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August  2016, 21(6): 1757-1774. doi: 10.3934/dcdsb.2016021

## Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping

 1 UR Analysis and Control of PDE UR13ES64, Department of Mathematics, Faculty of Sciences of Monastir University of Monastir, 5019 Monastir, Tunisia

Received  July 2015 Revised  February 2016 Published  June 2016

Let a fourth and a second order evolution equations be coupled via the interface by transmission conditions, and suppose that the first one is stabilized by a localized distributed feedback. What will then be the effect of such a partial stabilization on the decay of solutions at infinity? Is the behavior of the first component sufficient to stabilize the second one? The answer given in this paper is that sufficiently smooth solutions decay logarithmically at infinity even the feedback dissipation affects an arbitrarily small open subset of the interior. The method used, in this case, is based on a frequency method, and this by combining a contradiction argument with the Carleman estimates technique to carry out a special analysis for the resolvent.
Citation: Fathi Hassine. Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1757-1774. doi: 10.3934/dcdsb.2016021
##### References:
 [1] K. Ammari and S. Nicaise, Stabilization of a transmission wave/plate equation, Journal of Differential Equations, 249 (2010), 707-727. doi: 10.1016/j.jde.2010.03.007. [2] M. Alves, J. M. Rivera, M. Sepúlveda and O. V. Villágran, Exponential and the lack of exponential stability in transmission problems with localized Kelvin-Voigt dissipation, Acta Mechanica, 219 (2011), 145-167. [3] K. Ammari and G. Vodev, Boundary stabilization of the transmission problem for the Bernoulli-Euler plate equation, CUBO a mathematical journal, 11 (2009), 39-49. [4] C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, Journal of Evolution Equation, 8 (2008), 765-780. doi: 10.1007/s00028-008-0424-1. [5] M. Bellassoued, Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization, Asymptotic Anal., 35 (2003), 257-279. [6] N. Burq, Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonnance au voisinage du réel, (French) [Decay of the local energy of the wave equation for the exterior problem and absence of resonance near the real axis], Acta Math., 180 (1998), 1-29. doi: 10.1007/BF02392877. [7] S. Chen and K. Liu and Z. Liu, Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping, SIAM J. Appl. Math., 59 (1998), 651-668. [8] M. Daoulatli, Rate of decay of solutions of the wave equation with arbitrary localized nonlinear damping, Nonlinear Analysis, 73 (2010), 987-1003. doi: 10.1016/j.na.2010.04.026. [9] T. Duyckaerts, Optimal decay rates of the energy of a hyperbolic-parabolic system coupled by an interface, Asymptot. Anal., 51 (2007), 17-45. [10] M. Eller and D. Toundykov, Carleman estimates for elliptic boundary value problems with applications to the stabilization of hyperbolic systems, Evolution Equations and Control Theory, 1 (2012), 271-296. doi: 10.3934/eect.2012.1.271. [11] I. K. Fathallah, Logarithmic decay of the energy for an hyperbolic-parabolic coupled system, ESAIM-control Optimization and Calculus of Variations, 17 (2011), 801-835. doi: 10.1051/cocv/2010026. [12] F. Hassine, Energy decay estimates of elastic transmission wave/beam systems with a local Kelvin-Voigt damping, International Journal of Control, (just-accepted), (2015), 1-29. doi: 10.1080/00207179.2015.1135509. [13] F. Hassine, Remark on the pointwise stabilization of an elastic string equation, ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 96 (2016), 519-528. doi: 10.1002/zamm.201400260. [14] F. Hassine, Stability of elastic transmission systems with a local Kelvin-Voigt damping, European Journal of Control, 23 (2015), 84-93. doi: 10.1016/j.ejcon.2015.03.001. [15] G. Lebeau, Équation des ondes amorties, (French) [Damped wave equation], In Algebraic and geometric methods in mathematical physics (Kaciveli, 1993)of Math. Phys. Stud. Kluwer Acad. Publ. Dordrecht, 19 (1996), 73-109. [16] G. Lebeau and L. Robbiano, Contrôle exacte de l'équation de la chaleur, (French) [Exact control of the heat equation], Comm. Partial Differential Equations, 20 (1995), 335-356. doi: 10.1080/03605309508821097. [17] G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord, (French) [Stabilization of the wave equations by the boundary], Duke mathematical journal, 86 (1997), 465-491. doi: 10.1215/S0012-7094-97-08614-2. [18] G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Ration. Mech. Anal., 148 (1999), 179-231. doi: 10.1007/s002050050160. [19] K. Liu and Z. Liu, Exponential decay of the energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping, SIAM J. Control Optim., 36 (1998), 1086-1098. doi: 10.1137/S0363012996310703. [20] C. A. Raposo, W. D. Bastos and J. A. J. Avila, A transmission problem for Euler-Bernoulli beam with Kelvin-Voigt damping, Applied Mathematics and Information Sciences, 5 (2011), 17-28. [21] J. Le Rousseau and G. Lebeau, Introduction aux inégalités de Carleman pour les opérateurs elliptiques et paraboliques, Applications au prolongement unique et au contrôle des équations paraboliques, 2009. [22] J. Le Rousseau, K. Léautaud and L. Robbiano, Controllability of a parabolic system with a diffusive interface, In Séminaire Laurent Schwartz-Équations aux derivées partielles et applications, Année 2011-2012, Sémin. Équ. Déeriv. Partielles, pages Exp. No. XVII, 20. École Polytech., Palaiseau, (2013). [23] J. Le Rousseau and L. Robbiano, Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations, Arch. Rational Mech. Anal., 195 (2010), 953-990. doi: 10.1007/s00205-009-0242-9. [24] J. Rauch, X. Zhang and E. Zuazua, Polynomial decay for hyperbolic-parabolic coupled system, Math. Pures Appl., 84 (2005), 407-470. doi: 10.1016/j.matpur.2004.09.006. [25] L. Tebou, Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms, Mathematical control and related fields, 2 (2012), 45-60. doi: 10.3934/mcrf.2012.2.45. [26] M. Tucsnak and G. Weiss, Observation And Control For Operator Semigroups, Birkhäuser Verlag AG, 2009. doi: 10.1007/978-3-7643-8994-9. [27] J. T. Wolka, B. Rowley and B. Lawruk, Boundary Value Problems For Elliptic System, Cambridge University Press, Cambridge, 1995. [28] X. Zhang and E. Zuazua, Asymptotic behavior of a hyperbolic-parabolic coupled system arising in fluid-structure interaction, International Series of Numerical Mathematics, 154 (2007), 445-455. doi: 10.1007/978-3-7643-7719-9_43. [29] X. Zhang and E. Zuazua, Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction, Arch. Ration. Mech. Anal., 184 (2007), 49-120. doi: 10.1007/s00205-006-0020-x. [30] W. Zhang and Z. Zhang, Stabilization of transmission coupled wave and Euler-Bernoulli equations on Riemannian manifolds by nonlinear feedbacks, J. Math. Anal. Appl., 422 (2015), 1504-1526. doi: 10.1016/j.jmaa.2014.09.044.

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##### References:
 [1] K. Ammari and S. Nicaise, Stabilization of a transmission wave/plate equation, Journal of Differential Equations, 249 (2010), 707-727. doi: 10.1016/j.jde.2010.03.007. [2] M. Alves, J. M. Rivera, M. Sepúlveda and O. V. Villágran, Exponential and the lack of exponential stability in transmission problems with localized Kelvin-Voigt dissipation, Acta Mechanica, 219 (2011), 145-167. [3] K. Ammari and G. Vodev, Boundary stabilization of the transmission problem for the Bernoulli-Euler plate equation, CUBO a mathematical journal, 11 (2009), 39-49. [4] C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, Journal of Evolution Equation, 8 (2008), 765-780. doi: 10.1007/s00028-008-0424-1. [5] M. Bellassoued, Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization, Asymptotic Anal., 35 (2003), 257-279. [6] N. Burq, Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonnance au voisinage du réel, (French) [Decay of the local energy of the wave equation for the exterior problem and absence of resonance near the real axis], Acta Math., 180 (1998), 1-29. doi: 10.1007/BF02392877. [7] S. Chen and K. Liu and Z. Liu, Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping, SIAM J. Appl. Math., 59 (1998), 651-668. [8] M. Daoulatli, Rate of decay of solutions of the wave equation with arbitrary localized nonlinear damping, Nonlinear Analysis, 73 (2010), 987-1003. doi: 10.1016/j.na.2010.04.026. [9] T. Duyckaerts, Optimal decay rates of the energy of a hyperbolic-parabolic system coupled by an interface, Asymptot. Anal., 51 (2007), 17-45. [10] M. Eller and D. Toundykov, Carleman estimates for elliptic boundary value problems with applications to the stabilization of hyperbolic systems, Evolution Equations and Control Theory, 1 (2012), 271-296. doi: 10.3934/eect.2012.1.271. [11] I. K. Fathallah, Logarithmic decay of the energy for an hyperbolic-parabolic coupled system, ESAIM-control Optimization and Calculus of Variations, 17 (2011), 801-835. doi: 10.1051/cocv/2010026. [12] F. Hassine, Energy decay estimates of elastic transmission wave/beam systems with a local Kelvin-Voigt damping, International Journal of Control, (just-accepted), (2015), 1-29. doi: 10.1080/00207179.2015.1135509. [13] F. Hassine, Remark on the pointwise stabilization of an elastic string equation, ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 96 (2016), 519-528. doi: 10.1002/zamm.201400260. [14] F. Hassine, Stability of elastic transmission systems with a local Kelvin-Voigt damping, European Journal of Control, 23 (2015), 84-93. doi: 10.1016/j.ejcon.2015.03.001. [15] G. Lebeau, Équation des ondes amorties, (French) [Damped wave equation], In Algebraic and geometric methods in mathematical physics (Kaciveli, 1993)of Math. Phys. Stud. Kluwer Acad. Publ. Dordrecht, 19 (1996), 73-109. [16] G. Lebeau and L. Robbiano, Contrôle exacte de l'équation de la chaleur, (French) [Exact control of the heat equation], Comm. Partial Differential Equations, 20 (1995), 335-356. doi: 10.1080/03605309508821097. [17] G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord, (French) [Stabilization of the wave equations by the boundary], Duke mathematical journal, 86 (1997), 465-491. doi: 10.1215/S0012-7094-97-08614-2. [18] G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Ration. Mech. Anal., 148 (1999), 179-231. doi: 10.1007/s002050050160. [19] K. Liu and Z. Liu, Exponential decay of the energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping, SIAM J. Control Optim., 36 (1998), 1086-1098. doi: 10.1137/S0363012996310703. [20] C. A. Raposo, W. D. Bastos and J. A. J. Avila, A transmission problem for Euler-Bernoulli beam with Kelvin-Voigt damping, Applied Mathematics and Information Sciences, 5 (2011), 17-28. [21] J. Le Rousseau and G. Lebeau, Introduction aux inégalités de Carleman pour les opérateurs elliptiques et paraboliques, Applications au prolongement unique et au contrôle des équations paraboliques, 2009. [22] J. Le Rousseau, K. Léautaud and L. Robbiano, Controllability of a parabolic system with a diffusive interface, In Séminaire Laurent Schwartz-Équations aux derivées partielles et applications, Année 2011-2012, Sémin. Équ. Déeriv. Partielles, pages Exp. No. XVII, 20. École Polytech., Palaiseau, (2013). [23] J. Le Rousseau and L. Robbiano, Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations, Arch. Rational Mech. Anal., 195 (2010), 953-990. doi: 10.1007/s00205-009-0242-9. [24] J. Rauch, X. Zhang and E. Zuazua, Polynomial decay for hyperbolic-parabolic coupled system, Math. Pures Appl., 84 (2005), 407-470. doi: 10.1016/j.matpur.2004.09.006. [25] L. Tebou, Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms, Mathematical control and related fields, 2 (2012), 45-60. doi: 10.3934/mcrf.2012.2.45. [26] M. Tucsnak and G. Weiss, Observation And Control For Operator Semigroups, Birkhäuser Verlag AG, 2009. doi: 10.1007/978-3-7643-8994-9. [27] J. T. Wolka, B. Rowley and B. Lawruk, Boundary Value Problems For Elliptic System, Cambridge University Press, Cambridge, 1995. [28] X. Zhang and E. Zuazua, Asymptotic behavior of a hyperbolic-parabolic coupled system arising in fluid-structure interaction, International Series of Numerical Mathematics, 154 (2007), 445-455. doi: 10.1007/978-3-7643-7719-9_43. [29] X. Zhang and E. Zuazua, Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction, Arch. Ration. Mech. Anal., 184 (2007), 49-120. doi: 10.1007/s00205-006-0020-x. [30] W. Zhang and Z. Zhang, Stabilization of transmission coupled wave and Euler-Bernoulli equations on Riemannian manifolds by nonlinear feedbacks, J. Math. Anal. Appl., 422 (2015), 1504-1526. doi: 10.1016/j.jmaa.2014.09.044.
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