Fast Reaction, Slow Diffusion, and Curve Shortening

The reaction-diffusion problem \[ u_1 = ε Δ u - ε - 1 V_n ( u ), u( {x,0,ε } ) = g( x ),∂ _n u = 0 on ∂ Ω \] for a vector u( x,t,ε ) is considered in a domain Ω ∈ Rᵐ. An asymptotic solution is constructed for ε small. It shows that at each $x,u$ tends quickly to a minimum of $V( u )$. When V has several minima, When u tends to a piecewise constant function. Boundary layer expansions are constructed around the resulting surfaces of discontinuity or fronts. Each front is found to move along its normal with a constant velocity determined by the discontinuity $[ V ]$ in V across it. When $[ V ] = 0$, the front's normal velocity is ε κ, where κ is its mean curvature. The motion of fronts in this manner is studied for arcs in the plane which are normal to ∂ Ω at their endpoints, and for fronts that are closed curves. It is shown a front can shrink to a point in a finite time or tend to a locally shortest diameter of Ω. In the latter case, a nonconstant steady state u( x,∞ ,ε ) results.