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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

MSC:
 35K57 Reaction-diffusion equations 35B40 Asymptotic behavior of solutions to PDEs
Keywords:
diffusion; transition zone; interface