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Nonlocal Diffusion Elliptic System Modelling The Behaviour Of a Bacteria And a Living Nutrient

M. A. V. Costa, Y. B. C. Carranza, C. Morales-Rodrigo, A. Suarez

TL;DR

Bifurcation methods and the Implicit Function Theorem are employed to obtain the existence and uniqueness of positive solution for a class of non-local elliptic system.

Abstract

In this paper, we discuss the existence and uniqueness of coexistence states for a class of non-local elliptic system. This problem models the behaviour of a bacteria and a living nutrient, whose diffusion depends on the population of the bacteria in a non-local and nonlinear way. Mainly, we employ bifurcation methods and the Implicit Function Theorem to obtain the existence and uniqueness of positive solution.

Nonlocal Diffusion Elliptic System Modelling The Behaviour Of a Bacteria And a Living Nutrient

TL;DR

Bifurcation methods and the Implicit Function Theorem are employed to obtain the existence and uniqueness of positive solution for a class of non-local elliptic system.

Abstract

In this paper, we discuss the existence and uniqueness of coexistence states for a class of non-local elliptic system. This problem models the behaviour of a bacteria and a living nutrient, whose diffusion depends on the population of the bacteria in a non-local and nonlinear way. Mainly, we employ bifurcation methods and the Implicit Function Theorem to obtain the existence and uniqueness of positive solution.
Paper Structure (7 sections, 8 theorems, 126 equations, 1 figure)

This paper contains 7 sections, 8 theorems, 126 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and Notations
  3. A Priori Bounds and Non-Existence of Coexistence States
  4. Local Bifurcation Analysis
  5. Global Bifurcation Analysis
  6. Uniqueness result
  7. Case $a(0)=0$

Key Result

Theorem 1.1

Assume $a(0)>0$. From the trivial solution $(u,v)=(0,0)$ emanates an unbounded continuum $\mathcal{C}\subset \mathbb{R}\times C_0^1(\overline\Omega)\times C_0^1(\overline\Omega)$ of positive solutions of (P) at Moreover, if $b\leq 0$ or $b>0$ or $\rho$ is small or $\sigma$, or $b>0$ and $a$ verifies (2), then for some $\lambda_0\leq 0$. where $\hbox{Proj}_\mathbb{R}(\lambda,u,v)=\lambda$ for $(\

Figures (1)

  • Figure 1: Bifurcation diagrams of (P)

Theorems & Definitions (16)

  • Theorem 1.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 3.1
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • ...and 6 more