Front propagation in periodic excitable media

This paper is devoted to the study of pulsating travelling fronts for reaction‐diffusion‐advection equations in a general class of periodic domains with underlying periodic diffusion and velocity fields. Such fronts move in some arbitrarily given direction with an unknown effective speed. The notion of pulsating travelling fronts generalizes that of travelling fronts for planar or shear flows. Various existence, uniqueness and monotonicity results are proved for two classes of reaction terms. Firstly, for a combustion‐type nonlinearity, it is proved that the pulsating travelling front exists and that its speed is unique. Moreover, the front is increasing with respect to the time variable and unique up to translation in time. We also consider one class of monostable nonlinearity which arises either in combustion or biological models. Then, the set of possible speeds is a semi‐infinite interval, closed and bounded from below. For each possible speed, there exists a pulsating travelling front which is increasing in time. This result extends the classical Kolmogorov‐Petrovsky‐Piskunov case. Our study covers in particular the case of flows in all of space with periodic advections such as periodic shear flows or a periodic array of vortical cells. These results are also obtained for cylinders with oscillating boundaries or domains with a periodic array of holes. © 2002 Wiley Periodicals, Inc.