Mass concentration for the L^2-critical nonlinear Schrödinger equations of higher orders

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July 2011, 29(3): 909-928. doi: 10.3934/dcds.2011.29.909

Mass concentration for the $L^2$-critical nonlinear Schrödinger equations of higher orders

Myeongju Chae ^1, , Sunggeum Hong ^2, and Sanghyuk Lee ^3,

1.	Department of Applied Mathematics, Hankyong National University, Ansong 456-749, South Korea
2.	Department of Mathematics, Chosun University, Gwangju 501-759, South Korea
3.	School of Mathematical Sciences, Seoul National University, Seoul 151-742, South Korea

Received August 2009 Revised September 2010 Published November 2010

Abstract
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We consider the mass concentration phenomenon for the $L^2$-critical nonlinear Schrödinger equations of higher orders. We show that any solution $u$ to $iu_{t} + (-\Delta)^{\frac\alpha 2} u =\pm |u|^\frac{2\alpha}{d}u$, $u(0,\cdot)\in L^2$ for $\alpha >2$, which blows up in a finite time, satisfies a mass concentration phenomenon near the blow-up time. We verify that as $\alpha$ increases, the size of region capturing a mass concentration gets wider due to the stronger dispersive effect.

Keywords: Schrödinger equations, mass concentration., high orders, mixed norm blow-up.

Mathematics Subject Classification: Primary: 35B05, 35B30, 35B33; Secondary: 35Q55, 42B1.

Citation: Myeongju Chae, Sunggeum Hong, Sanghyuk Lee. Mass concentration for the $L^2$-critical nonlinear Schrödinger equations of higher orders. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 909-928. doi: 10.3934/dcds.2011.29.909

References:

[1]	P. Bégout and A. Vargas, Mass concentration Phenomena for the $L^2$-critical for the nonlinear Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 5257-5282. doi: 10.1090/S0002-9947-07-04250-X.
[2]	J. Bergh and J. Löfström, "Interpolation Spaces," Springer, New York, 1976.
[3]	J. Bourgain, Refinements of Strichartz' inequality and applications to $2$D-NLS with critical nonlinearity, Int. Math. Res. Not., 5 (1998), 253-283. doi: 10.1155/S1073792898000191.
[4]	M. Chae, S. Hong, J. Kim, S. Lee and C. W. Yang, On mass concentration for the $L^2$-critical nonlinear Schrödinger equations, Comm. Partial Differential Equations, 34 (2009), 486-505. doi: 10.1080/03605300902812426.
[5]	M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425. doi: 10.1006/jfan.2000.3687.
[6]	G. Fibich, B. Ilan and G. Papanicolaou, Self-focusing with fourth order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462. doi: 10.1137/S0036139901387241.
[7]	V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339. doi: 10.1103/PhysRevE.53.R1336.
[8]	V. I. Karpman and A. G. Shagalov, Stability of soliton described by nonlinear Schrödinger type equations with higher-order dispersion, Phys. D., 144 (2000), 194-210. doi: 10.1016/S0167-2789(00)00078-6.
[9]	M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.
[10]	S. Lee and A. Vargas, Sharp null form estimates for the wave equation, Amer. J. Math., 130 (2008), 1279-1326. doi: 10.1353/ajm.0.0024.
[11]	F. Merle and Y. Tsutsumi, $L^2$ concentration of blow up solutions for the nonlinear Schrödinger equation with critical power nonlinearity, J. Diff. Eq., 84 (1990), 205-214.
[12]	A. Moyua, A. Vargas and L. Vega, Schrödinger maximal function and restriction properties of the Fourier transform, Int. Math. Res. Not., 16 (1996), 793-815. doi: 10.1155/S1073792896000499.
[13]	H. Nawa, Mass concentration phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity, Funkcial. Ekvac., 35 (1992), 1-18.
[14]	B. Pausader, "Problèmes Bien Posés et Diffusion pour Des équations Non Linéaires Dispersives D'ordre Quatre," Ph.d dissertation, Université de Cergy Pontoise, 2008.
[15]	K. Rogers and A. Vargas, A refinement of the Strichartz inequality on the saddle and applications, J. Funct. Anal., 241 (2006), 212-231. doi: 10.1016/j.jfa.2006.04.026.
[16]	E. M. Stein, "Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals," Princeton Univ. Press, 1993.
[17]	T. Tao, A sharp bilinear restrictions estimate for paraboloids, Geom. Funct. Anal., 13 (2003), 1359-1384. doi: 10.1007/s00039-003-0449-0.
[18]	T. Tao, "Nonlinear Dispersive Equations," CBMS Regional Conf. Ser. in Math. Amer. Math. Soc., 106, 2006.
[19]	T. Tao, A. Vargas and L. Vega, A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc., 11 (1998), 967-1000. doi: 10.1090/S0894-0347-98-00278-1.

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References:

[1]	P. Bégout and A. Vargas, Mass concentration Phenomena for the $L^2$-critical for the nonlinear Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 5257-5282. doi: 10.1090/S0002-9947-07-04250-X.
[2]	J. Bergh and J. Löfström, "Interpolation Spaces," Springer, New York, 1976.
[3]	J. Bourgain, Refinements of Strichartz' inequality and applications to $2$D-NLS with critical nonlinearity, Int. Math. Res. Not., 5 (1998), 253-283. doi: 10.1155/S1073792898000191.
[4]	M. Chae, S. Hong, J. Kim, S. Lee and C. W. Yang, On mass concentration for the $L^2$-critical nonlinear Schrödinger equations, Comm. Partial Differential Equations, 34 (2009), 486-505. doi: 10.1080/03605300902812426.
[5]	M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425. doi: 10.1006/jfan.2000.3687.
[6]	G. Fibich, B. Ilan and G. Papanicolaou, Self-focusing with fourth order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462. doi: 10.1137/S0036139901387241.
[7]	V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339. doi: 10.1103/PhysRevE.53.R1336.
[8]	V. I. Karpman and A. G. Shagalov, Stability of soliton described by nonlinear Schrödinger type equations with higher-order dispersion, Phys. D., 144 (2000), 194-210. doi: 10.1016/S0167-2789(00)00078-6.
[9]	M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.
[10]	S. Lee and A. Vargas, Sharp null form estimates for the wave equation, Amer. J. Math., 130 (2008), 1279-1326. doi: 10.1353/ajm.0.0024.
[11]	F. Merle and Y. Tsutsumi, $L^2$ concentration of blow up solutions for the nonlinear Schrödinger equation with critical power nonlinearity, J. Diff. Eq., 84 (1990), 205-214.
[12]	A. Moyua, A. Vargas and L. Vega, Schrödinger maximal function and restriction properties of the Fourier transform, Int. Math. Res. Not., 16 (1996), 793-815. doi: 10.1155/S1073792896000499.
[13]	H. Nawa, Mass concentration phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity, Funkcial. Ekvac., 35 (1992), 1-18.
[14]	B. Pausader, "Problèmes Bien Posés et Diffusion pour Des équations Non Linéaires Dispersives D'ordre Quatre," Ph.d dissertation, Université de Cergy Pontoise, 2008.
[15]	K. Rogers and A. Vargas, A refinement of the Strichartz inequality on the saddle and applications, J. Funct. Anal., 241 (2006), 212-231. doi: 10.1016/j.jfa.2006.04.026.
[16]	E. M. Stein, "Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals," Princeton Univ. Press, 1993.
[17]	T. Tao, A sharp bilinear restrictions estimate for paraboloids, Geom. Funct. Anal., 13 (2003), 1359-1384. doi: 10.1007/s00039-003-0449-0.
[18]	T. Tao, "Nonlinear Dispersive Equations," CBMS Regional Conf. Ser. in Math. Amer. Math. Soc., 106, 2006.
[19]	T. Tao, A. Vargas and L. Vega, A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc., 11 (1998), 967-1000. doi: 10.1090/S0894-0347-98-00278-1.

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