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December  2021, 14(12): 4201-4211. doi: 10.3934/dcdss.2021114

Global well-posedness for the Cauchy problem of generalized Boussinesq equations in the control problem regarding initial data

1. 

Department of Electronic Information, Jiangsu University of Science and Technology, Zhenjiang 212003, China

2. 

School of Science and Technology, Cape Breton University, Sydney, Nova Scotia, B1P 6L2, Canada

* Corresponding author: Shaohua Chen

Received  August 2021 Revised  August 2021 Published  December 2021 Early access  October 2021

Fund Project: The first author was supported by Jiangsu key R & D plan(BE2018007) and the second author was supported by NSERC Grant RGPIN-2019-05940

The Cauchy problem of one dimensional generalized Boussinesq equation is treated by the approach of variational method in order to realize the control aim, which is the control problem reflecting the relationship between initial data and global dynamics of solution. For a class of more general nonlinearities we classify the initial data for the global solution and finite time blowup solution. The results generalize some existing conclusions related this problem.

Citation: Xiaoqiang Dai, Shaohua Chen. Global well-posedness for the Cauchy problem of generalized Boussinesq equations in the control problem regarding initial data. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4201-4211. doi: 10.3934/dcdss.2021114
References:
[1]

J. L. BonaM. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media Ⅰ: Derivation and the linear theory, J. Nonlinear Sci., 12 (2002), 283-318.  doi: 10.1007/s00332-002-0466-4.

[2]

J. L. Bona and R. L. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Commun. Math. Phys., 118 (1988), 15-29.  doi: 10.1007/BF01218475.

[3]

J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, et communiquant au liquide contene dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pure Appl., 17 (1872), 55-108. 

[4]

X. DaiC. YangS. HuangT. Yu and Y. Zhu, Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems, Electron. Res. Arch., 28 (2020), 91-102.  doi: 10.3934/era.2020006.

[5]

D.-A. Geba and E. Witz, Revisited bilinear Schrödinger estimates with applications to generalized Boussinesq equations, Electron, Res. Arch., 28 (2020), 627-649.  doi: 10.3934/era.2020033.

[6]

T.-E. GhoulV. T. Nguyen and H. Zaag, Construction of type Ⅰ blowup solutions for a higher order semilinear parabolic equation, Adv. Nonlinear. Anal., 9 (2020), 388-412.  doi: 10.1515/anona-2020-0006.

[7]

G. Hwang and B. Moon, Global existence and propagation speed for a Degasperis-Procesi equation with both dissipation and dispersion, Electron. Res. Arch., 28 (2020), 15-25.  doi: 10.3934/era.2020002.

[8]

W. LianM. S. Ahmed and R. Xu, Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity, Opuscula Math., 40 (2020), 111-130.  doi: 10.7494/OpMath.2020.40.1.111.

[9]

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 (2020), 613-632.  doi: 10.1515/anona-2020-0016.

[10]

Q. LinY. H. Wu and R. Loxton, On the Cauchy problem for a generalized Boussinesq equation, J. Math. Anal. Appl., 353 (2009), 186-195.  doi: 10.1016/j.jmaa.2008.12.002.

[11]

F. Linares, Global existence of small solutions for a generalized Boussinesq equation, J. Differential Equations, 106 (1993), 257-293.  doi: 10.1006/jdeq.1993.1108.

[12]

Y. Liu, Instability and blow-up of solutions to a generalized Boussinesq equation, SIAM J. Math. Anal., 26 (1995), 1527-1546.  doi: 10.1137/S0036141093258094.

[13]

Y. Liu, Instability of solitary waves for generalized Boussinesq equations, J. Dynamics Differential Equations, 5 (1993), 537-558.  doi: 10.1007/BF01053535.

[14]

Y. Liu, Strong instability of solitary-wave solutions of a generalized Boussinesq equation, J. Differential Equations, 164 (2000), 223-239.  doi: 10.1006/jdeq.2000.3765.

[15]

Y. Liu and R. Xu, Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation, Phys. D, 237 (2008), 721-731.  doi: 10.1016/j.physd.2007.09.028.

[16]

Y. Liu and R. Xu, A class of fourth order wave equations with dissipative and nonlinear strain terms, J. Differential Equations, 244 (2008), 200-228.  doi: 10.1016/j.jde.2007.10.015.

[17]

Y. Liu and R. Xu, Fourth order wave equations with nonlinear strain and source terms, J. Math. Anal. Appl., 331 (2007), 585-607.  doi: 10.1016/j.jmaa.2006.09.010.

[18]

T. LuoT. Tao and L. Zhang, Finite energy weak solutions of 2d Boussinesq equations with diffusive temperature, Discrete Contin. Dyn. Syst., 40 (2020), 3737-3765.  doi: 10.3934/dcds.2019230.

[19]

A. MohammedV. D. Rădulescu and A. Vitolo, Blow-up solutions for fully nonlinear equations: Existence, asymptotic estimates and uniqueness, Adv. Nonlinear. Anal., 9 (2020), 39-64.  doi: 10.1515/anona-2018-0134.

[20]

H. Qiu and Z.-A. Yao, The regularized Boussinesq equations with partial dissipations in dimension two, Electron. Res. Arch., 28 (2020), 1375-1393.  doi: 10.3934/era.2020073.

[21]

R. Xue, Local and global existence of solutions for- the Cauchy problem of a generalized Boussinesq equation, J. Math. Anal. Appl., 316 (2006), 307-327.  doi: 10.1016/j.jmaa.2005.04.041.

[22]

Y. YangM. Salik AhmedL. Qin and R. Xu, Global well-posedness of a class of fourth-order strongly damped nonlinear wave equations, Opuscula Math., 39 (2019), 297-313.  doi: 10.7494/OpMath.2019.39.2.297.

show all references

References:
[1]

J. L. BonaM. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media Ⅰ: Derivation and the linear theory, J. Nonlinear Sci., 12 (2002), 283-318.  doi: 10.1007/s00332-002-0466-4.

[2]

J. L. Bona and R. L. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Commun. Math. Phys., 118 (1988), 15-29.  doi: 10.1007/BF01218475.

[3]

J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, et communiquant au liquide contene dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pure Appl., 17 (1872), 55-108. 

[4]

X. DaiC. YangS. HuangT. Yu and Y. Zhu, Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems, Electron. Res. Arch., 28 (2020), 91-102.  doi: 10.3934/era.2020006.

[5]

D.-A. Geba and E. Witz, Revisited bilinear Schrödinger estimates with applications to generalized Boussinesq equations, Electron, Res. Arch., 28 (2020), 627-649.  doi: 10.3934/era.2020033.

[6]

T.-E. GhoulV. T. Nguyen and H. Zaag, Construction of type Ⅰ blowup solutions for a higher order semilinear parabolic equation, Adv. Nonlinear. Anal., 9 (2020), 388-412.  doi: 10.1515/anona-2020-0006.

[7]

G. Hwang and B. Moon, Global existence and propagation speed for a Degasperis-Procesi equation with both dissipation and dispersion, Electron. Res. Arch., 28 (2020), 15-25.  doi: 10.3934/era.2020002.

[8]

W. LianM. S. Ahmed and R. Xu, Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity, Opuscula Math., 40 (2020), 111-130.  doi: 10.7494/OpMath.2020.40.1.111.

[9]

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 (2020), 613-632.  doi: 10.1515/anona-2020-0016.

[10]

Q. LinY. H. Wu and R. Loxton, On the Cauchy problem for a generalized Boussinesq equation, J. Math. Anal. Appl., 353 (2009), 186-195.  doi: 10.1016/j.jmaa.2008.12.002.

[11]

F. Linares, Global existence of small solutions for a generalized Boussinesq equation, J. Differential Equations, 106 (1993), 257-293.  doi: 10.1006/jdeq.1993.1108.

[12]

Y. Liu, Instability and blow-up of solutions to a generalized Boussinesq equation, SIAM J. Math. Anal., 26 (1995), 1527-1546.  doi: 10.1137/S0036141093258094.

[13]

Y. Liu, Instability of solitary waves for generalized Boussinesq equations, J. Dynamics Differential Equations, 5 (1993), 537-558.  doi: 10.1007/BF01053535.

[14]

Y. Liu, Strong instability of solitary-wave solutions of a generalized Boussinesq equation, J. Differential Equations, 164 (2000), 223-239.  doi: 10.1006/jdeq.2000.3765.

[15]

Y. Liu and R. Xu, Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation, Phys. D, 237 (2008), 721-731.  doi: 10.1016/j.physd.2007.09.028.

[16]

Y. Liu and R. Xu, A class of fourth order wave equations with dissipative and nonlinear strain terms, J. Differential Equations, 244 (2008), 200-228.  doi: 10.1016/j.jde.2007.10.015.

[17]

Y. Liu and R. Xu, Fourth order wave equations with nonlinear strain and source terms, J. Math. Anal. Appl., 331 (2007), 585-607.  doi: 10.1016/j.jmaa.2006.09.010.

[18]

T. LuoT. Tao and L. Zhang, Finite energy weak solutions of 2d Boussinesq equations with diffusive temperature, Discrete Contin. Dyn. Syst., 40 (2020), 3737-3765.  doi: 10.3934/dcds.2019230.

[19]

A. MohammedV. D. Rădulescu and A. Vitolo, Blow-up solutions for fully nonlinear equations: Existence, asymptotic estimates and uniqueness, Adv. Nonlinear. Anal., 9 (2020), 39-64.  doi: 10.1515/anona-2018-0134.

[20]

H. Qiu and Z.-A. Yao, The regularized Boussinesq equations with partial dissipations in dimension two, Electron. Res. Arch., 28 (2020), 1375-1393.  doi: 10.3934/era.2020073.

[21]

R. Xue, Local and global existence of solutions for- the Cauchy problem of a generalized Boussinesq equation, J. Math. Anal. Appl., 316 (2006), 307-327.  doi: 10.1016/j.jmaa.2005.04.041.

[22]

Y. YangM. Salik AhmedL. Qin and R. Xu, Global well-posedness of a class of fourth-order strongly damped nonlinear wave equations, Opuscula Math., 39 (2019), 297-313.  doi: 10.7494/OpMath.2019.39.2.297.

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