Regularity and solutions for flame modelling in porous medium

Abstract

The presented article deals with a model of flame propagation in porous medium. We depart from previously reported models in flame propagation, and we propose a new modelling conception based on a p-Laplacian operator. Such an operator is intended to extend the conceptions for reproducing the diffusion given in porous media. In addition, we introduce a nonlinear reaction term that generalizes the classical KPP-Fisher model of typical use in combustion. Our analysis shows first the boundedness and uniqueness of weak solutions. Afterward, we reformulate the driving equations using the travelling wave technique; and we introduce a density and flux ansatz to analyse the stability of a critical point. We obtain asymptotic profiles of travelling wave solutions, by making use of the Geometric Perturbation Theory. Eventually, we obtain upper solutions under selfsimilar forms for finite support propagating flames. This is particularly applicable for flames departing from compactly supported initial pressure–temperature distributions.

Keywords

• Flame propagation
• p-Laplacian
• Generalized KPP-fisher
• Travelling wave
• Geometric perturbation theory
• Finite propagation