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Stability for a weakly coupled wave equations with a boundary dissipation of fractional derivative type

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Abstract

This paper deals with the stability for a weakly coupled wave equations with a boundary dissipation of fractional derivative type. We have proved well posedness and polynomial stability using the semigroup theory and a sharp result provided by Borichev and Tomilov.

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References

  1. Alabau, F., Cannarsa, P., Komornik, V.: Indirect internal stabilizationof weakly coupled evolution equations. J. Evol. Equ. 2, 127–150 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arendt, W., Batty, C.J.K.: Tauberian theorems and stability of one-parameter semigroups. Trans. Amer. Math. Soc. 306(2), 837–852 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bastos, W.D., Spezamiglio, A., Raposo, C.A.: On exact boundary controllability for linearly coupled wave equations. J. Math. Anal. Appl. (2011). Art. Id 15692

  4. Batkai, A., Engel, K.J., Schnaubelt, R.: Polynomial stability of operator semigroups. Math. Nachr. 279, 1425–1440 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Borichev, A., Tomilov, Y.: Optimal polynomial decay of functions and operator semigroups. Math. Ann. 347(2), 455–478 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Boussoira, F.A.: Stabilisation frontière indirecte de systèmes faiblement couplés. C.R. Acad. Sci. Paris, Sér. I 328 (1999) 1015-1020

  7. Boyadjiev, L., Kamenov, O., Kalla, S.L.: On the Lauwerier formulation of the temperature field problems in oil strata. International J. Math. Math. Sci. 10, 1577–1588 (2005)

    Article  MATH  Google Scholar 

  8. Caputo, M.: Linear models of dissipation whose \(Q\) is almost frequency independent Part II. Geophys. J. R. Astr. Soc. 13(5), 529–539 (1967)

    Article  Google Scholar 

  9. Cordeiro, S., Lobato, R.F.C., Raposo, C.A.: Optimal polynomial decay for a coupled system of wave with past history. Open J. Math. Anal. 4, 49–59 (2020)

    Article  Google Scholar 

  10. Choi, J., Maccamy, R.: Fractional order Volterra equations with applications to elasticity. J. Math. Anal. Appl. 139, 448–464 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, (2006)

  12. Kilbas, A., Trujillo, J.: Differential equation of fractional order: methods, results and Problems. Appl. Anal. Vol. I 78(2), 435–493 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kilbas, A., Trujillo, J.: Differential equation of fractional order: methods, results and problems. Appl. Anal. Vol. Vol. II 81(1–2), 153–192 (2001)

    MathSciNet  MATH  Google Scholar 

  14. Komornik, V., Bopeng, R.: Boundary stabilization of compactly coupled wave equations. Asymptotic Anal. 14, 339–359 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mbodje, B.: Wave energy decay under fractional derivative controls. IMA. IMA J. Math. Control Inf. 23, 237–257 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mbodje, B., Montseny, G.: Boundary fractional derivative control of the wave equation. IEEE Trans. Autom. Control. 40, 368–382 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Najafi, M.: Study of exponential stability of coupled wave systems via distributed stabilizer. Int. J. Math. Math. Sci. 28, 479–491 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Park, J.Y., Bae, J.J.: On coupled wave equation of Kirchhoff type with nonlinear boundary dam** and memory term. Appl. Math. Comput. 129, 87–105 (2002)

    MathSciNet  MATH  Google Scholar 

  19. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)

    Book  MATH  Google Scholar 

  20. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Spplications. Academic Press, Cambridge, MA, USA (1999)

    MATH  Google Scholar 

  21. Sabeur, M., Rachid, A.: Exponential stability of some wave coupled systems. J. Math.Anal. 4, 8–21 (2013)

    MathSciNet  MATH  Google Scholar 

  22. Samko, S., Kilbas, A., Marichev, O.: Integral and derivatives of fractional order. Gordon Breach, New York (1993)

    MATH  Google Scholar 

  23. Torvik, P.J., Bagley, R.L.: On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. 51(2), 294–298 (1984)

    Article  MATH  Google Scholar 

  24. Zarraga, O., Sarría, I., García-Barruetabeña, J., Cortés, F.: An analysis of the dynamical behaviour of systems with fractional dam** for mechanical engineering applications. Symmetry (2019)

    Article  Google Scholar 

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Acknowledgements

The authors thank the referees for their valuable considerations, which is improved this manuscript.

Funding

O.P.V. Villagran was partially supported by project FONDECYT/1191137. A.J.A. Ramos thanks CNPq/Brazil for financial support. Grant 310729/2019-0.

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Authors and Affiliations

  1. Department of Mathematics, University of Tarapaca, Casilla 7D, Arica, Chile

    O. P. V. Villagran

  2. Federal University of Bahia, Av. Adhemar de Barros, s/n, 40170-115, Salvador, BA, Brazil

    C. A. Nonato

  3. Mathematics Department, Federal University of São João del-Rei, Praça Frei Orlando, 170, 36307-352, São João del-Rei, MG, Brazil

    C. A. Raposo

  4. Faculty of Mathematics, Federal University of Pará, Rua Raimundo Santana, s/n, 68721-000, Salinópolis, PA, Brazil

    A. J. A. Ramos

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  1. O. P. V. Villagran
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  2. C. A. Nonato
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  3. C. A. Raposo
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  4. A. J. A. Ramos
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Correspondence to A. J. A. Ramos.

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Villagran, O.P.V., Nonato, C.A., Raposo, C.A. et al. Stability for a weakly coupled wave equations with a boundary dissipation of fractional derivative type. Rend. Circ. Mat. Palermo, II. Ser 72, 803–831 (2023). https://doi.org/10.1007/s12215-021-00703-w

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  • DOI: https://doi.org/10.1007/s12215-021-00703-w

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