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December  2021, 14(12): 4293-4320. doi: 10.3934/dcdss.2021122

On the well-posedness and stability for the fourth-order Schrödinger equation with nonlinear derivative term

1. 

School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

2. 

Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education, Institutes, Guangzhou University, Guangzhou 510006, China

* Corresponding author: dihuafei@yeah.net

Received  July 2021 Revised  September 2021 Published  December 2021 Early access  October 2021

Fund Project: Huafei Di is supported by the NSF of China (11801108, 11801495), the Scientific Program of Guangdong Province (2021A1515010314), and the College Scientific Research Project (YG2020005) of Guangzhou University

Considered herein is the well-posedness and stability for the Cauchy problem of the fourth-order Schrödinger equation with nonlinear derivative term $ iu_{t}+\Delta^2 u-u\Delta|u|^2+\lambda|u|^pu = 0 $, where $ t\in\mathbb{R} $ and $ x\in \mathbb{R}^n $. First of all, for initial data $ \varphi(x)\in H^2(\mathbb{R}^{n}) $, we establish the local well-poseness and finite time blow-up criterion of the solutions, and give a rough estimate of blow-up time and blow-up rate. Secondly, under a smallness assumption on the initial value $ \varphi(x) $, we demonstrate the global well-posedness of the solutions by applying two different methods, and at the same time give the scattering behavior of the solutions. Finally, based on founded a priori estimates, we investigate the stability of solutions by the short-time and long-time perturbation theories, respectively.

Citation: Kelin Li, Huafei Di. On the well-posedness and stability for the fourth-order Schrödinger equation with nonlinear derivative term. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4293-4320. doi: 10.3934/dcdss.2021122
References:
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B. Guo and B. Wang, The global Cauchy problem and scattering of solutions for nonlinear Schrödinger equations in $H^s$, Differ. Integral Equ., 15 (2002), 107-1083.   Google Scholar

[13]

R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys. B., 37 (1980), 83-87.  doi: 10.1007/BF01325508.  Google Scholar

[14]

Z. Huo and Y. Jia, The Cauchy problem for the fourth-order nonlinear Schrödinger equation related to the vortex filament, J. Differ. Equ., 214 (2005), 1-35.  doi: 10.1016/j.jde.2004.09.005.  Google Scholar

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W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2019), 613-632.  doi: 10.1515/anona-2020-0016.  Google Scholar

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[20]

C. MiaoH. Wu and J. Zhang, Scattering theory below energy for the cubic fourth-order Schrödinger equation, Math. Nachr., 288 (2015), 798-823.  doi: 10.1002/mana.201400012.  Google Scholar

[21]

C. MiaoG. Xu and L. Zhao, Global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case, J. Differ. Equ., 246 (2009), 3715-3749.  doi: 10.1016/j.jde.2008.11.011.  Google Scholar

[22]

C. Miao and B. Zhang, Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations, Discret. Contin. Dyn. Syst. A., 17 (2007), 181-200.  doi: 10.3934/dcds.2007.17.181.  Google Scholar

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B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dyn. Partial Differ. Equ., 4 (2007), 197-225.  doi: 10.4310/DPDE.2007.v4.n3.a1.  Google Scholar

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B. Pausader, The cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517.  doi: 10.1016/j.jfa.2008.11.009.  Google Scholar

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J. Segata, Remark on well-posedness for the fourth order nonlinear Schrödinger type equation, Proc. Amer. Math. Soc., 132 (2004), 3559-3568.  doi: 10.1090/S0002-9939-04-07620-8.  Google Scholar

[26]

J. Shu and J. Zhang, Sharp condition of global existence for second-order derivative nonlinear Schrödinger equations in two space dimensions, J. Math. Anal. Appl., 326 (2007), 1001-1006.  doi: 10.1016/j.jmaa.2006.03.055.  Google Scholar

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J. Simon, Compact sets in the space $L^{p}(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[28]

K. H. Spatschek and S. G. Tagary, Nonlinear propagation of ion-cyclotron modes, Phys. Fluids, 20 (1977), 1505-1509.  doi: 10.1063/1.862049.  Google Scholar

[29]

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[30]

R.-Z. Xu and C. Xu, Nonlinear Schrödinger equation with combined power-type nonlinearities and harmonic potential, Appl. Math. Mech., 31 (2010), 521-528.  doi: 10.1007/s10483-010-0412-7.  Google Scholar

[31]

R. Xu and C. Xu, Sharp conditions of global existence for second-order derivative nonlinear Schrödinger equations with combined power-type nonlinearities, Z. Angew. Math. Mech., 93 (2013), 29-37.  doi: 10.1002/zamm.201200083.  Google Scholar

[32]

F. YangZ.-H. Ning and L. Chen, Exponential stability of the nonlinear Schrödinger equation with locally distributed damping on compact Riemannian manifold, Adv. Nonlinear Anal., 10 (2021), 569-583.  doi: 10.1515/anona-2020-0149.  Google Scholar

[33]

H. Ye and Y. Yu, The existence of normalized solutions for $L^2$-critical quasilinear Schrödinger equations, J. Math. Anal. Appl., 497 (2021), 124829.  doi: 10.1016/j.jmaa.2020.124839.  Google Scholar

[34]

M.Y. Yu and P. K. Shuhla, On the formation of upper-hybrid solitons, Plasma Phys., 19 (1977), 889-893.  doi: 10.1088/0032-1028/19/9/008.  Google Scholar

[35]

J. Zhang and J. Zheng, Energy critical fourth-order Schrödinger equations with subcritical perturbations, Nonlinear Anal. Theory Methods Appl., 73 (2010), 1004-1014.  doi: 10.1016/j.na.2010.04.027.  Google Scholar

[36]

M. Zhang and M. S. Ahmed, Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential, Adv. Nonlinear Anal., 9 (2020), 882-894.  doi: 10.1515/anona-2020-0031.  Google Scholar

[37]

S. ZhuJ. Zhang and H. Yang, Limiting profile of the blow-up solutions for the fourth-order nonlinear Schrödinger equation, Dyn. Partial Differ. Equ., 7 (2010), 187-205.  doi: 10.4310/DPDE.2010.v7.n2.a4.  Google Scholar

[38]

S. ZhuJ. Zhang and H. Yang, Biharmonic nonlinear Schrödinger equation and the profile decomposition, Nonlinear Anal. Theory Methods Appl., 74 (2011), 6244-6255.  doi: 10.1016/j.na.2011.06.004.  Google Scholar

show all references

References:
[1]

D. BonheureS. Cingolani and S. Secchi, Concentration phenomena for the Schrödinger-Poisson system in $\mathbb{R}^2$, Discret. Contin. Dyn. Syst. S., 14 (2021), 1631-1648.  doi: 10.3934/dcdss.2020447.  Google Scholar

[2]

L. Cai and F. Zhang, The Brezis-Nirenberg type double critical problem for a class of Schrödinger equations, Electron. Res. Arch., 29 (2021), 2475-2488.  doi: 10.3934/era.2020125.  Google Scholar

[3]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, Courant Institute of Mathematical Sciences, American Mathematical Society, New York, 10 2003. doi: 10.1090/cln/010.  Google Scholar

[4]

J. Chen and B. Guo, Blow up and strong instability result for a quasilinear Schrödinger equation, Appl. Math. Model., 33 (2009), 4192-4200.  doi: 10.1016/j.apm.2009.03.003.  Google Scholar

[5]

M. Colin, On the local well-possedness on quasilinear Schrödinger equations in arbitrary space dimension, Commun. Partial Differ. Equ., 27 (2002), 325-354.  doi: 10.1081/PDE-120002789.  Google Scholar

[6]

S. Cuccagna and M. Maeda, A Survey on asymptotic stability of ground states of nonlinear Schrödinger equations II, Discret. Contin. Dyn. Syst. S., 14 (2021), 1693-1716.  doi: 10.3934/dcdss.2020450.  Google Scholar

[7]

G. DaiR. Tian and Z. Zhang, Global bifurcations and a priori bounds of positive solutions for coupled nonlinear Schrödinger Systems, Discret. Contin. Dyn. Syst. S., 12 (2019), 1905-1927.  doi: 10.3934/dcdss.2019125.  Google Scholar

[8]

V. D. Dinh, On blowup solutions to the focusing intercritical nonlinear fourth-order Schrödinger equation, J. Dyn. Differ. Equ., 31 (2019), 1793-1823.  doi: 10.1007/s10884-018-9690-y.  Google Scholar

[9]

G. FibichB. Ilan and G. Papanicolaou, Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462.  doi: 10.1137/S0036139901387241.  Google Scholar

[10]

Y. Fukumoto and H. K. Moffatt, Motion and expansion of a viscous vortex ring. Part I. A higher-order asymptotic formula for the velocity, J. Fluid. Mech., 417 (2000), 1-45.  doi: 10.1017/S0022112000008995.  Google Scholar

[11]

J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 2 (1985), 309-327.  doi: 10.1016/S0294-1449(16)30399-7.  Google Scholar

[12]

B. Guo and B. Wang, The global Cauchy problem and scattering of solutions for nonlinear Schrödinger equations in $H^s$, Differ. Integral Equ., 15 (2002), 107-1083.   Google Scholar

[13]

R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys. B., 37 (1980), 83-87.  doi: 10.1007/BF01325508.  Google Scholar

[14]

Z. Huo and Y. Jia, The Cauchy problem for the fourth-order nonlinear Schrödinger equation related to the vortex filament, J. Differ. Equ., 214 (2005), 1-35.  doi: 10.1016/j.jde.2004.09.005.  Google Scholar

[15]

V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth-order nonlinear Schrödinger-type equations, Phys. Rev. E., 53 (1996), 1336-1339.  doi: 10.1103/PhysRevE.53.R1336.  Google Scholar

[16]

V. I. Karpman and A. G. Shagalov, Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion, Physica D., 144 (2000), 194-210.  doi: 10.1016/S0167-2789(00)00078-6.  Google Scholar

[17]

T. Kato, On nonlinear schrödinger eqautions, Ann. Inst. Henri Poincaré Phys. Théor., 46 (1987), 113-129.   Google Scholar

[18]

W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2019), 613-632.  doi: 10.1515/anona-2020-0016.  Google Scholar

[19]

X. Liu and T. Zhang, $H^2$ blowup result for a Schrödinger equation with nonlinear source term, Electron. Res. Arch., 28 (2020), 777-794.  doi: 10.3934/era.2020039.  Google Scholar

[20]

C. MiaoH. Wu and J. Zhang, Scattering theory below energy for the cubic fourth-order Schrödinger equation, Math. Nachr., 288 (2015), 798-823.  doi: 10.1002/mana.201400012.  Google Scholar

[21]

C. MiaoG. Xu and L. Zhao, Global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case, J. Differ. Equ., 246 (2009), 3715-3749.  doi: 10.1016/j.jde.2008.11.011.  Google Scholar

[22]

C. Miao and B. Zhang, Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations, Discret. Contin. Dyn. Syst. A., 17 (2007), 181-200.  doi: 10.3934/dcds.2007.17.181.  Google Scholar

[23]

B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dyn. Partial Differ. Equ., 4 (2007), 197-225.  doi: 10.4310/DPDE.2007.v4.n3.a1.  Google Scholar

[24]

B. Pausader, The cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517.  doi: 10.1016/j.jfa.2008.11.009.  Google Scholar

[25]

J. Segata, Remark on well-posedness for the fourth order nonlinear Schrödinger type equation, Proc. Amer. Math. Soc., 132 (2004), 3559-3568.  doi: 10.1090/S0002-9939-04-07620-8.  Google Scholar

[26]

J. Shu and J. Zhang, Sharp condition of global existence for second-order derivative nonlinear Schrödinger equations in two space dimensions, J. Math. Anal. Appl., 326 (2007), 1001-1006.  doi: 10.1016/j.jmaa.2006.03.055.  Google Scholar

[27]

J. Simon, Compact sets in the space $L^{p}(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[28]

K. H. Spatschek and S. G. Tagary, Nonlinear propagation of ion-cyclotron modes, Phys. Fluids, 20 (1977), 1505-1509.  doi: 10.1063/1.862049.  Google Scholar

[29]

R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of sollutions of wave equations, Duke Math. J., 44 (1972), 705-714.  doi: 10.1215/S0012-7094-77-04430-1.  Google Scholar

[30]

R.-Z. Xu and C. Xu, Nonlinear Schrödinger equation with combined power-type nonlinearities and harmonic potential, Appl. Math. Mech., 31 (2010), 521-528.  doi: 10.1007/s10483-010-0412-7.  Google Scholar

[31]

R. Xu and C. Xu, Sharp conditions of global existence for second-order derivative nonlinear Schrödinger equations with combined power-type nonlinearities, Z. Angew. Math. Mech., 93 (2013), 29-37.  doi: 10.1002/zamm.201200083.  Google Scholar

[32]

F. YangZ.-H. Ning and L. Chen, Exponential stability of the nonlinear Schrödinger equation with locally distributed damping on compact Riemannian manifold, Adv. Nonlinear Anal., 10 (2021), 569-583.  doi: 10.1515/anona-2020-0149.  Google Scholar

[33]

H. Ye and Y. Yu, The existence of normalized solutions for $L^2$-critical quasilinear Schrödinger equations, J. Math. Anal. Appl., 497 (2021), 124829.  doi: 10.1016/j.jmaa.2020.124839.  Google Scholar

[34]

M.Y. Yu and P. K. Shuhla, On the formation of upper-hybrid solitons, Plasma Phys., 19 (1977), 889-893.  doi: 10.1088/0032-1028/19/9/008.  Google Scholar

[35]

J. Zhang and J. Zheng, Energy critical fourth-order Schrödinger equations with subcritical perturbations, Nonlinear Anal. Theory Methods Appl., 73 (2010), 1004-1014.  doi: 10.1016/j.na.2010.04.027.  Google Scholar

[36]

M. Zhang and M. S. Ahmed, Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential, Adv. Nonlinear Anal., 9 (2020), 882-894.  doi: 10.1515/anona-2020-0031.  Google Scholar

[37]

S. ZhuJ. Zhang and H. Yang, Limiting profile of the blow-up solutions for the fourth-order nonlinear Schrödinger equation, Dyn. Partial Differ. Equ., 7 (2010), 187-205.  doi: 10.4310/DPDE.2010.v7.n2.a4.  Google Scholar

[38]

S. ZhuJ. Zhang and H. Yang, Biharmonic nonlinear Schrödinger equation and the profile decomposition, Nonlinear Anal. Theory Methods Appl., 74 (2011), 6244-6255.  doi: 10.1016/j.na.2011.06.004.  Google Scholar

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