Propagation of Reactions in Inhomogeneous Media

Consider reaction‐diffusion equation u t =Δ u  +  f ( x,u ) with x ∈ ℝ dand 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 superlevel 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 ≥ 1 2}in the sense of Hausdorff distance of superlevel 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 counterexample 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 equilibriau − ≤ u +of the PDE and to solutions not necessarily satisfyingu − ≤ u ≤ u + . © 2016 Wiley Periodicals, Inc.