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July & August  2021, 20(7&8): 2789-2809. doi: 10.3934/cpaa.2021119

## Frequency domain approach to decay rates for a coupled hyperbolic-parabolic system

 1 Institut de Recherche Mathématique Avancée, Université de Strasbourg, 67084 Strasbourg, France 2 School of Mathematical Sciences, Qufu Normal University, 273165, Qufu, China 3 School of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, China

* Corresponding author

Received  January 2021 Revised  June 2021 Published  July & August 2021 Early access  July 2021

Fund Project: This work is partially supported by the NSF of China under grants 11931011, 11821001 and 11831011, and by the Science Development Project of Sichuan University under grant 2020SCUNL201

We consider the asymptotic behavior of a linear model arising in fluid-structure interactions. The system is formed by a heat equation and a wave equation in two distinct domains, which are coupled by atransmission condition along the interface of the domains. By means of the frequency domain approach, we establish some decay rates for the whole system. Our results also showthat the decay of the fluid-structure interaction depends not only on the transmission of the damping from the heat equation to the wave equation, but also on the location of the damping for the wave equation.

Citation: Bopeng Rao, Xu Zhang. Frequency domain approach to decay rates for a coupled hyperbolic-parabolic system. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2789-2809. doi: 10.3934/cpaa.2021119
##### References:
 [1] G. Avalos, I. Lasiecka and R. Triggiani, Heat-wave interaction in 2–3 dimensions: optimal rational decay rate, J. Math. Anal. Appl., 437 (2016), 782-815.  doi: 10.1016/j.jmaa.2015.12.051. [2] G. Avalos and R. Triggiani, Rational decay rates for a PDE heat-structure interaction: a frequency domain approach, Evol. Equ. Control Theory, 2 (2013), 233–253. doi: 10.3934/eect.2013.2.233. [3] C. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765–780. doi: 10.1007/s00028-008-0424-1. [4] C. Batty, L. Paunonen and D. Seifert, Optimal energy decay in a one-dimensional coupled wave-heat system, J. Evol. Equ., 16 (2016), 649–664. doi: 10.1007/s00028-015-0316-0. [5] C. Batty, L. Paunonen and D. Seifert, Optimal energy decay for the wave-heat system on a rectangular domain, SIAM J. Math. Anal., 51 (2019), 808–819. doi: 10.1137/18M1195796. [6] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455–478. doi: 10.1007/s00208-009-0439-0. [7] N. Burq, Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinagage du réel, Acta. Math., 180 (1998), 1-29.  doi: 10.1007/BF02392877. [8] S. Chen, Analysis of Singularities for Partial Differential Equations, Series in Applied and Computational Mathematics, 1. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. [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] F. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differ. Equ., 1 (1985), 43-56. [11] Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644.  doi: 10.1007/s00033-004-3073-4. [12] Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman and Hall/CRC, London, 1999. [13] P. Loreti and B. Rao, Optimal energy decay rate for partially damped systems by spectral compensation,, SIAM J. Control Optim., 45 (2006), 1612-1632.  doi: 10.1137/S0363012903437319. [14] J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Masson, Paris (1988). [15] A. C. S. Ng, Optimal Energy Decay in A One-Dimensional Wave-Heat-Wave System, Springer Proceedings in Mathematics and Statistics 325, Springer, Cham, 2020,293–314. [16] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [17] J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112. [18] J. Rauch, X. Zhang and E. Zuazua, Polynomial decay for a hyperbolic-parabolic coupled system, J. Math. Pures Appl., 84 (2005), 407-470.  doi: 10.1016/j.matpur.2004.09.006. [19] J. Rozendaal, D. Seifert and R. Stahn, Optimal rates of decay for operator semigroups on Hilbert spaces, Adv. Math., 346 (2019), 359-388.  doi: 10.1016/j.aim.2019.02.007. [20] X. Zhang and E. Zuazua, Control, observation and polynomial decay for a coupled heat-wave system, C. R. Math. Acad. Sci. Paris, 336 (2003), 823-828.  doi: 10.1016/S1631-073X(03)00204-8. [21] X. Zhang and E. Zuazua, Polynomial decay and control of a $1$-d hyperbolic-parabolic coupled system, J. Differ. Equ., 204 (2004), 380-438.  doi: 10.1016/j.jde.2004.02.004. [22] 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.

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##### References:
 [1] G. Avalos, I. Lasiecka and R. Triggiani, Heat-wave interaction in 2–3 dimensions: optimal rational decay rate, J. Math. Anal. Appl., 437 (2016), 782-815.  doi: 10.1016/j.jmaa.2015.12.051. [2] G. Avalos and R. Triggiani, Rational decay rates for a PDE heat-structure interaction: a frequency domain approach, Evol. Equ. Control Theory, 2 (2013), 233–253. doi: 10.3934/eect.2013.2.233. [3] C. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765–780. doi: 10.1007/s00028-008-0424-1. [4] C. Batty, L. Paunonen and D. Seifert, Optimal energy decay in a one-dimensional coupled wave-heat system, J. Evol. Equ., 16 (2016), 649–664. doi: 10.1007/s00028-015-0316-0. [5] C. Batty, L. Paunonen and D. Seifert, Optimal energy decay for the wave-heat system on a rectangular domain, SIAM J. Math. Anal., 51 (2019), 808–819. doi: 10.1137/18M1195796. [6] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455–478. doi: 10.1007/s00208-009-0439-0. [7] N. Burq, Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinagage du réel, Acta. Math., 180 (1998), 1-29.  doi: 10.1007/BF02392877. [8] S. Chen, Analysis of Singularities for Partial Differential Equations, Series in Applied and Computational Mathematics, 1. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. [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] F. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differ. Equ., 1 (1985), 43-56. [11] Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644.  doi: 10.1007/s00033-004-3073-4. [12] Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman and Hall/CRC, London, 1999. [13] P. Loreti and B. Rao, Optimal energy decay rate for partially damped systems by spectral compensation,, SIAM J. Control Optim., 45 (2006), 1612-1632.  doi: 10.1137/S0363012903437319. [14] J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Masson, Paris (1988). [15] A. C. S. Ng, Optimal Energy Decay in A One-Dimensional Wave-Heat-Wave System, Springer Proceedings in Mathematics and Statistics 325, Springer, Cham, 2020,293–314. [16] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [17] J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112. [18] J. Rauch, X. Zhang and E. Zuazua, Polynomial decay for a hyperbolic-parabolic coupled system, J. Math. Pures Appl., 84 (2005), 407-470.  doi: 10.1016/j.matpur.2004.09.006. [19] J. Rozendaal, D. Seifert and R. Stahn, Optimal rates of decay for operator semigroups on Hilbert spaces, Adv. Math., 346 (2019), 359-388.  doi: 10.1016/j.aim.2019.02.007. [20] X. Zhang and E. Zuazua, Control, observation and polynomial decay for a coupled heat-wave system, C. R. Math. Acad. Sci. Paris, 336 (2003), 823-828.  doi: 10.1016/S1631-073X(03)00204-8. [21] X. Zhang and E. Zuazua, Polynomial decay and control of a $1$-d hyperbolic-parabolic coupled system, J. Differ. Equ., 204 (2004), 380-438.  doi: 10.1016/j.jde.2004.02.004. [22] 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.
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