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Sharp wellposedness 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, T. Tao, Sharp wellposedness and illposedness results for a quadratic nonlinear Schrödinger equation, J. Funct. Anal., 233 (2006), 228259. 
[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), 107156. 
[3] 
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Sharp global wellposedness for KdV and modified KdV on $\mathbb{R}$ and $\mathbb{T}$, J. Amer. Math. Soc., 16 (2003), 705749. 
[4] 
B. L. Guo, B. X. Wang, The global Cauchy problem and scattering of solutions for nonlinear Schrödinger equations in $H^{s}$, Diff. Int. Eqns., 15 (2002), 10731083. 
[5] 
C. Hao, L. Hsiao, B. X. Wang, Wellposedness for the fourthorder Schrödinger equations, J. Math. Anal. Appl., 320 (2006), 246265. 
[6] 
C. Hao, L. Hsiao, B. X. Wang, Wellposedness of the Cauchy problem for the fourthorder Schrödinger equations in high dimensions, J. Math. Anal. Appl., 328 (2007), 5883. 
[7] 
V. I. Karpman, Stabilization of soliton instabilities by higherorder dispersion: Fourth order nonlinear Schrödingertype equations, Phys. Rev. E, 53 (1996), 13361339. 
[8] 
N. Kishimoto, Remark on the paper "Sharp wellposedness and illposedness results for a quadratic nonlinear Schrödinger equation" by I. Bejenaru and T. Tao, Atl. Electron. J. Math., 4 (2011), 3548. 
[9] 
B. A. Ivanov, A. M. Kosevich, Stable threedimensional smallamplitude soliton in magnetic materials, Sov. J. Low Temp. Phys., 9 (1983), 439442. 
[10] 
C. X. Miao, G. X. Xu, L. F. Zhao, Global wellposedness and scattering for the focusing energycritical nonlinear Schrödinger equations of fourth order in the radial case, J. Diff. Eqns., 246 (2009), 37153749. 
[11] 
C. X. Miao, J. Q. Zheng, Scattering theory for the defocusing fourthorder Schrödinger equation, Nonlinearity, 29 (2016), 692736. 
[12] 
B. Pausader, Global wellposedness for energy critical fourthorder Schrödinger equations in the radial case, Dyn. Partial Diff. Eqns., 4 (2007), 197225. 
[13] 
B. Pausader, The cubic fourthorder Schrödinger equation, J. Funct. Anal., 256 (2009), 24732517. 
[14] 
B. Pausader, The focusing energycritical fourthorder Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., 24 (2009), 12751292. 
[15] 
B. Pausader, S. L. Shao, The masscritical fourthorder Schrödinger equation in high dimensions, J. Hyperbolic Diff. Eqns., 7 (2010), 651705. 
[16] 
B. Pausader, S. X. Xia, Scattering theory for the fourthorder Schrödinger equation in low dimensions, Nonlinearity, 26 (2013), 21752191. 
[17] 
H. Pecher, W. von Wahl, Time dependent nonlinear Schrödinger equations, Manuscripta Math., 27 (1979), 125157. 
[18] 
J. Segata, Modified wave operators for the fourthorder nonlinear Schrödingertype equation with cubic nonlinearity, Math. Methods. Appl. Sci., 26 (2006), 17851800. 
[19] 
T. Tao, Multilinear weighted convolution of $L^{2}$ functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839908. 
[20] 
S. K. Turitsyn, Threedimensional dispersion of nonlinearity and stability of multidimentional solitons, Teoret. Mat. Fiz. , 64 (1985), 226232 (Russian). 
[21] 
J. Q. Zheng, Wellposedness for the fourthorder Schrödinger equations with quadratic nonlinearity, Adv. Diff. Eqns., 16 (2011), 467486. 
show all references
References:
[1] 
I. Bejenaru, T. Tao, Sharp wellposedness and illposedness results for a quadratic nonlinear Schrödinger equation, J. Funct. Anal., 233 (2006), 228259. 
[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), 107156. 
[3] 
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Sharp global wellposedness for KdV and modified KdV on $\mathbb{R}$ and $\mathbb{T}$, J. Amer. Math. Soc., 16 (2003), 705749. 
[4] 
B. L. Guo, B. X. Wang, The global Cauchy problem and scattering of solutions for nonlinear Schrödinger equations in $H^{s}$, Diff. Int. Eqns., 15 (2002), 10731083. 
[5] 
C. Hao, L. Hsiao, B. X. Wang, Wellposedness for the fourthorder Schrödinger equations, J. Math. Anal. Appl., 320 (2006), 246265. 
[6] 
C. Hao, L. Hsiao, B. X. Wang, Wellposedness of the Cauchy problem for the fourthorder Schrödinger equations in high dimensions, J. Math. Anal. Appl., 328 (2007), 5883. 
[7] 
V. I. Karpman, Stabilization of soliton instabilities by higherorder dispersion: Fourth order nonlinear Schrödingertype equations, Phys. Rev. E, 53 (1996), 13361339. 
[8] 
N. Kishimoto, Remark on the paper "Sharp wellposedness and illposedness results for a quadratic nonlinear Schrödinger equation" by I. Bejenaru and T. Tao, Atl. Electron. J. Math., 4 (2011), 3548. 
[9] 
B. A. Ivanov, A. M. Kosevich, Stable threedimensional smallamplitude soliton in magnetic materials, Sov. J. Low Temp. Phys., 9 (1983), 439442. 
[10] 
C. X. Miao, G. X. Xu, L. F. Zhao, Global wellposedness and scattering for the focusing energycritical nonlinear Schrödinger equations of fourth order in the radial case, J. Diff. Eqns., 246 (2009), 37153749. 
[11] 
C. X. Miao, J. Q. Zheng, Scattering theory for the defocusing fourthorder Schrödinger equation, Nonlinearity, 29 (2016), 692736. 
[12] 
B. Pausader, Global wellposedness for energy critical fourthorder Schrödinger equations in the radial case, Dyn. Partial Diff. Eqns., 4 (2007), 197225. 
[13] 
B. Pausader, The cubic fourthorder Schrödinger equation, J. Funct. Anal., 256 (2009), 24732517. 
[14] 
B. Pausader, The focusing energycritical fourthorder Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., 24 (2009), 12751292. 
[15] 
B. Pausader, S. L. Shao, The masscritical fourthorder Schrödinger equation in high dimensions, J. Hyperbolic Diff. Eqns., 7 (2010), 651705. 
[16] 
B. Pausader, S. X. Xia, Scattering theory for the fourthorder Schrödinger equation in low dimensions, Nonlinearity, 26 (2013), 21752191. 
[17] 
H. Pecher, W. von Wahl, Time dependent nonlinear Schrödinger equations, Manuscripta Math., 27 (1979), 125157. 
[18] 
J. Segata, Modified wave operators for the fourthorder nonlinear Schrödingertype equation with cubic nonlinearity, Math. Methods. Appl. Sci., 26 (2006), 17851800. 
[19] 
T. Tao, Multilinear weighted convolution of $L^{2}$ functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839908. 
[20] 
S. K. Turitsyn, Threedimensional dispersion of nonlinearity and stability of multidimentional solitons, Teoret. Mat. Fiz. , 64 (1985), 226232 (Russian). 
[21] 
J. Q. Zheng, Wellposedness for the fourthorder Schrödinger equations with quadratic nonlinearity, Adv. Diff. Eqns., 16 (2011), 467486. 
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