Geodesic
Sharp transition between extinction and propagation of reaction
Zlatoš, Andrej1
1Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Journal of the American Mathematical Society, Tome 19 (2006) no. 1, pp. 251-263

Voir la notice de l'article provenant de la source American Mathematical Society

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 $L$, 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.
DOI : 10.1090/S0894-0347-05-00504-7

Zlatoš, Andrej  1

1 Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706 
Zlatoš, Andrej. Sharp transition between extinction and propagation of reaction. Journal of the American Mathematical Society, Tome 19 (2006) no. 1, pp. 251-263. doi: 10.1090/S0894-0347-05-00504-7
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