4:00 pm Thursday, October 8, 2015

Andrej Zlatos (UW Madison)

Abstract: 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-diff usion equations with ignition reaction functions, under very general hypotheses.

Specifically, we show that in up to three spatial dimensions, the width (both in space and time) of the zone where the reaction occurs stays 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 higher dimensions. As an indication of the added difficulties, we also show that three dimensions is indeed the borderline case, and the result is in fact false for general inhomogeneous media in four and more dimensions.

We study fine details of spreading of reactive processes (e.g., combustion) in multi-dimensional inhomogeneous media. One typically observes a transition from one equilibrium (e.g., unburned fuel) to another (e.g., burned fuel) to happen on short spatial as well as temporal scales. We demonstrate that this phenomenon also occurs in one of the simplest models for reactive processes, reaction-diffusion equations with ignition reaction functions (as well as with some monostable, bistable, and mixed reaction functions, in a slightly weaker form), under very general hypotheses.

Specifically, we show that in up to three spatial dimensions, the width (both in space and time) of the zone where reaction occurs stays uniformly bounded in time for some fairly general classes of initial data, and this bound even becomes independent of the initial datum as well as the reaction function, after an initial time interval. Such results have recently been obtained in one dimension but are new in higher dimensions. As one indication of the added difficulties, we also show that three dimensions is indeed the borderline case, and the result is false for general inhomogeneous media in four and more dimensions.