Explicit decay rate for a degenerate hyperbolic-parabolic coupled system

Han, Zhong-Jie; Wang, Gengsheng; Wang, Jing

ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 116.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

This paper studies the stability of a 1-dim system which comprises a wave equation and a degenerate heat equation in two connected bounded intervals. The coupling between these two different components occurs at the interface with certain transmission conditions. We find an explicit polynomial decay rate for solutions of this system. This rate depends on the degree of the degeneration for the diffusion coefficient near the interface. Besides, the well-posedness of this degenerate coupled system is proved by the semigroup theory.

Classification : 35B40, 93D20
Mots-clés : Polynomial decay, $C_0$ semigroup, coupled heat-wave equation, degenerate coefficient

[1] F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability. J. Evol. Equ. 6 (2006) 161–204. | DOI | MR | Zbl

[2] 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 | MR | Zbl

[3] 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 | MR | Zbl

[4] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model. Indiana Univ. Math. J. 57 (2008) 1173–1207. | DOI | MR | Zbl

[5] E.N. Batuev and V.D. Stepanov, Weighted inequalities of Hardy type. Siberian Math. J. 30 (1989) 8–16. | DOI | MR | Zbl

[6] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups. Math. Ann. 347 (2010) 455–478. | DOI | MR | Zbl

[7] P. Cannarsa, P. Martinez and J. Vancostenoble, Null controllability of degenerate heat equations. Adv. Differ. Equ. 10 (2005) 153–190. | MR | Zbl

[8] P. Cannarsa and L. De Teresa, Controllability of 1-d coupled degenerate parabolic equations. Electron. J. Differ. Eq. 73 (2009) 1–21. | MR | Zbl

[9] G. Citti and M. Manfredini, A degenerate parabolic equation arising in image processing. Commun. Appl. Anal. 8 (2004) 125–141. | MR | Zbl

[10] Q. Du, M.D. Gunzburger, L.S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem. Discrete Contin. Dyn. Syst. 9 (2003) 633–650. | DOI | MR | Zbl

[11] T. Duyckaerts, Optimal decay rates of the energy of a hyperbolic-parabolic system coupled by an interface. Asymptot. Anal. 51 (2007) 17–45. | MR | Zbl

[12] S.N. Ethier, A class of degenerate diffusion processes occurring in population genetics. Comm. Pure Appl. Math. 29 (1976) 483–493. | DOI | MR | Zbl

[13] H. Gao, L. Li and Z. Liu, Stability of degenerate heat equation in non-cylindrical/ cylindrical domain. Z. Angew. Math. Phys. 70 (2019) 120. | DOI | MR | Zbl

[14] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a nonlinear fluid structure interaction system. J. Differ. Equ. 247 (2009) 1452–1478. | DOI | MR | Zbl

[15] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions for a fluid-structure interaction system. Adv. Differ. Equ. 15 (2010) 231–254. | MR | Zbl

[16] K. Liu, Z. Liu and Q. Zhang, Eventual differentiability of a string with local Kelvin-Voigt damping. ESAIM: COCV 23 (2017) 443–454. | Numdam | MR | Zbl

[17] 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 | MR | Zbl

[18] Z. Liu and Q. Zhang, Stability of a string with local Kelvin-Voigt damping and nonsmooth coefficient at interface. SIAM J. Control and Optim. 54 (2016) 1859–1871. | DOI | MR | Zbl

[19] Z. Liu and Q. Zhang, Stability and regularity of solution to the Timoshenko beam equation with local Kelvin-Voigt damping. SIAM J. Control and Optim. 56 (2018) 3919–3947. | DOI | MR | Zbl

[20] Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems. Chapman & Hall/ CRC, Boca Raton (1999). | MR | Zbl

[21] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983). | DOI | MR | Zbl

[22] J. Rauch, X. Zhang and E. Zuazua, Polynomial decay for a hyperbolic-parabolic coupled system. J. Math. Pures Appl. 84 (2005) 407–470. | MR | Zbl

[23] J.M. Wang, B. Ren and M. Krstic, Stabilization and Gevrey regularity of a Schrödinger equation in boundary feedback with a heat equation. IEEE Trans. Automat. Contr. 57 (2012) 179–185. | DOI | MR | Zbl

[24] 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 | MR | Zbl

[25] X. Zhang and E. Zuazua, Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction. Arch. Ration. Mech. An. 184 (2007) 49–120. | DOI | MR | Zbl

[26] X. Zhang and E. Zuazua, Control, observation and polynomial decay for a coupled heat-wave system. C. R. Acad. Sci. Paris, Ser. I 336 (2003) 823–828. | DOI | MR | Zbl

[27] E. Zuazua, Null control of a 1-d model of mixed hyperbolic-parabolic type, in Optimal Control and Partial Differential Equations, edited by J.L. Menaldi, et al., IOS Press, Amsterdam (2001), 198–210. | MR | Zbl