Interfacial Propagation in Inhomogeneous Medium, Some Results and Questions

We will present some results for the interfacial propagation in inhomogeneous medium. The prototype equation is motion by mean curvature. The key feature is the interaction between the mean curvature of the interface and the underlying spatial inhomogeneity. We will describe the transition between the pinning and de-pinning of the interface and the existence of pulsating waves. Some recent investigations on the pinning threshold and the front propagations between patterns will also be discussed.