Nikolai Nadirashvili

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.

Travelling Fronts and Entire Solutions¶of the Fisher-KPP Equation in ℝN

Abstract: This paper is devoted to time-global solutions of the Fisher-KPP equation in ℝN: where f is a C2 concave function on [0,1] such that f(0)=f(1)=0 and f>0 on (0,1). It is well known that this equation admits a finite-dimensional manifold of planar travelling-fronts solutions. By considering the mixing of any density of travelling fronts, we prove the existence of an infinite-dimensional manifold of solutions. In particular, there are infinite-dimensional manifolds of (nonplanar) travelling fronts and radial solutions. Furthermore, up to an additional assumption, a given solution u can be represented in terms of such a mixing of travelling fronts.

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.

Parallel and circular flows for the two-dimensional Euler equations

We consider steady flows of ideal incompressible fluids in two-dimensional domains. These flows solve the Euler equations with tangential boundary conditions. If such a flow has no stagnation point in the domain or at infinity, in the sense that the infimum of its norm over the domain is positive, then it inherits the geometric properties of the domain, for some simple classes of domains. Namely, if the domain is a strip or a half-plane, then such a flow turns out to be parallel to the boundary of the domain. If the domain is the plane, the flow is then a parallel flow, that is, its trajectories are parallel lines. If the domain is an annulus, then the flow is circular, that is, the streamlines are concentric circles. The results are based on qualitative properties and classification results for some semilinear elliptic equations satisfied by the stream function.

Existence et propriétés qualitatives de solutions entières de l'équation de KPP

Resume Cette Note porte sur les solutions definies pour tout temps, i.e. les solutions entieres, de: u t = u xx + ƒ ( u ), 0 u ( x , t ) x ∈ ℝ, t ∈ ℝ, ou ƒ est du type KPP sur [0,1]. Cette equation admet un nombre infini de solutions du type ondes progressives, ainsi que des solutions du type u ( t ). Nous avons construit quatre autres varietes de solutions, la plus grande etant de dimension 5. De plus, les ondes progressives sont sur le bord de ces quatre varietes. Nous avons aussi traite la question de l’unicite pour une certaine classe de solutions.This Note deals with the solutions defined for all time (i.e. entire) of: ut = uxx + ƒ (u), 0 < u (x, t) < 1, x ∈ ℝ, t ∈ ℝ, where ƒ is a KPP type nonlinearity on [0,1]. This equation admits infinitely many travelling waves type solutions as well as solutions of the type u(t). We have build four other manifolds of solutions, the biggest one being five-dimensional. Furthermore, the travelling waves are on the boundary of these four manifolds. We also answer the question of the uniqueness for a certain class of solutions.