In this talk, we concern the parabolic Allen-Cahn equation with a potential $K$ with slow diffusion and fast reaction. In particular, the convergence of solutions  to a generalized Brakke's mean curvature flow is established in the limit of a small parameter $ \epsilon \rightarrow 0$. More precisely, we show that a sequence of Radon measures, associated to energy density of solutions to the parabolic Allen-Cahn equation, converges to a weight measure of an integral varifold. Moreover, the limiting varifold evolves by a vector which is the difference between the mean curvature vector and the normal part of ${\nabla K}/{2K}$.

Dr. Gao-Feng Zheng is a professor and vice dean  in the School of Mathematics and Statistics at Central China Normal University. Currently, Dr. Zheng's research focuses are elliptic and parabolic partial differential equations, geometric evolution equations and geometric measure theory.