## Reaction-diffusion equations, interfaces and geometric flows

The subject of this post is a reaction-diffusion equation of the form $\partial_t u=\Delta u-\frac{1}{\epsilon^2} f'(u)$ for a real-valued function $u$ with $t\in {\bf R}^+$ and $x\in {\bf R}^n$. Here $\epsilon$ is a small parameter. This equation is often known as the Allen-Cahn equation and the phenomena discussed in the following were described in a 1979 paper of Allen and Cahn in Acta Metallurgica (which I have not seen). My primary source is a paper by Xinfu Chen (J. Diff. Eq. 96, 116) which includes both a nice introduction to the subject and rigorous proofs of some key theorems. The function $f$ in the equation is a smooth function with certainly qualitative properties: two minima $u_1^-$ and $u_2^-$ separated by a maximum $u^+$.

In physical terms there are two opposing effects at work in this system. There is reaction which drives the evolution to resemble that of solutions of the corresponding ODE (limit $\epsilon\to\infty$) and diffusion which causes the solution to spread out in space. Generic solutions of the ODE tend to one of the two minima of the potential at late times. If the diffusion is set to zero this leads to two spatial regions where the solution has almost constant values $u_1^-$ and $u_2^-$ respectively. They are separated by an narrow interface where the spatial derivatives of $u$ are large and the the values of $u$ are close to $u^+$. These ideas are captured mathematically by proving asympotic expansions in the parameter $\epsilon$ for the solutions on time intervals whose lengths depend on $\epsilon$ in a suitable way.

The following phenomena arise. Here I assume that the space dimension $n$ is at least two. The case $n=1$ is special. The first statement is that on a suitable time interval, which is short when $\epsilon$ is small, an interface forms. It can be described by a hypersurface. Without going into details it can be thought of as the level hypersurface $u=u^+$. The second describes how the interface moves in space on a longer timescale. If the function $f$ has unequal values at the minima then the interface moves in the direction of its normal towards the region corresponding to the smaller minimum with a velocity which is proportional to the difference of the two values. On the timescale on which this motion takes place the interface stands still in the case that the two values are equal. It can, however, be shown that in the case of equal minima the interface moves in a definite way on an even longer timescale. The interface moves in the direction of its normal with a velocity which is proportional to its mean curvature at the given point. In other words, the hypersurface is a solution of mean curvature flow, one of the best-known geometric flows.

To finish I will state a question which I have no answer to, not even vaguely. Are there any useful analogues of these results for the hyperbolic equation $\partial_t^2 u+\partial_t u=\Delta u-\frac{1}{\epsilon^2} f'(u)$? Here there is a potential interaction between reaction, damping and dispersion.