Propagation of Reactions in Inhomogeneous Media

Consider reaction-diffusion equation ut = ∆u+f(x, u) with x ∈ R and general inhomogeneous ignition reaction f ≥ 0 vanishing at u = 0, 1. Typical solutions 0 ≤ u ≤ 1 transition from 0 to 1 as time progresses, and we study them in the region where this transition occurs. Under fairly general qualitative hypotheses on f we show that in dimensions d ≤ 3, the Hausdorff distance of the super-level sets {u ≥ ε} and {u ≥ 1 − ε} remains uniformly bounded in time for each ε ∈ (0, 1). Thus, u remains uniformly in time close to the characteristic function of {u ≥ 12} in the sense of Hausdorff distance of super-level sets. We also show that each {u ≥ ε} expands with average speed (over any long enough time interval) between the two spreading speeds corresponding to any x-independent lower and upper bounds on f . On the other hand, these results turn out to be false in dimensions d ≥ 4, at least without further quantitative hypotheses on f . The proof for d ≤ 3 is based on showing that as the solution propagates, small values of u cannot escape far ahead of values close to 1. The proof for d ≥ 4 is via construction of a counter-example for which this fails. Such results were before known for d = 1 but are new for general non-periodic media in dimensions d ≥ 2 (some are also new for homogeneous and periodic media). They extend in a somewhat weaker sense to monostable, bistable, and mixed reaction types, as well as to transitions between general equilibria u− < u of the PDE, and to solutions not necessarily satisfying u− ≤ u ≤ u.