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On singular perturbation problems with Robin boundary condition

Henri BerestyckiJuncheng Wei — 2003

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider the following singularly perturbed elliptic problem ϵ 2 Δ u - u + f ( u ) = 0 , u > 0 in Ω , ϵ u ν + λ u = 0 on Ω , where f satisfies some growth conditions, 0 λ + , and Ω N ( N > 1 ) is a smooth and bounded domain. The cases λ = 0 (Neumann problem) and λ = + (Dirichlet problem) have been studied by many authors in recent years. We show that, there exists a generic constant λ * > 1 such that, as ϵ 0 , the least energy solution has a spike near the boundary if λ λ * , and has an interior spike near the innermost part of the domain if λ > λ * . Central to our study...

Speed-up of reaction-diffusion fronts by a line of fast diffusion

Henri BerestyckiAnne-Charline CoulonJean-Michel RoquejoffreLuca Rossi

Séminaire Laurent Schwartz — EDP et applications

In these notes, we discuss a new model, proposed by H. Berestycki, J.-M. Roquejoffre and L. Rossi, to describe biological invasions in the plane when a strong diffusion takes place on a line. This model seems relevant to account for the effects of roads on the spreading of invasive species. In what follows, the diffusion on the line will either be modelled by the Laplacian operator, or the fractional Laplacian of order less than 1. Of interest to us is the asymptotic speed of spreading in the direction...

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