zbMATH — the first resource for mathematics

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

35K57 Reaction-diffusion equations
35B25 Singular perturbations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI