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Homogenization and flame propagation in periodic excitable media: the asymptotic speed of propagation. (English) Zbl 1093.35010
The authors describe some qualitative properties of a flame propagation problem posed in a periodic medium. They indeed consider the problem $$\partial _{t}u=\Delta u-f(x/\varepsilon ) \beta _{\delta}(u)$$ posed in $$\mathbb{R}^{n}\times \mathbb{R}$$. Here $$f$$ is a 1-periodic function in all directions which satisfies $$0<\lambda \leq f(x) \leq \Lambda$$. $$\beta _{\delta}$$ is deduced from a Lipschitz continuous function $$\beta$$ through $$\beta _{\delta}(s) =\beta (s/\delta ) /\delta$$, $$\beta$$ being positive in $$(0,1)$$, equal to 0 elsewhere, and increasing on $$\left[ 0,b \right]$$ for some positive $$b$$. The small parameters $$\varepsilon$$ and $$\delta$$ are linked through $$\delta =\varepsilon \tau$$, for some positive $$\tau$$. The authors consider pulsating travelling fronts $$u^{\varepsilon ,\delta}$$ for the above problem, that is solutions satisfying $$u\rightarrow 0$$ (resp. 1) as $$x\cdot e\rightarrow-\infty$$ (resp. $$+\infty$$), where $$e\in S^{n-1}$$, and $$u(x+k,t) =u(x,t-k\cdot e/c^{\varepsilon ,\delta}(e) )$$, for some real $$c^{\varepsilon ,\delta}(e)$$. H. Berestycki and F. Hamel proved in [Commun. Pure Appl. Math. 55, No. 8, 949–1032 (2002; Zbl 1024.37054)] some existence and uniqueness result for $$c^{\varepsilon ,\delta}(e)$$ and $$u^{\varepsilon ,\delta}$$. Upper and lower bounds are given for $$\gamma ^{\varepsilon ,\delta}(e) =c^{\varepsilon ,\delta}(e)$$ in terms of planelike solutions of the associated stationary problem. Given $$\eta >0$$, the main result of the paper proves that the slope $$\gamma ^{\varepsilon ,\delta}(e)$$ belongs to $$(\gamma _{\min}^{\tau}(e)-\eta ,\gamma _{\min}^{\tau}(e) +\eta )$$, when $$\varepsilon$$ is small enough. When $$\tau$$ goes to 0, the quantity $$\gamma _{\min}^{\tau}(e)$$ converges to some $$\gamma _{\min}(e)$$ which can be computed in a few number of special cases, among which is the 1D case. The authors here extend previous results they obtained in [Arch. Ration. Mech. Anal. 172, No. 2, 153–190 (2004; Zbl 1058.76070)]. The proof is based on the qualitative properties of planelike solutions.

##### MSC:
 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35K55 Nonlinear parabolic equations 80A25 Combustion
##### Keywords:
Planelike solution; Upper and lower bounds
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##### References:
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