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Évolution géométrique d’interfaces. (Geometrical evolution of interfaces). (French) Zbl 0698.35078
Consider the semilinear parabolic equation \[ \partial u/\partial t-h^ 2\Delta u+\phi (u)=0,\quad x\in {\mathbb{R}}^ n,\quad t>0, \] with initial condition \(u(x,0)=u_ 0(x)\), \(x\in {\mathbb{R}}^ n\), where \(\phi =\Phi '\), \(\Phi (u)=(1/2)(1-u^ 2)^ 2\), and h is a small parameter. For small times, the solution u behaves as if there were no diffusion, therefore it tends to \(\pm 1\), according to the sign of initial data. The transition zone where \(u=0\) is called interface. If the initial data has an interface, it is given an asymptotic expansion of arbitrarily high order and error estimates valid up to the time \(O(h^ 2)\). At lowest order, the interface evolves normally, with a velocity proportional to the mean curvature.
Reviewer: D.Tătaru

35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs