February  2020, 19(2): 747-769. doi: 10.3934/cpaa.2020035

Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain

1. 

Universidad Privada del Norte, Campus Breña, Av. Tingo María 1122, Lima, Peru

2. 

Institute of Mathematics, Federal University of Rio de Janeiro, P.O. Box 68530, CEP 21941-909, Rio de Janeiro, RJ, Brazil

* Corresponding author

Received  October 2018 Revised  July 2019 Published  October 2019

In this paper we are concerned with a Boussinesq system for small-amplitude long waves arising in nonlinear dispersive media. Considerations will be given for the global well-posedness and the time decay rates of solutions when the model is posed on a periodic domain and a general class of damping operator acts in each equation. By means of spectral analysis and Fourier expansion, we prove that the solutions of the linearized system decay uniformly or not to zero, depending on the parameters of the damping operators. In the uniform decay case, the result is extended for the full system.

Citation: George J. Bautista, Ademir F. Pazoto. Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain. Communications on Pure & Applied Analysis, 2020, 19 (2) : 747-769. doi: 10.3934/cpaa.2020035
References:
[1]

D. K. Arrowsmith and C. M. Place, Dynamical Systems: Differential Equations, Maps and Chaotic Behaviour, Chapman and Hall, London, 1992. doi: 10.1007/978-94-011-2388-4.  Google Scholar

[2]

G. J. Bautista and A. F. Pazoto, Large-time red behavior of a linear Boussinesq system for the water waves, J. Dyn. Diff. Equ., 31 (2019), 959-978.  doi: 10.1007/s10884-018-9689-4.  Google Scholar

[3]

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

[4]

J. L. BonaM. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. Ⅱ: Nonlinear theory, Nonlinearity, 17 (2004), 925-9052.  doi: 10.1088/0951-7715/17/3/010.  Google Scholar

[5]

J. L. BonaG. W. Pritchard and L. R. Scott, An evaluation of a model equation for water waves, Philos. Trans. Roy. Soc. London Ser. A, 302 (1981), 457-510.  doi: 10.1098/rsta.1981.0178.  Google Scholar

[6]

J. L. Bona and J. Wu, Zero-dissipation limit for nonlinear waves, M2AN Math. Model. Numer. Anal., 34 (2000), 275-301.  doi: 10.1051/m2an:2000141.  Google Scholar

[7]

J. Boussinesq, Théorie de l'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire, Comptes Rendus de l'Académie de Sciences, 72 (1871), 755-759.   Google Scholar

[8]

R. A. Capistrano-Filho, A. F. Pazoto and L. Rosier, Control of a Boussinesq system of KdV-KdV type on a bounded interval, ESAIM Control Optim. Calc. Var., DOI: https://doi.org/10.1051/cocv/2018036. Google Scholar

[9]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 13, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[10]

J.-P. ChehabP. Garnier and Y. Mammeri, Long-time behavior of solutions of a BBM equation with generalized damping, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1897-1915.  doi: 10.3934/dcdsb.2015.20.1897.  Google Scholar

[11]

M. Chen and O. Goubet, Long-time asymptotic behavior of dissipative Boussinesq systems, Discrete Contin. Dyn. Syst., 17 (2007), 509-528.  doi: 10.3934/dcdss.2009.2.37.  Google Scholar

[12]

W. Littman and L. Markus, Some recent results on control and stabilization of flexible structures, Proc. COMCON on Stabilization of Flexible Structures (Montpellier, France), 1987, 151–161. Google Scholar

[13]

S. MicuJ. H. OrtegaL. Rosier and B.-Y. Zhang, Control and stabilization of a family of Boussinesq systems, Discrete Contin. Dyn. Syst., 24 (2009), 273-313.  doi: 10.3934/dcds.2009.24.273.  Google Scholar

[14]

S. Micu and A. F. Pazoto, Stabilization of a Boussinesq system with localized damping, J. Anal. Math., 137 (2019), 291-337.  doi: 10.1007/s11854-018-0074-3.  Google Scholar

[15]

S. Micu and A. F. Pazoto, Stabilization of a Boussinesq system with generalized damping, Systems Control Lett., 105 (2017), 62-69.  doi: 10.1016/j.sysconle.2017.04.012.  Google Scholar

[16]

A. F. Pazoto and L. Rosier, Stabilization of a Boussinesq system of KdV-KdV type, Systems Control Lett., 57 (2008), 595-601.  doi: 10.1016/j.sysconle.2007.12.009.  Google Scholar

[17]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, New York, 1987.  Google Scholar

show all references

References:
[1]

D. K. Arrowsmith and C. M. Place, Dynamical Systems: Differential Equations, Maps and Chaotic Behaviour, Chapman and Hall, London, 1992. doi: 10.1007/978-94-011-2388-4.  Google Scholar

[2]

G. J. Bautista and A. F. Pazoto, Large-time red behavior of a linear Boussinesq system for the water waves, J. Dyn. Diff. Equ., 31 (2019), 959-978.  doi: 10.1007/s10884-018-9689-4.  Google Scholar

[3]

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

[4]

J. L. BonaM. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. Ⅱ: Nonlinear theory, Nonlinearity, 17 (2004), 925-9052.  doi: 10.1088/0951-7715/17/3/010.  Google Scholar

[5]

J. L. BonaG. W. Pritchard and L. R. Scott, An evaluation of a model equation for water waves, Philos. Trans. Roy. Soc. London Ser. A, 302 (1981), 457-510.  doi: 10.1098/rsta.1981.0178.  Google Scholar

[6]

J. L. Bona and J. Wu, Zero-dissipation limit for nonlinear waves, M2AN Math. Model. Numer. Anal., 34 (2000), 275-301.  doi: 10.1051/m2an:2000141.  Google Scholar

[7]

J. Boussinesq, Théorie de l'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire, Comptes Rendus de l'Académie de Sciences, 72 (1871), 755-759.   Google Scholar

[8]

R. A. Capistrano-Filho, A. F. Pazoto and L. Rosier, Control of a Boussinesq system of KdV-KdV type on a bounded interval, ESAIM Control Optim. Calc. Var., DOI: https://doi.org/10.1051/cocv/2018036. Google Scholar

[9]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 13, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[10]

J.-P. ChehabP. Garnier and Y. Mammeri, Long-time behavior of solutions of a BBM equation with generalized damping, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1897-1915.  doi: 10.3934/dcdsb.2015.20.1897.  Google Scholar

[11]

M. Chen and O. Goubet, Long-time asymptotic behavior of dissipative Boussinesq systems, Discrete Contin. Dyn. Syst., 17 (2007), 509-528.  doi: 10.3934/dcdss.2009.2.37.  Google Scholar

[12]

W. Littman and L. Markus, Some recent results on control and stabilization of flexible structures, Proc. COMCON on Stabilization of Flexible Structures (Montpellier, France), 1987, 151–161. Google Scholar

[13]

S. MicuJ. H. OrtegaL. Rosier and B.-Y. Zhang, Control and stabilization of a family of Boussinesq systems, Discrete Contin. Dyn. Syst., 24 (2009), 273-313.  doi: 10.3934/dcds.2009.24.273.  Google Scholar

[14]

S. Micu and A. F. Pazoto, Stabilization of a Boussinesq system with localized damping, J. Anal. Math., 137 (2019), 291-337.  doi: 10.1007/s11854-018-0074-3.  Google Scholar

[15]

S. Micu and A. F. Pazoto, Stabilization of a Boussinesq system with generalized damping, Systems Control Lett., 105 (2017), 62-69.  doi: 10.1016/j.sysconle.2017.04.012.  Google Scholar

[16]

A. F. Pazoto and L. Rosier, Stabilization of a Boussinesq system of KdV-KdV type, Systems Control Lett., 57 (2008), 595-601.  doi: 10.1016/j.sysconle.2007.12.009.  Google Scholar

[17]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, New York, 1987.  Google Scholar

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