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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 and Continuous Dynamical Systems - S, 2021, 14 (12) : 4293-4320. doi: 10.3934/dcdss.2021122
References:
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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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[17]

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

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[17]

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

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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