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Asymptotic behavior of a linear higher-order BBM-system with damping

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Abstract

The article deals with a class of Boussinesq system coupling two higher-order Benjamin–Bona–Mahony (BBM)-type equations. Introducing appropriate damping mechanisms, we study the asymptotic behavior in time of the corresponding damped models. This is done both in the case of internal and boundary damping. We first prove the global well-posedness of the systems and the convergence towards a solution which is null on a band. Then, applying a unique continuation property it follows that the origin is asymptotically stable for the damped models. Our proofs rely on the approach developed in Rosier (J Math Study 49:195–204, 2016) to study similar problems for the scalar BBM equation.

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References

  1. Arrowsmith, D.K., Place, C.M.: Dynamical Systems: Differential Equations, Maps, and Chaotic Behavior, Chapman and Hall Mathematics Series. Chapman & Hall, London (1992)

    Book  MATH  Google Scholar 

  2. Bautista, G.J., Micu, S., Pazoto, A.F.: On the controllability of a model system for long waves in nonlinear dispersive media. Nonlinearity 34, 989–1013 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bautista, G.J., Pazoto, A.F.: Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain. Commun. Pure Appl. Anal. 19, 747–769 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bautista, G.J., Pazoto, A.F.: Large-time behavior of a linear Boussinesq system for the water waves. J. Dyn. Differ. Equ. 31, 959–978 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bona, J.L., Chen, M., Saut, J.-C.: Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: derivation and linear theory. J. Nonlinear Sci. 12, 283–318 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bona, J.L., Chen, M., Saut, J.-C.: Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II: nonlinear theory. Nonlinearity 17, 925–952 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Capistrano Filho, R.C., Pazoto, A.F., Rosier, L.: Control of Boussinesq system of KdV–KdV type on a bounded interval. ESAIM Control Optim. Calc. Var. 25, 55 (2019). (Art. 58)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, M., Goubet, O.: Long-time asymptotic behavior of two-dimensional dissipative Boussinesq systems. Discrete Contin. Dyn. Syst. Ser. S 2, 37–53 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Lions, J.L.: Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Tome 1, Contrôlabilité exacte, Collection de Recherches en Mathématiques Appliquées, 8. Masson, Paris (1988)

    MATH  Google Scholar 

  10. Micu, S.: On the controllability of the linearized Benjamin–Bona–Mahony equation. SIAM J. Control Optim 39, 1677–1696 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Micu, S., Ortega, J.H., Rosier, L., Zhang, B.-Y.: Control and stabilization of a family of Boussinesq systems. Discrete Contin. Dyn. Syst. 24, 273–313 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Micu, S., Pazoto, A.F.: Stabilization of a Boussinesq system with localized damping. J. Anal. Math. 137, 291–337 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  13. Micu, S., Pazoto, A.F.: Stabilization of a Boussinesq system with generalized damping. Syst. Control Lett. 105, 62–69 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Pazoto, A.F., Rosier, L.: Stabilization of a Boussinesq system of KdV–KdV type. Syst. Control Lett. 57, 595–601 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Rosier, L.: Exact boundary controllability for the Korteweg–de Vries equation on a bounded domain. ESAIM Control Optim. Calc. Var. 2, 33–55 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Rosier, L., Zhang, B.-Y.: Unique continuation property and control for the Benjamin–Bona–Mahony equation on a periodic domain. J. Differ. Equ. 254, 141–178 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Rosier, L.: On the Benjamin–Bona–Mahony equation with a localized damping. J. Math. Study 49, 195–204 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Yamamoto, M.: One unique continuation for a linearized Benjamin–Bona–Mahony equation. J. Inverse Ill-Posed Probl. 11, 537–543 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The first author was supported by Capes and CNPq (Brazil). The second author was partially supported by CNPq (Brazil).

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Authors and Affiliations

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

    Oscar A. Sierra Fonseca & Ademir F. Pazoto

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  1. Oscar A. Sierra Fonseca
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  2. Ademir F. Pazoto
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Correspondence to Ademir F. Pazoto.

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Sierra Fonseca, O.A., Pazoto, A.F. Asymptotic behavior of a linear higher-order BBM-system with damping. Z. Angew. Math. Phys. 73, 231 (2022). https://doi.org/10.1007/s00033-022-01861-2

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  • DOI: https://doi.org/10.1007/s00033-022-01861-2

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