In this talk I provide a proof that justifies the formal asymptotic argument developed by H. Fan and S. Ji for the long time behavior of the solutions to a reaction diffusion equation. The structure of the reaction term provides this equation with two stable constant equilibrium solutions u=-1,1 and an unstable equilibrium u=0. The solutions of this equation, as time goes to infinity, converge to a piecewise constant function attaining the values of the stable equilibria. The boundaries of the regions where the limiting function attains each constant value evolve with a normal velocity that depends on the geometry of these boundaries. I'll describe the evolution of these boundaries using the notion of generalized flows developed by G. Barles and T. Souganidis.
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