  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 $u^-\le u\le
u^+$.
