## Sharp transition between extinction and propagation of reaction

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**19**(2006), 251-263 Request permission

## 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.## References

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## Additional Information

**Andrej Zlatoš**- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
- Email: zlatos@math.wisc.edu
- Received by editor(s): April 15, 2005
- Published electronically: August 24, 2005
- © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**19**(2006), 251-263 - MSC (2000): Primary 35K57; Secondary 35K15
- DOI: https://doi.org/10.1090/S0894-0347-05-00504-7
- MathSciNet review: 2169048