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January  2021, 20(1): 77-99. doi: 10.3934/cpaa.2020258

Scattering of the focusing energy-critical NLS with inverse square potential in the radial case

Kai Yang

School of Mathematics, Southeast University, Nanjing, Jiangsu Province, 211189, China

Received  March 2020 Revised  August 2020 Published  January 2021 Early access  October 2020

Fund Project: K. Yang was supported in part by the "Jiangsu Shuang Chuang Doctoral Plan" and the Natural Science Foundation of Jiangsu Province(China): BK20200346 and BK20190323

We consider the Cauchy problem of the focusing energy-critical nonlinear Schrödinger equation with an inverse square potential. We prove that if any radial solution obeys the supreme of the kinetic energy over the maximal lifespan is below the kinetic energy of the ground state solution, then the solution exists globally in time and scatters in both time directions.

Keywords: Scattering, inverse square potential, ground state solution, energy critical, NLS.
Mathematics Subject Classification: Primary: 35Q55.
Citation: Kai Yang. Scattering of the focusing energy-critical NLS with inverse square potential in the radial case. Communications on Pure & Applied Analysis, 2021, 20 (1) : 77-99. doi: 10.3934/cpaa.2020258
References:
[1]

T. Aubin, Problémes isopérimétriques et espaces de Sobolev, J. Differ. Geom., 11 (1976), 573-598.   Google Scholar

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H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Am. J. Math., 121 (1999), 131-175.   Google Scholar

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J. Bourgain, Global well-posedness of defocusing 3D critical NLS in the radial case, J. Am. Math. Soc., 12 (1999), 145-171.  doi: 10.1090/S0894-0347-99-00283-0.  Google Scholar

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J. Bourgain, New global well-posedness results for nonlinear Schrödinger equations, AMS Colloquium Publications, 46, 1999. doi: 10.1090/coll/046.  Google Scholar

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N. Burq, F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal., 203 (2003), 519-549.  doi: 10.1016/S0022-1236(03)00238-6.  Google Scholar

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T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes, 10 (2003). doi: 10.1090/cln/010.  Google Scholar

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T. Cazenave and F. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^{s}$, Nonlinear Anal., 14 (1990), 807-836.  doi: 10.1016/0362-546X(90)90023-A.  Google Scholar

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Y. Chen, J. Lu and F. Meng, Focusing nonlinear Hartree equation with inverse-square potential, arXiv: 1907.12757. doi: 10.1063/1.5054167.  Google Scholar

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J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^{3}$, Annals of Math., 167 (2008), 767-865.  doi: 10.4007/annals.2008.167.767.  Google Scholar

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B. Dodson, Global well-posedness and scattering for the focusing, cubic Schrödinger equation in dimension $d = 4$, Ann. Sci. Éc. Norm. Supér, 52 (2019), 139-180.   Google Scholar

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G. Grillakis, On nonlinear Schrödinger equations, Commun. PDE, 25 (2000), 1827-1844.  doi: 10.1080/03605300008821569.  Google Scholar

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M. Keel and T. Tao, Endpoint Strichartz estimates, Am. J. Math., 120 (1998), 955-980.   Google Scholar

[13]

C. Kenig and F. Merle, Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

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C. Kenig and F. Merle, Scattering for bounded solutions to the cubic, defocusing NLS in 3 dimensions, T. Am. Math. Soc., 362 (2010), 1937-1962.  doi: 10.1090/S0002-9947-09-04722-9.  Google Scholar

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S. Keraani, On the blow up phenomenon of the critical nonlinear Schrödinger equation, J. Funct. Anal., 235 (2006), 171-192.  doi: 10.1016/j.jfa.2005.10.005.  Google Scholar

[16]

R. Killip, C. Miao, M. Visan, J. Zhang and J. Zheng, Sobolev spaces adapted to the Schrödinger operator with inverse-square potential, Math. Z., 288 (2018), 1273-1298. doi: 10.1007/s00209-017-1934-8.  Google Scholar

[17]

R. Killip, J. Murphy, M. Visan and J. Zheng, The focusing cubic NLS with inverse-square potential in three space dimensions, Differ. Integral Equ., 30 (2017), 161-206.   Google Scholar

[18]

R. Killip, C. Miao, M. Visan, J. Zhang and J. Zheng, The energy-critical NLS with inverse-square potential, Discrete Contin. Dyn. Syst., 37 (2017), 3831-3866. doi: 10.3934/dcds.2017162.  Google Scholar

[19]

R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Am. J. Math., 132 (2010), 361–424. doi: 10.1353/ajm.0.0107.  Google Scholar

[20]

R. Killip and M. Visan, Nonlinear Schrö dinger equations at critical regularity, Evol. Equ., (2009), 89–100. doi: 10.1007/s00208-013-0960-z.  Google Scholar

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R. Killip, M. Visan, and X. Zhang, Quintic NLS in the exterior of a strictly convex obstacle, Am. J. Math., 138 (2016), 1193-1346. doi: 10.1353/ajm.2016.0039.  Google Scholar

[22]

R. Killip, M. Visan and X. Zhang, Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on $\mathbb{R}^{2}$. arXiv: 1606.07738. Google Scholar

[23]

D. Li and X. Zhang, Dynamics for the energy critical nonlinear Schrödinger equation in high dimensions, J. Funct. Anal., 256 (2009), 1928-1961. doi: 10.1016/j.jfa.2008.12.007.  Google Scholar

[24]

D. Li and X. Zhang, Dynamics for the energy critical nonlinear Wave equation in high dimensions, Trans. AMS., 363 (2011), 1137-1160. doi: 10.1090/S0002-9947-2010-04999-2.  Google Scholar

[25]

J. Lu, C. Miao and J. Murphy, Scattering in $H^{1}$ for the intercritical NLS with an inverse square potential, J. Differ. Equ., 264 (2018), 3174-3211.  doi: 10.1016/j.jde.2017.11.015.  Google Scholar

[26]

F. Merle and L. Vega, Compactness at blow-up time for L2 solutions of the critical nonlinear Schrödinger equation in 2D, International Mathematics Research Notices, 1998 (1998), 399-425.  doi: 10.1155/S1073792898000270.  Google Scholar

[27]

C. Miao, J. Murphy and J. Zheng, The energy-critical nonlinear wave equation with an inverse-square potential, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 37 (2020), 417-456.  doi: 10.1016/j.anihpc.2019.09.004.  Google Scholar

[28]

F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh, Dispersive estimates for wave equation with the inverse-square potential, Discrete Contin. Dyn. Syst. Ser. A, 9 (2003), 1387-1400.  doi: 10.3934/dcds.2003.9.1387.  Google Scholar

[29]

E. Ryckman and M. Visan, Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $\mathbb{R}^{1+4}$, Am. J. Math., 129 (2007), 1-60.  doi: 10.1353/ajm.2007.0004.  Google Scholar

[30]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura. Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013.  Google Scholar

[31]

T. Tao, Global well-posedness and scattering for higher-dimensional energy-critical non-linear Schrödinger equation for radial data, New York J. Math., 11 (2005), 57-80.   Google Scholar

[32]

T. Tao and M. Visan, Stability of energy-critical nonlinear Schrödinger equations in high dimensions, Electron. J. Differ. Equ., 118 (2005), 1-28.   Google Scholar

[33]

J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153.  doi: 10.1006/jfan.1999.3556.  Google Scholar

[34]

M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., 138 (2007), 281-374.  doi: 10.1215/S0012-7094-07-13825-0.  Google Scholar

[35]

K. Yang, Dynamics of The Energy Critical Nonlinear Schrödinger Equation with Inverse Square Potential, PhD thesis, University of Iowa, 2017.  Google Scholar

[36]

K. Yang, The focusing NLS on exterior domains in three dimensions, Commun. Pure. Appl. Anal., 16 (2017), 2269-2297.  doi: 10.3934/cpaa.2017112.  Google Scholar

[37]

K. Yang, The symplectic non-squeezing properties of mass subcritical Hartree equations, J. Math. Anal. Appl., 449 (2017), 427-455.  doi: 10.1016/j.jmaa.2016.11.079.  Google Scholar

[38]

K. Yang, Scattering of the energy-critical NLS with inverse square potential, J. Math. Anal. Appl., 487 (2020), 124006. doi: 10.1016/j.jmaa.2020.124006.  Google Scholar

[39]

J. Zhang and J. Zheng, Strichartz estimates and wave equation in a conic singular space, Math. Ann., 376 (2020), 525-581. doi: 10.1007/s00208-019-01892-7.  Google Scholar

[40]

J. Zhang and J. Zheng, Scattering theory for nonlinear Schrödinger with inverse-square potential, J. Funct. Anal., 267 (2014), 2907-2932.  doi: 10.1016/j.jfa.2014.08.012.  Google Scholar

[41]

J. Zheng, Focusing NLS with inverse square potential, J. Math. Phys., 59 (2018), 111502, 14pp. doi: 10.1063/1.5054167.  Google Scholar

show all references

References:
[1]

T. Aubin, Problémes isopérimétriques et espaces de Sobolev, J. Differ. Geom., 11 (1976), 573-598.   Google Scholar

[2]

H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Am. J. Math., 121 (1999), 131-175.   Google Scholar

[3]

J. Bourgain, Global well-posedness of defocusing 3D critical NLS in the radial case, J. Am. Math. Soc., 12 (1999), 145-171.  doi: 10.1090/S0894-0347-99-00283-0.  Google Scholar

[4]

J. Bourgain, New global well-posedness results for nonlinear Schrödinger equations, AMS Colloquium Publications, 46, 1999. doi: 10.1090/coll/046.  Google Scholar

[5]

N. Burq, F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal., 203 (2003), 519-549.  doi: 10.1016/S0022-1236(03)00238-6.  Google Scholar

[6]

T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes, 10 (2003). doi: 10.1090/cln/010.  Google Scholar

[7]

T. Cazenave and F. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^{s}$, Nonlinear Anal., 14 (1990), 807-836.  doi: 10.1016/0362-546X(90)90023-A.  Google Scholar

[8]

Y. Chen, J. Lu and F. Meng, Focusing nonlinear Hartree equation with inverse-square potential, arXiv: 1907.12757. doi: 10.1063/1.5054167.  Google Scholar

[9]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^{3}$, Annals of Math., 167 (2008), 767-865.  doi: 10.4007/annals.2008.167.767.  Google Scholar

[10]

B. Dodson, Global well-posedness and scattering for the focusing, cubic Schrödinger equation in dimension $d = 4$, Ann. Sci. Éc. Norm. Supér, 52 (2019), 139-180.   Google Scholar

[11]

G. Grillakis, On nonlinear Schrödinger equations, Commun. PDE, 25 (2000), 1827-1844.  doi: 10.1080/03605300008821569.  Google Scholar

[12]

M. Keel and T. Tao, Endpoint Strichartz estimates, Am. J. Math., 120 (1998), 955-980.   Google Scholar

[13]

C. Kenig and F. Merle, Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

[14]

C. Kenig and F. Merle, Scattering for bounded solutions to the cubic, defocusing NLS in 3 dimensions, T. Am. Math. Soc., 362 (2010), 1937-1962.  doi: 10.1090/S0002-9947-09-04722-9.  Google Scholar

[15]

S. Keraani, On the blow up phenomenon of the critical nonlinear Schrödinger equation, J. Funct. Anal., 235 (2006), 171-192.  doi: 10.1016/j.jfa.2005.10.005.  Google Scholar

[16]

R. Killip, C. Miao, M. Visan, J. Zhang and J. Zheng, Sobolev spaces adapted to the Schrödinger operator with inverse-square potential, Math. Z., 288 (2018), 1273-1298. doi: 10.1007/s00209-017-1934-8.  Google Scholar

[17]

R. Killip, J. Murphy, M. Visan and J. Zheng, The focusing cubic NLS with inverse-square potential in three space dimensions, Differ. Integral Equ., 30 (2017), 161-206.   Google Scholar

[18]

R. Killip, C. Miao, M. Visan, J. Zhang and J. Zheng, The energy-critical NLS with inverse-square potential, Discrete Contin. Dyn. Syst., 37 (2017), 3831-3866. doi: 10.3934/dcds.2017162.  Google Scholar

[19]

R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Am. J. Math., 132 (2010), 361–424. doi: 10.1353/ajm.0.0107.  Google Scholar

[20]

R. Killip and M. Visan, Nonlinear Schrö dinger equations at critical regularity, Evol. Equ., (2009), 89–100. doi: 10.1007/s00208-013-0960-z.  Google Scholar

[21]

R. Killip, M. Visan, and X. Zhang, Quintic NLS in the exterior of a strictly convex obstacle, Am. J. Math., 138 (2016), 1193-1346. doi: 10.1353/ajm.2016.0039.  Google Scholar

[22]

R. Killip, M. Visan and X. Zhang, Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on $\mathbb{R}^{2}$. arXiv: 1606.07738. Google Scholar

[23]

D. Li and X. Zhang, Dynamics for the energy critical nonlinear Schrödinger equation in high dimensions, J. Funct. Anal., 256 (2009), 1928-1961. doi: 10.1016/j.jfa.2008.12.007.  Google Scholar

[24]

D. Li and X. Zhang, Dynamics for the energy critical nonlinear Wave equation in high dimensions, Trans. AMS., 363 (2011), 1137-1160. doi: 10.1090/S0002-9947-2010-04999-2.  Google Scholar

[25]

J. Lu, C. Miao and J. Murphy, Scattering in $H^{1}$ for the intercritical NLS with an inverse square potential, J. Differ. Equ., 264 (2018), 3174-3211.  doi: 10.1016/j.jde.2017.11.015.  Google Scholar

[26]

F. Merle and L. Vega, Compactness at blow-up time for L2 solutions of the critical nonlinear Schrödinger equation in 2D, International Mathematics Research Notices, 1998 (1998), 399-425.  doi: 10.1155/S1073792898000270.  Google Scholar

[27]

C. Miao, J. Murphy and J. Zheng, The energy-critical nonlinear wave equation with an inverse-square potential, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 37 (2020), 417-456.  doi: 10.1016/j.anihpc.2019.09.004.  Google Scholar

[28]

F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh, Dispersive estimates for wave equation with the inverse-square potential, Discrete Contin. Dyn. Syst. Ser. A, 9 (2003), 1387-1400.  doi: 10.3934/dcds.2003.9.1387.  Google Scholar

[29]

E. Ryckman and M. Visan, Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $\mathbb{R}^{1+4}$, Am. J. Math., 129 (2007), 1-60.  doi: 10.1353/ajm.2007.0004.  Google Scholar

[30]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura. Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013.  Google Scholar

[31]

T. Tao, Global well-posedness and scattering for higher-dimensional energy-critical non-linear Schrödinger equation for radial data, New York J. Math., 11 (2005), 57-80.   Google Scholar

[32]

T. Tao and M. Visan, Stability of energy-critical nonlinear Schrödinger equations in high dimensions, Electron. J. Differ. Equ., 118 (2005), 1-28.   Google Scholar

[33]

J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153.  doi: 10.1006/jfan.1999.3556.  Google Scholar

[34]

M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., 138 (2007), 281-374.  doi: 10.1215/S0012-7094-07-13825-0.  Google Scholar

[35]

K. Yang, Dynamics of The Energy Critical Nonlinear Schrödinger Equation with Inverse Square Potential, PhD thesis, University of Iowa, 2017.  Google Scholar

[36]

K. Yang, The focusing NLS on exterior domains in three dimensions, Commun. Pure. Appl. Anal., 16 (2017), 2269-2297.  doi: 10.3934/cpaa.2017112.  Google Scholar

[37]

K. Yang, The symplectic non-squeezing properties of mass subcritical Hartree equations, J. Math. Anal. Appl., 449 (2017), 427-455.  doi: 10.1016/j.jmaa.2016.11.079.  Google Scholar

[38]

K. Yang, Scattering of the energy-critical NLS with inverse square potential, J. Math. Anal. Appl., 487 (2020), 124006. doi: 10.1016/j.jmaa.2020.124006.  Google Scholar

[39]

J. Zhang and J. Zheng, Strichartz estimates and wave equation in a conic singular space, Math. Ann., 376 (2020), 525-581. doi: 10.1007/s00208-019-01892-7.  Google Scholar

[40]

J. Zhang and J. Zheng, Scattering theory for nonlinear Schrödinger with inverse-square potential, J. Funct. Anal., 267 (2014), 2907-2932.  doi: 10.1016/j.jfa.2014.08.012.  Google Scholar

[41]

J. Zheng, Focusing NLS with inverse square potential, J. Math. Phys., 59 (2018), 111502, 14pp. doi: 10.1063/1.5054167.  Google Scholar

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