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Speed of KPP fronts with a cut-off: rigorous results

By R. Benguria, M. C. Depassier, M. Loss and Católica Chile

Abstract

We study the reaction diffusion equation u t = u xx + f(u) with a cut-off in the reaction term (Fig 1). We show that for reaction terms of the form f(u) = u- ϕ(u) where |ϕ(u) | < K u p the asymptotic speed of the front satisfies where In the limit of small ǫ In the limit ǫ → 1, The asymptotic speed of the front, for arbitrary reaction terms f(u) such that f(0) = f(1) =0 can be derived from the variational principle where s 0 is an arbitrary parameter and where the supremum is taken over positive increasing functions u(s) such that u(0) =0, u(s 0) = 1. We apply the variational principle to a reaction term The upper bound is constructed observing that the reaction function f(u), (hence the corresponding F(u)) is smaller than the reaction term shown with a solid line in Fig. 1. For the reaction term shown with the solid line there is a function u(s) for which the supremum is attained. This function can be calculated explicitly, and the upper bound follows immediately. The lower bound is obtained using as a trial function the function u(s) described above which is given by wher

Year: 2013
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