In this paper, we study an abstract class of weakly dissipative second‐order systems with finite memory. We establish a new general decay rate for the solution of the system under some appropriate conditions on the memory kernel (relaxation function). Our result improves and generalizes many existing results in the literature. We also give some examples to illustrate our abstract result.

In this paper, we are concerned with a memory-type Timoshenko system with Dirichlet boundary conditions and a very general class of relaxation functions. We prove the existence and uniqueness of solutions of the system as well as some new decay results which generalize and improve many earlier ones in the literature. We consider the case of equal-speeds and the case of non-equal-speeds of propagation. We also give some numerical illustrations and related comparisons.

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This paper is concerned with the following memory-type Bresse system ρ 1 ϕtt − k 1 (ϕx + ψ + lw)x − lk 3 (wx − lϕ) = 0, ρ 2 ψtt − k 2 ψxx + k 1 (ϕx + ψ + lw) + t 0 g(t − s)ψxx(•, s)ds = 0, ρ 1 wtt − k 3 (wx − lϕ)x + lk 1 (ϕx + ψ + lw) = 0, with homogeneous Dirichlet-Neumann-Neumann boundary conditions, where (x, t) ∈ (0, L) × (0, ∞), g is a positive strictly increasing function satisfying, for some nonnegative functions ξ and H, g (t) ≤ −ξ(t)H(g(t)), ∀t ≥ 0. Under appropriate conditions on ξ and H, we prove, in cases of equal and non-equal speeds of wave propagation, some new decay results that generalize and improve the recent results in the literature.