Zero width limit of the heat equation on moving thin domains

Abstract

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 $\Gamma (t)$.