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Fast reaction, slow diffusion, and curve shortening. (English) Zbl 0701.35012
The paper considers a reaction-diffusion equation, containing a small parameter $$\epsilon$$, in a domain $$\Omega \subset {\mathbb{R}}^ n:$$ $u_ t=\epsilon \Delta u-\epsilon^{-1}V_ u(u),\quad u(x,0,\epsilon)=g(x),\quad \partial_ nu=0\text{ on } \partial \Omega.$ The asymptotic behavior of u as $$\epsilon\to 0$$ is studied from various viewpoints. By introducing a fast time variable $$\tau =t/\epsilon$$, u is expanded into a power series of $$\epsilon^ 2$$. Derived is an o.d. equation for the lowest term $$v_ 0(x,\tau)$$. The domain $$\Omega$$ is divided into subdomains according to basins of attraction associated with $$u_ 1,...,u_ K$$, at each of which V(u) attains a local minimum. As a result, $$v_ 0$$ approaches a piecewise constant function as $$\tau\to \infty$$. Since a similar o.d. equation for the second lowest term $$v_ 1$$ does not correctly describe the behavior of $$v_ 1$$ near the boundary, a boundary layer expansion is introduced. The above expansion is no more valid on the common boundary (front) of the resulting subdomains. Thus, another expansion in a boundary layer around the front is proposed with an assumption that the front also moves. By assuming that the lowest term approaches a traveling wave for $$\tau$$ large, the motion of the front is determined: The level set moves along its normal at constant speed determined by $$[V]=V(u_ i)-V(u_ j)$$. When $$[V]=0$$, the motion of the front on the slow time scale $$\eta =\epsilon t$$ is determined according to the equation describing flow by curvature of level sets. Finally, the motion of a front in a plane domain $$\Omega \subset {\mathbb{R}}^ 2$$ is studied. The front normally intersects $$\partial \Omega$$, or is a closed curve in $$\Omega$$. It is shown that the front approaches a locally shortest diameter of $$\Omega$$ with endpoints on $$\partial \Omega$$, or shrinks to a point in a finite time.
Reviewer: T.Nambu

##### MSC:
 35B25 Singular perturbations in context of PDEs 35K57 Reaction-diffusion equations 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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