Fronts under arrest II: analytical foundations

We study a class of minimal geometric partial differential equations that serves as a framework to understand the evolution of boundaries between states in different pattern forming systems. The framework combines normal growth, curvature flow and nonlocal interaction terms to track the motion of these interfaces. This approach was first developed to understand arrested fronts in a bacterial system. These are fronts that become stationary as they grow into each other. This paper establishes analytic foundations and geometric insight for studying this class of equations. In so doing, an efficient numerical scheme is developed and employed to gain further insight into the dynamics of these general pattern forming systems.