Pulsating front speed-up and quenching of reaction by fast advection

Abstract

We consider reaction–diffusion equations with combustion-type nonlinearities in two dimensions and study speed-up of their pulsating fronts by general periodic incompressible flows with a cellular structure. We show that the occurrence of front speed-up in the sense , with A the amplitude of the flow and c*(A) the (minimal) front speed, only depends on the geometry of the flow and not on the reaction function. In particular, front speed-up occurs for KPP reactions if and only if it does for ignition reactions. We provide a sharp characterization of the periodic symmetric flows which achieve this speed-up and also show that these are precisely those which, when scaled properly, are able to quench any ignition reaction.