QUENCHING AND PROPAGATION OF BISTABLE REACTION-DIFFUSION FRONTS IN MULTIDIMENSIONAL PERIODIC MEDIA

We study the front dynamics of the bistable reaction-diffusion equations with periodic diffusion and/or convection coefficients in several space dimensions. When traveling wave solutions exist, the solutions of the initial value problem behave as wave fronts propagating with the effective speeds of traveling waves under various initial conditions. Yet due to the bistable nature of the nonlinearity, traveling waves may not always exist when the medium variations from the mean states are large enough. Their existence is closely related to the detailed forms of diffusion and convection coefficients, more so in multidimension than in one. We present a simple sufficient condition for the nonexistence of traveling waves (quenching) using perturbation method. Our two dimensional finite difference numerical computations show a variety of front behaviors, such as: the propagation, quenching and retreat of fronts. We found numerically that quenching occurs in two space dimensions when diffusion is spatially uniform and convection field is a periodic array of rotating vortices if the root mean square of the convection field reaches a critical number.