Sharp Transition Between Extinction and Propagation of Reaction

We consider the reaction-diffusion equation $$T_t=T_{xx}+f(T)$$ on $\text{bbR}$ 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 $t\to \mathrm{\infty }$ when $L<L_0$, and $T\to 1$ when $L>L_1$. 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.