Rate of L^2-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential

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March 2011, 1(1): 119-127. doi: 10.3934/mcrf.2011.1.119

Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential

Jian Zhang ^1, , Shihui Zhu ^2, and Xiaoguang Li ^3,

1.	College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610068
2.	College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China
3.	College of Economics, Sichuan Normal University, Chengdu 610066, China

Received October 2010 Revised January 2011 Published March 2011

Abstract
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We consider the blow-up solutions of the Cauchy problem for the critical nonlinear Schrödinger equation with a repulsive harmonic potential. In terms of Merle and Tsutsumi's arguments as well as Carles' transform, the $L^2$-concentration property of radially symmetric blow-up solutions is obtained.

Keywords: $L^2$-concentration., repulsive harmonic potential, blow-up solution, Nonlinear Schrödinger equation.

Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B4.

Citation: Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control and Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119

References:

[1]	R. Carles, Nonlinear Schrödinger equations with repulsive harmonic potential and applications, SIAM J. Math. Anal., 35 (2003), 823-843. doi: 10.1137/S0036141002416936.
[2]	R. Carles, Critical nonlinear Schrödinger equations with and without harmonic potential, Math. Models Methods Appl. Sci., 12 (2002), 1513-1523. doi: 10.1142/S0218202502002215.
[3]	T. Cazenave, "Semilinear Schrödinger Equations," "Courant Lecture Notes in Mathematics," 10, NYU, CIMS, AMS, 2003.
[4]	M. J. Landam, G. C. Papanicolao, C. Sulem and P. L. Sulem, Rate of blowup for solutions of nonlinear Schrödinger equation at critical dimension, Phys. Rev. A., 38 (1988), 3837-3834. doi: 10.1103/PhysRevA.38.3837.
[5]	X. G. Li, J. Zhang and G. G. Chen, $L^{2}$-concentration of blow-up solutions for the nonlinear Schrödinger equations with harmonic potential, Chinese Ann. Math. Ser. A, 26 (2005), 31-38; (translation in Chinese J. Contemp. Math., 26 (2005), 35-42.
[6]	X. G. Li and J. Zhang, Limit behavior of blow-up solutions for critical nonlinear Schrödinger equation with harmonic potential, Differential Integral Equations, 19 (2006), 761-771.
[7]	M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u +u^p=0$ in $R^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
[8]	F. Merle and P. Raphaël, Blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. of Math., 161 (2005), 157-222. doi: 10.4007/annals.2005.161.157.
[9]	F. Merle and P. Raphaël, On a sharp lower bound on the blow-up rate for the $L^{2}$-critical nonlinear Schrödingerequation, J. Amer. Math. Soc., 19 (2005), 37-90. doi: 10.1090/S0894-0347-05-00499-6.
[10]	F. Merle and P. Raphaël, On universality of blow up profile for $L^{2}$-critical nonlinear Schrödinger equation, Invent. Math., 156 (2004), 565-672. doi: 10.1007/s00222-003-0346-z.
[11]	F. Merle and P. Raphaël, Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation, Comm. Math. Phys., 253 (2005), 675-704. doi: 10.1007/s00220-004-1198-0.
[12]	F. Merle and Y. Tsutsumi, $L^{2}$ concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity, J. Differential Equations, 84 (1990), 205-214. doi: 10.1016/0022-0396(90)90075-Z.
[13]	Y. G. Oh, Cauchy problem and Ehrenfest's law of nonlinear Schrödinger equations with potentials, J. Differential Equations, 81 (1989), 255-274. doi: 10.1016/0022-0396(89)90123-X.
[14]	P. Raphaël, Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation, Math. Ann., 331 (2005), 577-609. doi: 10.1007/s00208-004-0596-0.
[15]	W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517.
[16]	Y. Tsutsumi, Rate of $L^2$ concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power, Nonlinear Anal., 15 (1990), 719-724. doi: 10.1016/0362-546X(90)90088-X.
[17]	M. Wadati and T. Tsurumi, Critical number of atoms for the magnetically trapped Bose-Einstein condensate with negative s-wave scattering length, Phys. Lett. A, 247 (1998), 287-293. doi: 10.1016/S0375-9601(98)00583-0.
[18]	M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576. doi: 10.1007/BF01208265.
[19]	J. Zhang, Stability of attractive Bose-Einstein condensate, J. Statist. Phys., 101 (2000), 731-746. doi: 10.1023/A:1026437923987.
[20]	J. Zhang, Sharp threshold for blowup and global existence in nonlinear Schrödinger equations under a harmonic potential, Comm. Partial Differential Equations, 30 (2005), 1429-1443. doi: 10.1080/03605300500299539.

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References:

[1]	R. Carles, Nonlinear Schrödinger equations with repulsive harmonic potential and applications, SIAM J. Math. Anal., 35 (2003), 823-843. doi: 10.1137/S0036141002416936.
[2]	R. Carles, Critical nonlinear Schrödinger equations with and without harmonic potential, Math. Models Methods Appl. Sci., 12 (2002), 1513-1523. doi: 10.1142/S0218202502002215.
[3]	T. Cazenave, "Semilinear Schrödinger Equations," "Courant Lecture Notes in Mathematics," 10, NYU, CIMS, AMS, 2003.
[4]	M. J. Landam, G. C. Papanicolao, C. Sulem and P. L. Sulem, Rate of blowup for solutions of nonlinear Schrödinger equation at critical dimension, Phys. Rev. A., 38 (1988), 3837-3834. doi: 10.1103/PhysRevA.38.3837.
[5]	X. G. Li, J. Zhang and G. G. Chen, $L^{2}$-concentration of blow-up solutions for the nonlinear Schrödinger equations with harmonic potential, Chinese Ann. Math. Ser. A, 26 (2005), 31-38; (translation in Chinese J. Contemp. Math., 26 (2005), 35-42.
[6]	X. G. Li and J. Zhang, Limit behavior of blow-up solutions for critical nonlinear Schrödinger equation with harmonic potential, Differential Integral Equations, 19 (2006), 761-771.
[7]	M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u +u^p=0$ in $R^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
[8]	F. Merle and P. Raphaël, Blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. of Math., 161 (2005), 157-222. doi: 10.4007/annals.2005.161.157.
[9]	F. Merle and P. Raphaël, On a sharp lower bound on the blow-up rate for the $L^{2}$-critical nonlinear Schrödingerequation, J. Amer. Math. Soc., 19 (2005), 37-90. doi: 10.1090/S0894-0347-05-00499-6.
[10]	F. Merle and P. Raphaël, On universality of blow up profile for $L^{2}$-critical nonlinear Schrödinger equation, Invent. Math., 156 (2004), 565-672. doi: 10.1007/s00222-003-0346-z.
[11]	F. Merle and P. Raphaël, Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation, Comm. Math. Phys., 253 (2005), 675-704. doi: 10.1007/s00220-004-1198-0.
[12]	F. Merle and Y. Tsutsumi, $L^{2}$ concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity, J. Differential Equations, 84 (1990), 205-214. doi: 10.1016/0022-0396(90)90075-Z.
[13]	Y. G. Oh, Cauchy problem and Ehrenfest's law of nonlinear Schrödinger equations with potentials, J. Differential Equations, 81 (1989), 255-274. doi: 10.1016/0022-0396(89)90123-X.
[14]	P. Raphaël, Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation, Math. Ann., 331 (2005), 577-609. doi: 10.1007/s00208-004-0596-0.
[15]	W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517.
[16]	Y. Tsutsumi, Rate of $L^2$ concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power, Nonlinear Anal., 15 (1990), 719-724. doi: 10.1016/0362-546X(90)90088-X.
[17]	M. Wadati and T. Tsurumi, Critical number of atoms for the magnetically trapped Bose-Einstein condensate with negative s-wave scattering length, Phys. Lett. A, 247 (1998), 287-293. doi: 10.1016/S0375-9601(98)00583-0.
[18]	M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576. doi: 10.1007/BF01208265.
[19]	J. Zhang, Stability of attractive Bose-Einstein condensate, J. Statist. Phys., 101 (2000), 731-746. doi: 10.1023/A:1026437923987.
[20]	J. Zhang, Sharp threshold for blowup and global existence in nonlinear Schrödinger equations under a harmonic potential, Comm. Partial Differential Equations, 30 (2005), 1429-1443. doi: 10.1080/03605300500299539.

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