Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

This paper is devoted to the study of pulsating travelling fronts for reaction-diffusion-advection equations in a general class of periodic domains with underlying periodic diffusion and velocity fields. Such fronts move in some arbitrarily given direction with an unknown effective speed. The notion of pulsating travelling fronts generalizes that of travelling fronts for planar or shear flows. Various existence, uniqueness and monotonicity results are proved for two classes of reaction terms. Firstly, for a combustion-type nonlinearity, it is proved that the pulsating travelling front exists and that its speed is unique. Moreover, the front is increasing with respect to the time variable and unique up to translation in time. We also consider one class of monostable nonlinearity which arises either in combustion or biological models. Then, the set of possible speeds is a semi-infinite interval, closed and bounded from below. For each possible speed, there exists a pulsating travelling front which is increasing in time. This result extends the classical Kolmogorov-Petrovsky-Piskunov case. Our study covers in particular the case of flows in all of space with periodic advections such as periodic shear flows or a periodic array of vortical cells. These results are also obtained for cylinders with oscillating boundaries or domains with a periodic array of holes. © 2002 Wiley Periodicals, Inc.

Front propagation in periodic excitable media

In this paper we investigate a diffusive logistic model with a free boundary in one space dimension. We aim to use the dynamics of such a problem to describe the spreading of a new or invasive species, with the free boundary representing the expanding front. We prove a spreading-vanishing dichotomy for this model, namely the species either successfully spreads to all the new environment and stabilizes at a positive equilibrium state, or it fails to establish and dies out in the long run. Sharp criteria for spreading and vanishing are given. Moreover, we show that when spreading occurs, for large time, the expanding front moves at a constant speed. This spreading speed is uniquely determined by an elliptic problem induced from the original model.

Spreading-Vanishing Dichotomy in the Diffusive Logistic Model with a Free Boundary

This volume deals with the regularity theory for elliptic systems. We may find the origin of such a theory in two of the problems posed by David Hilbert in his celebrated lecture delivered on the occasion of the International Congress of Mathematicians in 1900 in Paris: 19th problem: are the solutions to regular problems in the Calculus of Variations always necessarily analytic? - 20th problem: does any variational problem have a solution, provided that certain assumptions regarding the given boundary conditions are satisfied, and provided that the notion of a solution is suitably extended? During the last century these two problems have generated a great deal of work, usually referred to as is in regularity theory, which makes this topic quite relevant in many fields and still very active for research. However, the purpose of this volume, addressed mainly to students, is much more limited. We aim to illustrate only some of the basic ideas and techniques introduced in this context, confining ourselves to important but simple situations and refraining from completeness. In fact some relevant topics are omitted. Topics covered include: harmonic functions, direct methods, Hilbert space methods and Sobolev spaces, energy estimates, Schauder and Lp-theory both with and without potential theory, including the Calderon Zygmund theorem, Harnack's and De Giorgi-Moser-Nash theorems in the scalar case and partial regularity theorems in the vector valued case; finally, harmonic maps and minimal graphs in codimension 1 and greater than 1.

An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs

This article gives an overview of the developments in controlled diffusion processes, emphasizing key results regarding existence of optimal controls and their characterization via dynamic programming for a variety of cost criteria and structural assumptions. Stochastic maximum principle and control under partial observations (equivalently, control of nonlinear filters) are also discussed. Several other related topics are briefly sketched.

Controlled diffusion processes

This paper is devoted to time-global solutions of the Fisher-KPP equation in ℝ
 N
 :
 where f is a C

 2 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.

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

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.

Entire solutions of the KPP equation

This paper is concerned with the asymptotic behaviour of the principal eigenvalue of some linear elliptic equations in the limit of high first-order coefficients. Roughly speaking, one of the main results says that the principal eigenvalue, with Dirichlet boundary conditions, is bounded as the amplitude of the coefficients of the first-order derivatives goes to infinity if and only if the associated dynamical system has a first integral, and the limiting eigenvalue is then determined through the minimization of the Dirichlet functional over all first integrals. A parabolic version of these results, as well as other results for more general equations, are given. Some of the main consequences concern the influence of high advection or drift on the speed of propagation of pulsating travelling fronts.

/pdf/elliptic-eigenvalue-problems-with-large-drift-and-4teqs88lvz.pdf

Elliptic Eigenvalue Problems with Large Drift and Applications to Nonlinear Propagation Phenomena

Rayleigh's conjecture on the principal frequency of the clamped plate

Consider a convex planar domain with two axes of symmetry. We show that the maximum and minimum of a Neumann eigenfunction with lowest nonzero eigenvalue occur at points on the boundary only. We deduce J. Rauch's "hot spots" conjecture in the following form. If the initial temperature distribution is not orthogonal to the first nonzero eigenspace, then the point at which the temperature achieves its maximum tends to the boundary. In fact the maximum point reaches the boundary in finite time if the boundary has positive curvature. Results of this type have already been proved by Bafiuelos and Burdzy [BB] using the heat equation and probabilistic methods to deform initial conditions to eigenfunctions. We introduce here a new technique based on deformation of the domain. An advantage of our method is that it works even in the case of multiple eigenvalues. On the way toward our results, we prove monotonicity properties for Neumann eigenfunctions for symmetric domains that need not be convex and deduce a sharp comparison of eigenvalues with the Dirichlet problem of independent interest.

/pdf/the-hot-spots-conjecture-for-domains-with-two-axes-of-4hz5i7u6yz.pdf

Nikolai Nadirashvili

Papers

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

Entire solutions of the KPP equation

Elliptic Eigenvalue Problems with Large Drift and Applications to Nonlinear Propagation Phenomena

Rayleigh's conjecture on the principal frequency of the clamped plate

The “hot spots” conjecture for domains with two axes of symmetry