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A global existence and blow-up threshold for Davey-Stewartson equations in $\mathbb{R}^3$
1. | School of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China |
2. | Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640 |
3. | College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007 |
References:
[1] |
D. Anker and N. C. Freeman, On the soliton solutions of the Davey-Stewartson equation for long waves, Proc. Roy. Soc. London, 360 (1978), 529-540.
doi: 10.1098/rspa.1978.0083. |
[2] |
M. J. Ablowitz and A. S. Fokas, On the inverse scattering transform of multidimensional nonlinear equations related to first-order systems in the plane, J. Math. Phys., 25 (1984), 2494-2505.
doi: 10.1063/1.526471. |
[3] |
R. Cipolatti, On the existence of standing waves for a Davey-Stewartson system, Commun. Partial Diff. Eqns., 17 (1992), 967-988.
doi: 10.1080/03605309208820872. |
[4] |
R. Cipolatti, On the instability of ground states for a Davey-Stewartson system, Ann. Inst. H. Poincaré, 58 (1993), 85-104. |
[5] |
V. D. Djordjevic and L. G. Redekopp, On two-dimensional packets of capillary-gravity waves, J. Fluid Mech., 79 (1977), 703-714.
doi: 10.1017/S0022112077000408. |
[6] |
A. Davey and S. K. Stewartson, On three-dimensional packets of surface waves, Proc. R. Soc. London, 338 (1974), 101-110.
doi: 10.1098/rspa.1974.0076. |
[7] |
Z. H. Gan and J. Zhang, Sharp threshold of global existence and instability of standing wave for a Davey-Stewartson system, Commun. Math. Phys., 283 (2008), 93-125.
doi: 10.1007/s00220-008-0456-y. |
[8] |
J. M. Ghidaglia and J. C. Saut, On the initial value problem for the Davey-Stewartson systems, Nonlinearity, 3 (1990), 475-506.
doi: 10.1088/0951-7715/3/2/010. |
[9] |
N. Godet, A lower bound on the blow-up rate for the Davey-Stewartson system on the torus, Ann. Inst. H. Poincaré - AN, 30 (2013), 691-703.
doi: 10.1016/j.anihpc.2012.12.001. |
[10] |
B. L. Guo and B. X. Wang, The Cauchy problem for Davey-Stewartson systems, Commun. Pure Appl. Math., 52 (1999), 1477-1490.
doi: 10.1002/(SICI)1097-0312(199912)52:12<1477::AID-CPA1>3.0.CO;2-N. |
[11] |
N. Hayashi, Local existence in time of small solutions to the Davey-Stewartson systems, Ann. Inst. H. Poincaré, 65 (1996), 313-366. |
[12] |
N. Hayashi and H. Hirata, Global existence and asymptotic behavior in time of small solutions to the elliptic-hyperbolic Davey-Stewartson system, Nonlinearity, 9 (1996), 1387-1409.
doi: 10.1088/0951-7715/9/6/001. |
[13] |
N. Hayashi and J. C. Saut, Global existence of small solutions to the Davey-Stewartson and the Ishimori systems, Diff. and Integ. Eqns., 8 (1995), 1657-1675. |
[14] |
T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, Intern. Math. Res. Notices, 46 (2005), 2815-2828.
doi: 10.1155/IMRN.2005.2815. |
[15] |
X. G. Li, J. Zhang, S. Y. Lai and Y. H. Wu, The sharp threshold and limiting profile of blow-up solutions for a Davey-Stewartson system, J. Diff. Eqns., 250 (2011), 2197-2226.
doi: 10.1016/j.jde.2010.10.022. |
[16] |
F. Linares and G. Ponce, On the Davey-Stewartson systems, Ann. Inst. H. Poincaré, 10 (1993), 523-548. |
[17] |
J. Lu and Y. F. Wu, Sharp threshold for scattering of a generalized Davey-Stewartson system in three dimension, Commun. Pure Appl. Anal., 14 (2015), 1641-1670.
doi: 10.3934/cpaa.2015.14.1641. |
[18] |
T. Ozawa, Exact blow-up solutions to Cauchy problem for the Davey-Stewartson system, Proc. R. Soc. Lond. Ser. A, 436 (1992), 345-349.
doi: 10.1098/rspa.1992.0022. |
[19] |
M. Ohta, Stability of standing waves for the generalized Davey-Stewartson system, J. Dyn. Diff. Eqns., 6 (1994), 325-334.
doi: 10.1007/BF02218533. |
[20] |
G. Richards, Mass concentration for the Davey-Stewartson system, Diff. and Integ. Eqns., 24 (2011), 261-280. |
[21] |
J. Shu and J. Zhang, Sharp conditions of global existence for the generalized Davey-Stewartson system, IMA J. Appl. Math., 72 (2007), 36-42.
doi: 10.1093/imamat/hxl029. |
[22] |
M. Tsutsumi, Decay of weak solutions to the Davey-Stewartson systems, J. Math. Anal. Appl., 182 (1994), 680-704.
doi: 10.1006/jmaa.1994.1113. |
[23] |
B. X. Wang and B. L. Guo, On the initial value problem and scattering of solutions for the generalized Davey-Stewartson systems, Sci. China Ser. A, 44 (2001), 994-1002.
doi: 10.1007/BF02878975. |
[24] |
M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., 87 (1983), 567-576. |
[25] |
H. Yang, X. M. Fan and S. H. Zhu, Global analysis for rough solutions to the Davey- Stewartson system, Abstract and Appl. Anal., 2012 (2012), Article ID 578701, 22pp. |
[26] |
J. Zhang and S. Zhu, Sharp blow-up criteria for the Davey-Stewartson system in $\mathbbR^3 $, Dyn. Partial Diff. Eqns., 8 (2011), 239-260.
doi: 10.4310/DPDE.2011.v8.n3.a4. |
[27] |
S. H. Zhu, Blow-up dynamics of $L^2$ solutions for the Davey-Stewartson system, Acta Math. Sinica, English Ser., 31 (2015), 411-429.
doi: 10.1007/s10114-015-4349-7. |
show all references
References:
[1] |
D. Anker and N. C. Freeman, On the soliton solutions of the Davey-Stewartson equation for long waves, Proc. Roy. Soc. London, 360 (1978), 529-540.
doi: 10.1098/rspa.1978.0083. |
[2] |
M. J. Ablowitz and A. S. Fokas, On the inverse scattering transform of multidimensional nonlinear equations related to first-order systems in the plane, J. Math. Phys., 25 (1984), 2494-2505.
doi: 10.1063/1.526471. |
[3] |
R. Cipolatti, On the existence of standing waves for a Davey-Stewartson system, Commun. Partial Diff. Eqns., 17 (1992), 967-988.
doi: 10.1080/03605309208820872. |
[4] |
R. Cipolatti, On the instability of ground states for a Davey-Stewartson system, Ann. Inst. H. Poincaré, 58 (1993), 85-104. |
[5] |
V. D. Djordjevic and L. G. Redekopp, On two-dimensional packets of capillary-gravity waves, J. Fluid Mech., 79 (1977), 703-714.
doi: 10.1017/S0022112077000408. |
[6] |
A. Davey and S. K. Stewartson, On three-dimensional packets of surface waves, Proc. R. Soc. London, 338 (1974), 101-110.
doi: 10.1098/rspa.1974.0076. |
[7] |
Z. H. Gan and J. Zhang, Sharp threshold of global existence and instability of standing wave for a Davey-Stewartson system, Commun. Math. Phys., 283 (2008), 93-125.
doi: 10.1007/s00220-008-0456-y. |
[8] |
J. M. Ghidaglia and J. C. Saut, On the initial value problem for the Davey-Stewartson systems, Nonlinearity, 3 (1990), 475-506.
doi: 10.1088/0951-7715/3/2/010. |
[9] |
N. Godet, A lower bound on the blow-up rate for the Davey-Stewartson system on the torus, Ann. Inst. H. Poincaré - AN, 30 (2013), 691-703.
doi: 10.1016/j.anihpc.2012.12.001. |
[10] |
B. L. Guo and B. X. Wang, The Cauchy problem for Davey-Stewartson systems, Commun. Pure Appl. Math., 52 (1999), 1477-1490.
doi: 10.1002/(SICI)1097-0312(199912)52:12<1477::AID-CPA1>3.0.CO;2-N. |
[11] |
N. Hayashi, Local existence in time of small solutions to the Davey-Stewartson systems, Ann. Inst. H. Poincaré, 65 (1996), 313-366. |
[12] |
N. Hayashi and H. Hirata, Global existence and asymptotic behavior in time of small solutions to the elliptic-hyperbolic Davey-Stewartson system, Nonlinearity, 9 (1996), 1387-1409.
doi: 10.1088/0951-7715/9/6/001. |
[13] |
N. Hayashi and J. C. Saut, Global existence of small solutions to the Davey-Stewartson and the Ishimori systems, Diff. and Integ. Eqns., 8 (1995), 1657-1675. |
[14] |
T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, Intern. Math. Res. Notices, 46 (2005), 2815-2828.
doi: 10.1155/IMRN.2005.2815. |
[15] |
X. G. Li, J. Zhang, S. Y. Lai and Y. H. Wu, The sharp threshold and limiting profile of blow-up solutions for a Davey-Stewartson system, J. Diff. Eqns., 250 (2011), 2197-2226.
doi: 10.1016/j.jde.2010.10.022. |
[16] |
F. Linares and G. Ponce, On the Davey-Stewartson systems, Ann. Inst. H. Poincaré, 10 (1993), 523-548. |
[17] |
J. Lu and Y. F. Wu, Sharp threshold for scattering of a generalized Davey-Stewartson system in three dimension, Commun. Pure Appl. Anal., 14 (2015), 1641-1670.
doi: 10.3934/cpaa.2015.14.1641. |
[18] |
T. Ozawa, Exact blow-up solutions to Cauchy problem for the Davey-Stewartson system, Proc. R. Soc. Lond. Ser. A, 436 (1992), 345-349.
doi: 10.1098/rspa.1992.0022. |
[19] |
M. Ohta, Stability of standing waves for the generalized Davey-Stewartson system, J. Dyn. Diff. Eqns., 6 (1994), 325-334.
doi: 10.1007/BF02218533. |
[20] |
G. Richards, Mass concentration for the Davey-Stewartson system, Diff. and Integ. Eqns., 24 (2011), 261-280. |
[21] |
J. Shu and J. Zhang, Sharp conditions of global existence for the generalized Davey-Stewartson system, IMA J. Appl. Math., 72 (2007), 36-42.
doi: 10.1093/imamat/hxl029. |
[22] |
M. Tsutsumi, Decay of weak solutions to the Davey-Stewartson systems, J. Math. Anal. Appl., 182 (1994), 680-704.
doi: 10.1006/jmaa.1994.1113. |
[23] |
B. X. Wang and B. L. Guo, On the initial value problem and scattering of solutions for the generalized Davey-Stewartson systems, Sci. China Ser. A, 44 (2001), 994-1002.
doi: 10.1007/BF02878975. |
[24] |
M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., 87 (1983), 567-576. |
[25] |
H. Yang, X. M. Fan and S. H. Zhu, Global analysis for rough solutions to the Davey- Stewartson system, Abstract and Appl. Anal., 2012 (2012), Article ID 578701, 22pp. |
[26] |
J. Zhang and S. Zhu, Sharp blow-up criteria for the Davey-Stewartson system in $\mathbbR^3 $, Dyn. Partial Diff. Eqns., 8 (2011), 239-260.
doi: 10.4310/DPDE.2011.v8.n3.a4. |
[27] |
S. H. Zhu, Blow-up dynamics of $L^2$ solutions for the Davey-Stewartson system, Acta Math. Sinica, English Ser., 31 (2015), 411-429.
doi: 10.1007/s10114-015-4349-7. |
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