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Article Dans Une Revue Journal of Differential Equations Année : 2021

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

Résumé

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 < α < 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-Lipschitz vector field in dimension 2. The latter is treated by qualitative topological methods based on the Poincaré-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 α = 1, but in accordance with α = 0. Finally, the integro-differential system is solved via a fixed-point method.
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Dates et versions

hal-02979187 , version 1 (27-10-2020)

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Claude-Michel Brauner, Robert Roussarie, Peipei Shang, Linwan Zhang. Existence of a traveling wave solution in a free interface problem with fractional order kinetics. Journal of Differential Equations, 2021, 281, pp.105-147. ⟨10.1016/j.jde.2021.01.034⟩. ⟨hal-02979187⟩
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