Propagation of Reactions in Inhomogeneous Media

Abstract

Consider reaction-diffusion equation $u_t=\Delta u + f(x,u)$ with $x\in\mathbb{R}^d$ and general inhomogeneous ignition reaction $f\ge 0$ vanishing at $u=0,1$. Typical solutions $0\le u\le 1$ 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 $d\le 3$, the Hausdorff distance of the super-level sets $\{u\ge\epsilon\}$ and $\{u\ge 1-\epsilon\}$ remains uniformly bounded in time for each $\epsilon\in(0,1)$. Thus, $u$ remains uniformly in time close to the characteristic function of $\{u\ge\tfrac 12\}$ in the sense of Hausdorff distance of super-level sets. We also show that $\{u\ge\tfrac 12\}$ 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 $d\ge 4$, at least without further quantitative hypotheses on $f$. The proof for $d\le 3$ 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 $d\ge 4$ involves 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 $d\ge 2$ (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 $u^-\le u\le u^+$.