# American Institute of Mathematical Sciences

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

 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.
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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