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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

Congming Peng 1,2,, and  Dun Zhao 1,

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.
Keywords: Inhomogeneous fractional nonlinear Schrödinger equation, global existence, blow-up solutions, localized virial estimate.
Mathematics Subject Classification: 35Q51, 35Q55.
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:
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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J. Krieger, E. 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.  Google Scholar

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

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

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  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.jfa.2016.08.011.  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. doi: 10.1090/cln/010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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.  Google Scholar

[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.  Google Scholar

[9]

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: 10.3934/cpaa.2014.13.1267.  Google Scholar

[10]

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

[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.  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.jmaa.2017.11.060.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

J. Krieger, E. 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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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