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Apalara, Tijani A.; Messaoudi, Salim A.; Keddi, Ahmed A.

On the decay rates of Timoshenko system with second sound. (English) Zbl 1353.35056

Math. Methods Appl. Sci. 39, No. 10, 2671-2684 (2016).
Considered is an extension of the classical Timoshenko system of two coupled hyperbolic equations to a one dimensional linear system of four PDE by adding a heat equation modeled by the Cattaneo’s law and a coupling equation. Some initial and boundary conditions are defined. The well-posedness of the problem is discussed first. Further one proves a result on the exponential stability of the energy of the solution and one investigates the polynomial stability and cases when a lack of exponential decay occurs.
Reviewer: Claudia Simionescu-Badea (Wien)

Cited in 34 Documents

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs
35L55 Higher-order hyperbolic systems
35L57 Initial-boundary value problems for higher-order hyperbolic systems
74D05 Linear constitutive equations for materials with memory
93D15 Stabilization of systems by feedback
93D20 Asymptotic stability in control theory
35G45 Boundary value problems for systems of linear higher-order PDEs

Keywords:

well-posedness; exponential stability; polynomial stability; Cattaneo’s law
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References:

[1] TimoshenkoSP. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philosophical Magazine Series1921; 6(41):744-746.
[2] KimJU, RenardyY. Boundary control of the Timoshenko beam. SIAM Journal on Control and Optimization1987; 25(6):1417-1429. · Zbl 0632.93057
[3] MessaoudiSA, MustafaMI. On the internal and boundary stabilization of Timoshenko beams. Nonlinear Differential Equations and Applications2008; 15(6):655-671. · Zbl 1163.74037
[4] SoufyaneA, WhebeA. Uniform stabilization for the Timoshenko beam by a locally distributed damping. Electronic Journal of Differential Equations2003; 2003(29):1-14. · Zbl 1012.35053
[5] GuesmiaA, MessaoudiSA. On the control of a viscoelastic damped Timoshenko‐type system. Applied Mathematics and Computation2008; 206(2):589-597. · Zbl 1154.74030
[6] GuesmiaA, MessaoudiSA. General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping. Mathematical Methods in the Applied Sciences2009; 32(16):2102-2122. · Zbl 1183.35036
[7] GuesmiaA, MessaoudiSA. On the stabilization of Timoshenko systems with memory and different speeds of wave propagation. Applied Mathematics and Computation2013; 219(17):9424-9437. · Zbl 1290.74019
[8] Muñoz RiveraJE, Fernández SareHD. Stability of Timoshenko systems with past history. Journal of Mathematical Analysis and Applications2008; 339(1):482-502. · Zbl 1132.45008
[9] Muñoz RiveraJE, RackeR. Timoshenko systems with indefinite damping. Journal of Mathematical Analysis and Applications2008; 341(2):1068-1083. · Zbl 1139.35023
[10] MessaoudiSA, MustafaMI. On the stabilization of the Timoshenko system by a weak nonlinear dissipation. Mathematical Methods in the Applied Sciences2009; 32(4):454-469. · Zbl 1171.35014
[11] MessaoudiSA, MustafaMI. A stability result in a memory‐type Timoshenko system. Dynamic Systems and Applications2009; 18(3):457-468. · Zbl 1183.35194
[12] MessaoudiSA, Said‐HouariB. Uniform decay in a Timoshenko‐type system with past history. Journal of Mathematical Analysis and Applications2009; 360(2):459-475. · Zbl 1183.35040
[13] Almeida JúniorDS, SantosML, Muñoz RiveraJE. Stability to weakly dissipative Timoshenko systems. Mathematical Methods in the Applied Sciences2013; 36(14):1965-1976. · Zbl 1273.74072
[14] Muñoz RiveraJE, RackeR. Mildly dissipative nonlinear Timoshenko systems—global existence and exponential stability. Journal of Mathematical Analysis and Applications2002; 276(1):248-278. · Zbl 1106.35333
[15] GuesmiaA, MessaoudiSA, Soufyane A. Stabilization of a linear Timoshenko system with infinite history and applications to the Timoshenko‐heat systems. Electronic Journal of Differential Equations2012; 2012(193):1-45. · Zbl 1295.35088
[16] Almeida JúniorDS, SantosML, Muñoz RiveraJE. Stability to 1‐D thermoelastic Timoshenko beam acting on shear force. Zeitschrift für angewandte Mathematik und Physik2013: 1-17.
[17] JosephDD, PreziosiL. Heat waves. Reviews of Modern Physics1989; 61(1):41-73. · Zbl 1129.80300
[18] Fernández SareHD, RackeR. On the stability of damped Timoshenko systems: Cattaneo versus Fourier law. Archive for Rational Mechanics and Analysis2009; 194(1):221-251. · Zbl 1251.74011
[19] MessaoudiSA, PokojovyM, SaidHouariB. Nonlinear damped Timoshenko systems with second sound-global existence and exponential stability. Mathematical Methods in the Applied Sciences2009; 32(5):505-534. · Zbl 1156.35321
[20] SantosML, Almeida JúniorDS, Muńoz RiveraJE. The stability number of the Timoshenko system with second sound. Journal of Differential Equations2012; 253(9):2715-2733. · Zbl 1250.74013
[21] Said‐HouariB, KasimovA. Damping by heat conduction in the Timoshenko system: Fourier and Cattaneo are the same. Journal of Differential Equations2013; 225(4):611-632. · Zbl 1282.35074
[22] RackeR, Said‐HouariB. Global existence and decay property of the Timoshenko system in thermoelasticity with second sound. Nonlinear Analysis: Theory, Methods and Applications2012; 75(13):4957-4973. · Zbl 1254.35217
[23] Said‐HouariB, KasimovA. Decay property of Timoshenko system in thermoelasticity. Mathematical Methods in the Applied Sciences2012; 35(3):314-333. · Zbl 1243.35025
[24] ColemanBD, HrusaWJ, OwenDR. Stability of equilibrium for a nonlinear hyperbolic system describing heat propagation by second sound in solids. Archive for Rational Mechanics and Analysis1986; 94(3):267-289. · Zbl 0621.73132
[25] RackeR. Thermoelasticity with second sound—exponential stability in linear and nonlinear 1−d. Mathematical Methods in the Applied Sciences2002; 25(5):409-441. · Zbl 1008.74027
[26] BrezisH. Functional analysis, Sobolev spaces and partial differential equations. Springer: Dordrecht, Heidelberg, New York, London, 2010. · Zbl 1218.46002
[27] GearhartL. Spectral theory for contraction semigroups on Hilbert space. Transactions of the American Mathematical Society1978; 236: 385-394. · Zbl 0326.47038
[28] HuangF. Characteristic condition for exponential stability of linear dynamical systems in Hilbert space. Annals of Differential Equations1985; 1(1):43-56. · Zbl 0593.34048
[29] PrüssJ. On the spectrum of C_0‐semigroups. Transactions of the American Mathematical Society1984; 284: 847-857. · Zbl 0572.47030
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