Seminár z kvalitatívnej teórie diferenciálnych rovníc

Seminar on Qualitative Theory of Differential Equations

Tuesday 26.8.2014 at 10:00 Lecture room F1-108

Andrej Zlatoš (University of Wisconsin):

Reactive Processes in Inhomogeneous Media

Abstract:

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 spatial 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.