Monatshefte für Mathematik

, Volume 188, Issue 1, pp 131–162 | Cite as

The Cauchy problem for a combustion model in a porous medium with two layers

  • J. C. da Mota
  • M. M. SantosEmail author
  • R. A. Santos


We prove the existence of local and global in time solutions of the Cauchy problem for a combustion model in a porous medium with two layers. The model is a system of four equations, consisting of two nonlinear reaction–convection–diffusion equations coupled with two ordinary differential equations, with the coupling occurring in both the reaction functions and in the differential operator coefficients. To obtain the local solution, we first construct an iteration scheme of approximate solutions to the system. Using the continuous dependence of solutions for parabolic equations with respect to the coefficients of the equations, we show that the constructed iteration scheme contains a sequence which converges to a local solution of the system, under the assumption that the initial data are Lipschitz continuous, bounded and non negative. We show that this solution can be extended globally, if the initial data are additionally in the Lebesgue space \(L^p\), for some \(p\in (1,\infty )\).


Reaction–diffusion system Combustion model Iterative monotone method Parametrix method Auxiliary functions 

Mathematics Subject Classification

35K15 35K45 35K57 35K60 80A25 


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© Springer-Verlag GmbH Austria 2017

Authors and Affiliations

  1. 1.Departamento de MatemáticaIME-UFG (Instituto de Matemática e Estatística–Universidade Federal de Goiás)GoiâniaBrazil
  2. 2.Departamento de MatemáticaIMECC-UNICAMP (Instituto de Matemática, Estatística e Computação Científica–Universidade Estadual de Campinas)CampinasBrazil

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