Zero width limit of the heat equation on moving thin domains

We study the behavior of a variational solution to the Neumann type problem of the heat equation on a moving thin domain Omega(epsilon)(t ) that converges to an evolving surface Gamma(t) as the width of Omega(epsilon)(t ) goes to zero. We show that, under suitable assumptions, the average in the normal direction of Gamma(t) of a variational solution to the heat equation converges weakly in a function space on Gamma(t) as the width of Omega(epsilon)(t) goes to zero, and that the limit is a unique variational solution to a limit equation on Gamma(t), which is a new type of linear diffusion equation involving the mean curvature and the normal velocity of Gamma(t) We also estimate the difference between variational solutions to the heat equation on Omega(epsilon)(t) and the limit equation on Gamma(t).