GENERATION AND PROPAGATION OF INTERFACES FOR REACTION DIFFUSION-EQUATIONS

This paper is concerned with the asymptotic behavior as ε ↖ 0 of the solution uε of the reaction-diffusion equation in RN × R+:ut - δ +( 1 ∈2)= 0 where φ is the derivative of a bistable potential. We show that if the initial data u(·, 0) has values in both domains of attraction of the potential, then an interface will develop in a short time O(ε2|lnε|). We also show that if the wells of the potential are of equal depth, then this interface will propagate with normal velocity equal to its mean curvature. Our result is valid as long as the interface remains smooth. If the initial interface is compact and N ≥ 2 then the interface will disappear in a finite time (but not if N = 1). In case the depths of the wells are not equal then in order to obtain get "reasonable" results, we must work on the scaled time s = t ε (slower time scale). In this scale we show that the interface moves with a constant speed proportional to the difference of the depths of the two wells, along the normal, towards the domain of the deeper well; this result is valid for all s ε{lunate} (0, ∞) and does not actually depend on the regularity of the interface. We also extend the above results to the homogeneous second initial-boundary value problem. In case the depths of the wells are equal and the initial interface is orthogonal to the boundary, we prove that the interface moves with normal velocity equal to its mean curvature provided that there is a family of hypersurfaces which moves according to their mean curvature and intersect the boundary orthogonally. © 1992.