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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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