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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  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} $ 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. 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.  [1] Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. 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