ENTIRE SOLUTIONS FOR LOTKA–VOLTERRA COMPETITION-DIFFUSION MODEL

2013 ◽  
Vol 06 (04) ◽  
pp. 1350020 ◽  
Author(s):  
XIAOHUAN WANG ◽  
GUANGYING LV
Keyword(s):  
Classical Solution ◽  
Diffusion Model ◽  
Entire Solution ◽  
Entire Solutions ◽  
Wave Fronts ◽  
Space And Time ◽  
Time Variables ◽  
Competition Diffusion

This paper is concerned with the existence of entire solutions of Lotka–Volterra competition-diffusion model. Using the comparing argument and sub-super solutions method, we obtain the existence of entire solutions which behave as two wave fronts coming from the both sides of x-axis, where an entire solution is meant by a classical solution defined for all space and time variables.

Entire solutions for some reaction–diffusion systems

2015 ◽  
Vol 08 (04) ◽  
pp. 1550052 ◽  
Author(s):  
Guangying Lv ◽  
Dang Luo
Keyword(s):  
Classical Solution ◽  
General Model ◽  
Reaction Diffusion ◽  
Entire Solution ◽  
Reaction Model ◽  
Entire Solutions ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Time Variables ◽  
Belousov Zhabotinskii Reaction

This paper is concerned with the existence of entire solutions of some reaction–diffusion systems. We first consider Belousov–Zhabotinskii reaction model. Then we study a general model. Using the comparing argument and sub-super-solutions method, we obtain the existence of entire solutions which behave as two wavefronts coming from the both sides of x-axis, where an entire solution is meant by a classical solution defined for all space and time variables. At last, we give some examples to explain our results for the general models.

ENTIRE SOLUTIONS OF A CURVATURE FLOW IN AN UNDULATING CYLINDER

2018 ◽  
Vol 99 (1) ◽  
pp. 137-147
Author(s):  
LIXIA YUAN ◽  
BENDONG LOU
Keyword(s):  
Curvature Flow ◽  
Entire Solution ◽  
Travelling Wave ◽  
Normal Velocity ◽  
Two Dimensional ◽  
Entire Solutions ◽  
Smooth Functions ◽  
Periodic Traveling Waves ◽  
Periodic Travelling Wave ◽  
Positive Functions

We consider a curvature flow $V=\unicode[STIX]{x1D705}+A$ in a two-dimensional undulating cylinder $\unicode[STIX]{x1D6FA}$ described by $\unicode[STIX]{x1D6FA}:=\{(x,y)\in \mathbb{R}^{2}\mid -g_{1}(y)<x<g_{2}(y),y\in \mathbb{R}\}$, where $V$ is the normal velocity of a moving curve contacting the boundaries of $\unicode[STIX]{x1D6FA}$ perpendicularly, $\unicode[STIX]{x1D705}$ is its curvature, $A>0$ is a constant and $g_{1}(y),g_{2}(y)$ are positive smooth functions. If $g_{1}$ and $g_{2}$ are periodic functions and there are no stationary curves, Matano et al. [‘Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit’, Netw. Heterog. Media1 (2006), 537–568] proved the existence of a periodic travelling wave. We consider the case where $g_{1},g_{2}$ are general nonperiodic positive functions and the problem has some stationary curves. For each stationary curve $\unicode[STIX]{x1D6E4}$ unstable from above/below, we construct an entire solution growing out of it, that is, a solution curve $\unicode[STIX]{x1D6E4}_{t}$ which increases/decreases monotonically, converging to $\unicode[STIX]{x1D6E4}$ as $t\rightarrow -\infty$ and converging to another stationary curve or to $+\infty /-\infty$ as $t\rightarrow \infty$.

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 A Class of functional equations which have entire solutions

1988 ◽  
Vol 38 (3) ◽  
pp. 351-356 ◽  
Author(s):  
Peter L. Walker
Keyword(s):  
Functional Equation ◽  
Entire Function ◽  
Wide Class ◽  
Functional Equations ◽  
Entire Functions ◽  
Inverse Function ◽  
Entire Solution ◽  
Entire Solutions ◽  
Image Position

We consider the Abelian functional equationwhere φ is a given entire function and g is to be found. The inverse function f = g−1 (if one exists) must satisfyWe show that for a wide class of entire functions, which includes φ(z) = ez − 1, the latter equation has a non-constant entire solution.

THE $W^{(2)}_3$ CONFORMAL ALGEBRA AND THE BOUSSINESQ HIERARCHY

1991 ◽  
Vol 06 (26) ◽  
pp. 2397-2409 ◽  
Author(s):  
P. MATHIEU ◽  
W. OEVEL
Keyword(s):  
Nonlinear Equations ◽  
Poisson Structure ◽  
Boussinesq Equation ◽  
Free Field ◽  
Conformal Algebra ◽  
Integrable Hierarchy ◽  
Classical Formula ◽  
Field Representation ◽  
Space And Time ◽  
Time Variables

The classical [Formula: see text] algebra Polyakov is shown to be equivalent to the second Poisson structure of a new integrable hierarchy of nonlinear equations. The hierarchy is related to the Boussinesq hierarchy by interhcanging the roles of the space and time variables x and t in the Boussinesq equation. From this relation the Miura map, relating the new hierarchy to its modified version, can be derived systematically. It is found to be equivalent to the known free field representation of the [Formula: see text] algebra.

Deterministic epidemic waves

1976 ◽  
Vol 80 (2) ◽  
pp. 315-330 ◽  
Author(s):  
C. Atkinson ◽  
G. E. H. Reuter
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Deterministic Model ◽  
Removal Rate ◽  
Uniform Density ◽  
Space And Time ◽  
Time Variables ◽  
Image Position

In the well-known deterministic model for the spread of an epidemic, one considers a population of uniform density along a line and divides the population into three classes: susceptible but uninfected, infected and infectious, infected but removed. If we denote space and time variables by s, t and let x(s, t), y(s, t), z(s, t) be the proportions of the population at (s, t) in these three classes, then x + y + z = 1 and we suppose thatHere Ῡ(s, t) denotes a space average ∫ y(s + σ) p(σ) dσ, where p is a probability density function; b is the removal rate; the scale of t has been adjusted to remove a constant that would otherwise occur in (1).

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