This paper is concerned with a parabolic equation with a non-local term defined on a compact two-dimensional Riemannian surface Ω. If the total mass of the solution, λ, is equal to 8π and Ω is the standard sphere S2, it is a Hamilton’s normalized Ricci flow. We obtain the global in time existence of the solution to this problem for 0<λ≤8π. If 0<λ<8π, the orbit is compact while for λ=8π, there is a time sequence along which the solution converges to a stationary solution.