an:02113848
Zbl 1058.76070
Caffarelli, Luis A.; Lee, Ki-Ahm; Mellet, Antoine
Singular limit and homogenization for flame propagation in periodic excitable media
EN
Arch. Ration. Mech. Anal. 172, No. 2, 153-190 (2004).
00107289
2004
j
76V05 76M50 35B27 80A25
pulsating travelling fronts; singularly perturbed equation; free boundary problem
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. \textit{H. Berestycki} and \textit{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).
Abdolrahman Razani (Qazvin)
Zbl 1024.37054