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 & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119
References:
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References:
 [1] R. Carles, Nonlinear Schrödinger equations with repulsive harmonic potential and applications,, SIAM J. Math. Anal., 35 (2003), 823. 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. doi: 10.1142/S0218202502002215. Google Scholar [3] T. Cazenave, "Semilinear Schrödinger Equations,", in, 10 (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/BF01208265. Google Scholar [19] J. Zhang, Stability of attractive Bose-Einstein condensate,, J. Statist. Phys., 101 (2000), 731. 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. doi: 10.1080/03605300500299539. Google Scholar
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