nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 827-846. doi: 10.3934/dcds.2012.32.827 |
[6] |
Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure and Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034 |
[7] |
Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259 |
[8] |
Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085 |
[9] |
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 |
[10] |
Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure and Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027 |
[11] |
Van Duong Dinh. Blow-up criteria for linearly damped nonlinear Schrödinger equations. Evolution Equations and Control Theory, 2021, 10 (3) : 599-617. doi: 10.3934/eect.2020082 |
[12] |
Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072 |
[13] |
Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations and Control Theory, 2022, 11 (2) : 515-536. doi: 10.3934/eect.2021011 |
[14] |
Cristophe Besse, Rémi Carles, Norbert J. Mauser, Hans Peter Stimming. Monotonicity properties of the blow-up time for nonlinear Schrödinger equations: Numerical evidence. Discrete and Continuous Dynamical Systems - B, 2008, 9 (1) : 11-36. doi: 10.3934/dcdsb.2008.9.11 |
[15] |
Pierre Roux, Delphine Salort. Towards a further understanding of the dynamics in the excitatory NNLIF neuron model: Blow-up and global existence. Kinetic and Related Models, 2021, 14 (5) : 819-846. doi: 10.3934/krm.2021025 |
[16] |
Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1 |
[17] |
Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058 |
[18] |
Shuyin Wu, Joachim Escher, Zhaoyang Yin. Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation. Discrete and Continuous Dynamical Systems - B, 2009, 12 (3) : 633-645. doi: 10.3934/dcdsb.2009.12.633 |
[19] |
Xiaoliang Li, Baiyu Liu. Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3093-3112. doi: 10.3934/cpaa.2020134 |
[20] |
Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]