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In this paper we consider a semilinear parabolic equation in an infinite cylinder. The spatially varying nonlinearity is such that it connects two (spatially independent) bistable nonlinearities in a compact set in space. We prove that, given such a setting, a traveling wave obeying the equation with the one bistable nonlinearity and starting at the respective side of the cylinder, will converge to a traveling wave solution prescribed by the nonlinearity on the other side.

We consider the reaction-diffusion equation \[ T_t = T_{xx} + f(T) \] on $\bbR$ with $T_0(x) \equiv \chi_{[-L,L]} (x)$ and $f(0)=f(1)=0$. In 1964 Kanel' proved that if $f$ is an ignition non-linearity, then $T\to 0$ as $t\to\infty$ when $LL_1$. We answer the open question of relation of $L_0$ and $L_1$ by showing that $L_0=L_1$. We also determine the large time limit of $T$ in the critical case $L=L_0$, thus providing the phase portrait for the above PDE with respect to a 1-parameter family of initial data. Analogous results for combustion and bistable non-linearities are proved as well., Comment: 15 pages

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^, Comment: 56 pages