Reaction-Diffusion Processes and Evolution to Harmonic Maps

Initial-boundary value problems are considered for the reaction-diffusion equation $u_1 = \varepsilon \Delta u - \varepsilon ^{ - 1} f( u )$ with x in a domain $\Omega $ in $R''$ and u in $R'''$. First the asymptotic behavior of u is determined for $\varepsilon $ small when $f( u ) = 0$ on a connected manifold M of stable equilibrium points. It is found that a tends rapidly to M, being driven by reaction. Then u evolves slowly by diffusion restricted toM. It tends ultimately to a limit that is a harmonic map of $\Omega $ into M Next, the case where $f( u )$ has stable equilibrium points on two manifolds $M_1 $ and $M_2 $ is treated. In this case a front develops in $\Omega $, It separates the regions where u is close to $M_1 $, from the regions whereu is close to $M_2 $. For $f( u ) = V_n ( u )$ a boundary layer solution is constructed for u near the front, and the velocity of the front is found to be proportional to the jump in V across it, to leading order in $\varepsilon $. When $V( u )$ has the same value on $M_1 $ and $M_2 $, this term is zero and the front velocity is$\varepsilon $ times its mean curvature. The case of a spherically symmetric potential $V( {| u |} )$ and the case $M = S^1 $ are presented to illustrate the results.