Conditional stability of front solutions

The paper addresses the question of asymptotic stability for front solutions corresponding to certain models of phase transition, invasions in population genetics, or nonlinear dynamics of perturbations of partial differential equations. The structure of front solutions for these equations is discussed, with emphasis on the relationship between the monotone front with minimum velocity and known front speed results. For a class of scalar reaction-diffusion equations, a Lyapunov functional in a travelling frame of reference is derived. Solutions which are minimal for the Lyapunov functional in certain directions of function space are stable for perturbations in those directions. The well-known minimal monotonic front solution turns out to be a minimum for the Lyapunov functional. A description of the class of perturbations to which this front is stable is derived