Extensive molecular-dynamics simulations of melting processes of copper clusters, Cu₅₅, Cu₁₄₇ and Cu₃₀₉ with closed-shell icosahedral structure, were performed using an environment-dependent classical tight-binding potential. The results reveal that the cluster may transit completely from pure solid to pure liquid phase with temperature via a dynamics-coexistence (DC) process, depending upon the latent heat and vibrational entropy. In the DC regime, no other than the first-order solid-liquid phase transition with time is related to the specific structure of the cluster.