François Hamel

The speed of propagation for KPP type problems. I: periodic framework

This paper is devoted to some nonlinear propagation phenomena in periodic and more general domains, for reaction-diffusion equations with Kolmogorov-Petrovsky-Piskunov (KPP) type nonlinearities. The case of periodic domains with periodic underlying excitable media is a follow-up of the article \cite{bh}. It is proved that the minimal speed of pulsating fronts is given by a variational formula involving linear eigenvalue problems. Some consequences concerning the influence of the geometry of the domain, of the reaction, advection and diffusion coefficients are given. The last section deals with the notion of asymptotic spreading speed. The main properties of the spreading speed are given. Some of them are based on some new Liouville type results for nonlinear elliptic equations in unbounded domains.

Entire Solutions of the KPP Equation

This paper deals with the solutions defined for all time of the KPP equation ut = uxx+ f(u); 0 0, f 0 (1) 0i n(0; 1), and f 0 (s) f 0 (0) in [0; 1]. This equation admits infinitely many traveling-wave-type solutions, increasing or decreasing in x .I t also admits solutions that depend only on t. In this paper, we build four other manifolds of solutions: One is 5-dimensional, one is 4-dimensional, and two are 3-dimensional. Some of these new solutions are obtained by considering two traveling waves that come from both sides of the real axis and mix. Furthermore, the traveling-wave solutions are on the boundary of these four manifolds. c 1999 John Wiley & Sons, Inc.

One-dimensional symmetry of bounded entire solutions of some elliptic equations

This paper is about one-dimensional symmetry properties for some entire and bounded solutions of ∆u + f(u) = 0 in IR. We consider solutions u such that −1 < u < 1 and u(x1, · · · , xn) → ±1 as xn → ±∞, uniformly with respect to x1, · · · , xn−1. Under some conditions on f , we prove that the solutions only depend on the variable xn. We also discuss more general elliptic operators. The qualitative properties then strongly depend on the coefficients of the operator. These results extend to higher dimensions and to more general operators a result of Ghoussoub and Gui [21] proved for n ≤ 3. AMS Classification : 35B05, 35B40, 35B50, 35J60.

Allee effect promotes diversity in traveling waves of colonization

Most mathematical studies on expanding populations have focused on the rate of range expansion of a population. However, the genetic consequences of population expansion remain an understudied body of theory. Describing an expanding population as a traveling wave solution derived from a classical reaction-diffusion model, we analyze the spatio-temporal evolution of its genetic structure. We show that the presence of an Allee effect (i.e., a lower per capita growth rate at low densities) drastically modifies genetic diversity, both in the colonization front and behind it. With an Allee effect (i.e., pushed colonization waves), all of the genetic diversity of a population is conserved in the colonization front. In the absence of an Allee effect (i.e., pulled waves), only the furthest forward members of the initial population persist in the colonization front, indicating a strong erosion of the diversity in this population. These results counteract commonly held notions that the Allee effect generally has adverse consequences. Our study contributes new knowledge to the surfing phenomenon in continuous models without random genetic drift. It also provides insight into the dynamics of traveling wave solutions and leads to a new interpretation of the mathematical notions of pulled and pushed waves.

Uniqueness and stability properties of monostable pulsating fronts

In this paper, we prove the uniqueness, up to shifts, of pulsating traveling fronts for reaction-diffusion equations in periodic media with Kolmogorov-Petrovsky-Piskunov type nonlinearities. These results provide in particular a complete classification of all KPP pulsating fronts. Furthermore, in the more general case of monostable nonlineari-ties, we also derive several global stability properties and convergence to the pulsating fronts for the solutions of the Cauchy problem with front-like initial data. In particular, we prove the stability of KPP pulsating fronts with minimal speed, which is a new result even in the case when the medium is invariant in the direction of propagation.

The speed of propagation for KPP type problems. II: general domains

A modular ramp system includes a ramp component and a connector releasably joined to the ramp component. The ramp component has first and second inclined surfaces and a connecting portion positioned between the first and second inclined surfaces. The connector has a bottom side formed with first and second coupling sections connectible to the connecting portion. The connector is positionable in a first position in which one of the first and second coupling sections is releasably joined to the connecting portion of the ramp component, and the other of the first and second coupling sections is adapted to releasably couple to a ramp or an external structure, and is positionable in a second position in which the first and second coupling sections are releasably joined to the connecting portion of the ramp component.

A short proof of the logarithmic Bramson correction in Fisher-KPP equations

In this paper, we explain in simple PDE terms a famous result of Bramson about the loga- rithmic delay of the position of the solutions u(t, x) of Fisher-KPP reaction-diffusion equations in R, with respect to the position of the travelling front with minimal speed. Our proof is based on the comparison of u to the solutions of linearized equations with Dirichlet boundary conditions at the position of the minimal front, with and without the logarithmic delay. Our analysis also yields the large-time convergence of the solutions u along their level sets to the profile of the minimal travelling front.

Solutions of Semilinear Elliptic Equations in with Conical&Shaped Level Sets

This article deals with the questions of the existence, of the uniqueness and of the qulitative properties of solutions of semiliner elliptic equations. Three types of conical conditions at infinity are successively considered. this defines three frameworks: the weak framework, the strong are based on different kinds of sliding methods and, following the ideas of Berestycki, Nirenberg and Vega, on comparison principles in cones

On the nonlocal Fisher–KPP equation: steady states, spreading speed and global bounds

We consider the Fisher–KPP (for Kolmogorov–Petrovsky–Piskunov) equation with a nonlocal interaction term. We establish a condition on the interaction that allows for existence of non-constant periodic solutions, and prove uniform upper bounds for the solutions of the Cauchy problem, as well as upper and lower bounds on the spreading rate of the solutions with compactly supported initial data.

The logarithmic delay of KPP fronts in a periodic medium

We consider solutions of the KPP-type equations with a periodically varying reaction rate, and compactly supported initial data. It has been shown by M. Bramson in the case of the constant reaction rate that the lag between the position of such solutions and that of the traveling waves grows as (3/2) log(t). We generalize this result to the periodic case