Power series solutions for the KPP equation

In this paper we solve the KPP equation by a non numerical method. To this end we find power series solutions where the coefficients are computed recursively. We also prove convergence of the series and illustrate the method by few examples.      

Entire solutions to the Monge-Ampère equation

We consider the Monge-Ampère equation det(D²u)=Ψ(x,u,Du) in Rn, n?3, where Ψ is a positive function in C²(Rn×R×Rn). We prove the existence of convex solutions, provided there exist a subsolution of the form and a superharmonic bounded positive function φ satisfying: .     

Entire solutions for a discrete diffusive equation

We study entire solutions of a discrete diffusive equation with bistable nonlinearity. It is well known that there are three different wavefronts connecting any two of those three equilibria, say, 0,a,1. We construct three different types of entire solutions. The first one is a solution which behaves as two opposite wavefronts (connecting 0 and 1) of the same positive speed approaching each other from both sides of the real line. The second one is a solution which behaves as two different wavefronts (connecting a and one of {0,1}) approaching each other from both sides of the real line and converging to the wavefront connecting 0 and 1. The third one is a solution which behaves as a wavefront connecting a and 0 and a wavefront connecting 0 and 1 approaching each other from both sides of the real line.     

New travelling wave solutions for the Fisher–KPP equation with general exponents

The Fisher–KPP equation with general nonlinear diffusion and arbitrary kinetic orders in the reaction terms is considered. The existence of oscillatory travelling wave solutions is proved for this model. Conditions for the existence of such solutions are provided.     

Entire positive solutions of the singular Emden-Fowler equation with nonlinear gradient term

Let p and q be locally Hölder functions in ?^N, p > 0 and q ≥ 0. We study the Emden-Fowler equation $-\triangle u+q(x)\mid \nabla u\mid^\alpha=p(x)u^{-\gamma}$ in ?^N, where α and γ are positive numbers. Our main result establishes that the above equation has a unique positive solutions decaying to zero at infinity. Our proof is elementary and it combines the maximum principle for elliptic equations with a theorem of Crandall, Rabinowitz and Tartar.     

On stationary distributions for the KPP equation with branching noise

This paper applies the method of Harris's convergence theorem for additive particle systems to a stochastic PDE that arises as the limit of long range contact processes. This is used to study the uniqueness of a translation invariant stationary distribution and its domain of attraction.

Résumé

On applique la méthode conduisant au théorème de convergence de Harris pour les systèmes de particules additifs à une EDP stochastique qui apparaît comme limite d'un processus de contact à longue portée. On étudie ainsi l'unicité de la mesure stationnaire invariante par translation et son domaine d'attraction.     

Entire solutions of the Euler-Poisson equations

All entire solutions of Euler-Poisson equations are presented.Published in Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 5, pp. 677–686, May, 2004.     

Entire solution of the degenerate Fisher equation

This paper deals with the solutions defined for all time of the degenerate Fisher equation. Some solutions are obtained by considering two traveling fronts with critical speed that come from both sides of the X-axis and mix. Unfortunately, the entire solutions which behave as two opposite wave fronts with non-critical speed approaching each other from both sides of the X-axis can not be obtained, because the essential difficulty originates from the algebraic decay rate of the fronts with non-critical speed.     

Entire solutions of integrodifference equations

This paper is concerned with the existence of nontrivial entire solutions of integrodifference equations. By constructing proper auxiliary integrodifference equations and functions, we present the existence of nontrivial entire solutions even if the birth function is not monotone, which are different from the well-studied travelling wave solutions. The convergence of entire solutions is also given.     

Entire solutions of the abstract cauchy problem

We introduce a family of operators that we will callentire C-groups, and apply them to the first and second order abstract Cauchy problem, for a large class of linear operators on a Banach space. This produces unique solutions, for all initial data in a large (often dense) set, eachof which extends to an entire function, with continuous dependence on the initial data. Applications include the backward heat equation and the Cauchy problem for the Laplace equation.     

Entire solutions of eiconal type equations

The paper characterizes entire solutions to partial differential equations for polynomials p in . Received: 14 August 2006     

Fisher–KPP equation with advection on the half‐line

We consider the Fisher–KPP equation with advection: u_t=u_xx?βu_x+f(u) on the half‐line x∈(0,∞), with no‐flux boundary condition u_x?βu = 0 at x = 0. We study the influence of the advection coefficient ?β on the long time behavior of the solutions. We show that for any compactly supported, nonnegative initial data, (i) when β∈(0,c₀), the solution converges locally uniformly to a strictly increasing positive stationary solution, (ii) when β∈[c₀,∞), the solution converges locally uniformly to 0, here c₀ is the minimal speed of the traveling waves of the classical Fisher–KPP equation. Moreover, (i) when β > 0, the asymptotic positions of the level sets on the right side of the solution are (β + c₀)t + o(t), and (ii) when β≥c₀, the asymptotic positions of the level sets on the left side are (β ? c₀)t + o(t). Copyright © 2016 John Wiley & Sons, Ltd.     

Entire solutions of quasilinear elliptic equations

We study entire solutions of non-homogeneous quasilinear elliptic equations, with Eqs. (1) and (2) below being typical. A particular special case of interest is the following: Let u be an entire distribution solution of the equation Δpu=|u|^q−1u, where p>1. If q>p−1 then u≡0. On the other hand, if 0<q<p−1 and u(x)=o(|x|^{p/(p−q−1)}) as |x|→∞, then again u≡0. If q=p−1 then u≡0 for all solutions with at most algebraic growth at infinity.