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Singular limit and homogenization for flame propagation in periodic excitable media. (English) Zbl 1058.76070
The authors focus on front propagation phenomena for a class of one-phase free boundary problems describing laminar flames: $u_t+q(x).\nabla u=\triangle u \quad \text{in}\;\Omega(u):=\{u>0\},$ $| \nabla u| ^2=2f(x)M\quad \text{on}\;\partial \Omega(u).\;\tag{1}$ Such an equation naturally arises as the asymptotic limit ($$\varepsilon$$ goes to zero) of the following advection-reaction-diffusion equation: $u_t+q(x)\nabla u(x)=\triangle u-f(x)\beta_{\varepsilon}(x), \tag{2}$ where the reaction term is defined by $$\beta_{\varepsilon}(s)=\frac{1}{\varepsilon}\beta(\frac{s}{\varepsilon})$$, with $$\beta(s)$$ being a Lipschitz function and satisfying $\beta(s)>0,\quad \text{in}\;(0,1),\;\beta(s)=0\;\text{otherwise, and}\;M=\int_0^1\beta(s)\,ds.$ The authors are concerned with equation (2) when the advection term $$q(x)$$ and the reaction term $$f(x)$$ are no longer constant, but have some periodicity. In this framework, the notion of travelling waves can be replaced by a more general notion of pulsating travelling fronts. H. Berestycki and F. Henri [Commun. Pure Appl. Math. 55, No. 8, 949–1032 (2002; Zbl 1024.37054)] proved existence and uniqueness of pulsating travelling fronts for singularly perturbed equation (2). In the paper under review, the authors investigate the behaviour of pulsating travelling fronts when $$\varepsilon \leq \underline{\varepsilon}| L| \ll 1$$ for some constant $$\underline{\varepsilon}$$, where singular parameters $$\varepsilon >0$$ is related to the activation energy, and $$| L|$$ is the period of the medium for the combustion of premixed gases in periodic media. More precisely, they establish that pulsating travelling fronts behave like travelling waves, when the period $$| L|$$ is small and $$\varepsilon\leq \underline{\varepsilon}| L|$$. They also study the convergence of the solution, as $$\varepsilon$$ goes to zero and $$| L|$$ is fixed, to a solution of free boundary problem (1).

##### MSC:
 76V05 Reaction effects in flows 76M50 Homogenization applied to problems in fluid mechanics 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 80A25 Combustion
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