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 & 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.  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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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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

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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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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

[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.   Google Scholar

[26]

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

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[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.   Google Scholar

[26]

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

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

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

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