无点火温度截止时燃烧的熄灭和传播
Quenching and Propagation of Combustion Without Ignition Temperature Cutoff
摘要：
本文研究了具有燃烧型反应项、无点火截止温度和周期流动的圆柱$Ω= \mathbb{R}\times\mathbb{T}^m$中的反应扩散方程。我们表明，如果反应函数衰减为大于3的$T$的幂作为$T\to 0$和初始数据是小的，那么火焰熄灭-解决方案淬灭。另一方面，如果衰变功率小于3或初始数据很大，则不会发生猝灭，燃烧区域随时间线性扩展。这推广了Aronson-Weinberger关于无流情况的结果。我们还考虑剪切流与大振幅，并表明，如果反应幂律衰减大于3和流量只有小高原（连接域，它是恒定的），然后任何compensated支持的初始数据淬火时，流量振幅足够大（这是不正确的，如果功率是小于3或在一个大的平台的存在）。这推广了Constantin-Kiselev-Ryzhik关于具有点火温度截止的燃烧的结果。我们的工作进行的情况下，$Ω= \mathbb{R}^n\times\mathbb{T}^m$，当临界功率是$1 + 2/n$，以及某些非周期性流。
英文摘要：
We study a reaction-diffusion equation in the cylinder $Ω= \mathbb{R}\times\mathbb{T}^m$, with combustion-type reaction term without ignition temperature cutoff, and in the presence of a periodic flow. We show that if the reaction function decays as a power of $T$ larger than three as $T\to 0$ and the initial datum is small, then the flame is extinguished -- the solution quenches. If, on the other hand, the power of decay is smaller than three or initial datum is large, then quenching does not happen, and the burning region spreads linearly in time. This extends results of Aronson-Weinberger for the no-flow case. We also consider shear flows with large amplitude and show that if the reaction power-law decay is larger than three and the flow has only small plateaux (connected domains where it is constant), then any compactly supported initial datum is quenched when the flow amplitude is large enough (which is not true if the power is smaller than three or in the presence of a large plateau). This extends results of Constantin-Kiselev-Ryzhik for combustion with ignition temperature cutoff. Our work carries over to the case $Ω= \mathbb{R}^n\times\mathbb{T}^m$, when the critical power is $1 + 2/n$, as well as to certain non-periodic flows.
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