Fast Reaction, Slow Diffusion, and Curve Shortening

Source abstract

The reaction-diffusion problem $$u_1 = \varepsilon \Delta u - \varepsilon ^{ - 1} V_n ( u ),\quad u( {x,0,\varepsilon } ) = g( x ),\quad\partial _n u = 0\text{ on }\partial \Omega$$ for a vector $u( x,t,\varepsilon )$ is considered in a domain $\Omega \in R^m$. An asymptotic solution is constructed for $\varepsilon$ 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 $\varepsilon \kappa$, where $\kappa$ is its mean curvature. The motion of fronts in this manner is studied for arcs in the plane which are normal to $\partial \Omega$ 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 $\Omega$. In the latter case, a nonconstant steady state $u( x,\infty ,\varepsilon )$ results.