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arXiv · 2010.15685

Existence of a traveling wave solution in a free interface problem with fractional order kinetics

Claude-Michel Brauner·Robert Roussarie·Peipei Shang·Linwan Zhang
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Abstract

In this paper we consider a system of two reaction-diffusion equations that models diffusional-thermal combustion with stepwise ignition-temperature kinetics and fractional reaction order 0 < $\alpha$ < 1. We turn the free interface problem into a scalar free boundary problem coupled with an integral equation. The main intermediary step is to reduce the scalar problem to the study of a non-C 1 vector field in dimension 2. The latter is treated by qualitative topo-logical methods based on the Poincar{\'e}-Bendixson Theorem. The phase portrait is determined and the existence of a stable manifold at the origin is proved. A significant result is that the settling time to reach the origin is finite, meaning that the trailing interface is finite in contrast to the case $\alpha$ = 1, but in accordance with $\alpha$ = 0. Finally, the integro-differential system is solved via a fixed-point method.

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Claude-Michel Brauner, Robert Roussarie, Peipei Shang, Linwan Zhang. 2020-10-29. Existence of a traveling wave solution in a free interface problem with fractional order kinetics. https://arxiv.org/abs/2010.15685

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