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.
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.
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