Existence and general stabilization of the Timoshenko system of thermo-viscoelasticity of type III with frictional damping and delay terms

Advances in Nonlinear Analysis - Tập 7 Số 4 - Trang 547-569 - 2018
Miaomiao Chen1, Wenjun Liu1, Weican Zhou1
1College of Mathematics and Statistics , Nanjing University of Information Science and Technology , Nanjing 210044 , P. R. China

Tóm tắt

Abstract In this paper, we consider the following Timoshenko system of thermo-viscoelasticity of type III with frictional damping and delay terms: { ρ 1 ⁢ φ t ⁢ t - K ⁢ ( φ x + ψ ) x = 0 , ( x , t ) ∈ ( 0 , 1 ) × ( 0 , ∞ ) , ρ 2 ⁢ ψ t ⁢ t - b ⁢ ψ x ⁢ x + K ⁢ ( φ x + ψ ) + β ⁢ θ x = 0 , ( x , t ) ∈ ( 0 , 1 ) × ( 0 , ∞ ) , ρ 3 ⁢ θ t ⁢ t - δ ⁢ θ x ⁢ x + γ ⁢ ψ t ⁢ t ⁢ x + ∫ 0 t g ⁢ ( t - s ) ⁢ θ x ⁢ x ⁢ ( s ) ⁢ d s + μ 1 ⁢ θ t ⁢ ( x , t ) + μ 2 ⁢ θ t ⁢ ( x , t - τ ) = 0 , ( x , t ) ∈ ( 0 , 1 ) × ( 0 , ∞ ) , \left\{\begin{aligned} &\displaystyle\rho_{1}\varphi_{tt}-K(\varphi_{x}+\psi)_% {x}=0,&&\displaystyle(x,t)\in(0,1)\times(0,\infty),\\ &\displaystyle\rho_{2}\psi_{tt}-b\psi_{xx}+K(\varphi_{x}+\psi)+\beta\theta_{x}% =0,&&\displaystyle(x,t)\in(0,1)\times(0,\infty),\\ &\displaystyle\rho_{3}\theta_{tt}-\delta\theta_{xx}+\gamma\psi_{ttx}+\int_{0}^% {t}g(t-s)\theta_{xx}(s)\,\mathrm{d}s+\mu_{1}\theta_{t}(x,t)+\mu_{2}\theta_{t}(% x,t-\tau)=0,&&\displaystyle(x,t)\in(0,1)\times(0,\infty),\end{aligned}\right. together with initial datum and boundary conditions of Dirichlet type, where g is a positive non-increasing relaxation function and μ 1 , μ 2 {\mu_{1},\mu_{2}} are positive constants. Under a hypothesis between the weight of the delay term and the weight of the friction damping term, we prove the global existence of solutions by using the Faedo–Galerkin approximations together with some energy estimates. Then, by introducing appropriate Lyapunov functionals, under the imposed constrain on the above two weights, we establish a general energy decay result from which the exponential and polynomial types of decay are only special cases.

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Tài liệu tham khảo

F. Boulanouar and S. Drabla, General boundary stabilization result of memory-type thermoelasticity with second sound, Electron. J. Differential Equations 2014 (2014), Paper No. 202.

M. M. Cavalcanti, Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping, Differential Integral Equations 14 (2001), no. 1, 85–116.

M. M. Cavalcanti, Uniform decay rates for the energy of Timoshenko system with the arbitrary speeds of propagation and localized nonlinear damping, Z. Angew. Math. Phys. 65 (2014), no. 6, 1189–1206. 10.1007/s00033-013-0380-7

M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electron. J. Differential Equations 2002 (2002), Paper No. 44.

M. M. Cavalcanti and A. Guesmia, General decay rates of solutions to a nonlinear wave equation with boundary condition of memory type, Differential Integral Equations 18 (2005), no. 5, 583–600.

M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim. 42 (2003), no. 4, 1310–1324. 10.1137/S0363012902408010

M. de Lima Santos, Decay rates for solutions of a Timoshenko system with a memory condition at the boundary, Abstr. Appl. Anal. 7 (2002), no. 10, 531–546. 10.1155/S1085337502204133

A. Djebabla and N. Tatar, Exponential stabilization of the Timoshenko system by a thermo-viscoelastic damping, J. Dyn. Control Syst. 16 (2010), no. 2, 189–210. 10.1007/s10883-010-9089-5

A. Djebabla and N. Tatar, Stabilization of the Timoshenko beam by thermal effect, Mediterr. J. Math. 7 (2010), no. 3, 373–385. 10.1007/s00009-010-0058-8

A. Djebabla and N. Tatar, Exponential stabilization of the Timoshenko system by a thermal effect with an oscillating kernel, Math. Comput. Modelling 54 (2011), no. 1–2, 301–314. 10.1016/j.mcm.2011.02.013

D. Feng, D. Shi and W. Zhang, Boundary feedback stabilization of Timoshenko beam with boundary dissipation, Sci. China Ser. A 41 (1998), no. 5, 483–490. 10.1007/BF02879936

J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM J. Control Optim. 25 (1987), no. 6, 1417–1429. 10.1137/0325078

M. Kirane and B. Said-Houari, Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys. 62 (2011), no. 6, 1065–1082. 10.1007/s00033-011-0145-0

M. Kirane, B. Said-Houari and M. N. Anwar, Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks, Commun. Pure Appl. Anal. 10 (2011), no. 2, 667–686.

G. Li, D. Wang and B. Zhu, Well-posedness and decay of solutions for a transmission problem with history and delay, Electron. J. Differential Equations 2016 (2016), Paper No. 23.

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.

W. Liu, General decay of the solution for a viscoelastic wave equation with a time-varying delay term in the internal feedback, J. Math. Phys. 54 (2013), no. 4, Article ID 043504.

W. Liu and K. Chen, Existence and general decay for nondissipative hyperbolic differential inclusions with acoustic/memory boundary conditions, Math. Nachr. 289 (2016), no. 2–3, 300–320. 10.1002/mana.201400343

W. Liu, K. Chen and J. Yu, Existence and general decay for the full von Kármán beam with a thermo-viscoelastic damping, frictional dampings and a delay term, IMA J. Math. Control Inform. (2015), 10.1093/imamci/dnv056.

W. Liu, Y. Sun and G. Li, On decay and blow-up of solutions for a singular nonlocal viscoelastic problem with a nonlinear source term, preprint (2013), http://arxiv.org/abs/1303.4246; to appear in Topol. Methods Nonlinear Anal.

Z. Liu and C. Peng, Exponential stability of a viscoelastic Timoshenko beam, Adv. Math. Sci. Appl. 8 (1998), no. 1, 343–351.

S. A. Messaoudi and A. Fareh, Energy decay in a Timoshenko-type system of thermoelasticity of type III with different wave-propagation speeds, Arab. J. Math. (Springer) 2 (2013), no. 2, 199–207. 10.1007/s40065-012-0061-y

S. A. Messaoudi, M. Pokojovy and B. Said-Houari, Nonlinear damped Timoshenko systems with second sound—Global existence and exponential stability, Math. Methods Appl. Sci. 32 (2009), no. 5, 505–534. 10.1002/mma.1049

S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system of thermoelasticity of type III, J. Math. Anal. Appl. 348 (2008), no. 1, 298–307. 10.1016/j.jmaa.2008.07.036

J. E. Muñoz Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability, J. Math. Anal. Appl. 276 (2002), no. 1, 248–278. 10.1016/S0022-247X(02)00436-5

J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems, Discrete Contin. Dyn. Syst. 9 (2003), no. 6, 1625–1639. 10.3934/dcds.2003.9.1625

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim. 45 (2006), no. 5, 1561–1585. 10.1137/060648891

S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations 21 (2008), no. 9–10, 935–958.

C. A. Raposo, Exponential stability for the Timoshenko system with two weak dampings, Appl. Math. Lett. 18 (2005), no. 5, 535–541. 10.1016/j.aml.2004.03.017

B. Said-Houari and Y. Laskri, A stability result of a Timoshenko system with a delay term in the internal feedback, Appl. Math. Comput. 217 (2010), no. 6, 2857–2869.

B. Said-Houari and R. Rahali, A stability result for a Timoshenko system with past history and a delay term in the internal feedback, Dynam. Systems Appl. 20 (2011), no. 2–3, 327–353.

D.-H. Shi, S. H. Hou and D.-X. Feng, Feedback stabilization of a Timoshenko beam with an end mass, Internat. J. Control 69 (1998), no. 2, 285–300. 10.1080/002071798222848

M. A. Shubov, Asymptotic and spectral analysis of the spatially nonhomogeneous Timoshenko beam model, Math. Nachr. 241 (2002), 125–162. 10.1002/1522-2616(200207)241:1<125::AID-MANA125>3.0.CO;2-3

A. Soufyane and A. Wehbe, Uniform stabilization for the Timoshenko beam by a locally distributed damping, Electron. J. Differential Equations 2003 (2003), Paper No. 29.

F. Tahamtani and A. Peyravi, Asymptotic behavior and blow-up of solutions for a nonlinear viscoelastic wave equation with boundary dissipation, Taiwanese J. Math. 17 (2013), no. 6, 1921–1943. 10.11650/tjm.17.2013.3034

S. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismaticbars, Philos. Mag. 41 (1921), 744–746. 10.1080/14786442108636264

D. Wang, G. Li and B. Zhu, Well-posedness and general decay of solution for a transmission problem with viscoelastic term and delay, J. Nonlinear Sci. Appl. 9 (2016), no. 3, 1202–1215. 10.22436/jnsa.009.03.46

S.-T. Wu, Asymptotic behavior for a viscoelastic wave equation with a delay term, Taiwanese J. Math. 17 (2013), no. 3, 765–784. 10.11650/tjm.17.2013.2517

S.-T. Wu, General decay of solutions for a nonlinear system of viscoelastic wave equations with degenerate damping and source terms, J. Math. Anal. Appl. 406 (2013), no. 1, 34–48. 10.1016/j.jmaa.2013.04.029

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Advances in Nonlinear Analysis

Tập 7 Số 4

547-569

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