The model under consideration appears to be reasonably generic. It arises in describing such diverse phenomena as solid-state combustion, rapid solidification, and barodiffusion driven detonation in porous media. In the combustion context, the simplest version of the model consists of two heat equations for the temperature in two domains ("cold" and "burnt" material), with three boundary conditions at the free boundary. The free-boundary conditions represent the heat balance and the kinetic relation between the temperature and the interface velocity. Dynamics of the front are determined by the subtle interaction between the energy release by the combustion process and energy dissipation by the media. For a natural class of kinetic functions the one-dimensional model problem is well-posed globally in time, with uniformly bounded solutions. Numerical simulations on the model reveal an amazing variety of dynamical patterns (even in one spatial dimension), including sequences of period doubling bifurcations leading to chaos, infinite period bifurcations, spinning waves etc.
This work is a joint work with Michael Frankel of Indiana Un iversity-Purdue University at Indianapolis