# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  ; ; Uniform estimates for regularization of free-boundary problems. Analysis and partial differential equations, 567–619. Lectures Notes in Pure and Applied Mathematics, 122. Dekker, New York, 1990.  Berestycki, Comm Pure Appl Math 55 pp 949– (2002)  Caffarelli, Differential Integral Equations 8 pp 1585– (1995)  Caffarelli, Comm Pure Appl Math 54 pp 1403– (2001)  Caffarelli, Amer J Math 120 pp 391– (1998)  ; Homogenizations of nonvariational viscosity solutions. Preprint, 2005.  ; Homogenization of the osciillating free boundaries: the elliptic case. Preprint, 2005.  Caffarelli, Arch Ration Mech Anal 172 pp 153– (2004)  Daskalopoulos, Duke Math J 108 pp 295– (2001)  Daskalopoulos, Comm Pure Appl Math 55 pp 633– (2002)  Daskalopoulos, Comm Partial Differential Equations 29 pp 71– (2004)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.