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.

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