- Research Article
- Open Access
Exponential Stability and Global Attractors for a Thermoelastic Bresse System
© Zhiyong Ma. 2010
- Received: 13 September 2010
- Accepted: 29 October 2010
- Published: 21 November 2010
We consider the stability properties for thermoelastic Bresse system which describes the motion of a linear planar shearable thermoelastic beam. The system consists of three wave equations and two heat equations coupled in certain pattern. The two wave equations about the longitudinal displacement and shear angle displacement are effectively damped by the dissipation from the two heat equations. We use multiplier techniques to prove the exponential stability result when the wave speed of the vertical displacement coincides with the wave speed of the longitudinal or of the shear angle displacement. Moreover, the existence of the global attractor is firstly achieved.
- Wave Speed
- Exponential Stability
- Global Attractor
- Longitudinal Displacement
- Global Weak Solution
where , , and are the longitudinal, vertical, and shear angle displacement, and are the temperature deviations from the along the longitudinal and vertical directions, , , , , , , , and are positive constants for the elastic and thermal material properties.
they obtained exponential stability for the thermoelastic Timoshenko system (1.8) when ; later, they proved energy decay for a Timoshenko-type system with history in thermoelasticity of type III , and this paper is similar to  with an extra damping that comes from the presence of a history term; it improves the result of  in the sense that the case of nonequal wave speed has been considered and the relaxation function g is allowed to decay exponentially or polynomially. We refer the reader to [4–10] for the Timoshenko system with other kinds of damping mechanisms such as viscous damping, viscoelastic damping of Boltzmann type acting on the motion equation of or . In all these cases, the rotational displacement of the Timoshenko system is effectively damped due to the thermal energy dissipation. In fact, the energy associated with this component of motion decays exponentially. The transverse displacement is only indirectly damped through the coupling, which can be observed from (1.2). The effectiveness of this damping depends on the type of coupling and the wave speeds. When the wave speeds are the same ( ), the indirect damping is actually strong enough to induce exponential stability for the Timoshenko system, but when the wave speeds are different, the Timoshenko system loses the exponential stability. This phenomenon has been observed for partially damped second-order evolution equations. We would like to mention other works in [11–15] for other related models.
Recently, Liu and Rao  considered a similar system; they used semigroup method and showed that the exponentially decay rate is preserved when the wave speed of the vertical displacement coincides with the wave speed of longitudinal displacement or of the shear angle displacement. Otherwise, only a polynomial-type decay rate can be obtained; their main tools are the frequency-domain characterization of exponential decay obtained by Prüss  and Huang  and of polynomial decay obtained recently by Muñoz Rivera and Fernández Sare . For the attractors, we refer to [19–24].
where is the Poisson's ratio. Thus, the exponential stability for the case of is only mathematically sound. However, it does provide useful insight into the study of similar models arising from other applications.
Here we state and prove a decay result in the case of equal wave speeds propagation.
We are now ready to state our main stability result.
The proof of our result will be established through several lemmas.
and Young's inequality, the assertion of the lemma follows.
The assertion of the lemma then follows, using Young's and Poincaré's inequalities.
Then, using Young's and Poincaré's inequalities, we can obtain the assertion.
Then, using Young's inequality, we can obtain the assertion.
Then, noticing , again, from the above two equalities and Young's inequality, we can obtain the assertion.
Then, insert (2.28) and (2.29) into (2.27), and the assertion of the lemma follows.
Then, using Young's and Poincaré's inequalities, we can obtain the assertion.
We are now ready to prove Theorem 2.2.
Proof of Theorem 2.2.
Integrating (2.39) over and using (2.38) lead to (2.6). This completes the proof of Theorem 2.2.
In this section, we establish the existence of the global attractor for system (1.1)–(1.5).
Recall that the global attractor of acting on is a compact set enjoying the following properties:
(1) is fully invariant for , that is, for every ;
(2) is an attracting set, namely, for any bounded set ,
where denotes the Hausdorff semidistance on .
for every .
it is clear that is still a bounded absorbing set which is also invariant for , that is, for every .
where stands for -inner product on .
is a connected and compact global attractor of . Therefore, we have proved the following result.
Under the assumption of , problem (3.1) possesses a unique global attractor .
- Bresse JAC: Cours de Mécanique Appliquée. Mallet Bachelier, Paris, France; 1859.MATHGoogle Scholar
- Messaoudi SA, Said-Houari B: Energy decay in a Timoshenko-type system of thermoelasticity of type III. Journal of Mathematical Analysis and Applications 2008,348(1):298-307. 10.1016/j.jmaa.2008.07.036MathSciNetView ArticleMATHGoogle Scholar
- Messaoudi SA, Said-Houari B: Energy decay in a Timoshenko-type system with history in thermoelasticity of type III. Advances in Differential Equations 2009,14(3-4):375-400.MathSciNetMATHGoogle Scholar
- Ammar-Khodja F, Benabdallah A, Muñoz Rivera JE, Racke R: Energy decay for Timoshenko systems of memory type. Journal of Differential Equations 2003,194(1):82-115. 10.1016/S0022-0396(03)00185-2MathSciNetView ArticleMATHGoogle Scholar
- Muñoz Rivera JE, Fernández Sare HD: Stability of Timoshenko systems with past history. Journal of Mathematical Analysis and Applications 2008,339(1):482-502. 10.1016/j.jmaa.2007.07.012MathSciNetView ArticleMATHGoogle Scholar
- Fernández Sare HD, Racke R: On the stability of damped Timoshenko systems: Cattaneo versus Fourier law. Archive for Rational Mechanics and Analysis 2009,194(1):221-251. 10.1007/s00205-009-0220-2MathSciNetView ArticleMATHGoogle Scholar
- Liu K, Liu Z, Rao B: Exponential stability of an abstract nondissipative linear system. SIAM Journal on Control and Optimization 2001,40(1):149-165. 10.1137/S0363012999364930MathSciNetView ArticleMATHGoogle Scholar
- Messaoudi SA, Pokojovy M, Said-Houari B: Nonlinear damped Timoshenko systems with second sound—global existence and exponential stability. Mathematical Methods in the Applied Sciences 2009,32(5):505-534. 10.1002/mma.1049MathSciNetView ArticleMATHGoogle Scholar
- Muñoz Rivera JE, Racke R: Mildly dissipative nonlinear Timoshenko systems—global existence and exponential stability. Journal of Mathematical Analysis and Applications 2002,276(1):248-278. 10.1016/S0022-247X(02)00436-5MathSciNetView ArticleMATHGoogle Scholar
- Soufyane A: Stabilisation de la poutre de Timoshenko. Comptes Rendus de l'Académie des Sciences. Série I 1999,328(8):731-734.MathSciNetMATHGoogle Scholar
- Avalos G, Lasiecka I: Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation. SIAM Journal on Mathematical Analysis 1998,29(1):155-182. 10.1137/S0036141096300823MathSciNetView ArticleMATHGoogle Scholar
- Avalos G, Lasiecka I: Exponential stability of a thermoelastic system without mechanical dissipation. Rendiconti dell'Istituto di Matematica dell'Università di Trieste 1996, 28: 1-28.MathSciNetMATHGoogle Scholar
- Chueshov I, Lasiecka I: Inertial manifolds for von Kármán plate equations. Applied Mathematics and Optimization 2002,46(2-3):179-206.MathSciNetMATHGoogle Scholar
- Chueshov I, Lasiecka I: Long-time behavior of second order evolution equations with nonlinear damping. Memoirs of the American Mathematical Society 2008.,195(912):Google Scholar
- Chueshov I, Lasiecka I: Von Karman Evolution Equations: Well-posedness and Long Time Dynamics, Springer Monographs in Mathematics. Springer, New York, NY, USA; 2010:xiv+766.View ArticleMATHGoogle Scholar
- Liu Z, Rao B: Energy decay rate of the thermoelastic Bresse system. Zeitschrift für Angewandte Mathematik und Physik 2009,60(1):54-69. 10.1007/s00033-008-6122-6MathSciNetView ArticleMATHGoogle Scholar
- Prüss J:On the spectrum of -semigroups. Transactions of the American Mathematical Society 1984,284(2):847-857. 10.2307/1999112MathSciNetView ArticleMATHGoogle Scholar
- Huang FL: Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Annals of Differential Equations 1985,1(1):43-56.MathSciNetMATHGoogle Scholar
- Borini S, Pata V: Uniform attractors for a strongly damped wave equation with linear memory. Asymptotic Analysis 1999,20(3-4):263-277.MathSciNetMATHGoogle Scholar
- Chueshov I, Lasiecka I: Attractors and long time behavior of von Karman thermoelastic plates. Applied Mathematics and Optimization 2008,58(2):195-241. 10.1007/s00245-007-9031-8MathSciNetView ArticleMATHGoogle Scholar
- Giorgi C, Muñoz Rivera JE, Pata V: Global attractors for a semilinear hyperbolic equation in viscoelasticity. Journal of Mathematical Analysis and Applications 2001,260(1):83-99. 10.1006/jmaa.2001.7437MathSciNetView ArticleMATHGoogle Scholar
- Pata V, Zucchi A: Attractors for a damped hyperbolic equation with linear memory. Advances in Mathematical Sciences and Applications 2001,11(2):505-529.MathSciNetMATHGoogle Scholar
- Qin Y: Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors, Operator Theory: Advances and Applications. Volume 184. Birkhäuser, Basel, Switzerland; 2008:xvi+465.Google Scholar
- Zheng S, Qin Y: Maximal attractor for the system of one-dimensional polytropic viscous ideal gas. Quarterly of Applied Mathematics 2001,59(3):579-599.MathSciNetMATHGoogle Scholar
- Liu Z, Zheng S: Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC Research Notes in Mathematics. Volume 398. Chapman & Hall/CRC, Boca Raton, Fla, USA; 1999:x+206.Google Scholar
- Pazy A: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences. Volume 44. Springer, New York, NY, USA; 1983:viii+279.View ArticleGoogle Scholar
- Temam R: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences. Volume 68. Springer, New York, NY, USA; 1988:xvi+500.View ArticleMATHGoogle Scholar
- Hale JK: Asymptotic behaviour and dynamics in infinite dimensions. In Nonlinear Differential Equations. Volume 132. Edited by: Hale J, Martinez-Amores P. Pitman, Boston, Mass, USA; 1985:1-42.Google Scholar
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