On fractional nonlinear Schrödinger equation with combined power-type nonlinearities

Previous Article
Chern-Simons gauged sigma model into $ \mathbb{H}^2 $ and its self-dual equations
DCDS Home
This Issue
Next Article
2D incompressible Euler equations: New explicit solutions

August 2019, 39(8): 4565-4612. doi: 10.3934/dcds.2019188

On fractional nonlinear Schrödinger equation with combined power-type nonlinearities

Van Duong Dinh ^1,2, and Binhua Feng ^3,,

1.	Laboratoire Paul Painlevé UMR 8524, Université de Lille CNRS, 59655 Villeneuve d'Ascq Cedex, France
2.	Department of Mathematics, HCMC University of Pedagogy, 280 An Duong Vuong, Ho Chi Minh, Vietnam
3.	Department of Mathematics, Northwest Normal University, Lanzhou, China

^* Corresponding author: Binhua Feng

Received September 2018 Revised February 2019 Published May 2019

We undertake a comprehensive study for the fractional nonlinear Schrödinger equation

$ i\partial_t u - (-\Delta)^s u = \mu_1 |u|^{\alpha_1} u + \mu_2 |u|^{\alpha_2} u, \quad u(0) = u_0, $

where

$ \frac{d}{2d-1} \leq s <1 $

,

$ 0 < \alpha_1 <\alpha_2 < \frac{4s}{d-2s} $

. Firstly, we establish the local and global well-posedness results for non-radial and radial

$ H^s $

initial data, radial

$ \dot{H}^{s_c}\cap \dot{H}^s $

initial data, where

$ s_c = \frac{d}{2}-\frac{2s}{\alpha_2} $

. Secondly, we study the asymptotic behavior of global radial

$ H^s $

solutions. Of particular interest is the

$ L^2 $

-critical case and the results in this case are conditional on a conjectured global existence and spacetime estimate for the

$ L^2 $

-critical fractional nonlinear Schrödinger equation. Thirdly, we obtain sufficient conditions about existence of blow-up radial

$ \dot{H}^{s_c} \cap \dot{H}^s $

solutions, and derive the sharp threshold mass of blow-up and global existence for this equation with

$ L^2 $

-critical and

$ L^2 $

-subcritical nonlinearities. Finally, we obtain the dynamical behaviour of blow-up solutions in both

$ L^2 $

-critical and

$ L^2 $

-supercritical cases, including mass-concentration and limiting profile.

Keywords: Fractional nonlinear Schrödinger equation, global existence, scattering, blow-up criteria, blow-up dynamics.

Mathematics Subject Classification: Primary: 35B44; Secondary: 35Q55.

Citation: Van Duong Dinh, Binhua Feng. On fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4565-4612. doi: 10.3934/dcds.2019188

References:

[1]	J. Bergh and J. Löfstöm, Interpolation Spaces-An Introduction, Springer-Verlag, Berlin, 1976.
[2]	S. Bhattarai, On fractional Schrödinger systems of Choquard type, J. Differential Equations, 263 (2017), 3197-3229. doi: 10.1016/j.jde.2017.04.034.
[3]	T. Boulenger, D. Himmelsbach and E. Lenzmann, Blowup for fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603. doi: 10.1016/j.jfa.2016.08.011.
[4]	T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A.
[5]	T. Cazenave, Semilinear Schrödinger Equations, Courant Lectures Notes in Mathematics 10, New York, AMS, 2003. doi: 10.1090/cln/010.
[6]	X. Cheng, C. Miao and L. Zhao, Global well-posedness and scattering for nonlinear Schrödinger equations with combined nonlinearities in the radial case, J. Differential Equations, 261 (2016), 2881-2934. doi: 10.1016/j.jde.2016.04.031.
[7]	Y. Cho, Short-range scattering of Hartree type fractional NLS, J. Differential Equations, 262 (2017), 116-144. doi: 10.1016/j.jde.2016.09.025.
[8]	Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa, On the Cauchy problem of fractional Schrödinger equations with Hartree type nonlimearity, Funkcial. Ekvac., 56 (2013), 193-224. doi: 10.1619/fesi.56.193.
[9]	Y. Cho, G. Hwang, S. Kwon and S. Lee, Profile decompositions and blow-up phenomena of mass critical fractional Schrödinger equations, Nonlinear Anal., 86 (2013), 12-29. doi: 10.1016/j.na.2013.03.002.
[10]	Y. Cho, G. Hwang, S. Kwon and S. Lee, On finite time blow-up for the mass-critical Hartree equations, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 467-479. doi: 10.1017/S030821051300142X.
[11]	Y. Cho, G. Hwang, S. Kwon and S. Lee, Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete Contin. Dyn. Syst., 35 (2015), 2863-2880. doi: 10.3934/dcds.2015.35.2863.
[12]	Y. Cho, G. Hwang, S. Kwon and S. Lee, Profile decompositions of fractional Schrödinger equations with angularly regular data, J. Differential Equations, 256 (2014), 3011-3037. doi: 10.1016/j.jde.2014.01.030.
[13]	Y. Cho, G. Hwang and T. Ozawa, On the focusing energy-critical fractional nonlinear Schrödinger equations, Adv. Differential Equations, 23 (2018), 161-192.
[14]	Y. Cho and S. Lee, Strichartz estimates in spherical coordinates, Indiana Univ. Math. J., 62 (2013), 991-1020. doi: 10.1512/iumj.2013.62.4970.
[15]	Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365. doi: 10.1142/S0219199709003399.
[16]	Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128. doi: 10.3934/cpaa.2011.10.1121.
[17]	M. Christ and 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.
[18]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $ \mathbb R^3$, Comm. Pure Appl. Math., 57 (2004), 987-1014. doi: 10.1002/cpa.20029.
[19]	V. D. Dinh, Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev spaces, Int. J. Appl. Math., 31 (2018), 483-525.
[20]	V. D. Dinh, On blowup solutions to the focusing mass-critical nonlinear fractional Schrödinger equation, Commun. Pure Appl. Anal., 18 (2019), 689-708.
[21]	V. D. Dinh, A study on blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schrödinger equation, J. Math. Phys., 59 (2018), 071506, 25pp. doi: 10.1063/1.5027713.
[22]	B. Feng, On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities, J. Evol. Equ., 18 (2018), 203-220. doi: 10.1007/s00028-017-0397-z.
[23]	B. Feng, On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities, Commun. Pure Appl. Anal., 17 (2018), 1785-1804. doi: 10.3934/cpaa.2018085.
[24]	B. Feng and H. Zhang, Stability of standing waves for the fractional Schrödinger-Hartree equation, J. Math. Anal. Appl., 460 (2018), 352-364. doi: 10.1016/j.jmaa.2017.11.060.
[25]	B. Feng and 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.
[26]	R. L. Frank and E. Lenzmann, Uniqueness of nonlinear ground states for fractional Laplacians in $ \mathbb R$, Acta Math..
[27]	R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726. doi: 10.1002/cpa.21591.
[28]	J. Fröhlich, G. Jonsson and E. Lenzmann, Boson stars as solitary waves, Comm. Math. Phys., 274 (2007), 1-30. doi: 10.1007/s00220-007-0272-9.
[29]	J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z., 189 (1985), 487-505. doi: 10.1007/BF01168155.
[30]	Z. Guo, Y. Sire, Y. Wang and L. Zhao, On the energy-critical fractional Schrödinger equation in the radial case, Dyn. Partial Differ. Equ., 15 (2018), 265-282. doi: 10.4310/DPDE.2018.v15.n4.a2.
[31]	Z. Guo and Y. Wang, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations, J. Anal. Math., 124 (2014), 1-38. doi: 10.1007/s11854-014-0025-6.
[32]	Q. Guo and S. Zhu, Sharp threshold of blow-up and scattering for the fractional Hartree equation, J. Differential Equations, 264 (2018), 2802-2832. doi: 10.1016/j.jde.2017.11.001.
[33]	Q. Guo and S. Zhu, Sharp criteria of scattering for the fractional NLS, preprint, arXiv: 1706.02549.
[34]	Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14 (2015), 2265-2282. doi: 10.3934/cpaa.2015.14.2265.
[35]	Y. Ke, Remark on the Strichartz estimates in the radial case, J. Math. Anal. Appl., 387 (2012), 857-861. doi: 10.1016/j.jmaa.2011.09.039.
[36]	K. Kirkpatrick, E. Lenzmann and G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys., 317 (2013), 563-591. doi: 10.1007/s00220-012-1621-x.
[37]	N. Laskin, Fractional Quantum Mechanics and Lévy Path Integrals, Physics Letter A, 268 (2000), 298-304. doi: 10.1016/S0375-9601(00)00201-2.
[38]	N. Laskin, Fractional Schrödinger equations, Physics Review E, 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108.
[39]	F. Merle and P. Raphaël, Blow up of critical norm for some radial $L^2$ super critical nonlinear Schrödinger equations, Amer. J. Math., 130 (2008), 945-978. doi: 10.1353/ajm.0.0012.
[40]	C. Miao, G. Xu and L. Zhao, The dynamics of the 3D radial NLS with combined terms, Comm. Math. Phys., 318 (2013), 767-808. doi: 10.1007/s00220-013-1677-2.
[41]	C. Miao, T. Zhao and J. Zheng, On the 4D nonlinear Schrödinger equation with combined terms under the energy threshold, Calc. Var. Partial Differential Equations, 56 (2017), Art. 179, 39 pp. doi: 10.1007/s00526-017-1264-z.
[42]	C. Peng and Q. Shi, Stability of standing waves for the fractional nonlinear Schrödinger equation, J. Math. Phys., 59 (2018), 011508, 11pp. doi: 10.1063/1.5021689.
[43]	C. Sun, H. Wang, X. Yao and J. Zheng, Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., 38 (2018), 2207-2228. doi: 10.3934/dcds.2018091.
[44]	T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343. doi: 10.1080/03605300701588805.
[45]	H. Triebel, Theory of Function Spaces, Basel Birkhäuser, 1983. doi: 10.1007/978-3-0346-0416-1.
[46]	G. X. Xu and J. W. Yang, Long time dynamics of the 3D radial NLS with the combined terms, Acta Math. Sin. (Engl. Ser.), 32 (2016), 521-540. doi: 10.1007/s10114-016-5401-y.
[47]	J. Zhang and S. Zhu, Stability of standing waves for the nonlinear fractional Schrödinger equation, J. Dynam. Differential Equations, 29 (2017), 1017-1030. doi: 10.1007/s10884-015-9477-3.
[48]	S. Zhu, On the blow-up solutions for the nonlinear fractional Schrödinger equation, J. Differential Equations, 261 (2016), 1506-1531. doi: 10.1016/j.jde.2016.04.007.
[49]	S. Zhu, Existence of stable standing waves for the fractional Schrödinger equations with combined nonlinearities, J. Evol. Equ., 17 (2017), 1003-1021. doi: 10.1007/s00028-016-0363-1.

show all references

References:

[1]	J. Bergh and J. Löfstöm, Interpolation Spaces-An Introduction, Springer-Verlag, Berlin, 1976.
[2]	S. Bhattarai, On fractional Schrödinger systems of Choquard type, J. Differential Equations, 263 (2017), 3197-3229. doi: 10.1016/j.jde.2017.04.034.
[3]	T. Boulenger, D. Himmelsbach and E. Lenzmann, Blowup for fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603. doi: 10.1016/j.jfa.2016.08.011.
[4]	T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A.
[5]	T. Cazenave, Semilinear Schrödinger Equations, Courant Lectures Notes in Mathematics 10, New York, AMS, 2003. doi: 10.1090/cln/010.
[6]	X. Cheng, C. Miao and L. Zhao, Global well-posedness and scattering for nonlinear Schrödinger equations with combined nonlinearities in the radial case, J. Differential Equations, 261 (2016), 2881-2934. doi: 10.1016/j.jde.2016.04.031.
[7]	Y. Cho, Short-range scattering of Hartree type fractional NLS, J. Differential Equations, 262 (2017), 116-144. doi: 10.1016/j.jde.2016.09.025.
[8]	Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa, On the Cauchy problem of fractional Schrödinger equations with Hartree type nonlimearity, Funkcial. Ekvac., 56 (2013), 193-224. doi: 10.1619/fesi.56.193.
[9]	Y. Cho, G. Hwang, S. Kwon and S. Lee, Profile decompositions and blow-up phenomena of mass critical fractional Schrödinger equations, Nonlinear Anal., 86 (2013), 12-29. doi: 10.1016/j.na.2013.03.002.
[10]	Y. Cho, G. Hwang, S. Kwon and S. Lee, On finite time blow-up for the mass-critical Hartree equations, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 467-479. doi: 10.1017/S030821051300142X.
[11]	Y. Cho, G. Hwang, S. Kwon and S. Lee, Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete Contin. Dyn. Syst., 35 (2015), 2863-2880. doi: 10.3934/dcds.2015.35.2863.
[12]	Y. Cho, G. Hwang, S. Kwon and S. Lee, Profile decompositions of fractional Schrödinger equations with angularly regular data, J. Differential Equations, 256 (2014), 3011-3037. doi: 10.1016/j.jde.2014.01.030.
[13]	Y. Cho, G. Hwang and T. Ozawa, On the focusing energy-critical fractional nonlinear Schrödinger equations, Adv. Differential Equations, 23 (2018), 161-192.
[14]	Y. Cho and S. Lee, Strichartz estimates in spherical coordinates, Indiana Univ. Math. J., 62 (2013), 991-1020. doi: 10.1512/iumj.2013.62.4970.
[15]	Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365. doi: 10.1142/S0219199709003399.
[16]	Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128. doi: 10.3934/cpaa.2011.10.1121.
[17]	M. Christ and 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.
[18]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $ \mathbb R^3$, Comm. Pure Appl. Math., 57 (2004), 987-1014. doi: 10.1002/cpa.20029.
[19]	V. D. Dinh, Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev spaces, Int. J. Appl. Math., 31 (2018), 483-525.
[20]	V. D. Dinh, On blowup solutions to the focusing mass-critical nonlinear fractional Schrödinger equation, Commun. Pure Appl. Anal., 18 (2019), 689-708.
[21]	V. D. Dinh, A study on blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schrödinger equation, J. Math. Phys., 59 (2018), 071506, 25pp. doi: 10.1063/1.5027713.
[22]	B. Feng, On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities, J. Evol. Equ., 18 (2018), 203-220. doi: 10.1007/s00028-017-0397-z.
[23]	B. Feng, On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities, Commun. Pure Appl. Anal., 17 (2018), 1785-1804. doi: 10.3934/cpaa.2018085.
[24]	B. Feng and H. Zhang, Stability of standing waves for the fractional Schrödinger-Hartree equation, J. Math. Anal. Appl., 460 (2018), 352-364. doi: 10.1016/j.jmaa.2017.11.060.
[25]	B. Feng and 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.
[26]	R. L. Frank and E. Lenzmann, Uniqueness of nonlinear ground states for fractional Laplacians in $ \mathbb R$, Acta Math..
[27]	R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726. doi: 10.1002/cpa.21591.
[28]	J. Fröhlich, G. Jonsson and E. Lenzmann, Boson stars as solitary waves, Comm. Math. Phys., 274 (2007), 1-30. doi: 10.1007/s00220-007-0272-9.
[29]	J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z., 189 (1985), 487-505. doi: 10.1007/BF01168155.
[30]	Z. Guo, Y. Sire, Y. Wang and L. Zhao, On the energy-critical fractional Schrödinger equation in the radial case, Dyn. Partial Differ. Equ., 15 (2018), 265-282. doi: 10.4310/DPDE.2018.v15.n4.a2.
[31]	Z. Guo and Y. Wang, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations, J. Anal. Math., 124 (2014), 1-38. doi: 10.1007/s11854-014-0025-6.
[32]	Q. Guo and S. Zhu, Sharp threshold of blow-up and scattering for the fractional Hartree equation, J. Differential Equations, 264 (2018), 2802-2832. doi: 10.1016/j.jde.2017.11.001.
[33]	Q. Guo and S. Zhu, Sharp criteria of scattering for the fractional NLS, preprint, arXiv: 1706.02549.
[34]	Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14 (2015), 2265-2282. doi: 10.3934/cpaa.2015.14.2265.
[35]	Y. Ke, Remark on the Strichartz estimates in the radial case, J. Math. Anal. Appl., 387 (2012), 857-861. doi: 10.1016/j.jmaa.2011.09.039.
[36]	K. Kirkpatrick, E. Lenzmann and G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys., 317 (2013), 563-591. doi: 10.1007/s00220-012-1621-x.
[37]	N. Laskin, Fractional Quantum Mechanics and Lévy Path Integrals, Physics Letter A, 268 (2000), 298-304. doi: 10.1016/S0375-9601(00)00201-2.
[38]	N. Laskin, Fractional Schrödinger equations, Physics Review E, 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108.
[39]	F. Merle and P. Raphaël, Blow up of critical norm for some radial $L^2$ super critical nonlinear Schrödinger equations, Amer. J. Math., 130 (2008), 945-978. doi: 10.1353/ajm.0.0012.
[40]	C. Miao, G. Xu and L. Zhao, The dynamics of the 3D radial NLS with combined terms, Comm. Math. Phys., 318 (2013), 767-808. doi: 10.1007/s00220-013-1677-2.
[41]	C. Miao, T. Zhao and J. Zheng, On the 4D nonlinear Schrödinger equation with combined terms under the energy threshold, Calc. Var. Partial Differential Equations, 56 (2017), Art. 179, 39 pp. doi: 10.1007/s00526-017-1264-z.
[42]	C. Peng and Q. Shi, Stability of standing waves for the fractional nonlinear Schrödinger equation, J. Math. Phys., 59 (2018), 011508, 11pp. doi: 10.1063/1.5021689.
[43]	C. Sun, H. Wang, X. Yao and J. Zheng, Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., 38 (2018), 2207-2228. doi: 10.3934/dcds.2018091.
[44]	T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343. doi: 10.1080/03605300701588805.
[45]	H. Triebel, Theory of Function Spaces, Basel Birkhäuser, 1983. doi: 10.1007/978-3-0346-0416-1.
[46]	G. X. Xu and J. W. Yang, Long time dynamics of the 3D radial NLS with the combined terms, Acta Math. Sin. (Engl. Ser.), 32 (2016), 521-540. doi: 10.1007/s10114-016-5401-y.
[47]	J. Zhang and S. Zhu, Stability of standing waves for the nonlinear fractional Schrödinger equation, J. Dynam. Differential Equations, 29 (2017), 1017-1030. doi: 10.1007/s10884-015-9477-3.
[48]	S. Zhu, On the blow-up solutions for the nonlinear fractional Schrödinger equation, J. Differential Equations, 261 (2016), 1506-1531. doi: 10.1016/j.jde.2016.04.007.
[49]	S. Zhu, Existence of stable standing waves for the fractional Schrödinger equations with combined nonlinearities, J. Evol. Equ., 17 (2017), 1003-1021. doi: 10.1007/s00028-016-0363-1.

[1]	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
[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]	Jinmyong An, Roesong Jang, Jinmyong Kim. Global existence and blow-up for the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022111
[4]	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
[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]	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
[8]	Nadjat Doudi, Salah Boulaaras, Nadia Mezouar, Rashid Jan. Global existence, general decay and blow-up for a nonlinear wave equation with logarithmic source term and fractional boundary dissipation. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022106
[9]	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
[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]	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
[12]	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
[13]	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
[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]	Wenjun Liu, Jiangyong Yu, Gang Li. Global existence, exponential decay and blow-up of solutions for a class of fractional pseudo-parabolic equations with logarithmic nonlinearity. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4337-4366. doi: 10.3934/dcdss.2021121
[16]	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
[17]	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
[18]	Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
[19]	Lili Du, Zheng-An Yao. Localization of blow-up points for a nonlinear nonlocal porous medium equation. Communications on Pure and Applied Analysis, 2007, 6 (1) : 183-190. doi: 10.3934/cpaa.2007.6.183
[20]	Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Van Duong Dinh Binhua Feng
on Google Scholar
Van Duong Dinh Binhua Feng

2021 Impact Factor: 1.588

Tools

Article outline

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Van Duong Dinh Binhua Feng
on Google Scholar
Van Duong Dinh Binhua Feng

[Back to Top]