Journal of Integral Equations and Applications

General and optimal decay in a memory-type Timoshenko system

Salim A. Messaoudi and Jamilu Hashim Hassan

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This paper is concerned with the following memory-type Timoshenko system \[ \rho _1\varphi _{tt}-K(\varphi _x+\psi )_x=0 \] \[ \rho _2\psi _{tt}-b\psi _{xx}+K(\varphi _x+\psi )+ \displaystyle \int _0^tg(t-s)\psi _{xx}(s)\,ds=0, \] $(x,t)\in (0,L)\times (0,\infty )$, with Dirichlet boundary conditions, where $g$ is a positive non-increasing function satisfying, for some constant $1\leq p\lt {3}/{2}$, \[ g'(t)\leq -\xi (t)g^p(t),\quad \mbox {for all }t\geq 0. \] We prove some decay results which generalize and improve many earlier results in the literature. In particular, our result gives the optimal decay for the case of polynomial stability.

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J. Integral Equations Applications, Volume 30, Number 1 (2018), 117-145.

First available in Project Euclid: 10 April 2018

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Zentralblatt MATH identifier

Primary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35L05: Wave equation 35L15: Initial value problems for second-order hyperbolic equations 35L70: Nonlinear second-order hyperbolic equations

Timoshenko system viscoelastic relaxation function general decay optimal decay equal and non-equal speeds of wave propagation


Messaoudi, Salim A.; Hassan, Jamilu Hashim. General and optimal decay in a memory-type Timoshenko system. J. Integral Equations Applications 30 (2018), no. 1, 117--145. doi:10.1216/JIE-2018-30-1-117.

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