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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).

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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