Large time behavior of fronts governed by eikonal equations

  • Guy Barles

    Université de Tours, France
  • Jean-Michel Roquejoffre

    Université Paul Sabatier, Toulouse, France


Motivated by a model of solid combustion in heterogeneous media, we investigate the time-asymptotic behaviour of flame fronts evolving with a periodic space-dependent normal velocity; using the so-called “level set approach” we are led to study the large time behaviour of solutions of eikonal equations. We first provide a general approach which shows that the asymptotic normal velocity of such a flame front depends only on its normal direction and is given by the homogenized Hamiltonian of the eikonal equation. Then we turn to a more precise study of the asymptotic behaviour of the flame front when the initial front is a graph of a periodic function: in this case, the front moves asymptotically with a constant normal velocity and we are able to prove that, in coordinates moving with this constant velocity, the front has a time-periodic asymptotic behaviour in the following two cases: (i) when there is a straight line of maximal speed, and (ii) when the space dimension is 2. These results are obtained by using homogenization, control theory and dynamical systems methods (Aubry–Mather sets).

Cite this article

Guy Barles, Jean-Michel Roquejoffre, Large time behavior of fronts governed by eikonal equations. Interfaces Free Bound. 5 (2003), no. 1, pp. 83–102

DOI 10.4171/IFB/73