Taiwanese Journal of Mathematics

GLOBAL EXISTENCE AND ENERGY DECAY OF SOLUTIONS TO A NONLINEAR TIMOSHENKO BEAM SYSTEM WITH A DELAY TERM

Abbes Benaissa and Mounir Bahlil

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Abstract

We consider the Timoshenko system in boundeddomain with a delay term in the nonlinear internal feedback $$\begin{cases} \rho_{1} \varphi_{tt}(x,t) - K(\varphi_{x}+\psi)_{x}(x,t) = 0, \\ \rho_{2} \psi_{tt}(x,t) - b \psi_{xx}(x,t) + K(\varphi_{x}+\psi)(x,t) \\ \qquad\qquad + \mu_1 g_1(\psi_{t}(x,t)) + \mu_2 g_2(\psi_{t}(x,t-\tau)) = 0, \end{cases}$$ and prove the global existence of its solutions in Sobolev spaces by means of the energy method combined with the Faedo-Galerkin procedureunder a condition between the weight of the delay term in the feedback and the weight of the term without delay. Furthermore, we establish a decay rate estimate for the energy by introducing suitable Lyapunov functionals.

Article information

Source
Taiwanese J. Math., Volume 18, Number 5 (2014), 1411-1437.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706519

Digital Object Identifier
doi:10.11650/tjm.18.2014.3586

Mathematical Reviews number (MathSciNet)
MR3265070

Zentralblatt MATH identifier
1357.35261

Subjects
Primary: 35B40: Asymptotic behavior of solutions 35L70: Nonlinear second-order hyperbolic equations 49K25 93D15: Stabilization of systems by feedback

Keywords
nonlinear Timoshenko system delay term decay rate Lyapunov functionals

Citation

Benaissa, Abbes; Bahlil, Mounir. GLOBAL EXISTENCE AND ENERGY DECAY OF SOLUTIONS TO A NONLINEAR TIMOSHENKO BEAM SYSTEM WITH A DELAY TERM. Taiwanese J. Math. 18 (2014), no. 5, 1411--1437. doi:10.11650/tjm.18.2014.3586. https://projecteuclid.org/euclid.twjm/1499706519


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