Sharp threshold for scattering of a generalized Davey-Stewartson system in three dimension

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September 2015, 14(5): 1641-1670. doi: 10.3934/cpaa.2015.14.1641

Sharp threshold for scattering of a generalized Davey-Stewartson system in three dimension

1.	China Academy of Engineering Physics, Beijing 100088, China
2.	School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 1000875 , China

Received November 2013 Revised December 2014 Published June 2015

Abstract
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In this paper, we consider the Cauchy problem for the generalized Davey-Stewartson system \begin{eqnarray} &i\partial_t u + \Delta u =-a|u|^{p-1}u+b_1uv_{x_1}, (t,x)\in R \times R^3,\\ &-\Delta v=b_2(|u|^2)_{x_1}, \end{eqnarray} where $a>0,b_1b_2>0$, $\frac{4}{3}+1< p< 5$. We first use a variational approach to give a dichotomy of blow-up and scattering for the solution of mass supercritical equation with the initial data satisfying $J(u_0)

Keywords: scatter, concentration-compactness., Davey-Stewartson system, blow up, global well-posedness.

Mathematics Subject Classification: Primary 35Q53; Secondary 47J3.

Citation: Jing Lu, Yifei Wu. Sharp threshold for scattering of a generalized Davey-Stewartson system in three dimension. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1641-1670. doi: 10.3934/cpaa.2015.14.1641

References:

[1]	H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations,, \emph{Amer. J. Math.}, 121 (1999), 131. Google Scholar
[2]	T. Cazenave, An Introduction to Nonlinear Schrödinger Equations,, Textos de M\'etedos Matem\'aticos, 22 (1989). Google Scholar
[3]	R. Cipolatti, On the existence of standing waves for a Davey-Stewartson system,, \emph{Comm. Part. Diff. Eq.}, 17 (1992), 967. doi: 10.1080/03605309208820872. Google Scholar
[4]	R. Cipolatti, On the instability of ground states for a Davey-Stewartson system,, \emph{Annales de l'I. H. P., 58 (1993), 85. Google Scholar
[5]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified Kdv on $R$ and $T$,, \emph{J. Amer. Math. Soc.}, 16 (2003), 705. doi: 10.1090/S0894-0347-03-00421-1. Google Scholar
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[7]	A. Davey and K. Stewartson, On 3-dimensional packets of surface waves,, \emph{Proc. R. Soc. London A, 338 (1974), 101. Google Scholar
[8]	T. Duyckaerts, J. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation,, \emph{Math. Res. Lett.}, 15 (2008), 1233. doi: 10.4310/MRL.2008.v15.n6.a13. Google Scholar
[9]	D. Du, Y. Wu and K. Zhang, On blow-up criterion for the nonlinear Schrödinger equation,, Preprint., (). Google Scholar
[10]	D. Foschi, Inhomogeneous Strichartz estimates,, \emph{J. Hyper. Diff. Eq.}, 2 (2005), 1. doi: 10.1142/S0219891605000361. Google Scholar
[11]	Z. Gan and J. Zhang, Sharp threshold of global existence and instability of standing wave for a Davey-Stewartson system,, \emph{Comm. Math. Phys.}, 283 (2008), 93. doi: 10.1007/s00220-008-0456-y. Google Scholar
[12]	J-M. Ghidaglia and J. C. Saut, On the initial value problem for the Davey-Stewartson systems,, \emph{Nonlinearity}, 3 (1990), 475. Google Scholar
[13]	P. Gérard, Oscillations and concentration effects in semilinear dispersive wave equations,, \emph{J. Funct. Anal.}, 141 (1996), 60. doi: 10.1006/jfan.1996.0122. Google Scholar
[14]	J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3d cubic nonlinear Schrödinger equation,, \emph{Comm. Math. Phys.}, 282 (2008), 435. doi: 10.1007/s00220-008-0529-y. Google Scholar
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[16]	S. Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation,, \emph{Analysis & PDE}, 4-3 (2011), 4. doi: 10.2140/apde.2011.4.405. Google Scholar
[17]	C. Kenig and F. Merle, Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear Schrödinger equation in the radial case,, \emph{Invent. Math.}, 166 (2006), 645. doi: 10.1007/s00222-006-0011-4. Google Scholar
[18]	C. Kenig, G. Ponce and L. Vega, On the Zakharov and Zakharov-Shulman systems,, \emph{J. Funct. Anal.}, 127 (1995), 204. doi: 10.1006/jfan.1995.1009. Google Scholar
[19]	S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equation,, \emph{J. Diff. Eq.}, 175 (2001), 353. doi: 10.1006/jdeq.2000.3951. Google Scholar
[20]	R. Killip, M. Visan and X. Zhang, The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher,, \emph{Anal. PDE}, 1 (2008), 229. doi: 10.2140/apde.2008.1.229. Google Scholar
[21]	C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data,, \emph{J. Funct. Anal.}, 253 (2007), 605. doi: 10.1016/j.jfa.2007.09.008. Google Scholar
[22]	C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the defocusing $H^{\frac12}$-subcritical Hartree equation in $R^d$,, \emph{Ann. I. H. Poincar$\acutee$-NA}, 26 (2009), 1831. doi: 10.1016/j.anihpc.2009.01.003. Google Scholar
[23]	C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the mass-critical Hartree equation with radial data,, \emph{J. Math. Pures Appl.}, 91 (2009), 49. doi: 10.1016/j.matpur.2008.09.003. Google Scholar
[24]	C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation in $\R^{1+n}$,, \emph{Comm. Partial. Diff. Eqn.}, 36 (2011), 729. doi: 10.1080/03605302.2010.531073. Google Scholar
[25]	K. Nishinari, K. Abe and J. Satsuma, Multidimensional behavior of an eletrostatic ion wave in a magnetized plasma,, \emph{Phys. Plasmas}, 1 (1994), 2559. Google Scholar
[26]	M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system,, \emph{Annales de l'I. H. P., 62 (1995), 69. Google Scholar
[27]	G. C. Papanicolaou, C. Sulem, P-L. Sulem and X. P. Wang, The focusing singularity of the Davey-Stewartson equations for gravity-capillary surface waves,, \emph{Physica D}, 72 (1994), 61. doi: 10.1016/0167-2789(94)90167-8. Google Scholar
[28]	C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Self-focusing and Wave Collapse,, Springer-Verlag, (1999). Google Scholar
[29]	M. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation,, \emph{Tran. Amer. Math. Soc.}, 359 (2007), 2123. doi: 10.1090/S0002-9947-06-04099-2. Google Scholar
[30]	V. Zakharov and E. Schulman, Integrability of nonlinear systems and perturbation theory,, in \emph{what is integrability?}(Zakharov, (): 189. Google Scholar

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References:

[1]	H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations,, \emph{Amer. J. Math.}, 121 (1999), 131. Google Scholar
[2]	T. Cazenave, An Introduction to Nonlinear Schrödinger Equations,, Textos de M\'etedos Matem\'aticos, 22 (1989). Google Scholar
[3]	R. Cipolatti, On the existence of standing waves for a Davey-Stewartson system,, \emph{Comm. Part. Diff. Eq.}, 17 (1992), 967. doi: 10.1080/03605309208820872. Google Scholar
[4]	R. Cipolatti, On the instability of ground states for a Davey-Stewartson system,, \emph{Annales de l'I. H. P., 58 (1993), 85. Google Scholar
[5]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified Kdv on $R$ and $T$,, \emph{J. Amer. Math. Soc.}, 16 (2003), 705. doi: 10.1090/S0894-0347-03-00421-1. Google Scholar
[6]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $R^3$,, \emph{Annals of Math.}, 167 (2008), 767. doi: 10.4007/annals.2008.167.767. Google Scholar
[7]	A. Davey and K. Stewartson, On 3-dimensional packets of surface waves,, \emph{Proc. R. Soc. London A, 338 (1974), 101. Google Scholar
[8]	T. Duyckaerts, J. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation,, \emph{Math. Res. Lett.}, 15 (2008), 1233. doi: 10.4310/MRL.2008.v15.n6.a13. Google Scholar
[9]	D. Du, Y. Wu and K. Zhang, On blow-up criterion for the nonlinear Schrödinger equation,, Preprint., (). Google Scholar
[10]	D. Foschi, Inhomogeneous Strichartz estimates,, \emph{J. Hyper. Diff. Eq.}, 2 (2005), 1. doi: 10.1142/S0219891605000361. Google Scholar
[11]	Z. Gan and J. Zhang, Sharp threshold of global existence and instability of standing wave for a Davey-Stewartson system,, \emph{Comm. Math. Phys.}, 283 (2008), 93. doi: 10.1007/s00220-008-0456-y. Google Scholar
[12]	J-M. Ghidaglia and J. C. Saut, On the initial value problem for the Davey-Stewartson systems,, \emph{Nonlinearity}, 3 (1990), 475. Google Scholar
[13]	P. Gérard, Oscillations and concentration effects in semilinear dispersive wave equations,, \emph{J. Funct. Anal.}, 141 (1996), 60. doi: 10.1006/jfan.1996.0122. Google Scholar
[14]	J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3d cubic nonlinear Schrödinger equation,, \emph{Comm. Math. Phys.}, 282 (2008), 435. doi: 10.1007/s00220-008-0529-y. Google Scholar
[15]	J. Holmer and S. Roudenko, Divergence of infinite-variance nonradial solutions to the 3d NLS equation,, \emph{Comm, 35 (2010), 875. doi: 10.1080/03605301003646713. Google Scholar
[16]	S. Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation,, \emph{Analysis & PDE}, 4-3 (2011), 4. doi: 10.2140/apde.2011.4.405. Google Scholar
[17]	C. Kenig and F. Merle, Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear Schrödinger equation in the radial case,, \emph{Invent. Math.}, 166 (2006), 645. doi: 10.1007/s00222-006-0011-4. Google Scholar
[18]	C. Kenig, G. Ponce and L. Vega, On the Zakharov and Zakharov-Shulman systems,, \emph{J. Funct. Anal.}, 127 (1995), 204. doi: 10.1006/jfan.1995.1009. Google Scholar
[19]	S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equation,, \emph{J. Diff. Eq.}, 175 (2001), 353. doi: 10.1006/jdeq.2000.3951. Google Scholar
[20]	R. Killip, M. Visan and X. Zhang, The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher,, \emph{Anal. PDE}, 1 (2008), 229. doi: 10.2140/apde.2008.1.229. Google Scholar
[21]	C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data,, \emph{J. Funct. Anal.}, 253 (2007), 605. doi: 10.1016/j.jfa.2007.09.008. Google Scholar
[22]	C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the defocusing $H^{\frac12}$-subcritical Hartree equation in $R^d$,, \emph{Ann. I. H. Poincar$\acutee$-NA}, 26 (2009), 1831. doi: 10.1016/j.anihpc.2009.01.003. Google Scholar
[23]	C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the mass-critical Hartree equation with radial data,, \emph{J. Math. Pures Appl.}, 91 (2009), 49. doi: 10.1016/j.matpur.2008.09.003. Google Scholar
[24]	C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation in $\R^{1+n}$,, \emph{Comm. Partial. Diff. Eqn.}, 36 (2011), 729. doi: 10.1080/03605302.2010.531073. Google Scholar
[25]	K. Nishinari, K. Abe and J. Satsuma, Multidimensional behavior of an eletrostatic ion wave in a magnetized plasma,, \emph{Phys. Plasmas}, 1 (1994), 2559. Google Scholar
[26]	M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system,, \emph{Annales de l'I. H. P., 62 (1995), 69. Google Scholar
[27]	G. C. Papanicolaou, C. Sulem, P-L. Sulem and X. P. Wang, The focusing singularity of the Davey-Stewartson equations for gravity-capillary surface waves,, \emph{Physica D}, 72 (1994), 61. doi: 10.1016/0167-2789(94)90167-8. Google Scholar
[28]	C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Self-focusing and Wave Collapse,, Springer-Verlag, (1999). Google Scholar
[29]	M. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation,, \emph{Tran. Amer. Math. Soc.}, 359 (2007), 2123. doi: 10.1090/S0002-9947-06-04099-2. Google Scholar
[30]	V. Zakharov and E. Schulman, Integrability of nonlinear systems and perturbation theory,, in \emph{what is integrability?}(Zakharov, (): 189. Google Scholar

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