scholarly journals A one-dimensional symmetry result for entire solutions to the Fisher-KPP equation

2020 ◽  
pp. 1
Author(s):  
Christos Sourdis
Keyword(s):  
Entire Solutions ◽  
Kpp Equation ◽  
One Dimensional ◽  
Symmetry Result

Uniqueness and Symmetry for the Mean Field Equation on Arbitrary Flat Tori

2020 ◽  
Author(s):  
Guangze Gu ◽  
Changfeng Gui ◽  
Yeyao Hu ◽  
Qinfeng Li
Keyword(s):  
Field Equation ◽  
Level Sets ◽  
Mean Field ◽  
Flat Torus ◽  
One Dimensional ◽  
Uniqueness Of The Solution ◽  
Flat Tori ◽  
The Mean ◽  
Symmetry Result ◽  
Mean Field Equation

Abstract We study the following mean field equation on a flat torus $T:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau )$: $$\begin{equation*} \varDelta u + \rho \left(\frac{e^{u}}{\int_{T}e^u}-\frac{1}{|T|}\right)=0, \end{equation*}$$where $ \tau \in \mathbb{C}, \mbox{Im}\ \tau>0$, and $|T|$ denotes the total area of the torus. We first prove that the solutions are evenly symmetric about any critical point of $u$ provided that $\rho \leq 8\pi $. Based on this crucial symmetry result, we are able to establish further the uniqueness of the solution if $\rho \leq \min{\{8\pi ,\lambda _1(T)|T|\}}$. Furthermore, we also classify all one-dimensional solutions by showing that the level sets must be closed geodesics.

scholarly journals Entire solutions of the Fisher–KPP equation on the half line

2019 ◽  
Vol 31 (3) ◽  
pp. 407-422 ◽  
Author(s):  
BENDONG LOU ◽  
JUNFAN LU ◽  
YOSHIHISA MORITA
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Traveling Wave ◽  
Wave Solution ◽  
Dirichlet Boundary ◽  
Entire Solution ◽  
Traveling Wave Solution ◽  
Half Line ◽  
Entire Solutions ◽  
Kpp Equation

In this paper, we study the entire solutions of the Fisher–KPP (Kolmogorov–Petrovsky–Piskunov) equation ut = uxx + f(u) on the half line [0, ∞) with Dirichlet boundary condition at x = 0. (1) For any $c \ge 2\sqrt {f'(0)} $, we show the existence of an entire solution ${{\cal U}^c}(x,t)$ which connects the traveling wave solution φc(x + ct) at t = −∞ and the unique positive stationary solution V(x) at t = +∞; (2) We also construct an entire solution ${{\cal U}}(x,t)$ which connects the solution of ηt = f(η) at t = −∞ and V(x) at t = +∞.

scholarly journals Existence and nonexistence of entire solutions to the logistic differential equation

2003 ◽  
Vol 2003 (17) ◽  
pp. 995-1003 ◽  
Author(s):  
Marius Ghergu ◽  
Vicentiu Radulescu
Keyword(s):  
Blow Up ◽  
Entire Solutions ◽  
One Dimensional ◽  
Nonexistence Result ◽  
Nonnegative Solutions ◽  
Existence And Nonexistence ◽  
Logistic Problem ◽  
Whole Space ◽  
The Mean ◽  
The One

We consider the one-dimensional logistic problem(rαA(|u′|)u′)′=rαp(r)f(u)on(0,∞),u(0)>0,u′(0)=0, whereαis a positive constant andAis a continuous function such that the mappingtA(|t|)is increasing on(0,∞). The framework includes the case wherefandpare continuous and positive on(0,∞),f(0)=0, andfis nondecreasing. Our first purpose is to establish a general nonexistence result for this problem. Then we consider the case of solutions that blow up at infinity and we prove several existence and nonexistence results depending on the growth ofpandA. As a consequence, we deduce that the mean curvature inequality problem on the whole space does not have nonnegative solutions, excepting the trivial one.

scholarly journals A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane

2017 ◽  
Vol 34 (2) ◽  
pp. 469-482 ◽  
Author(s):  
François Hamel ◽  
Xavier Ros-Oton ◽  
Yannick Sire ◽  
Enrico Valdinoci
Keyword(s):  
One Dimensional ◽  
Semilinear Equations ◽  
Symmetry Result

Entire solutions to advective Fisher-KPP equation on the half line

2021 ◽  
Vol 305 ◽  
pp. 103-120
Author(s):  
Bendong Lou ◽  
Jinzhe Suo ◽  
Kaiyuan Tan
Keyword(s):  
Half Line ◽  
Entire Solutions ◽  
Kpp Equation

scholarly journals An application of the Maslov complex germ method to the one-dimensional nonlocal Fisher–KPP equation

2018 ◽  
Vol 15 (06) ◽  
pp. 1850102 ◽  
Author(s):  
A. V. Shapovalov ◽  
A. Yu. Trifonov
Keyword(s):  
Nonlinear Equation ◽  
Linear Equation ◽  
Algebraic Equations ◽  
Semiclassical Asymptotics ◽  
Complex Germ ◽  
Kpp Equation ◽  
One Dimensional ◽  
Maslov Complex Germ ◽  
The One ◽  
Complex Germ Method

A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the one-dimensional nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov (Fisher–KPP) equation under the supposition of weak diffusion. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. The solutions of the linear problem are found with the use of symmetry operators. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval. We construct asymptotics which are different from the semiclassical ones and can describe evolution of the solutions of the Fisher–KPP equation at large times. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as an example of a space structure.

Entire solutions in the Fisher-KPP equation with nonlocal dispersal

2010 ◽  
Vol 11 (4) ◽  
pp. 2302-2313 ◽  
Author(s):  
Wan-Tong Li ◽  
Yu-Juan Sun ◽  
Zhi-Cheng Wang
Keyword(s):  
Entire Solutions ◽  
Nonlocal Dispersal ◽  
Kpp Equation

Entire solutions of time periodic Fisher–KPP equation on the half line

2020 ◽  
Vol 200 ◽  
pp. 112005
Author(s):  
Jingjing Cai ◽  
Li Xu ◽  
Yuan Chai
Keyword(s):  
Half Line ◽  
Entire Solutions ◽  
Kpp Equation ◽  
Time Periodic
Sign in / Sign up

Export Citation Format

Share Document