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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 and 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. 

[2]

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

[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.

[4]

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

[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.

[6]

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

[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.

[8]

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

[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.

[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. 

[11]

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

[12]

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

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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. 

[32]

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

[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.

[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.

[35]

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

[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.

[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.

[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.

[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.

[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.

[41]

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

show all references

References:
[1]

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

[2]

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

[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.

[4]

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

[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.

[6]

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

[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.

[8]

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

[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.

[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. 

[11]

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

[12]

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

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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. 

[32]

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

[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.

[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.

[35]

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

[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.

[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.

[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.

[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.

[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.

[41]

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

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