Andrej Zlatos


University of Wisconsin at Madison


Thursday, March 31, 2016 - 4:00pm to 5:00pm




We study fine details of spreading of reactive processes in multidimensional
inhomogeneous media. In the real world, one often observes a transition from one equilibrium (such as unburned areas in forest fires) to another (burned areas)to happen over short spatial as well as temporal distances. We demonstrate that this phenomenon also occurs in one of the simplest models of reactive processes, reaction-diffusion equations with ignition reaction functions, under very general hypotheses.

Specifically, in up to three spatial dimensions, the width (both in space and time) of the zone where the reaction occurs turns out to remain uniformly bounded in time for fairly general classes of initial data. This bound even becomes independent of the initial data and of the reaction function after an initial time interval. Such results have recently been obtained in one dimension, in which one can even completely characterize the long term dynamics of general solutions to the equation, but are new in dimensions two and three. An indication of the added difficulties is the fact that three dimensions turns out to indeed be the borderline case, as the bounded-width result is in fact false for general inhomogeneous media in four and more dimensions.