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Sharp Transition Between Extinction and Propagation of Reaction

Andrej Zlatos

TL;DR

The paper analyzes extinction versus propagation for the 1D reaction-diffusion equation $T_t = T_{xx} + f(T)$ with compactly supported initial data, covering ignition, combustion, and bistable nonlinearities. It develops a scaling- and comparison-based framework to identify a unique critical length $L_0$ separating quenching from propagation, establishing that $L_0 = L_1$ in the ignition setting. At criticality, the long-time limit at the origin is a fixed point of $f$ (for ignition) or a nontrivial stationary bell-shaped profile $U$ solving $0=U''+f(U)$ (for bistable/combustion), yielding a complete phase portrait for the 1-parameter initial data family. The results extend classical Kanel’ theory, show robustness to advection, and provide a unified approach to extinction thresholds across nonlinearities.

Abstract

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 $L<L_0$, 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.

Sharp Transition Between Extinction and Propagation of Reaction

TL;DR

The paper analyzes extinction versus propagation for the 1D reaction-diffusion equation with compactly supported initial data, covering ignition, combustion, and bistable nonlinearities. It develops a scaling- and comparison-based framework to identify a unique critical length separating quenching from propagation, establishing that in the ignition setting. At criticality, the long-time limit at the origin is a fixed point of (for ignition) or a nontrivial stationary bell-shaped profile solving (for bistable/combustion), yielding a complete phase portrait for the 1-parameter initial data family. The results extend classical Kanel’ theory, show robustness to advection, and provide a unified approach to extinction thresholds across nonlinearities.

Abstract

We consider the reaction-diffusion equation on with and . In 1964 Kanel' proved that if is an ignition non-linearity, then as when , and when . We answer the open question of relation of and by showing that . We also determine the large time limit of in the critical case , 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.

Paper Structure

This paper contains 4 sections, 4 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Preliminary Lemmas
  3. Proof of Theorem \ref{['T.1.1']}
  4. Proof of Theorem \ref{['T.1.2']}

Key Result

Theorem 1

Let \theta_0\in[0,1) and f:[0,1]\to{\mathbb{R}} be Lipschitz with f(\theta)=0 when \theta\in[0,\theta_0], f(\theta)>0 when \theta\in(\theta_0,1), and f(1)=0. If \theta_0>0 then assume in addition that f is non-decreasing on [\theta_0,\theta_0+\delta] for some \delta>0. Let T:[0,\infty)\times{\mathbb Then there is L_0\ge 0 such that

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof