scholarly journals Positive Solutions of a System Arising from Angiogenesis

2008 ◽  
Vol 8 (4) ◽  
Author(s):  
Manuel Delgado ◽  
Antonio Suárez
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Bifurcation Theory ◽  
Time Dependent ◽  
System Of Equations ◽  
Time Dependent Problem ◽  
Loss Of Regularity

AbstractWe study a system of equations arising from angiogenesis which contains a non- regular term that vanishes below a certain threshold. We are forced to modify the usual methods of bifurcation theory because of this loss of regularity. Nevertheless, we obtain results on the existence, uniqueness and permanence of a positive solution for the time-dependent problem; and the existence and uniqueness of a positive solution for the stationary one.

scholarly journals Existence and Exact Asymptotic Behavior of Positive Solutions for a Fractional Boundary Value Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Habib Mâagli ◽  
Noureddine Mhadhebi ◽  
Noureddine Zeddini
Keyword(s):  
Asymptotic Behavior ◽  
Continuous Function ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Exact Asymptotic Behavior ◽  
Nonnegative Continuous Function

We establish the existence and uniqueness of a positive solution for the fractional boundary value problem , with the condition , where , and is a nonnegative continuous function on that may be singular at or .

scholarly journals Existence and Global Behavior of Positive Solutions for Some Fourth-Order Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Ramzi S. Alsaedi
Keyword(s):  
Continuous Function ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Positive Solutions ◽  
Fourth Order ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Global Behavior ◽  
Nonnegative Continuous Function

We establish the existence and uniqueness of a positive solution to the following fourth-order value problem:u(4)(x)=a(x)uσ(x),x∈(0,1)with the boundary conditionsu(0)=u(1)=u'(0)=u'(1)=0, whereσ∈(-1,1)andais a nonnegative continuous function on (0, 1) that may be singular atx=0orx=1. We also give the global behavior of such a solution.

GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR FOR A SEMICONDUCTOR DRIFT-DIFFUSION-POISSON MODEL

2008 ◽  
Vol 18 (03) ◽  
pp. 443-487 ◽  
Author(s):  
HAO WU ◽  
PETER A. MARKOWICH ◽  
SONGMU ZHENG
Keyword(s):  
Global Existence ◽  
Existence And Uniqueness ◽  
Poisson Model ◽  
Stationary Problem ◽  
Time Dependent ◽  
Unique Equilibrium ◽  
Poisson System ◽  
Drift Diffusion ◽  
Global Existence And Uniqueness ◽  
Time Dependent Problem

In this paper a time-dependent as well as a stationary drift-diffusion-Poisson system for semiconductors are studied. Global existence and uniqueness of weak solution of the time-dependent problem are proven and we also prove the existence and uniqueness of the steady state. It is shown that as time tends to infinity, the solution of the time-dependent problem will converge to a unique equilibrium. Due to the presence of recombination-generation rate R in our drift-diffusion-Poisson model, the work of this paper in some sense extends the results in the previous literature (on both time-dependent problem and stationary problem).

Positive solutions for a multi-order fractional nonlinear system with variable delays

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6155-6166
Author(s):  
Asma Bouaziz ◽  
Mohamed Kerker
Keyword(s):  
Fixed Point ◽  
Nonlinear System ◽  
Fractional Derivative ◽  
Positive Solution ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Variable Delays

This paper is concerned with the existence and uniqueness of the positive solution for a multiorder fractional nonlinear system with variable delays. The fractional derivative will be in the Caputo sense. The obtained results are based on some fixed point theorems.

Existence and uniqueness of positive solutions of nonlinear Schrödinger systems

2015 ◽  
Vol 145 (2) ◽  
pp. 365-390 ◽  
Author(s):  
Haidong Liu ◽  
Zhaoli Liu ◽  
Jinyong Chang
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Ground State ◽  
Existence And Uniqueness ◽  
Unique Positive Solution ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger Systems ◽  
Schrödinger Systems ◽  
Least Energy ◽  
Schrödinger System

We prove that the Schrödinger systemwhere n = 1, 2, 3, N ≥ 2, λ1 = λ2 = … = λN = 1, βij = βji > 0 for i, j = 1, …, N, has a unique positive solution up to translation if the βij (i ≠ j) are comparatively large with respect to the βjj. The same conclusion holds if n = 1 and if the βij (i ≠ j) are comparatively small with respect to the βjj. Moreover, this solution is a ground state in the sense that it has the least energy among all non-zero solutions provided that the βij (i ≠ j) are comparatively large with respect to the βjj, and it has the least energy among all non-trivial solutions provided that n = 1 and the βij (i ≠ j) are comparatively small with respect to the βjj. In particular, these conclusions hold if βij = (i ≠ j) for some β and either β > max{β11, β22, …, βNN} or n = 1 and 0 < β < min{β11, β22, …, βNN}.

On the Structure of Positive Solutions for the Shigesada–Kawasaki–Teramoto Model with Large Interspecific Competition Rate

2020 ◽  
Vol 30 (01) ◽  
pp. 2050001
Author(s):  
Yukio Kan-on
Keyword(s):  
Positive Solution ◽  
Interspecific Competition ◽  
Positive Solutions ◽  
Comparison Principle ◽  
Nonlinear Diffusion ◽  
Bifurcation Theory ◽  
Diffusion System ◽  
Bifurcation Structure ◽  
Diffusion Term ◽  
Competition Diffusion

In this paper, we treat the competition-diffusion system with nonlinear diffusion term, which was proposed by Shigesada et al. [1979], and discuss the bifurcation structure of positive solution for the system when the interspecific competition rate is sufficiently large. To do this, we derive two kinds of limiting systems as the interspecific competition rate tends to infinity, and study the bifurcation structure of positive solution for each limiting system by employing the comparison principle and the bifurcation theory.

Global unique existence of a positive solution for a system of equations in electrochemistry

1998 ◽  
Vol 128 (3) ◽  
pp. 507-524
Author(s):  
Dongho Chae ◽  
Oleg Yu. Imanuvilov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Positive Solution ◽  
Existence And Uniqueness ◽  
Smooth Boundary ◽  
Nonlinear Partial Differential Equations ◽  
System Of Equations ◽  
Unique Existence ◽  
Positive Integers ◽  
Time Existence

In this paper we prove global-in-time existence and uniqueness of a positive solution for the system of nonlinear partial differential equations arising from an electrochemistry model. The powers of nonlinearity are allowed to be arbitrary positive integers, and our domain is any bounded subdomain of ℝ2 with a smooth boundary.

scholarly journals Existence and uniqueness of positive solutions for nonlinear Caputo-Hadamard fractional differential equations

2021 ◽  
Vol 40 (1) ◽  
pp. 139-152
Author(s):  
Abdelouaheb Ardjouni
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Positive Solution ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Upper And Lower Solutions ◽  
Hadamard Fractional Differential Equations ◽  
Fractional Differential

We prove the existence and uniqueness of a positive solution of nonlinear Caputo-Hadamard fractional differential equations. In the process we employ the Schauder and Banach fixed point theorems and the method of upper and lower solutions to show the existence and uniqueness of a positive solution. Finally, an example is given to illustrate our results.

Estimate, existence and nonexistence of positive solutions of Hardy–Hénon equations

2021 ◽  
pp. 1-24
Author(s):  
Xiyou Cheng ◽  
Lei Wei ◽  
Yimin Zhang
Keyword(s):  
Boundary Condition ◽  
Dirichlet Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Dirichlet Boundary Condition ◽  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
Existence And Nonexistence ◽  
Hénon Equation ◽  
Image Position

We consider the boundary Hardy–Hénon equation \[ -\Delta u=(1-|x|)^{\alpha} u^{p},\ \ x\in B_1(0), \] where $B_1(0)\subset \mathbb {R}^{N}$   $(N\geq 3)$ is a ball of radial $1$ centred at $0$ , $p>0$ and $\alpha \in \mathbb {R}$ . We are concerned with the estimate, existence and nonexistence of positive solutions of the equation, in particular, the equation with Dirichlet boundary condition. For the case $0< p<({N+2})/({N-2})$ , we establish the estimate of positive solutions. When $\alpha \leq -2$ and $p>1$ , we give some conclusions with respect to nonexistence. When $\alpha >-2$ and $1< p<({N+2})/({N-2})$ , we obtain the existence of positive solution for the corresponding Dirichlet problem. When $0< p\leq 1$ and $\alpha \leq -2$ , we show the nonexistence of positive solutions. When $0< p<1$ , $\alpha >-2$ , we give some results with respect to existence and uniqueness of positive solutions.

On Positive Solutions for Some Nonlinear Semipositone Elliptic Boundary Value

2006 ◽  
Vol 11 (4) ◽  
pp. 323-329 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Boundary Value Problems ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Laplacian Operator ◽  
Smooth Bounded Domain ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

This study concerns the existence of positive solutions to classes of boundary value problems of the form−∆u = g(x,u), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,where ∆ denote the Laplacian operator, Ω is a smooth bounded domain in RN (N ≥ 2) with ∂Ω of class C2, and connected, and g(x, 0) < 0 for some x ∈ Ω (semipositone problems). By using the method of sub-super solutions we prove the existence of positive solution to special types of g(x,u).

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