# American Institute of Mathematical Sciences

January  2012, 11(1): 275-305. doi: 10.3934/cpaa.2012.11.275

## Traveling wave solutions of a 3-component reaction-diffusion model in smoldering combustion

 1 Graduate School of Advanced Mathematical Science, Meiji University, Kawasaki, 214-8571 2 Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, Kawasaki, 214-8571

Received  August 2010 Revised  January 2011 Published  September 2011

We are concerned with a reaction diffusion model describing slow smoldering combustion. The process consists of a sheet of paper ignited on one side and in the presence of a flow of air confined in a narrow gap above the paper. It is observed that thermal-diffusion instability generates diverse spatial patterns in combustion front propagation, depending on flow velocity of gas supply. Particularly, if the velocity is rather fast, planar front propagating with almost constant velocity appears. Motivated by this observation, we discuss the existence and stability of $1$ dimensional traveling wave solutions of the model.
Citation: Kota Ikeda, Masayasu Mimura. Traveling wave solutions of a 3-component reaction-diffusion model in smoldering combustion. Communications on Pure & Applied Analysis, 2012, 11 (1) : 275-305. doi: 10.3934/cpaa.2012.11.275
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##### References:
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