November  2010, 14(4): 1601-1620. doi: 10.3934/dcdsb.2010.14.1601

Stabilization of some coupled hyperbolic/parabolic equations


Department of Mathematics & Statistics, Florida International University, Miami, FL 33199, United States

Received  August 2009 Revised  January 2010 Published  August 2010

First, we consider a coupled system consisting of the wave equation and the heat equation in a bounded domain. The coupling involves an operator parametrized by a real number $\mu$ lying in the interval [0,1]. We show that for $0\leq\mu<1$, the associated semigroup is not uniformly stable. Then we propose an explicit non-uniform decay rate. For $\mu=1$, the coupled system reduces to the thermoelasticity equations discussed by Lebeau and Zuazua [23], and subsequently by Albano and Tataru [1]; we show that in this case, the corresponding semigroup is exponentially stable but not analytic. Afterwards, we discuss some extensions of our results. Second, we consider partially clamped Kirchhoff thermoelastic plate without mechanical feedback controls, and we prove that the underlying semigroup is exponentially stable uniformly with respect to the rotational inertia. We use a constructive frequency domain method to prove the stabilization result, and we obtain an explicit decay rate by showing that the real part of the spectrum is uniformly bounded by a negative number that depends on the parameters of the system other than the rotational inertia; our approach is an alternative to the energy method applied by Avalos and Lasiecka [6].
Citation: Louis Tebou. Stabilization of some coupled hyperbolic/parabolic equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1601-1620. doi: 10.3934/dcdsb.2010.14.1601

Pedro Roberto de Lima, Hugo D. Fernández Sare. General condition for exponential stability of thermoelastic Bresse systems with Cattaneo's law. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3575-3596. doi: 10.3934/cpaa.2020156


Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020108


Bopeng Rao, Xu Zhang. Frequency domain approach to decay rates for a coupled hyperbolic-parabolic system. Communications on Pure & Applied Analysis, 2021, 20 (7&8) : 2789-2809. doi: 10.3934/cpaa.2021119


Yaping Wu, Niannian Yan. Stability of traveling waves for autocatalytic reaction systems with strong decay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1601-1633. doi: 10.3934/dcdsb.2017033


Moncef Aouadi, Alain Miranville. Quasi-stability and global attractor in nonlinear thermoelastic diffusion plate with memory. Evolution Equations & Control Theory, 2015, 4 (3) : 241-263. doi: 10.3934/eect.2015.4.241


Salim A. Messaoudi, Abdelfeteh Fareh. Exponential decay for linear damped porous thermoelastic systems with second sound. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 599-612. doi: 10.3934/dcdsb.2015.20.599


Ramon Quintanilla, Reinhard Racke. Stability in thermoelasticity of type III. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 383-400. doi: 10.3934/dcdsb.2003.3.383


Abdallah Ben Abdallah, Farhat Shel. Exponential stability of a general network of 1-d thermoelastic rods. Mathematical Control & Related Fields, 2012, 2 (1) : 1-16. doi: 10.3934/mcrf.2012.2.1


Yi Cheng, Zhihui Dong, Donal O' Regan. Exponential stability of axially moving Kirchhoff-beam systems with nonlinear boundary damping and disturbance. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021230


Hichem Kasri, Amar Heminna. Exponential stability of a coupled system with Wentzell conditions. Evolution Equations & Control Theory, 2016, 5 (2) : 235-250. doi: 10.3934/eect.2016003


Francesca Bucci, Igor Chueshov. Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations. Discrete & Continuous Dynamical Systems, 2008, 22 (3) : 557-586. doi: 10.3934/dcds.2008.22.557


Lei Wang, Zhong-Jie Han, Gen-Qi Xu. Exponential-stability and super-stability of a thermoelastic system of type II with boundary damping. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2733-2750. doi: 10.3934/dcdsb.2015.20.2733


Yong Ren, Huijin Yang, Wensheng Yin. Weighted exponential stability of stochastic coupled systems on networks with delay driven by $ G $-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3379-3393. doi: 10.3934/dcdsb.2018325


Luis Barreira, Claudia Valls. Delay equations and nonuniform exponential stability. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 219-223. doi: 10.3934/dcdss.2008.1.219


Adriana Flores de Almeida, Marcelo Moreira Cavalcanti, Janaina Pedroso Zanchetta. Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping. Evolution Equations & Control Theory, 2019, 8 (4) : 847-865. doi: 10.3934/eect.2019041


Farah Abdallah, Denis Mercier, Serge Nicaise. Spectral analysis and exponential or polynomial stability of some indefinite sign damped problems. Evolution Equations & Control Theory, 2013, 2 (1) : 1-33. doi: 10.3934/eect.2013.2.1


Alaa Hayek, Serge Nicaise, Zaynab Salloum, Ali Wehbe. Exponential and polynomial stability results for networks of elastic and thermo-elastic rods. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021142


Ramón Quintanilla, Reinhard Racke. Stability for thermoelastic plates with two temperatures. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6333-6352. doi: 10.3934/dcds.2017274


Margareth S. Alves, Rodrigo N. Monteiro. Stability of non-classical thermoelasticity mixture problems. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4879-4898. doi: 10.3934/cpaa.2020216


Yijing Sun, Yuxin Tan. Kirchhoff type equations with strong singularities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 181-193. doi: 10.3934/cpaa.2019010

2020 Impact Factor: 1.327


  • PDF downloads (70)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]