Homogenization and flame propagation in periodic excitable media: the asymptotic speed of propagation.

*(English)*Zbl 1093.35010The 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.

Reviewer: Alain Brillard (Mulhouse)

##### MSC:

35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |

35K55 | Nonlinear parabolic equations |

80A25 | Combustion |

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\textit{L. A. Caffarelli} et al., Commun. Pure Appl. Math. 59, No. 4, 501--525 (2006; Zbl 1093.35010)

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##### References:

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