Global existence, blow-up and mass concentration for the inhomogeneous nonlinear Schrödinger equation with inverse-square potential

Hui Jian; Min Gong; Meixia Cai; Hui Jian; Min Gong; Meixia Cai

doi:10.3934/era.2023375

Electronic Research Archive

2023, Volume 31, Issue 12: 7427-7451. doi: 10.3934/era.2023375

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Research article Special Issues

Global existence, blow-up and mass concentration for the inhomogeneous nonlinear Schrödinger equation with inverse-square potential

School of Science, East China Jiaotong University, Nanchang 330013, China

Received: 23 July 2023 Revised: 23 October 2023 Accepted: 05 November 2023 Published: 22 November 2023

In the current paper, the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation including inverse-square potential is considered. First, some criteria of global existence and finite-time blow-up in the mass-critical and mass-supercritical settings with $ 0 < c\leq c^{*} $ are obtained. Then, by utilizing the potential well method and the sharp Sobolev constant, the sharp condition of blow-up is derived in the energy-critical case with $ 0 < c < \frac{N^{2}+4N}{(N+2)^{2}}c^{*} $. Finally, we establish the mass concentration property of explosive solutions, as well as the dynamic behaviors of the minimal-mass blow-up solutions in the $ L^{2} $-critical setting for $ 0 < c < c^{*} $, by means of the variational characterization of the ground-state solution to the elliptic equation, scaling techniques and a suitable refined compactness lemma. Our results generalize and supplement the ones of some previous works.
- inhomogeneous nonlinear Schrödinger equation,
- inverse-square potential,
- blow-up,
- global existence,
- mass concentration
Citation: Hui Jian, Min Gong, Meixia Cai. Global existence, blow-up and mass concentration for the inhomogeneous nonlinear Schrödinger equation with inverse-square potential[J]. Electronic Research Archive, 2023, 31(12): 7427-7451. doi: 10.3934/era.2023375

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Abstract

In the current paper, the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation including inverse-square potential is considered. First, some criteria of global existence and finite-time blow-up in the mass-critical and mass-supercritical settings with $ 0 < c\leq c^{*} $ are obtained. Then, by utilizing the potential well method and the sharp Sobolev constant, the sharp condition of blow-up is derived in the energy-critical case with $ 0 < c < \frac{N^{2}+4N}{(N+2)^{2}}c^{*} $. Finally, we establish the mass concentration property of explosive solutions, as well as the dynamic behaviors of the minimal-mass blow-up solutions in the $ L^{2} $-critical setting for $ 0 < c < c^{*} $, by means of the variational characterization of the ground-state solution to the elliptic equation, scaling techniques and a suitable refined compactness lemma. Our results generalize and supplement the ones of some previous works.

References

[1]	H. E. Camblong, L. N. Epele, H. Fanchiotti, C. A. G. Canal, Quantum anomaly in molecular physics, Phys. Rev. Lett., 87 (2001), 220402. https://doi.org/10.1103/PhysRevLett.87.220402 doi: 10.1103/PhysRevLett.87.220402
[2]	K. M. Case, Singular potentials, Phys. Rev., 80 (1950), 797–806. https://doi.org/10.1103/PhysRev.80.797 doi: 10.1103/PhysRev.80.797
[3]	H. Kalf, U. W. Schmincke, J. Walter, R. Wüst, On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials, in Spectral Theory and Differential Equations, Springer press, 448 (1975), 182–226. https://doi.org/10.1007/BFB0067087
[4]	G. E. Astrakharchik, B. A. Malomed, Quantum versus mean-field collapse in a many-body system, Phys. Rev. A, 92 (2015), 043632. https://doi.org/10.1103/PhysRevA.92.043632 doi: 10.1103/PhysRevA.92.043632
[5]	H. Sakaguchi, B. A. Malomed, Suppression of the quantum-mechanical collapse by repulsive interactions in a quantum gas, Phys. Rev. A, 83 (2011), 013607. https://doi.org/10.1103/PhysRevA.83.013607 doi: 10.1103/PhysRevA.83.013607
[6]	H. Sakaguchi, B. A. Malomed, Suppression of the quantum collapse in binary bosonic gases, Phys. Rev. A, 88 (2013), 043638. https://doi.org/10.1103/PhysRevA.88.043638 doi: 10.1103/PhysRevA.88.043638
[7]	M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., 87 (1983), 567–576. https://doi.org/10.1007/BF01208265 doi: 10.1007/BF01208265
[8]	F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Math. J., 69 (1993), 427–454. https://doi.org/10.1215/S0012-7094-93-06919-0 doi: 10.1215/S0012-7094-93-06919-0
[9]	T. Hmidi, S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, Int. Math. Res. Not., 2005 (2005), 2815–2818. https://doi.org/10.1155/IMRN.2005.2815 doi: 10.1155/IMRN.2005.2815
[10]	F. Merle, Nonexistence of minimal blow-up solutions of equations $iu_t = -\Delta u-k(x)\|u\|^{\frac{4}{N}}u$ in $\mathbb{R}^N$, Ann. Inst. Henri Poincare, 64 (1996), 33–85.
[11]	J. Shu, J. Zhang, Sharp criterion of global existence for a class of nonlinear Schrödinger equation with critical exponent, Appl. Math. Comput., 182 (2006), 1482–1487. https://doi.org/10.1016/j.amc.2006.05.036 doi: 10.1016/j.amc.2006.05.036
[12]	Z. Liu, On a class of inhomogeneous, energy-critical, focusing, nonlinear Schrödinger equations, Acta Math. Sci., 33 (2013), 1522–1530. https://doi.org/10.1016/S0252-9602(13)60101-0 doi: 10.1016/S0252-9602(13)60101-0
[13]	J. Lu, C. X. Miao, J. Murphy, Scattering in $H^{1}$ for the intercritical NLS with an inverse-square potential, J. Differ. Equations, 264 (2018), 3174–3211. https://doi.org/10.1016/J.JDE.2017.11.015 doi: 10.1016/J.JDE.2017.11.015
[14]	K. Yang, Scattering of the energy-critical NLS with inverse square potential, J. Math. Anal. Appl., 487 (2020), 124006. https://doi.org/10.1016/j.jmaa.2020.124006 doi: 10.1016/j.jmaa.2020.124006
[15]	V. D. Dinh, Global existence and blowup for a class of the focusing nonlinear Schrödinger equation with inverse-square potential, J. Math. Anal. Appl., 468 (2018), 270–303. https://doi.org/10.1016/j.jmaa.2018.08.006 doi: 10.1016/j.jmaa.2018.08.006
[16]	X. F. Li, Global existence and blowup for Choquard equations with an inverse-square potential, J. Differ. Equations, 268 (2020), 4276–4319. https://doi.org/10.1016/j.jde.2019.10.028 doi: 10.1016/j.jde.2019.10.028
[17]	E. Csobo, F. Genoud, Minimal mass blow-up solutions for the $L^{2}$ critical NLS with inverse-square potential, Nonlinear Anal., 168 (2018), 110–129. https://doi.org/10.1016/j.na.2017.11.008 doi: 10.1016/j.na.2017.11.008
[18]	D. Mukherjee, P. T. Nam, P. T. Nguyen, Uniqueness of ground state and minimal-mass blow-up solutions for focusing NLS with Hardy potential, J. Funct. Anal., 281 (2021), 109092. https://doi.org/10.1016/j.jfa.2021.109092 doi: 10.1016/j.jfa.2021.109092
[19]	A. Bensouilah, $L^{2}$ concentration of blow-up solutions for the mass-critical NLS with inverse-square potential, Bull. Belg. Math. Soc. Simon Stevin, 26 (2019), 759–771. https://doi.org/10.36045/bbms/1579402821 doi: 10.36045/bbms/1579402821
[20]	J. J. Pan, J. Zhang, On the minimal mass blow-up solutions for the nonlinear Schrödinger equation with Hardy potential, Nonlinear Anal., 197 (2020), 111829. https://doi.org/10.1016/j.na.2020.111829 doi: 10.1016/j.na.2020.111829
[21]	A. Bensouilah, V. D. Dinh, S. H. Zhu, On stability and instability of standing waves for the nonlinear Schrödinger equation with an inverse-square potential, J. Math. Phys., 59 (2018), 101505. https://doi.org/10.1063/1.5038041 doi: 10.1063/1.5038041
[22]	V. D. Dinh, On the instability of standing waves for the nonlinear Schrödinger equation with inverse-square potential, Complex Var. Elliptic Equations, 66 (2021), 1699–1716. https://doi.org/10.1080/17476933.2020.1779235 doi: 10.1080/17476933.2020.1779235
[23]	S. X. Xia, Energy-critical nonlinear Schrödinger equation with inverse square potential and subcritical perturbation, J. Math. Anal. Appl., 487 (2020), 123955. https://doi.org/10.1016/j.jmaa.2020.123955 doi: 10.1016/j.jmaa.2020.123955
[24]	L. J. Cao, Existence of stable standing waves for the nonlinear Schrödinger equation with the Hardy potential, Discrete Contin. Dyn. Syst. - Ser. B, 28 (2023), 1342–1366. https://doi.org/10.3934/dcdsb.2022125 doi: 10.3934/dcdsb.2022125
[25]	H. W. Li, W. M. Zou, Normalized ground state for the Sobolev critical Schrödinger equation involving Hardy term with combined nonlinearities, Math. Nachr., 296 (2023), 2440–2466. https://doi.org/10.1002/mana.202000481 doi: 10.1002/mana.202000481
[26]	O. Goubet, I. Manoubi, Standing waves for semilinear Schrödinger equations with discontinuous dispersion, Rend. Circ. Mat. Palermo Ser. 2, 71 (2022), 1159–1171. https://doi.org/10.1007/s12215-022-00782-3 doi: 10.1007/s12215-022-00782-3
[27]	J. Zuo, C. Liu, C. Vetro, Normalized solutions to the fractional Schrödinger equation with potential, Mediterr. J. Math., 20 (2023), 216. https://doi.org/10.1007/s00009-023-02422-1 doi: 10.1007/s00009-023-02422-1
[28]	L. Campos, C. M. Guzmán, On the inhomogeneous NLS with inverse-square potential, Z. Angew. Math. Phys., 72 (2021), 143. https://doi.org/10.1007/s00033-021-01560-4 doi: 10.1007/s00033-021-01560-4
[29]	J. An, R. Jang, J. Kim, Global existence and blow-up for the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential, Discrete Contin. Dyn. Syst. - Ser. B, 28 (2023), 1046–1067. https://doi.org/10.3934/dcdsb.2022111 doi: 10.3934/dcdsb.2022111
[30]	J. J. Pan, J. Zhang, Blow-up solutions with minimal mass for the nonlinear Schrödinger equation with variable poteantial, Adv. Nonlinear Anal., 11 (2022), 58–71. https://doi.org/10.1515/anona-2020-0185 doi: 10.1515/anona-2020-0185
[31]	N. Okazawa, T. Suzuki, T. Yokota, Energy methods for abstract nonlinear Schrödinger equations, Evol. Equations Control Theory, 1 (2012), 337–354. https://doi.org/10.3934/eect.2012.1.337 doi: 10.3934/eect.2012.1.337
[32]	R. Killip, J. Murphy, M. Visan, J. Q. Zheng, The focusing cubic NLS with inverse-square potential in three space dimensions, Differ. Integr. Equations, 30 (2017), 161–206. https://doi.org/10.57262/die/1487386822 doi: 10.57262/die/1487386822
[33]	R. Killip, C. X. Miao, M. Visan, J. Y. Zhang, J. Q. Zheng, The energy-critical NLS with inverse-square potential, Discrete Contin. Dyn. Syst., 37 (2017), 3831–3866. https://doi.org/10.3934/dcds.2017162 doi: 10.3934/dcds.2017162
[34]	T. Cazenave, Semilinear Schrödinger Equations, in Courant Lecture Notes in Mathematics, American Mathematical Society Press, USA, 2003. https://doi.org/10.1090/cln/010
[35]	J. Holmer, S. Roudenko, On blow-up solutions to the 3D cubic nonlinear Schrödinger equation, Appl. Math. Res. eXpress, 2007 (2007), abm004. https://doi.org/10.1093/amrx/abm004 doi: 10.1093/amrx/abm004

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