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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  July 2019 Early access  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 and 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.

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