Uniform decay in a Timoshenko-type system with past history

Fernández Sare and Rivera [H.D. Fernández Sare, J.E. Muñoz Rivera, Stability of Timoshenko systems with past history, J. Math. Anal. Appl. 339 (1) (2008) 482–502] considered the following Timoshenko-type system ρ 1 φ t t − K ( φ x + ψ ) x = 0 , ρ 2 ψ t t − b ψ x x + ∫ 0 ∞ g ( t ) ψ x x ( t − s , . ) d s + K ( φ x + ψ ) = 0 , where g is a positive differentiable exponentially decaying function. They established an exponential decay result in the case of equal wave-speed propagation and a polynomial decay result in the case of nonequal wave-speed propagation. In this paper, we study the same system, for g decaying polynomially, and prove polynomial stability results for the equal and nonequal wave-speed propagation. Our results are established under conditions on the relaxation function weaker than those in [H.D. Fernández Sare, J.E. Muñoz Rivera, Stability of Timoshenko systems with past history, J. Math. Anal. Appl. 339 (1) (2008) 482–502].