December  2016, 9(6): 1899-1912. doi: 10.3934/dcdss.2016077

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

Received  July 2015 Revised  September 2016 Published  November 2016

In this paper we study the threshold of global existence and blow-up for the solutions to the generalized 3D Davey-Stewartson equations \begin{equation*} \left\{ \begin{aligned} & iu_t + \Delta u + |u|^{p-1} u + E_1(|u|^2)u = 0, \quad t > 0, \ \ x\in \mathbb{R}^3, \\ & u(0,x) = u_0(x) \in H^1(\mathbb{R}^3), \end{aligned} \right. \end{equation*} where $1 < p < \frac{7}{3}$ and the operator $E_1$ is given by $ E_1(f) = \mathcal {F}^{-1} \left( \frac{\xi_1^2}{|\xi|^2} \mathcal{F}(f) \right) $. We construct two kinds of invariant sets under the evolution flow by analyzing the property of the upper bound function of the energy. Then we show that the solution exists globally for the initial function $u_0$ in first kind of the invariant sets, while the solution blows up in finite time for $u_0$ in another kind. We remark that the exponent $ p $ is subcritical for the nonlinear Schrödinger equations for which blow-up solutions would not occur. The result shows that the occurrence of blow-up phenomenon is caused by the coupling mechanics of the Davey-Stewartson equations.
Citation: Shiming Li, Yongsheng Li, Wei Yan. A global existence and blow-up threshold for Davey-Stewartson equations in $\mathbb{R}^3$. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1899-1912. doi: 10.3934/dcdss.2016077
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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