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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

T. Cazenave, "Semilinear Schrödinger Equations," "Courant Lecture Notes in Mathematics," 10, NYU, CIMS, AMS, 2003. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576. doi: 10.1007/BF01208265.  Google Scholar

[19]

J. Zhang, Stability of attractive Bose-Einstein condensate, J. Statist. Phys., 101 (2000), 731-746. doi: 10.1023/A:1026437923987.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

T. Cazenave, "Semilinear Schrödinger Equations," "Courant Lecture Notes in Mathematics," 10, NYU, CIMS, AMS, 2003. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576. doi: 10.1007/BF01208265.  Google Scholar

[19]

J. Zhang, Stability of attractive Bose-Einstein condensate, J. Statist. Phys., 101 (2000), 731-746. doi: 10.1023/A:1026437923987.  Google Scholar

[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.  Google Scholar

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