Mass conserving Allen–Cahn equation and volume preserving mean curvature flow

We consider a mass conserving Allen–Cahn equation u_t = Δu + ε–2(f(u) – ελ(t)) in a bounded domain with no flux boundary condition, where ελ(t) is the average of f(u(·,t)) and –f is the derivative of a double equal well potential. Given a smooth hypersurface γ_0 contained in the domain, we show that the solution u^ε with appropriate initial data tends, as ε ↘ 0 , to a limit which takes only two values, with the jump occurring at the hypersurface obtained from the volume preserving mean curvature flow starting from γ_0 .