• Previous Article
    On the global convergence of frequency synchronization for Kuramoto and Winfree oscillators
  • DCDS-B Home
  • This Issue
  • Next Article
    Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source
July 2019, 24(7): 3335-3356. doi: 10.3934/dcdsb.2018323

Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation

1. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730070, China

2. 

School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741000, China

* Corresponding author: Congming Peng

Received  March 2018 Published  January 2019

Fund Project: This work is supported by NSFC Grants (No. 11475073, No. 11561059)

In this article, we study the initial-value problem for inhomogeneous fractional nonlinear Schrödinger equation
$ i\partial_{t}u = (-\Delta)^{s}u-|x|^{-b}|u|^{2\sigma}u, \, \, \, (t, x)\in \mathbb{R} \times \mathbb{R}^{N}, $
where
$ \frac{1}{2}<s<1, $
$ N\geq2 $
and
$ \frac{2s-b}{N}\leq \sigma<\frac{2s-b}{N-2s} $
. We prove a Gagliardo-Nirenberg-type estimate and use it to establish sufficient conditions for global existence in
$ H^{s}(\mathbb{R}^{N}) $
. In addition, we derive a localized Virial estimate for inhomogeneous fractional nonlinear Schrödinger equation in
$ \mathbb{R}^{N} $
, which uses Balakrishnan's formula for the fractional Laplacian
$ (-\Delta)^{s} $
from semigroup theory. By these estimates, we give the blowup criterion of radial solutions in
$ \mathbb{R}^{N} $
for
$ L^{2} $
-critical,
$ L^{2} $
-supercritical and
$ H^{s} $
-subcritical power.
Citation: Congming Peng, Dun Zhao. Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3335-3356. doi: 10.3934/dcdsb.2018323
References:
[1]

T. BoulengerD. Himmelsbach and E. Lenzmann, Blowup for fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603. doi: 10.1016/j.jfa.2016.08.011.

[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. doi: 10.1090/cln/010.

[3]

J. Q. Chen and B. L. Guo, Sharp constant of improved Gagliardo-Nirenberg inequality and its application, Annali di Matematica, 190 (2011), 341-354. doi: 10.1007/s10231-010-0152-3.

[4]

J. Q. Chen and B. L. Guo, Sharp global existence and blowing up results for inhomogeneous Schrödinger equations, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 357-367. doi: 10.3934/dcdsb.2007.8.357.

[5]

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.

[6]

Y. Cho and T. Ozawa, Short-range scattering of Hartree type fractional NLS Ⅱ, Nonlinear Anal., 157 (2017), 62-75. doi: 10.1016/j.na.2017.03.005.

[7]

Y. ChoG. HwangS. 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.

[8]

Y. ChoH. HajaiejG. 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. ChoH. HajaiejG. Hwang and T. Ozawa, On the orbital stability of fractional Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 1267-1282. doi: 10.3934/cpaa.2014.13.1267.

[10]

Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365. doi: 10.1142/S0219199709003399.

[11]

L. G. Farah, Global well-posedness and blow-up on the energy space for the inhomogeneous nonlinear Schrödinger equation, J. Evol. Equ., 16 (2016), 193-208. doi: 10.1007/s00028-015-0298-y.

[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.jmaa.2017.11.060.

[13]

B. H. Feng and H. Z. Zhang, Ground states for the fractional Schrödinger equation, Electron. J. Differ. Eq., 127 (2013), 11pp.

[14]

B. H. 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.

[15]

B. H. Feng and H. Z. 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.

[16]

F. Genoud, An Inhomogeneous L2-critical nonlinear Schrödinger equations, Z.anal.anwend., 31 (2012), 283-290. doi: 10.4171/ZAA/1460.

[17]

B. L. Guo and Z. H. Huo, Global well-posedness for the fractional nonlinear Schrödinger equations, Comm.Partial Differential Equations, 36 (2010), 247-255. doi: 10.1080/03605302.2010.503769.

[18]

Z. H. Guo and Y. Z. Wang, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equation, Journal d'Analyse Mathématique, 124 (2014), 1-38. doi: 10.1007/s11854-014-0025-6.

[19]

J. Holmer and S. Roudenko, On blow-up solutions to the 3D cubic nonlinear Schrödinger equation, AMRX.Appl.Math.Res.Express, (2007), Art. ID abm004, 31pp.

[20]

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.

[21]

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, 26pp. doi: 10.1098/rspa.2014.0364.

[22]

J. KriegerE. Lenzmann and P. Raphaël, Nondispersive solutions to the L2-critical half-wave equation, Arch. Ration. Mech. Anal., 209 (2013), 61-129. doi: 10.1007/s00205-013-0620-1.

[23]

N. Laskin, Fractional quantum mechanics and Lèvy path integrals, Physics Letter A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2.

[24]

N. Laskin, Fractional Schrödinger equations, Physics Review E, 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108.

[25]

F. Merle, Nonxistence of minimal blow up solutions of equation $i\partial_{t}u = -\Delta u-k(x)|u|^\frac{4}{N}u$ in $\mathbb{R}^{N}$, Ann.Inst.Henri Poincaré, 64 (1996), 33–85.

[26]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330. doi: 10.1016/0022-0396(91)90052-B.

[27]

C. M. Peng and Q. H. Shi, Stability of standing wave for the fractional nonlinear Schrödinger equation, J. Math. Phys., 59 (2018), 011508, 11pp. doi: 10.1063/1.5021689.

[28]

P. Raphaël and S. Jermeie, Existence and uniqueness of minimal blow up solutions to an inhomogeneous mass-critical NLS, J. Amer. Math. Soc., 24 (2011), 471-546. doi: 10.1090/S0894-0347-2010-00688-1.

[29]

T. Saanouni, Remark on the inhomogeneous fractional nonlinear Schrödinger equations, J. Math. Phys., 57 (2016), 081503, 14 pp. doi: 10.1063/1.4960045.

[30]

Q. H. Shi and S. Wang, Nonrelativistic approximation in the energy space for KGS system, J. Math. Anal. Appl., 462 (2018), 1242-1253. doi: 10.1016/j.jmaa.2018.02.039.

[31]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576. doi: 10.1007/BF01208265.

[32]

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.

[33]

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.

[34]

S. H. 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]

T. BoulengerD. Himmelsbach and E. Lenzmann, Blowup for fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603. doi: 10.1016/j.jfa.2016.08.011.

[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. doi: 10.1090/cln/010.

[3]

J. Q. Chen and B. L. Guo, Sharp constant of improved Gagliardo-Nirenberg inequality and its application, Annali di Matematica, 190 (2011), 341-354. doi: 10.1007/s10231-010-0152-3.

[4]

J. Q. Chen and B. L. Guo, Sharp global existence and blowing up results for inhomogeneous Schrödinger equations, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 357-367. doi: 10.3934/dcdsb.2007.8.357.

[5]

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.

[6]

Y. Cho and T. Ozawa, Short-range scattering of Hartree type fractional NLS Ⅱ, Nonlinear Anal., 157 (2017), 62-75. doi: 10.1016/j.na.2017.03.005.

[7]

Y. ChoG. HwangS. 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.

[8]

Y. ChoH. HajaiejG. 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. ChoH. HajaiejG. Hwang and T. Ozawa, On the orbital stability of fractional Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 1267-1282. doi: 10.3934/cpaa.2014.13.1267.

[10]

Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365. doi: 10.1142/S0219199709003399.

[11]

L. G. Farah, Global well-posedness and blow-up on the energy space for the inhomogeneous nonlinear Schrödinger equation, J. Evol. Equ., 16 (2016), 193-208. doi: 10.1007/s00028-015-0298-y.

[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.jmaa.2017.11.060.

[13]

B. H. Feng and H. Z. Zhang, Ground states for the fractional Schrödinger equation, Electron. J. Differ. Eq., 127 (2013), 11pp.

[14]

B. H. 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.

[15]

B. H. Feng and H. Z. 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.

[16]

F. Genoud, An Inhomogeneous L2-critical nonlinear Schrödinger equations, Z.anal.anwend., 31 (2012), 283-290. doi: 10.4171/ZAA/1460.

[17]

B. L. Guo and Z. H. Huo, Global well-posedness for the fractional nonlinear Schrödinger equations, Comm.Partial Differential Equations, 36 (2010), 247-255. doi: 10.1080/03605302.2010.503769.

[18]

Z. H. Guo and Y. Z. Wang, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equation, Journal d'Analyse Mathématique, 124 (2014), 1-38. doi: 10.1007/s11854-014-0025-6.

[19]

J. Holmer and S. Roudenko, On blow-up solutions to the 3D cubic nonlinear Schrödinger equation, AMRX.Appl.Math.Res.Express, (2007), Art. ID abm004, 31pp.

[20]

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.

[21]

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, 26pp. doi: 10.1098/rspa.2014.0364.

[22]

J. KriegerE. Lenzmann and P. Raphaël, Nondispersive solutions to the L2-critical half-wave equation, Arch. Ration. Mech. Anal., 209 (2013), 61-129. doi: 10.1007/s00205-013-0620-1.

[23]

N. Laskin, Fractional quantum mechanics and Lèvy path integrals, Physics Letter A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2.

[24]

N. Laskin, Fractional Schrödinger equations, Physics Review E, 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108.

[25]

F. Merle, Nonxistence of minimal blow up solutions of equation $i\partial_{t}u = -\Delta u-k(x)|u|^\frac{4}{N}u$ in $\mathbb{R}^{N}$, Ann.Inst.Henri Poincaré, 64 (1996), 33–85.

[26]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330. doi: 10.1016/0022-0396(91)90052-B.

[27]

C. M. Peng and Q. H. Shi, Stability of standing wave for the fractional nonlinear Schrödinger equation, J. Math. Phys., 59 (2018), 011508, 11pp. doi: 10.1063/1.5021689.

[28]

P. Raphaël and S. Jermeie, Existence and uniqueness of minimal blow up solutions to an inhomogeneous mass-critical NLS, J. Amer. Math. Soc., 24 (2011), 471-546. doi: 10.1090/S0894-0347-2010-00688-1.

[29]

T. Saanouni, Remark on the inhomogeneous fractional nonlinear Schrödinger equations, J. Math. Phys., 57 (2016), 081503, 14 pp. doi: 10.1063/1.4960045.

[30]

Q. H. Shi and S. Wang, Nonrelativistic approximation in the energy space for KGS system, J. Math. Anal. Appl., 462 (2018), 1242-1253. doi: 10.1016/j.jmaa.2018.02.039.

[31]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576. doi: 10.1007/BF01208265.

[32]

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.

[33]

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.

[34]

S. H. 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]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

Huiling Li, Mingxin Wang. Properties of blow-up solutions to a parabolic system with nonlinear localized terms. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 683-700. doi: 10.3934/dcds.2005.13.683

[7]

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

[8]

Jianqing Chen, Boling Guo. Sharp global existence and blowing up results for inhomogeneous Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 357-367. doi: 10.3934/dcdsb.2007.8.357

[9]

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

[10]

Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058

[11]

Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072

[12]

Yue Liu. Existence of unstable standing waves for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (1) : 193-209. doi: 10.3934/cpaa.2008.7.193

[13]

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

[14]

Guangyu Xu, Jun Zhou. Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1805-1820. doi: 10.3934/cpaa.2018086

[15]

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

[16]

Lili Du, Chunlai Mu, Zhaoyin Xiang. Global existence and blow-up to a reaction-diffusion system with nonlinear memory. Communications on Pure & Applied Analysis, 2005, 4 (4) : 721-733. doi: 10.3934/cpaa.2005.4.721

[17]

Monica Marras, Stella Vernier Piro. On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients. Conference Publications, 2013, 2013 (special) : 535-544. doi: 10.3934/proc.2013.2013.535

[18]

Shuyin Wu, Joachim Escher, Zhaoyang Yin. Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 633-645. doi: 10.3934/dcdsb.2009.12.633

[19]

Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519

[20]

Hua Chen, Huiyang Xu. Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1185-1203. doi: 10.3934/dcds.2019051

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (71)
  • HTML views (316)
  • Cited by (0)

Other articles
by authors

[Back to Top]