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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 & 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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