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December  2021, 14(12): 4321-4335. doi: 10.3934/dcdss.2021108

Blow-up phenomena for the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source

1. 

Department of Mathematics, Yibin University, Yibin, Sichuan 644000, China

2. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: Jun Zhou

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

This paper deals with the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source. By using some ordinary differential inequalities, the conditions on finite time blow-up of solutions are given with suitable assumptions on initial values. Moreover, the upper and lower bounds of the blow-up time are also investigated.

Citation: Jinxing Liu, Xiongrui Wang, Jun Zhou, Huan Zhang. Blow-up phenomena for the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4321-4335. doi: 10.3934/dcdss.2021108
References:
[1]

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.

[2]

J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl., 17 (1872), 55–108. http://dialnet.unirioja.es/descarga/articulo/4887986.pdf.

[3]

H. Chen and H. Xu, Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity, Discrete Contin. Dyn. Syst., 39 (2019), 1185-1203.  doi: 10.3934/dcds.2019051.

[4]

C. I. ChristovG. A. Maugin and A. V. Porubov, On Boussinesq's paradigm in nonlinear wave propagation, C. R. Mécanique, 335 (2007), 521-535.  doi: 10.1016/j.crme.2007.08.006.

[5]

C. I. ChristovG. A. Maugin and M. G. Velarde, Well-posed boussinesq paradigm with purely spatial higher-order derivatives, Phys. Rev. E Statistical Physics Plasmas Fluids and Related Interdisciplinary Topics, 54 (1996), 3621-3638.  doi: 10.1103/PhysRevE.54.3621.

[6]

P. A. ClarksonR. J. Leveque and R. Saxton, Solitary-wave interactions in elastic rods, Stud. Appl. Math., 75 (1986), 95-121.  doi: 10.1002/sapm198675295.

[7]

P. Daripa, Higher-order Boussinesq equations for two-way propagation of shallow water waves, Eur. J. Mech. B Fluids, 25 (2006), 1008-1021.  doi: 10.1016/j.euromechflu.2006.02.003.

[8]

P. Daripa and W. Hua, A numerical study of an ill-posed Boussinesq equation arising in water waves and nonlinear lattices: Filtering and regularization techniques, Appl. Math. Comput., 101 (1999), 159-207.  doi: 10.1016/S0096-3003(98)10070-X.

[9]

S. H. Deng, Generalized multi-hump wave solutions of KDV-KDV system of Boussinesq equations, Discrete Contin. Dyn. Syst., 39 (2019), 3671-3716.  doi: 10.3934/dcds.2019150.

[10]

A. Dé Godefroy, Existence, decay and blow-up for solutions to the sixth-order generalized Boussinesq equation, Discrete Contin. Dyn. Syst., 35 (2015), 117-137.  doi: 10.3934/dcds.2015.35.117.

[11]

A. Esfahani and L. G. Farah, Local well-posedness for the sixth-order boussinesq equation, J. Math. Anal. Appl., 385 (2012), 230-242.  doi: 10.1016/j.jmaa.2011.06.038.

[12]

J. A. Esquivel-Avila, Blow-up in damped abstract nonlinear equations, Electron. Res. Arch., 28 (2020), 347-267.  doi: 10.3934/era.2020020.

[13]

C. Guo and S. Fang, Global existence and pointwise estimates of solutions for the generalized sixth-order Boussinesq equation, Commun. Math. Sci., 15 (2017), 1457-1487.  doi: 10.4310/CMS.2017.v15.n5.a11.

[14]

V. Komornik, Exact Controllability and Stabilization, RAM: Research in Applied Mathematics. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. The multiplier method.

[15]

H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_tt = -Au+ F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.  doi: 10.1090/S0002-9947-1974-0344697-2.

[16]

H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146.  doi: 10.1137/0505015.

[17]

M.-R. Li and L.-Y. Tsai, Existence and nonexistence of global solutions of some system of semilinear wave equations, Nonlinear Anal., 54 (2003), 1397-1415.  doi: 10.1016/S0362-546X(03)00192-5.

[18]

W. LianJ. Wang and R. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differential Equations, 269 (2020), 4914-4959.  doi: 10.1016/j.jde.2020.03.047.

[19]

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.

[20]

M. LiaoQ. Liu and H. Ye, Global existence and blow-up of weak solutions for a class of fractional $p$-Laplacian evolution equations, Adv. Nonlinear Anal., 9 (2020), 1569-1591.  doi: 10.1515/anona-2020-0066.

[21]

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.

[22]

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.

[23]

G. Liu, The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term, Electron. Res. Arch., 28 (2020), 263-289.  doi: 10.3934/era.2020016.

[24]

X. Liu and J. Zhou, Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity, Electron. Res. Arch., 28 (2020), 599-625.  doi: 10.3934/era.2020032.

[25]

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.

[26]

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

[27]

V. G. Makhan'kov, Dynamics of classical solitons (in non-integrable systems), Phys. Reports, 35 (1978), 1-128.  doi: 10.1016/0370-1573(78)90074-1.

[28] G. A. Maugin, Nonlinear Waves in Elastic Crystals, Oxford Mathematical Monographs. Oxford University Press, Oxford, 1999. 
[29]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.

[30]

X. Su and S. Wang, The initial-boundary value problem for the generalized double dispersion equation, Z. Angew. Math. Phys., 68 (2017), Paper No. 53, 21 pp. doi: 10.1007/s00033-017-0798-4.

[31]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, volume 68 of Applied Mathematical Sciences, Springer-Verlag, New York, second edition, 1997.

[32]

S. Wang and G. Chen, The Cauchy problem for the generalized IMBq equation in $W^{s, p}(\mathbb{R}^n)$, J. Math. Anal. Appl., 266 (2002), 38-54.  doi: 10.1006/jmaa.2001.7670.

[33]

X. Wang and R. Xu, Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation, Adv. Nonlinear Anal., 10 (2021), 261-288.  doi: 10.1515/anona-2020-0141.

[34]

R. Xu, Cauchy problem of generalized Boussinesq equation with combined power-type nonlinearities, Math. Meth. Appl. Sci., 34 (2011), 2318-2328.  doi: 10.1002/mma.1536.

[35]

R. XuW. Lian and Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math., 63 (2020), 321-356.  doi: 10.1007/s11425-017-9280-x.

[36]

R. Xu and Y. Yang, Low regularity of solutions to the Rotation-Camassa-Holm type equation with the Coriolis effect, Discrete Contin. Dyn. Syst., 40 (2020), 6507-6527.  doi: 10.3934/dcds.2020288.

[37]

R. XuM. ZhangS. ChenY. Yang and J. Shen, The initial-boundary value problems for a class of sixth order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), 5631-5649.  doi: 10.3934/dcds.2017244.

[38]

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.

[39]

H. Zhang and J. Zhou, Asymptotic behaviors of solutions to a sixth-order Boussinesq equation with logarithmic nonlinearity, Comm. Pur. Appl. Anal., 20 (2021), 1601-1631.  doi: 10.3934/cpaa.2021034.

[40]

J. Zhou, Global existence and blow-up of solutions for a Kirchhoff type plate equation with damping, Appl. Math. Comput., 265 (2015), 807-818.  doi: 10.1016/j.amc.2015.05.098.

[41]

J. Zhou, Initial boundary value problem for a inhomogeneous pseudo-parabolic equation, Electron. Res. Arch., 28 (2020), 67-90.  doi: 10.3934/era.2020005.

[42]

J. Zhou and H. Zhang, Well-posedness of solutions for the sixth-order Boussinesq equation with linear strong damping and nonlinear source, J. Nonlinear Sci., 31 (2021), Paper No. 76, 61 pp. doi: 10.1007/s00332-021-09730-4.

show all references

References:
[1]

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.

[2]

J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl., 17 (1872), 55–108. http://dialnet.unirioja.es/descarga/articulo/4887986.pdf.

[3]

H. Chen and H. Xu, Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity, Discrete Contin. Dyn. Syst., 39 (2019), 1185-1203.  doi: 10.3934/dcds.2019051.

[4]

C. I. ChristovG. A. Maugin and A. V. Porubov, On Boussinesq's paradigm in nonlinear wave propagation, C. R. Mécanique, 335 (2007), 521-535.  doi: 10.1016/j.crme.2007.08.006.

[5]

C. I. ChristovG. A. Maugin and M. G. Velarde, Well-posed boussinesq paradigm with purely spatial higher-order derivatives, Phys. Rev. E Statistical Physics Plasmas Fluids and Related Interdisciplinary Topics, 54 (1996), 3621-3638.  doi: 10.1103/PhysRevE.54.3621.

[6]

P. A. ClarksonR. J. Leveque and R. Saxton, Solitary-wave interactions in elastic rods, Stud. Appl. Math., 75 (1986), 95-121.  doi: 10.1002/sapm198675295.

[7]

P. Daripa, Higher-order Boussinesq equations for two-way propagation of shallow water waves, Eur. J. Mech. B Fluids, 25 (2006), 1008-1021.  doi: 10.1016/j.euromechflu.2006.02.003.

[8]

P. Daripa and W. Hua, A numerical study of an ill-posed Boussinesq equation arising in water waves and nonlinear lattices: Filtering and regularization techniques, Appl. Math. Comput., 101 (1999), 159-207.  doi: 10.1016/S0096-3003(98)10070-X.

[9]

S. H. Deng, Generalized multi-hump wave solutions of KDV-KDV system of Boussinesq equations, Discrete Contin. Dyn. Syst., 39 (2019), 3671-3716.  doi: 10.3934/dcds.2019150.

[10]

A. Dé Godefroy, Existence, decay and blow-up for solutions to the sixth-order generalized Boussinesq equation, Discrete Contin. Dyn. Syst., 35 (2015), 117-137.  doi: 10.3934/dcds.2015.35.117.

[11]

A. Esfahani and L. G. Farah, Local well-posedness for the sixth-order boussinesq equation, J. Math. Anal. Appl., 385 (2012), 230-242.  doi: 10.1016/j.jmaa.2011.06.038.

[12]

J. A. Esquivel-Avila, Blow-up in damped abstract nonlinear equations, Electron. Res. Arch., 28 (2020), 347-267.  doi: 10.3934/era.2020020.

[13]

C. Guo and S. Fang, Global existence and pointwise estimates of solutions for the generalized sixth-order Boussinesq equation, Commun. Math. Sci., 15 (2017), 1457-1487.  doi: 10.4310/CMS.2017.v15.n5.a11.

[14]

V. Komornik, Exact Controllability and Stabilization, RAM: Research in Applied Mathematics. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. The multiplier method.

[15]

H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_tt = -Au+ F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.  doi: 10.1090/S0002-9947-1974-0344697-2.

[16]

H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146.  doi: 10.1137/0505015.

[17]

M.-R. Li and L.-Y. Tsai, Existence and nonexistence of global solutions of some system of semilinear wave equations, Nonlinear Anal., 54 (2003), 1397-1415.  doi: 10.1016/S0362-546X(03)00192-5.

[18]

W. LianJ. Wang and R. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differential Equations, 269 (2020), 4914-4959.  doi: 10.1016/j.jde.2020.03.047.

[19]

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.

[20]

M. LiaoQ. Liu and H. Ye, Global existence and blow-up of weak solutions for a class of fractional $p$-Laplacian evolution equations, Adv. Nonlinear Anal., 9 (2020), 1569-1591.  doi: 10.1515/anona-2020-0066.

[21]

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.

[22]

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.

[23]

G. Liu, The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term, Electron. Res. Arch., 28 (2020), 263-289.  doi: 10.3934/era.2020016.

[24]

X. Liu and J. Zhou, Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity, Electron. Res. Arch., 28 (2020), 599-625.  doi: 10.3934/era.2020032.

[25]

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.

[26]

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

[27]

V. G. Makhan'kov, Dynamics of classical solitons (in non-integrable systems), Phys. Reports, 35 (1978), 1-128.  doi: 10.1016/0370-1573(78)90074-1.

[28] G. A. Maugin, Nonlinear Waves in Elastic Crystals, Oxford Mathematical Monographs. Oxford University Press, Oxford, 1999. 
[29]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.

[30]

X. Su and S. Wang, The initial-boundary value problem for the generalized double dispersion equation, Z. Angew. Math. Phys., 68 (2017), Paper No. 53, 21 pp. doi: 10.1007/s00033-017-0798-4.

[31]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, volume 68 of Applied Mathematical Sciences, Springer-Verlag, New York, second edition, 1997.

[32]

S. Wang and G. Chen, The Cauchy problem for the generalized IMBq equation in $W^{s, p}(\mathbb{R}^n)$, J. Math. Anal. Appl., 266 (2002), 38-54.  doi: 10.1006/jmaa.2001.7670.

[33]

X. Wang and R. Xu, Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation, Adv. Nonlinear Anal., 10 (2021), 261-288.  doi: 10.1515/anona-2020-0141.

[34]

R. Xu, Cauchy problem of generalized Boussinesq equation with combined power-type nonlinearities, Math. Meth. Appl. Sci., 34 (2011), 2318-2328.  doi: 10.1002/mma.1536.

[35]

R. XuW. Lian and Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math., 63 (2020), 321-356.  doi: 10.1007/s11425-017-9280-x.

[36]

R. Xu and Y. Yang, Low regularity of solutions to the Rotation-Camassa-Holm type equation with the Coriolis effect, Discrete Contin. Dyn. Syst., 40 (2020), 6507-6527.  doi: 10.3934/dcds.2020288.

[37]

R. XuM. ZhangS. ChenY. Yang and J. Shen, The initial-boundary value problems for a class of sixth order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), 5631-5649.  doi: 10.3934/dcds.2017244.

[38]

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.

[39]

H. Zhang and J. Zhou, Asymptotic behaviors of solutions to a sixth-order Boussinesq equation with logarithmic nonlinearity, Comm. Pur. Appl. Anal., 20 (2021), 1601-1631.  doi: 10.3934/cpaa.2021034.

[40]

J. Zhou, Global existence and blow-up of solutions for a Kirchhoff type plate equation with damping, Appl. Math. Comput., 265 (2015), 807-818.  doi: 10.1016/j.amc.2015.05.098.

[41]

J. Zhou, Initial boundary value problem for a inhomogeneous pseudo-parabolic equation, Electron. Res. Arch., 28 (2020), 67-90.  doi: 10.3934/era.2020005.

[42]

J. Zhou and H. Zhang, Well-posedness of solutions for the sixth-order Boussinesq equation with linear strong damping and nonlinear source, J. Nonlinear Sci., 31 (2021), Paper No. 76, 61 pp. doi: 10.1007/s00332-021-09730-4.

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