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Reaction-diffusion processes and evolution to harmonic maps. (English) Zbl 0702.35128
The authors study the initial-boundary value problems for the reaction- diffusion equations $$u_ t=\epsilon \Delta_ n-\epsilon^{-1}f(u)$$ with $$x\in \Omega$$, ($$\Omega$$ domain in $${\mathbb{R}}^ n)$$, $$u\in {\mathbb{R}}^ m$$. Using the two time method on (the “fast time” $$\tau =t/\epsilon$$ and the slow one $$\eta =\epsilon t)$$ they study the asymptotic form of u, under the assumption that f(u) vanishes on a manifold M of attracting equilibrium values. They show that at the end of the initial interval u is in M, except a possibly thin layer at the boundary of $$\Omega$$. Under suitable conditions, as $$\epsilon\to 0$$, the solution u tends to a $$u^ 0\in M$$, which is an harmonic map from $$\Omega$$ to M (so that $$n^ 0$$ satisfies the diffusion equation); when $$M=S^ 1$$ the resulting harmonic map is analized.
When there are two manifolds $$M_ 1$$ and $$M_ 2$$ of attracting equilibrium points; a front develops in $$\Omega$$, which separates the regions when u is close to M, and respectively u is close to $$M_ 2$$. In the neighbourhood of this front both reaction and diffusion are important. When $$f(u)=V_ u(u)$$ this front moves with a velocity proportional to $$V_ 1-V_ 2$$, when $$V_ 1$$ and $$V_ 2$$ are the values of V on $$M_ 1$$, respectively $$M_ 2$$. When $$V_ 1=V_ 2$$ the front moves with a velocity proportional to $$\epsilon$$ times the mean curvature of the front.This situation is illustrated by the case where $$V(u)=V(| u|)$$ is spherically symmetric.
Reviewer: G.Gussi

##### MSC:
 35K57 Reaction-diffusion equations 35B25 Singular perturbations in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs
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