Well-posedness and general decay for a Lamé viscoelastic system with logarithmic damping

Purpose We study a Lamé-type viscoelastic system with logarithmic velocity damping and quantify its stabilizing effect. The dissipation is weak near zero (σ(v) ⋅ v ∼|v|4 as |v| → 0) yet becomes stronger than linear for large speeds. Our goal is to prove global well-posedness and quantify the long-time decay of the natural energy. Design/methodology/approach We combine the Faedo–Galerkin scheme, Aubin–Lions compactness, and Minty's monotonicity to construct weak solutions and derive an energy identity. A Lyapunov function adapted to the logarithmic dissipation links the decay of the relaxation kernel g to the mechanical energy, assuming g ≥ 0 is nonincreasing and −g′(t) ≥ ξ(t)g(t) with nonincreasing ξ. Findings We obtain global existence and uniqueness of weak solutions and a general decay estimate E(t)≤Cexp−κ∫0tξ(s)ds. Hence, exponential decay holds when inft≥0 ξ(t) 0, whereas polynomial rates follow if ξ(t) ∼ c(1 + t)−1. The results clarify the combined role of Lamé ellipticity, hereditary memory, and logarithmic damping in stabilization. Originality/value Prior Lamé–viscoelastic studies used logarithmic terms mainly as sources; here the damping itself is logarithmic. This reveals a distinct stabilization mechanism and unifies exponential and polynomial regimes.