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The logarithmic delay of KPP fronts in a periodic medium

François Hamel, James Nolen, Jean-Michel Roquejoffre, Lenya Ryzhik (2016)

Journal of the European Mathematical Society

We extend, to parabolic equations of the KPP type in periodic media, a result of Bramson which asserts that, in the case of a spatially homogeneous reaction rate, the time lag between the position of an initially compactly supported solution and that of a traveling wave grows logarithmically in time.

Travelling graphs for the forced mean curvature motion in an arbitrary space dimension

Régis Monneau, Jean-Michel Roquejoffre, Violaine Roussier-Michon (2013)

Annales scientifiques de l'École Normale Supérieure

We construct travelling wave graphs of the form z = - c t + φ ( x ) , φ : x N - 1 φ ( x ) , N 2 , solutions to the N -dimensional forced mean curvature motion V n = - c 0 + κ ( c c 0 ) with prescribed asymptotics. For any 1 -homogeneous function φ , viscosity solution to the eikonal equation | D φ | = ( c / c 0 ) 2 - 1 , we exhibit a smooth concave solution to the forced mean curvature motion whose asymptotics is driven by  φ . We also describe φ in terms of a probability measure on  § N - 2 .

Travelling Waves in Near-Degenerate Bistable Competition Models

E.O. Alzahrani, F.A. Davidson, N. Dodds (2010)

Mathematical Modelling of Natural Phenomena

We study a class of bistable reaction-diffusion systems used to model two competing species. Systems in this class possess two uniform stable steady states representing semi-trivial solutions. Principally, we are interested in the case where the ratio of the diffusion coefficients is small, i.e. in the near-degenerate case. First, limiting arguments are presented to relate solutions to such systems to those of the degenerate case where one species...

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