Positive radial solutions of a nonlinear boundary value problem
CPAA Home
Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy
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, $
$ 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}$
$p_2 = \frac{2s}{N}$
. 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
N. Laskin, Fractional Schrödinger equations, Physics Review E, 66 (2002), 056108. doi: 10.1016/j.jde.2003.12.002. Google Scholar
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Van Duong Dinh, Binhua Feng. On fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4565-4612. doi: 10.3934/dcds.2019188
Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027
Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034
Dapeng Du, Yifei Wu, Kaijun Zhang. On blow-up criterion for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3639-3650. doi: 10.3934/dcds.2016.36.3639
Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903
J. Cuevas, J. C. Eilbeck, N. I. Karachalios. Thresholds for breather solutions of the discrete nonlinear Schrödinger equation with saturable and power nonlinearity. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 445-475. doi: 10.3934/dcds.2008.21.445
Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1875-1897. doi: 10.3934/cpaa.2018089
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 & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119
Frank Merle, Hatem Zaag. O.D.E. type behavior of blow-up solutions of nonlinear heat equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 435-450. doi: 10.3934/dcds.2002.8.435
Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
Qiong Chen, Chunlai Mu, Zhaoyin Xiang. Blow-up and asymptotic behavior of solutions to a semilinear integrodifferential system. Communications on Pure & Applied Analysis, 2006, 5 (3) : 435-446. doi: 10.3934/cpaa.2006.5.435
Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881
Jianqing Chen. Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1613-1628. doi: 10.3934/dcdss.2016066
Laurent Di Menza, Olivier Goubet. Stabilizing blow up solutions to nonlinear schrÖdinger equations. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1059-1082. doi: 10.3934/cpaa.2017051
István Győri, Yukihiko Nakata, Gergely Röst. Unbounded and blow-up solutions for a delay logistic equation with positive feedback. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2845-2854. doi: 10.3934/cpaa.2018134
Cristophe Besse, Rémi Carles, Norbert J. Mauser, Hans Peter Stimming. Monotonicity properties of the blow-up time for nonlinear Schrödinger equations: Numerical evidence. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 11-36. doi: 10.3934/dcdsb.2008.9.11
Zaihui Gan, Jian Zhang. Blow-up, global existence and standing waves for the magnetic nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 827-846. doi: 10.3934/dcds.2012.32.827
Yūki Naito, Takasi Senba. Blow-up behavior of solutions to a parabolic-elliptic system on higher dimensional domains. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3691-3713. doi: 10.3934/dcds.2012.32.3691
Nicola Abatangelo. Large $s$-harmonic functions and boundary blow-up solutions for the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5555-5607. doi: 10.3934/dcds.2015.35.5555
Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843
Binhua Feng