# 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 $J(u_0) Citation: Jing Lu, Yifei Wu. Sharp threshold for scattering of a generalized Davey-Stewartson system in three dimension. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1641-1670. doi: 10.3934/cpaa.2015.14.1641 ##### References:  [1] H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175. [2] T. Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Métedos Matemáticos, Vol. 22, I.M.U.F.R.J., Rio de Janiero, 1989. [3] R. Cipolatti, On the existence of standing waves for a Davey-Stewartson system, Comm. Part. Diff. Eq., 17 (1992), 967-988. doi: 10.1080/03605309208820872. [4] R. Cipolatti, On the instability of ground states for a Davey-Stewartson system, Annales de l'I. H. P., section A, 58 (1993), 85-104. [5] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified Kdv on$\mathbb{R}^{N}$and$T$, J. Amer. Math. Soc., 16 (2003), 705-749. doi: 10.1090/S0894-0347-03-00421-1. [6] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in$R^3$, Annals of Math., 167 (2008), 767-865. doi: 10.4007/annals.2008.167.767. [7] A. Davey and K. Stewartson, On 3-dimensional packets of surface waves, Proc. R. Soc. London A, 338 (1974), 101-110. [8] T. Duyckaerts, J. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250. doi: 10.4310/MRL.2008.v15.n6.a13. [9] D. Du, Y. Wu and K. Zhang, On blow-up criterion for the nonlinear Schrödinger equation, Preprint. [10] D. Foschi, Inhomogeneous Strichartz estimates, J. Hyper. Diff. Eq., 2 (2005), 1-24. doi: 10.1142/S0219891605000361. [11] Z. Gan and J. Zhang, Sharp threshold of global existence and instability of standing wave for a Davey-Stewartson system, Comm. Math. Phys., 283 (2008), 93-125. doi: 10.1007/s00220-008-0456-y. [12] J-M. Ghidaglia and J. C. Saut, On the initial value problem for the Davey-Stewartson systems, Nonlinearity, 3 (1990), 475-506. [13] P. Gérard, Oscillations and concentration effects in semilinear dispersive wave equations, J. Funct. Anal., 141 (1996), 60-98. doi: 10.1006/jfan.1996.0122. [14] J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3d cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467. doi: 10.1007/s00220-008-0529-y. [15] J. Holmer and S. Roudenko, Divergence of infinite-variance nonradial solutions to the 3d NLS equation, Comm, Partial Differ. Eqns, 35, (2010), 875-905. doi: 10.1080/03605301003646713. [16] S. Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation, Analysis & PDE, 4-3 (2011), 405-460. doi: 10.2140/apde.2011.4.405. [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, Invent. Math., 166 (2006), 645-675. doi: 10.1007/s00222-006-0011-4. [18] C. Kenig, G. Ponce and L. Vega, On the Zakharov and Zakharov-Shulman systems, J. Funct. Anal., 127 (1995), 204-234. doi: 10.1006/jfan.1995.1009. [19] S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equation, J. Diff. Eq., 175 (2001), 353-392. doi: 10.1006/jdeq.2000.3951. [20] R. Killip, M. Visan and X. Zhang, The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Anal. PDE, 1 (2008), 229-266. doi: 10.2140/apde.2008.1.229. [21] C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data, J. Funct. Anal., 253 (2007), 605-627. doi: 10.1016/j.jfa.2007.09.008. [22] C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the defocusing$H^{\frac12}$-subcritical Hartree equation in$mathbb{R}^{d}$, Ann. I. H. Poincaré-NA, 26 (2009), 1831-1852. doi: 10.1016/j.anihpc.2009.01.003. [23] C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the mass-critical Hartree equation with radial data, J. Math. Pures Appl., 91 (2009), 49-79. doi: 10.1016/j.matpur.2008.09.003. [24] C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation in$\R^{1+n}$, Comm. Partial. Diff. Eqn., 36 (2011), 729-776. doi: 10.1080/03605302.2010.531073. [25] K. Nishinari, K. Abe and J. Satsuma, Multidimensional behavior of an eletrostatic ion wave in a magnetized plasma, Phys. Plasmas, 1 (1994), 2559-2565. [26] M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system, Annales de l'I. H. P., section A, 62 (1995), 69-80. [27] G. C. Papanicolaou, C. Sulem, P-L. Sulem and X. P. Wang, The focusing singularity of the Davey-Stewartson equations for gravity-capillary surface waves, Physica D, 72 (1994), 61-86. doi: 10.1016/0167-2789(94)90167-8. [28] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Self-focusing and Wave Collapse, Springer-Verlag, New York, 1999. [29] M. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation, Tran. Amer. Math. Soc., 359 (2007), 2123-2136. doi: 10.1090/S0002-9947-06-04099-2. [30] V. Zakharov and E. Schulman, Integrability of nonlinear systems and perturbation theory, in what is integrability?(Zakharov, ed.) 189-250, Springer Series on Nonlinear Dynamics, Springer-Verlag. show all references ##### References:  [1] H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175. [2] T. Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Métedos Matemáticos, Vol. 22, I.M.U.F.R.J., Rio de Janiero, 1989. [3] R. Cipolatti, On the existence of standing waves for a Davey-Stewartson system, Comm. Part. Diff. Eq., 17 (1992), 967-988. doi: 10.1080/03605309208820872. [4] R. Cipolatti, On the instability of ground states for a Davey-Stewartson system, Annales de l'I. H. P., section A, 58 (1993), 85-104. [5] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified Kdv on$\mathbb{R}^{N}$and$T$, J. Amer. Math. Soc., 16 (2003), 705-749. doi: 10.1090/S0894-0347-03-00421-1. [6] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in$R^3$, Annals of Math., 167 (2008), 767-865. doi: 10.4007/annals.2008.167.767. [7] A. Davey and K. Stewartson, On 3-dimensional packets of surface waves, Proc. R. Soc. London A, 338 (1974), 101-110. [8] T. Duyckaerts, J. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250. doi: 10.4310/MRL.2008.v15.n6.a13. [9] D. Du, Y. Wu and K. Zhang, On blow-up criterion for the nonlinear Schrödinger equation, Preprint. [10] D. Foschi, Inhomogeneous Strichartz estimates, J. Hyper. Diff. Eq., 2 (2005), 1-24. doi: 10.1142/S0219891605000361. [11] Z. Gan and J. Zhang, Sharp threshold of global existence and instability of standing wave for a Davey-Stewartson system, Comm. Math. Phys., 283 (2008), 93-125. doi: 10.1007/s00220-008-0456-y. [12] J-M. Ghidaglia and J. C. Saut, On the initial value problem for the Davey-Stewartson systems, Nonlinearity, 3 (1990), 475-506. [13] P. Gérard, Oscillations and concentration effects in semilinear dispersive wave equations, J. Funct. Anal., 141 (1996), 60-98. doi: 10.1006/jfan.1996.0122. [14] J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3d cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467. doi: 10.1007/s00220-008-0529-y. [15] J. Holmer and S. Roudenko, Divergence of infinite-variance nonradial solutions to the 3d NLS equation, Comm, Partial Differ. Eqns, 35, (2010), 875-905. doi: 10.1080/03605301003646713. [16] S. Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation, Analysis & PDE, 4-3 (2011), 405-460. doi: 10.2140/apde.2011.4.405. [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, Invent. Math., 166 (2006), 645-675. doi: 10.1007/s00222-006-0011-4. [18] C. Kenig, G. Ponce and L. Vega, On the Zakharov and Zakharov-Shulman systems, J. Funct. Anal., 127 (1995), 204-234. doi: 10.1006/jfan.1995.1009. [19] S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equation, J. Diff. Eq., 175 (2001), 353-392. doi: 10.1006/jdeq.2000.3951. [20] R. Killip, M. Visan and X. Zhang, The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Anal. PDE, 1 (2008), 229-266. doi: 10.2140/apde.2008.1.229. [21] C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data, J. Funct. Anal., 253 (2007), 605-627. doi: 10.1016/j.jfa.2007.09.008. [22] C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the defocusing$H^{\frac12}$-subcritical Hartree equation in$mathbb{R}^{d}$, Ann. I. H. Poincaré-NA, 26 (2009), 1831-1852. doi: 10.1016/j.anihpc.2009.01.003. [23] C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the mass-critical Hartree equation with radial data, J. Math. Pures Appl., 91 (2009), 49-79. doi: 10.1016/j.matpur.2008.09.003. [24] C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation in$\R^{1+n}$, Comm. Partial. Diff. Eqn., 36 (2011), 729-776. doi: 10.1080/03605302.2010.531073. [25] K. Nishinari, K. Abe and J. Satsuma, Multidimensional behavior of an eletrostatic ion wave in a magnetized plasma, Phys. Plasmas, 1 (1994), 2559-2565. [26] M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system, Annales de l'I. H. P., section A, 62 (1995), 69-80. [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, Physica D, 72 (1994), 61-86. doi: 10.1016/0167-2789(94)90167-8. [28] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Self-focusing and Wave Collapse, Springer-Verlag, New York, 1999. [29] M. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation, Tran. Amer. Math. Soc., 359 (2007), 2123-2136. doi: 10.1090/S0002-9947-06-04099-2. [30] V. Zakharov and E. Schulman, Integrability of nonlinear systems and perturbation theory, in what is integrability?(Zakharov, ed.) 189-250, Springer Series on Nonlinear Dynamics, Springer-Verlag.  [1] Christian Klein, Benson Muite, Kristelle Roidot. Numerical study of blow-up in the Davey-Stewartson system. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1361-1387. doi: 10.3934/dcdsb.2013.18.1361 [2] Shiming Li, Yongsheng Li, Wei Yan. 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