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September  2018, 17(5): 1785-1804. doi: 10.3934/cpaa.2018085

On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities

Binhua Feng ,

Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China

* Corresponding author: Binhua Feng

Received  May 2017 Revised  December 2017 Published  April 2018

Fund Project: This work is supported by NSFC Grants (No. 11601435, No. 11475073), Gansu Provincial Natural Science Foundation (1606RJZA010) and NWNU-LKQN-14-6

This paper is devoted to the analysis of blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities
$i\partial_t u-(-Δ)^su+λ_1|u|^{2p_1}u+λ_2|u|^{2p_2}u = 0, $
where
$ 0<p_1<p_2<\frac{2s}{N-2s}$
. Firstly, we obtain some sufficient conditions about existence of blow-up solutions, and then derive some sharp thresholds of blow-up and global existence by constructing some new estimates. Moreover, we find the sharp threshold mass of blow-up and global existence in the case
$ 0<p_1<\frac{2s}{N}$
and
$p_2 = \frac{2s}{N}$
. Finally, we investigate the dynamical properties of blow-up solutions, including
$L^2$
-concentration, blow-up rate and limiting profile.
Keywords: The fractional Schrödinger equation, Blow-up solutions, Combined power-type nonlinearities, Sharp thresholds, The dynamical behavior.
Mathematics Subject Classification: 35Q51, 35Q55.
Citation: Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085
References:
[1]

T. Boulenger, D. Himmelsbach and E. Lenzmann, Blowup for fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[2]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. Google Scholar

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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: 10.1016/j.jde.2003.12.002.  Google Scholar

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Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa, On the orbital stability of fractional Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 1267-1282.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

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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.jde.2003.12.002.  Google Scholar

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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: 10.1016/j.jde.2003.12.002.  Google Scholar

[7]

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: 10.1016/j.jde.2003.12.002.  Google Scholar

[8]

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.jde.2003.12.002.  Google Scholar

[9]

B. Feng, On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities, Journal of Evolution Equations, 18 (2018), 203-220.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[10]

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.jde.2003.12.002.  Google Scholar

[11]

B. Feng and X. Yuan, On the Cauchy problem for the Schrödinger-Hartree equation, Evol. Equ. Control Theory, 4 (2015), 431-445.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[12]

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.jde.2003.12.002.  Google Scholar

[13]

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.jde.2003.12.002.  Google Scholar

[14]

B. Feng, D. Zhao and C. Sun, On the Cauchy problem for the nonlinear Schrödinger equations with time-dependent linear loss/gain, J. Math. Anal. Appl., 416 (2014), 901-923.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[15]

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/j.jde.2003.12.002.  Google Scholar

[16]

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Q. Guo and S. Zhu, Sharp threshold of blow-up and scattering for the fractional Hartree equation, Journal of Differential Equations, 264 (2018), 2802-2832.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[18]

T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, International Mathematics Research Notices, 46 (2005), 2815-2828.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[19]

Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14 (2015), 2265-2282.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[20]

C. Klein, C. Sparber and P. Markowich, Numerical study of fractional nonlinear Schrödinger equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 20140364. doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[21]

E. A. Kuznetsov, J. J. Rasmussen, K. Rypdal and S. K. Turitsyn, Sharper criteria for the wave collapse, Physica D, 87 (1995), 273-284.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[22]

N. Laskin, Fractional Quantum Mechanics and Lévy Path Integrals, Physics Letter A, 268 (2000), 298-304.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[23]

N. Laskin, Fractional Schrödinger equations, Physics Review E, 66 (2002), 056108. doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[24]

J. Liu and A. Qian, Ground state solution for a Schrödinger-Poisson equation with critical growth, Nonlinear Anal. Real World Appl., 40 (2018), 428-443.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[25]

A. Mao, L. Yang, A. Qian and S. Luan, Shixia Existence and concentration of solutions of Schrödinger-Poisson system, Appl. Math. Lett., 68 (2017), 8-12.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[26]

F. Merle and P. Raphaël, On universality of blow-up profile for $L^2$ critical nonlinear Schrödinger equation, Invent. Math., 156 (2004), 565-572.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[27]

F. Merle and P. Raphaël, Blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. Math., 16 (2005), 157-222.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[28]

F. Merle and P. Raphaël, On a sharp lower bound on the blow-up rate for the $L^2$ critical nonlinear Schrödinger equation, J. Amer. Soc., 19 (2006), 37-90.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[29]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[30]

C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation, Applied Mathematical Sciences, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[31]

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: 10.1016/j.jde.2003.12.002.  Google Scholar

[32]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[33]

M. I. Weinstein, On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations, Comm. Partial Differential Equations, 11 (1986), 545-565.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[34]

J. Zhang, Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations, Nonlinear Anal., 48 (2002), 191-207.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[35]

J. Zhang and S. Zhu, Stability of standing waves for the nonlinear fractional Schrödinger equation, J. Dynam. Differential Equations, 29 (2017), 1017-1030 doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[36]

J. Zhang and S. Zhu, Sharp blow-up criteria for the Davey-Stewartson system in $\mathbb{R}^3$, Dynamics of PDE, 8 (2011), 239-260.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[37]

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.2003.12.002.  Google Scholar

[38]

S. Zhu, On the Davey-Stewartson system with competing nonlinearities, J. Math. Phys., 57 (2016), 031501. doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[39]

S. Zhu, Existence of stable standing waves for the fractional Schrödinger equations with combined nonlinearities, J. Evol. Equ., 17 (2017), 1003-1021 doi: 10.1016/j.jde.2003.12.002.  Google Scholar

show all references

References:
[1]

T. Boulenger, D. Himmelsbach and E. Lenzmann, Blowup for fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[2]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. Google Scholar

[3]

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: 10.1016/j.jde.2003.12.002.  Google Scholar

[4]

Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa, On the orbital stability of fractional Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 1267-1282.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[5]

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.jde.2003.12.002.  Google Scholar

[6]

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: 10.1016/j.jde.2003.12.002.  Google Scholar

[7]

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: 10.1016/j.jde.2003.12.002.  Google Scholar

[8]

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.jde.2003.12.002.  Google Scholar

[9]

B. Feng, On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities, Journal of Evolution Equations, 18 (2018), 203-220.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[10]

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.jde.2003.12.002.  Google Scholar

[11]

B. Feng and X. Yuan, On the Cauchy problem for the Schrödinger-Hartree equation, Evol. Equ. Control Theory, 4 (2015), 431-445.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[12]

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.jde.2003.12.002.  Google Scholar

[13]

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.jde.2003.12.002.  Google Scholar

[14]

B. Feng, D. Zhao and C. Sun, On the Cauchy problem for the nonlinear Schrödinger equations with time-dependent linear loss/gain, J. Math. Anal. Appl., 416 (2014), 901-923.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[15]

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/j.jde.2003.12.002.  Google Scholar

[16]

R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., 18 (1977), 1794-1797.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[17]

Q. Guo and S. Zhu, Sharp threshold of blow-up and scattering for the fractional Hartree equation, Journal of Differential Equations, 264 (2018), 2802-2832.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[18]

T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, International Mathematics Research Notices, 46 (2005), 2815-2828.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[19]

Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14 (2015), 2265-2282.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[20]

C. Klein, C. Sparber and P. Markowich, Numerical study of fractional nonlinear Schrödinger equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 20140364. doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[21]

E. A. Kuznetsov, J. J. Rasmussen, K. Rypdal and S. K. Turitsyn, Sharper criteria for the wave collapse, Physica D, 87 (1995), 273-284.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[22]

N. Laskin, Fractional Quantum Mechanics and Lévy Path Integrals, Physics Letter A, 268 (2000), 298-304.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[23]

N. Laskin, Fractional Schrödinger equations, Physics Review E, 66 (2002), 056108. doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[24]

J. Liu and A. Qian, Ground state solution for a Schrödinger-Poisson equation with critical growth, Nonlinear Anal. Real World Appl., 40 (2018), 428-443.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[25]

A. Mao, L. Yang, A. Qian and S. Luan, Shixia Existence and concentration of solutions of Schrödinger-Poisson system, Appl. Math. Lett., 68 (2017), 8-12.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[26]

F. Merle and P. Raphaël, On universality of blow-up profile for $L^2$ critical nonlinear Schrödinger equation, Invent. Math., 156 (2004), 565-572.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[27]

F. Merle and P. Raphaël, Blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. Math., 16 (2005), 157-222.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[28]

F. Merle and P. Raphaël, On a sharp lower bound on the blow-up rate for the $L^2$ critical nonlinear Schrödinger equation, J. Amer. Soc., 19 (2006), 37-90.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[29]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[30]

C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation, Applied Mathematical Sciences, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[31]

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: 10.1016/j.jde.2003.12.002.  Google Scholar

[32]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[33]

M. I. Weinstein, On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations, Comm. Partial Differential Equations, 11 (1986), 545-565.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[34]

J. Zhang, Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations, Nonlinear Anal., 48 (2002), 191-207.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[35]

J. Zhang and S. Zhu, Stability of standing waves for the nonlinear fractional Schrödinger equation, J. Dynam. Differential Equations, 29 (2017), 1017-1030 doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[36]

J. Zhang and S. Zhu, Sharp blow-up criteria for the Davey-Stewartson system in $\mathbb{R}^3$, Dynamics of PDE, 8 (2011), 239-260.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[37]

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.2003.12.002.  Google Scholar

[38]

S. Zhu, On the Davey-Stewartson system with competing nonlinearities, J. Math. Phys., 57 (2016), 031501. doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[39]

S. Zhu, Existence of stable standing waves for the fractional Schrödinger equations with combined nonlinearities, J. Evol. Equ., 17 (2017), 1003-1021 doi: 10.1016/j.jde.2003.12.002.  Google Scholar

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