March  2018, 17(2): 487-504. doi: 10.3934/cpaa.2018027

Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation

1. 

School of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China

2. 

College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007, China

Received  January 2017 Revised  September 2017 Published  March 2018

Fund Project: This work is supported by NSFC under grant numbers 11571118, 11771127 and 11401180, and also by the Fundamental Research Funds for the Central Universities of China under the grant number 2017ZD094.

In this paper, we investigate the Cauchy problem for the fourth order nonlinear Schrödinger equation
$i \partial_{t}u+\partial_{x}^{4}u=u^{2},\ \ (t,x)∈[0,T]× \mathbb{R}.$
Zheng (Adv. Differential Equations, 16(2011), 467-486.) has proved that the problem is locally well-posed in
$H^{s}(\mathbb{R})$
with
$-\frac{7}{4} <s≤q 0.$
In this paper, we aim at extending Zheng's work to a lower regularity index. We prove that the equation is locally well-posed in
$H^{s}(\mathbb{R})$
when
$s≥q -2$
and ill-posed when
$s < -2$
in the sense that the solution map is discontinuous for
$s <-2$
. The key ingredient used in this paper is Besov-type space introduced by Bejenaru and Tao (Journal of Functional Analysis, 233(2006), 228-259.).
Citation: Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027
References:
[1]

I. Bejenaru and T. Tao, Sharp well-posedness and ill-posedness results for a quadratic nonlinear Schrödinger equation, J. Funct. Anal., 233 (2006), 228-259. 

[2]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I. Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156. 

[3]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}$ and $\mathbb{T}$, J. Amer. Math. Soc., 16 (2003), 705-749. 

[4]

B. L. Guo and B. X. Wang, The global Cauchy problem and scattering of solutions for nonlinear Schrödinger equations in $H^{s}$, Diff. Int. Eqns., 15 (2002), 1073-1083. 

[5]

C. HaoL. Hsiao and B. X. Wang, Well-posedness for the fourth-order Schrödinger equations, J. Math. Anal. Appl., 320 (2006), 246-265. 

[6]

C. HaoL. Hsiao and B. X. Wang, Well-posedness of the Cauchy problem for the fourth-order Schrödinger equations in high dimensions, J. Math. Anal. Appl., 328 (2007), 58-83. 

[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. 

[8]

N. Kishimoto, Remark on the paper "Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation" by I. Bejenaru and T. Tao, Atl. Electron. J. Math., 4 (2011), 35-48. 

[9]

B. A. Ivanov and A. M. Kosevich, Stable three-dimensional small-amplitude soliton in magnetic materials, Sov. J. Low Temp. Phys., 9 (1983), 439-442. 

[10]

C. X. MiaoG. X. Xu and L. F. Zhao, Global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case, J. Diff. Eqns., 246 (2009), 3715-3749. 

[11]

C. X. Miao and J. Q. Zheng, Scattering theory for the defocusing fourth-order Schrödinger equation, Nonlinearity, 29 (2016), 692-736. 

[12]

B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dyn. Partial Diff. Eqns., 4 (2007), 197-225. 

[13]

B. Pausader, The cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517. 

[14]

B. Pausader, The focusing energy-critical fourth-order Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., 24 (2009), 1275-1292. 

[15]

B. Pausader and S. L. Shao, The mass-critical fourth-order Schrödinger equation in high dimensions, J. Hyperbolic Diff. Eqns., 7 (2010), 651-705. 

[16]

B. Pausader and S. X. Xia, Scattering theory for the fourth-order Schrödinger equation in low dimensions, Nonlinearity, 26 (2013), 2175-2191. 

[17]

H. Pecher and W. von Wahl, Time dependent nonlinear Schrödinger equations, Manuscripta Math., 27 (1979), 125-157. 

[18]

J. Segata, Modified wave operators for the fourth-order nonlinear Schrödinger-type equation with cubic non-linearity, Math. Methods. Appl. Sci., 26 (2006), 1785-1800. 

[19]

T. Tao, Multilinear weighted convolution of $L^{2}$ functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908. 

[20]

S. K. Turitsyn, Three-dimensional dispersion of nonlinearity and stability of multidimentional solitons, Teoret. Mat. Fiz. , 64 (1985), 226-232 (Russian).

[21]

J. Q. Zheng, Well-posedness for the fourth-order Schrödinger equations with quadratic nonlinearity, Adv. Diff. Eqns., 16 (2011), 467-486. 

show all references

References:
[1]

I. Bejenaru and T. Tao, Sharp well-posedness and ill-posedness results for a quadratic nonlinear Schrödinger equation, J. Funct. Anal., 233 (2006), 228-259. 

[2]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I. Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156. 

[3]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}$ and $\mathbb{T}$, J. Amer. Math. Soc., 16 (2003), 705-749. 

[4]

B. L. Guo and B. X. Wang, The global Cauchy problem and scattering of solutions for nonlinear Schrödinger equations in $H^{s}$, Diff. Int. Eqns., 15 (2002), 1073-1083. 

[5]

C. HaoL. Hsiao and B. X. Wang, Well-posedness for the fourth-order Schrödinger equations, J. Math. Anal. Appl., 320 (2006), 246-265. 

[6]

C. HaoL. Hsiao and B. X. Wang, Well-posedness of the Cauchy problem for the fourth-order Schrödinger equations in high dimensions, J. Math. Anal. Appl., 328 (2007), 58-83. 

[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. 

[8]

N. Kishimoto, Remark on the paper "Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation" by I. Bejenaru and T. Tao, Atl. Electron. J. Math., 4 (2011), 35-48. 

[9]

B. A. Ivanov and A. M. Kosevich, Stable three-dimensional small-amplitude soliton in magnetic materials, Sov. J. Low Temp. Phys., 9 (1983), 439-442. 

[10]

C. X. MiaoG. X. Xu and L. F. Zhao, Global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case, J. Diff. Eqns., 246 (2009), 3715-3749. 

[11]

C. X. Miao and J. Q. Zheng, Scattering theory for the defocusing fourth-order Schrödinger equation, Nonlinearity, 29 (2016), 692-736. 

[12]

B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dyn. Partial Diff. Eqns., 4 (2007), 197-225. 

[13]

B. Pausader, The cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517. 

[14]

B. Pausader, The focusing energy-critical fourth-order Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., 24 (2009), 1275-1292. 

[15]

B. Pausader and S. L. Shao, The mass-critical fourth-order Schrödinger equation in high dimensions, J. Hyperbolic Diff. Eqns., 7 (2010), 651-705. 

[16]

B. Pausader and S. X. Xia, Scattering theory for the fourth-order Schrödinger equation in low dimensions, Nonlinearity, 26 (2013), 2175-2191. 

[17]

H. Pecher and W. von Wahl, Time dependent nonlinear Schrödinger equations, Manuscripta Math., 27 (1979), 125-157. 

[18]

J. Segata, Modified wave operators for the fourth-order nonlinear Schrödinger-type equation with cubic non-linearity, Math. Methods. Appl. Sci., 26 (2006), 1785-1800. 

[19]

T. Tao, Multilinear weighted convolution of $L^{2}$ functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908. 

[20]

S. K. Turitsyn, Three-dimensional dispersion of nonlinearity and stability of multidimentional solitons, Teoret. Mat. Fiz. , 64 (1985), 226-232 (Russian).

[21]

J. Q. Zheng, Well-posedness for the fourth-order Schrödinger equations with quadratic nonlinearity, Adv. Diff. Eqns., 16 (2011), 467-486. 

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