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

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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 and Continuous Dynamical Systems - B, 2019, 24 (7) : 3335-3356. doi: 10.3934/dcdsb.2018323

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.
[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. 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.
[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.
[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.
[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. 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.
[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. 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.
[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. 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.
[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.
[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.
[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. 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.
[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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