April  2021, 20(4): 1601-1631. doi: 10.3934/cpaa.2021034

Asymptotic behaviors of solutions to a sixth-order Boussinesq equation with logarithmic nonlinearity

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

* Corresponding author

Received  August 2020 Revised  January 2021 Published  April 2021 Early access  March 2021

To understand the characteristics of dynamical behavior especially the kinetic evolution for logarithmic nonlinearity, we aim to study a sixth-order Boussinesq equation with logarithmic nonlinearity in a bounded domain $ \Omega\subset \mathbb{R}^n $ ($ n\geq1 $ is an integer) with smooth boundary $ \partial\Omega $, where the dispersive and the strong damping are taken into account. Based on the Faedo-Galërkin method, the logarithmic Sobolev inequality, and the potential well method, the main ingredient of this paper is to construct several conditions for initial data leading to the solution global existence or infinite time blow-up, and to study the polynomial decay and the exponential decay of the energy of the system.

Citation: Huan Zhang, Jun Zhou. Asymptotic behaviors of solutions to a sixth-order Boussinesq equation with logarithmic nonlinearity. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1601-1631. doi: 10.3934/cpaa.2021034
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. 

[3]

T. Cazenave and A. Haraux, Équations d'évolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse Math., 2 (1980), 21-51. 

[4]

H. ChenP. Luo and G. W. Liu, Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 422 (2015), 84-98.  doi: 10.1016/j.jmaa.2014.08.030.

[5]

H. Chen and S. Y. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differ. Equ., 258 (2015), 4424-4442.  doi: 10.1016/j.jde.2015.01.038.

[6]

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

[7]

C. I. ChristovG. A. Maugin and A. V. Porubov, On boussinesq${{\rm{\ddot s}}}$ paradigm in nonlinear wave propagation, Comptes Rendus Mécanique, 335 (2007), 521-535.  doi: 10.1016/j.crme.2007.08.006.

[8]

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

[9]

L. C. Evans, Graduate studies in mathematics, in Partial Differ. Equ., Am. Math. Soc., 1998. doi: 10.2307/3618751.

[10]

L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083.  doi: 10.2307/2373688.

[11]

Q. Y. Hu and H. W. Zhang, Initial boundary value problem for generalized logarithmic improved Boussinesq equation, Math. Methods Appl. Sci., 40 (2017), 3687-3697.  doi: 10.1002/mma.4255.

[12]

Q. Y. HuH. W. Zhang and G. W. Liu, Global existence and exponential growth of solution for the logarithmic Boussinesq-type equation, J. Math. Anal. Appl., 436 (2016), 990-1001.  doi: 10.1016/j.jmaa.2015.11.082.

[13]

V. Komornik, Exact Controllability and Stabilization: the Multiplier Method, Wiley Chichester, 1994. doi: 10.1090/S0273-0979-97-00717-9.

[14]

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.

[15]

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

[16]

J. L. Lions, Quelques méthodes de Résolution des Problemes aux Limites Nonlinéaires, 1969.

[17]

M. Liu and W. K. Wang, Global existence and pointwise estimates of solutions for the multidimensional generalized Boussinesq-type equation, Commun. Pure Appl. Anal., 13 (2014), 1203-1222.  doi: 10.3934/cpaa.2014.13.1203.

[18]

Y. C. Liu and R. Z. 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.

[19]

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.

[20]

L. W. Ma and Z. B. Fang, Energy decay estimates and infinite blow-up phenomena for a strongly damped semilinear wave equation with logarithmic nonlinear source, Math. Methods Appl. Sci., 41 (2018), 2639-2653.  doi: 10.1002/mma.4766.

[21]

R. L. Pego and M. I. Weinstein, Eigenvalues, and instabilities of solitary waves, Philos. Trans. Roy. Soc. London Ser. A, 340 (1992), 47-94.  doi: 10.1098/rsta.1992.0055.

[22]

N. Polat and A. Ertaş, Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation, J. Math. Anal. Appl., 349 (2009), 10-20.  doi: 10.1016/j.jmaa.2008.08.025.

[23]

R. Temam, Applied Mathematical Sciences, Springer-Verlag, New York, second edition, 1997. doi: 10.1088/0951-7715/18/5/013.

[24]

M. Tsutsumi and T. Matahashi, On the Cauchy problem for the Boussinesq type equation, Math. Japon., 36 (1991), 371-379. 

[25]

V. Varlamov, Existence and uniqueness of a solution to the Cauchy problem for the damped Boussinesq equation, Math. Methods Appl. Sci., 19 (1996), 639-649. 

[26]

V. Varlamov, On the Cauchy problem for the damped Boussinesq equation, Differ. Integral Equ., 9 (1996), 619-634. 

[27]

V. V. Varlamov, On spatially periodic solutions of the damped Boussinesq equation, Differ. Integral Equ., 10 (1997), 1197-1211. 

[28]

V. V. Varlamov, On the initial-boundary value problem for the damped Boussinesq equation, Discrete Contin. Dyn. S., 4 (1998), 431-444.  doi: 10.3934/dcds.1998.4.431.

[29]

V. V. Varlamov, Asymptotic behavior of solutions of the damped Boussinesq equation in two space dimensions, Int. J. Math. Math. Sci., 22 (1999), 131-145.  doi: 10.1155/S016117129922131X.

[30]

A. M. Wazwaz, Gaussian solitary waves for the logarithmic Boussinesq equation and the logarithmic regularized Boussinesq equation, Ocean Eng., 94 (2015), 111-115.  doi: 10.1016/j.oceaneng.2014.11.024.

[31]

S. B. Wang and X. Su, Global existence and nonexistence of the initial-boundary value problem for the dissipative Boussinesq equation, Nonlinear Anal., 134 (2016), 164-188.  doi: 10.1016/j.na.2016.01.004.

[32]

S. B. Wang and X. Su, The Cauchy problem for the dissipative Boussinesq equation, Nonlinear Anal. Real World Appl., 45 (2019), 116-141.  doi: 10.1016/j.nonrwa.2018.06.012.

[33]

Y. Wang, Existence and blow-up of solutions for the sixth-order damped Boussinesq equation, Bull. Iranian Math. Soc., 43 (2017), 1057-1071. 

[34]

Y. X. Wang, Existence and asymptotic behavior of solutions to the generalized damped Boussinesq equation, Electron J. Differ. Equ., 96 (2012), 11 pp. doi: 10.1155/2013/364165.

[35]

Y. X. Wang, Asymptotic decay estimate of solutions to the generalized damped Bq equation, J. Inequal. Appl., 323 (2013), 12 pp. doi: 10.1186/1029-242X-2013-323.

[36]

Y. Z. Wang, Y. S. Li and Q. H. Hu, Asymptotic behavior of the sixth-order Boussinesq equation with fourth-order dispersion term, Electron J. Differ. Equ., 161 (2018), 14 pp.

[37]

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

[38]

R. Z. XuY. B. LuoJ. H. Shen and S. B. Huang, Global existence and blow up for damped generalized Boussinesq equation, Acta Math. Appl. Sin. Engl. Ser., 33 (2017), 251-262.  doi: 10.1007/s10255-017-0655-4.

[39]

R. Y. 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.

[40]

S. M. Zheng, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2004.

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. 

[3]

T. Cazenave and A. Haraux, Équations d'évolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse Math., 2 (1980), 21-51. 

[4]

H. ChenP. Luo and G. W. Liu, Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 422 (2015), 84-98.  doi: 10.1016/j.jmaa.2014.08.030.

[5]

H. Chen and S. Y. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differ. Equ., 258 (2015), 4424-4442.  doi: 10.1016/j.jde.2015.01.038.

[6]

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

[7]

C. I. ChristovG. A. Maugin and A. V. Porubov, On boussinesq${{\rm{\ddot s}}}$ paradigm in nonlinear wave propagation, Comptes Rendus Mécanique, 335 (2007), 521-535.  doi: 10.1016/j.crme.2007.08.006.

[8]

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

[9]

L. C. Evans, Graduate studies in mathematics, in Partial Differ. Equ., Am. Math. Soc., 1998. doi: 10.2307/3618751.

[10]

L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083.  doi: 10.2307/2373688.

[11]

Q. Y. Hu and H. W. Zhang, Initial boundary value problem for generalized logarithmic improved Boussinesq equation, Math. Methods Appl. Sci., 40 (2017), 3687-3697.  doi: 10.1002/mma.4255.

[12]

Q. Y. HuH. W. Zhang and G. W. Liu, Global existence and exponential growth of solution for the logarithmic Boussinesq-type equation, J. Math. Anal. Appl., 436 (2016), 990-1001.  doi: 10.1016/j.jmaa.2015.11.082.

[13]

V. Komornik, Exact Controllability and Stabilization: the Multiplier Method, Wiley Chichester, 1994. doi: 10.1090/S0273-0979-97-00717-9.

[14]

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.

[15]

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

[16]

J. L. Lions, Quelques méthodes de Résolution des Problemes aux Limites Nonlinéaires, 1969.

[17]

M. Liu and W. K. Wang, Global existence and pointwise estimates of solutions for the multidimensional generalized Boussinesq-type equation, Commun. Pure Appl. Anal., 13 (2014), 1203-1222.  doi: 10.3934/cpaa.2014.13.1203.

[18]

Y. C. Liu and R. Z. 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.

[19]

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.

[20]

L. W. Ma and Z. B. Fang, Energy decay estimates and infinite blow-up phenomena for a strongly damped semilinear wave equation with logarithmic nonlinear source, Math. Methods Appl. Sci., 41 (2018), 2639-2653.  doi: 10.1002/mma.4766.

[21]

R. L. Pego and M. I. Weinstein, Eigenvalues, and instabilities of solitary waves, Philos. Trans. Roy. Soc. London Ser. A, 340 (1992), 47-94.  doi: 10.1098/rsta.1992.0055.

[22]

N. Polat and A. Ertaş, Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation, J. Math. Anal. Appl., 349 (2009), 10-20.  doi: 10.1016/j.jmaa.2008.08.025.

[23]

R. Temam, Applied Mathematical Sciences, Springer-Verlag, New York, second edition, 1997. doi: 10.1088/0951-7715/18/5/013.

[24]

M. Tsutsumi and T. Matahashi, On the Cauchy problem for the Boussinesq type equation, Math. Japon., 36 (1991), 371-379. 

[25]

V. Varlamov, Existence and uniqueness of a solution to the Cauchy problem for the damped Boussinesq equation, Math. Methods Appl. Sci., 19 (1996), 639-649. 

[26]

V. Varlamov, On the Cauchy problem for the damped Boussinesq equation, Differ. Integral Equ., 9 (1996), 619-634. 

[27]

V. V. Varlamov, On spatially periodic solutions of the damped Boussinesq equation, Differ. Integral Equ., 10 (1997), 1197-1211. 

[28]

V. V. Varlamov, On the initial-boundary value problem for the damped Boussinesq equation, Discrete Contin. Dyn. S., 4 (1998), 431-444.  doi: 10.3934/dcds.1998.4.431.

[29]

V. V. Varlamov, Asymptotic behavior of solutions of the damped Boussinesq equation in two space dimensions, Int. J. Math. Math. Sci., 22 (1999), 131-145.  doi: 10.1155/S016117129922131X.

[30]

A. M. Wazwaz, Gaussian solitary waves for the logarithmic Boussinesq equation and the logarithmic regularized Boussinesq equation, Ocean Eng., 94 (2015), 111-115.  doi: 10.1016/j.oceaneng.2014.11.024.

[31]

S. B. Wang and X. Su, Global existence and nonexistence of the initial-boundary value problem for the dissipative Boussinesq equation, Nonlinear Anal., 134 (2016), 164-188.  doi: 10.1016/j.na.2016.01.004.

[32]

S. B. Wang and X. Su, The Cauchy problem for the dissipative Boussinesq equation, Nonlinear Anal. Real World Appl., 45 (2019), 116-141.  doi: 10.1016/j.nonrwa.2018.06.012.

[33]

Y. Wang, Existence and blow-up of solutions for the sixth-order damped Boussinesq equation, Bull. Iranian Math. Soc., 43 (2017), 1057-1071. 

[34]

Y. X. Wang, Existence and asymptotic behavior of solutions to the generalized damped Boussinesq equation, Electron J. Differ. Equ., 96 (2012), 11 pp. doi: 10.1155/2013/364165.

[35]

Y. X. Wang, Asymptotic decay estimate of solutions to the generalized damped Bq equation, J. Inequal. Appl., 323 (2013), 12 pp. doi: 10.1186/1029-242X-2013-323.

[36]

Y. Z. Wang, Y. S. Li and Q. H. Hu, Asymptotic behavior of the sixth-order Boussinesq equation with fourth-order dispersion term, Electron J. Differ. Equ., 161 (2018), 14 pp.

[37]

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

[38]

R. Z. XuY. B. LuoJ. H. Shen and S. B. Huang, Global existence and blow up for damped generalized Boussinesq equation, Acta Math. Appl. Sin. Engl. Ser., 33 (2017), 251-262.  doi: 10.1007/s10255-017-0655-4.

[39]

R. Y. 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.

[40]

S. M. Zheng, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2004.

[1]

Akmel Dé Godefroy. Existence, decay and blow-up for solutions to the sixth-order generalized Boussinesq equation. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 117-137. doi: 10.3934/dcds.2015.35.117

[2]

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

[3]

Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042

[4]

Hua Chen, Huiyang Xu. Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1185-1203. doi: 10.3934/dcds.2019051

[5]

Wenjun Liu, Jiangyong Yu, Gang Li. Global existence, exponential decay and blow-up of solutions for a class of fractional pseudo-parabolic equations with logarithmic nonlinearity. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4337-4366. doi: 10.3934/dcdss.2021121

[6]

Nadjat Doudi, Salah Boulaaras, Nadia Mezouar, Rashid Jan. Global existence, general decay and blow-up for a nonlinear wave equation with logarithmic source term and fractional boundary dissipation. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022106

[7]

Xiaoliang Li, Baiyu Liu. Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3093-3112. doi: 10.3934/cpaa.2020134

[8]

Yue Pang, Xingchang Wang, Furong Wu. Global existence and blowup in infinite time for a fourth order wave equation with damping and logarithmic strain terms. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4439-4463. doi: 10.3934/dcdss.2021115

[9]

M. A. Christou, C. I. Christov. Fourier-Galerkin method for localized solutions of the Sixth-Order Generalized Boussinesq Equation. Conference Publications, 2001, 2001 (Special) : 121-130. doi: 10.3934/proc.2001.2001.121

[10]

Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072

[11]

Gongwei Liu. The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term. Electronic Research Archive, 2020, 28 (1) : 263-289. doi: 10.3934/era.2020016

[12]

Pablo Álvarez-Caudevilla, Jonathan D. Evans, Victor A. Galaktionov. Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3913-3938. doi: 10.3934/dcds.2018170

[13]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843

[14]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[15]

Yanpeng Jin, Ying Fu. Global Carleman estimate and its applications for a sixth-order equation related to thin solid films. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2775-2797. doi: 10.3934/cpaa.2022072

[16]

Monica Marras, Stella Vernier Piro. On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients. Conference Publications, 2013, 2013 (special) : 535-544. doi: 10.3934/proc.2013.2013.535

[17]

Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021

[18]

Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058

[19]

Shuyin Wu, Joachim Escher, Zhaoyang Yin. Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation. Discrete and Continuous Dynamical Systems - B, 2009, 12 (3) : 633-645. doi: 10.3934/dcdsb.2009.12.633

[20]

Jianbo Cui, Jialin Hong, Liying Sun. On global existence and blow-up for damped stochastic nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6837-6854. doi: 10.3934/dcdsb.2019169

2021 Impact Factor: 1.273

Metrics

  • PDF downloads (253)
  • HTML views (126)
  • Cited by (0)

Other articles
by authors

[Back to Top]