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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
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[1] ; ; 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.
[2] Berestycki, Comm Pure Appl Math 55 pp 949– (2002)
[3] Caffarelli, Differential Integral Equations 8 pp 1585– (1995)
[4] Caffarelli, Comm Pure Appl Math 54 pp 1403– (2001)
[5] Caffarelli, Amer J Math 120 pp 391– (1998)
[6] ; Homogenizations of nonvariational viscosity solutions. Preprint, 2005.
[7] ; Homogenization of the osciillating free boundaries: the elliptic case. Preprint, 2005.
[8] Caffarelli, Arch Ration Mech Anal 172 pp 153– (2004)
[9] Daskalopoulos, Duke Math J 108 pp 295– (2001)
[10] Daskalopoulos, Comm Pure Appl Math 55 pp 633– (2002)
[11] Daskalopoulos, Comm Partial Differential Equations 29 pp 71– (2004)
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