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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 |
$i \partial_{t}u+\partial_{x}^{4}u=u^{2},\ \ (t,x)∈[0,T]× \mathbb{R}.$ |
$H^{s}(\mathbb{R})$ |
$-\frac{7}{4} <s≤q 0.$ |
$H^{s}(\mathbb{R})$ |
$s≥q -2$ |
$s < -2$ |
$s <-2$ |
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. Colliander, M. Keel, G. Staffilani, H. 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. Hao, L. Hsiao and B. X. Wang,
Well-posedness for the fourth-order Schrödinger equations, J. Math. Anal. Appl., 320 (2006), 246-265.
|
[6] |
C. Hao, L. 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. Miao, G. 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. Colliander, M. Keel, G. Staffilani, H. 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. Hao, L. Hsiao and B. X. Wang,
Well-posedness for the fourth-order Schrödinger equations, J. Math. Anal. Appl., 320 (2006), 246-265.
|
[6] |
C. Hao, L. 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. Miao, G. 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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