Sharp transition between extinction and propagation of reaction

Abstract:

We consider the reaction-diffusion equation \[ T_t = T_{xx} + f(T) \] on ${\mathbb {R}}$ with $T_0(x) \equiv \chi _{[-L,L]} (x)$ and $f(0)=f(1)=0$. In 1964 Kanel$^{\prime }$ proved that if $f$ is an ignition non-linearity, then $T\to 0$ as $t\to \infty$ when $L<L_0$, and $T\to 1$ when $L>L_1$. We answer the open question of the 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.