A gradient-like property is established for second order semilinear conservative systems in presence of a linear damping term which is asymptotically weak for large times. The result is obtained under the condition that the only critical points of the potential are absolute minima. The damping term may vanish on large intervals for arbitrarily large times and tends to 0 at infinity, but not too fast (in a non-integrable way). When the potential satisfies an adapted, uniform, Lojasiewicz gradient inequality, convergence to equilibrium of all bounded solutions is shown, with examples in both analytic and non-analytic cases.