# American Institute of Mathematical Sciences

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

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 $J(u_0) Citation: Jing Lu, Yifei Wu. Sharp threshold for scattering of a generalized Davey-Stewartson system in three dimension. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1641-1670. doi: 10.3934/cpaa.2015.14.1641 ##### References:  [1] H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations,, \emph{Amer. J. Math.}, 121 (1999), 131. Google Scholar [2] T. Cazenave, An Introduction to Nonlinear Schrödinger Equations,, Textos de M\'etedos Matem\'aticos, 22 (1989). Google Scholar [3] R. Cipolatti, On the existence of standing waves for a Davey-Stewartson system,, \emph{Comm. Part. Diff. Eq.}, 17 (1992), 967. doi: 10.1080/03605309208820872. Google Scholar [4] R. Cipolatti, On the instability of ground states for a Davey-Stewartson system,, \emph{Annales de l'I. H. P., 58 (1993), 85. Google Scholar [5] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified Kdv on$R$and$T$,, \emph{J. Amer. Math. Soc.}, 16 (2003), 705. doi: 10.1090/S0894-0347-03-00421-1. Google Scholar [6] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in$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}$,, \emph{Comm. Partial. Diff. Eqn.}, 36 (2011), 729. doi: 10.1080/03605302.2010.531073. Google Scholar [25] K. Nishinari, K. Abe and J. Satsuma, Multidimensional behavior of an eletrostatic ion wave in a magnetized plasma,, \emph{Phys. Plasmas}, 1 (1994), 2559. Google Scholar [26] M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system,, \emph{Annales de l'I. H. P., 62 (1995), 69. Google Scholar [27] G. C. Papanicolaou, C. Sulem, P-L. Sulem and X. P. Wang, The focusing singularity of the Davey-Stewartson equations for gravity-capillary surface waves,, \emph{Physica D}, 72 (1994), 61. doi: 10.1016/0167-2789(94)90167-8. Google Scholar [28] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Self-focusing and Wave Collapse,, Springer-Verlag, (1999). Google Scholar [29] M. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation,, \emph{Tran. Amer. Math. Soc.}, 359 (2007), 2123. doi: 10.1090/S0002-9947-06-04099-2. Google Scholar [30] V. Zakharov and E. Schulman, Integrability of nonlinear systems and perturbation theory,, in \emph{what is integrability?}(Zakharov, (): 189. Google Scholar show all references ##### References:  [1] H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations,, \emph{Amer. J. Math.}, 121 (1999), 131. Google Scholar [2] T. Cazenave, An Introduction to Nonlinear Schrödinger Equations,, Textos de M\'etedos Matem\'aticos, 22 (1989). Google Scholar [3] R. Cipolatti, On the existence of standing waves for a Davey-Stewartson system,, \emph{Comm. Part. Diff. Eq.}, 17 (1992), 967. doi: 10.1080/03605309208820872. Google Scholar [4] R. Cipolatti, On the instability of ground states for a Davey-Stewartson system,, \emph{Annales de l'I. H. P., 58 (1993), 85. Google Scholar [5] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified Kdv on$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  [1] Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020382 [2] Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248 [3] Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. 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