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January  2020, 40(1): 423-439. doi: 10.3934/dcds.2020016

On the radius of spatial analyticity for defocusing nonlinear Schrödinger equations

Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea

* Corresponding author: Ihyeok Seo

Received  February 2019 Revised  July 2019 Published  October 2019

Fund Project: This research was supported by NRF-2019R1F1A1061316

In this paper we study spatial analyticity of solutions to the defocusing nonlinear Schrödinger equations $ iu_t + \Delta u = |u|^{p-1}u $, given initial data which is analytic with fixed radius. It is shown that the uniform radius of spatial analyticity of solutions at later time $ t $ cannot decay faster than $ 1/|t| $ as $ |t|\rightarrow\infty $. This extends the previous work of Tesfahun [19] for the cubic case $ p = 3 $ to the cases where $ p $ is any odd integer greater than $ 3 $.

Citation: Jaeseop Ahn, Jimyeong Kim, Ihyeok Seo. On the radius of spatial analyticity for defocusing nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 423-439. doi: 10.3934/dcds.2020016
References:
[1]

J. B. Baillon, T. Cazenave and M. Figueira, Équation de Schrödinger nonlinéaire, C. R. Acad. Sci., Paris 284 (1977), 869–872.  Google Scholar

[2]

J. L. Bona and Z. Grujić, Spatial analyticity properties of nonlinear waves, Math. Models Methods Appl. Sci., 13 (2003), 345-360.  doi: 10.1142/S0218202503002532.  Google Scholar

[3]

J. L. BonaZ. Grujić and H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 783-797.  doi: 10.1016/j.anihpc.2004.12.004.  Google Scholar

[4]

J. L. BonaZ. Grujić and H. Kalisch, Global solutions of the derivative Schrödinger equations in a class of functions analytic in a strip, J. Differential Equations, 229 (2006), 186-203.  doi: 10.1016/j.jde.2006.04.013.  Google Scholar

[5]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Almost conservation laws and global laws and global rough solutions to a nonlinear Schrödinger equation, Math. Res. Lett., 9 (2002), 659-682.  doi: 10.4310/MRL.2002.v9.n5.a9.  Google Scholar

[6]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Funct. Anal., 211 (2004), 173-218.  doi: 10.1016/S0022-1236(03)00218-0.  Google Scholar

[7]

Y. F. Fang and M. G. Grillakis, On the global existence of rough solutions of the cubic defocusing Schrödinger equation in $\mathbb{R}^{2+1}$, J. Hyperbolic Differ. Equ., 4 (2007), 233-257.  doi: 10.1142/S0219891607001161.  Google Scholar

[8]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I: The Cauchy problem, J. Funct. Anal., 32 (1979), 1-32.  doi: 10.1016/0022-1236(79)90076-4.  Google Scholar

[9]

J. Huang and M. Wang, New lower bounds on ther radius of spatial analyticity for the KdV equation, J. Differential Equations, 266 (2019), 5278-5317.  doi: 10.1016/j.jde.2018.10.025.  Google Scholar

[10]

T. Kato and K. Masuda, Nonlinear evolution equations and analyticity. I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 455-467.  doi: 10.1016/S0294-1449(16)30377-8.  Google Scholar

[11]

Y. Katznelson, An Introduction to Harmonic Analysis, Corrected ed., Dover Publications, Inc., New York, 1976.  Google Scholar

[12]

C. E. KenigG. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.  doi: 10.1215/S0012-7094-93-07101-3.  Google Scholar

[13]

S. Panizzi, On the domain of analyticity of solutions to semilinear Klein-Gordon equations, Nonlinear Anal., 75 (2012), 2841-2850.  doi: 10.1016/j.na.2011.11.031.  Google Scholar

[14]

S. Selberg, Spatial analyticity of solutions to nonlinear dispersive PDE, Non-linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis, 437–454, EMS Ser. Congr. Rep., Eur. Math. Soc., Zurich, 2018.  Google Scholar

[15]

S. Selberg, On the radius of spatial analyticity for solutions of the Dirac-Klein-Gordon equations in two space dimensions, Ann. I. H. Poincaré - AN, 36 (2019), 1311-1330.  doi: 10.1016/j.anihpc.2018.12.002.  Google Scholar

[16]

S. Selberg and D. O. da Silva, Lower bounds on the radius of spatial analyticity for the KdV equation, Ann. Henri Poincaré, 18 (2017), 1009-1023.  doi: 10.1007/s00023-016-0498-1.  Google Scholar

[17]

S. Selberg and A. Tesfahun, On the radius of spatial analyticity for the 1d Dirac-Klein-Gordon equations, J. Differential Equations, 259 (2015), 4732-4744.  doi: 10.1016/j.jde.2015.06.007.  Google Scholar

[18]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics, vol. 106, American Mathematical Society, Providence, RI, 2006, published for the Conference Board of the Mathematical Sciences, Washington, DC. doi: 10.1090/cbms/106.  Google Scholar

[19]

A. Tesfahun, On the radius of spatial analyticity for cubic nonlinear Schrödinger equations, J. Differential Equations, 263 (2017), 7496-7512.  doi: 10.1016/j.jde.2017.08.009.  Google Scholar

[20]

A. Tesfahun, Asymptotic lower bound for the radius of spatial analyticty to solutions of KdV equation, Commun. Contemp. Math.. doi: 10.1142/S021919971850061X.  Google Scholar

[21]

Y. Tsutsumi, L2 solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.   Google Scholar

show all references

References:
[1]

J. B. Baillon, T. Cazenave and M. Figueira, Équation de Schrödinger nonlinéaire, C. R. Acad. Sci., Paris 284 (1977), 869–872.  Google Scholar

[2]

J. L. Bona and Z. Grujić, Spatial analyticity properties of nonlinear waves, Math. Models Methods Appl. Sci., 13 (2003), 345-360.  doi: 10.1142/S0218202503002532.  Google Scholar

[3]

J. L. BonaZ. Grujić and H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 783-797.  doi: 10.1016/j.anihpc.2004.12.004.  Google Scholar

[4]

J. L. BonaZ. Grujić and H. Kalisch, Global solutions of the derivative Schrödinger equations in a class of functions analytic in a strip, J. Differential Equations, 229 (2006), 186-203.  doi: 10.1016/j.jde.2006.04.013.  Google Scholar

[5]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Almost conservation laws and global laws and global rough solutions to a nonlinear Schrödinger equation, Math. Res. Lett., 9 (2002), 659-682.  doi: 10.4310/MRL.2002.v9.n5.a9.  Google Scholar

[6]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Funct. Anal., 211 (2004), 173-218.  doi: 10.1016/S0022-1236(03)00218-0.  Google Scholar

[7]

Y. F. Fang and M. G. Grillakis, On the global existence of rough solutions of the cubic defocusing Schrödinger equation in $\mathbb{R}^{2+1}$, J. Hyperbolic Differ. Equ., 4 (2007), 233-257.  doi: 10.1142/S0219891607001161.  Google Scholar

[8]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I: The Cauchy problem, J. Funct. Anal., 32 (1979), 1-32.  doi: 10.1016/0022-1236(79)90076-4.  Google Scholar

[9]

J. Huang and M. Wang, New lower bounds on ther radius of spatial analyticity for the KdV equation, J. Differential Equations, 266 (2019), 5278-5317.  doi: 10.1016/j.jde.2018.10.025.  Google Scholar

[10]

T. Kato and K. Masuda, Nonlinear evolution equations and analyticity. I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 455-467.  doi: 10.1016/S0294-1449(16)30377-8.  Google Scholar

[11]

Y. Katznelson, An Introduction to Harmonic Analysis, Corrected ed., Dover Publications, Inc., New York, 1976.  Google Scholar

[12]

C. E. KenigG. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.  doi: 10.1215/S0012-7094-93-07101-3.  Google Scholar

[13]

S. Panizzi, On the domain of analyticity of solutions to semilinear Klein-Gordon equations, Nonlinear Anal., 75 (2012), 2841-2850.  doi: 10.1016/j.na.2011.11.031.  Google Scholar

[14]

S. Selberg, Spatial analyticity of solutions to nonlinear dispersive PDE, Non-linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis, 437–454, EMS Ser. Congr. Rep., Eur. Math. Soc., Zurich, 2018.  Google Scholar

[15]

S. Selberg, On the radius of spatial analyticity for solutions of the Dirac-Klein-Gordon equations in two space dimensions, Ann. I. H. Poincaré - AN, 36 (2019), 1311-1330.  doi: 10.1016/j.anihpc.2018.12.002.  Google Scholar

[16]

S. Selberg and D. O. da Silva, Lower bounds on the radius of spatial analyticity for the KdV equation, Ann. Henri Poincaré, 18 (2017), 1009-1023.  doi: 10.1007/s00023-016-0498-1.  Google Scholar

[17]

S. Selberg and A. Tesfahun, On the radius of spatial analyticity for the 1d Dirac-Klein-Gordon equations, J. Differential Equations, 259 (2015), 4732-4744.  doi: 10.1016/j.jde.2015.06.007.  Google Scholar

[18]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics, vol. 106, American Mathematical Society, Providence, RI, 2006, published for the Conference Board of the Mathematical Sciences, Washington, DC. doi: 10.1090/cbms/106.  Google Scholar

[19]

A. Tesfahun, On the radius of spatial analyticity for cubic nonlinear Schrödinger equations, J. Differential Equations, 263 (2017), 7496-7512.  doi: 10.1016/j.jde.2017.08.009.  Google Scholar

[20]

A. Tesfahun, Asymptotic lower bound for the radius of spatial analyticty to solutions of KdV equation, Commun. Contemp. Math.. doi: 10.1142/S021919971850061X.  Google Scholar

[21]

Y. Tsutsumi, L2 solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.   Google Scholar

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