Global Solutions in the Species Competitive Chemotaxis System with Inequal Diffusion Rates

Huaihuo Cao

doi:10.1155/2016/5015246

Abstract

This paper is devoted to studying the two-species competitive chemotaxis system with signal-dependent chemotactic sensitivities and inequal diffusion rates u1t=Δu1-∇·u1χ1v∇v+μ1u11-u1-a1u2, x∈Ω, t>0, u2t=Δu2-∇·u2χ2v∇v+μ2u21-a2u1-u2, x∈Ω, t>0, vt=τΔv-γv+u1+u2, x∈Ω, t>0, under homogeneous Neumann boundary conditions in a bounded and regular domain Ω⊂Rn (n≥1). If the nonnegative initial date (u10,u20,v0)∈(C1(Ω¯))3 and v0∈(v_,v¯) where the constants v¯>v_≥0, the system possesses a unique global solution that is uniformly bounded under some suitable assumptions on the chemotaxis sensitivity functions χ1(v), χ2(v) and linear chemical production function -γv+u1+u2.

Highlights

From a biological point of view, when χi(V) > 0, populations exhibit a tendency to move towards higher signal concentrations, while the choice χi(V) < 0 leads to a model for chemorepulsion, where populations prefer to move away from the chemical in question [2]
Denote h(u1, u2, V) = −γV + u1 + u2 representing the balance between the production of the chemical substance by the populations themselves and its natural degradation
The classical chemotaxis model was first introduced by Keller and Segel using a mathematical model of two parabolic

Summary

Introduction

By extending the method in [5] (see [9, 10]), the first step is to estimate some associated weighted functions which depend on signal density, and the second step is to obtain L∞-bounds of solutions from Lp-bounds using the variation-of-constants representation and a series of standard semigroup arguments (see [2, 5, 11, 12]) They proved that, if the nonnegative initial date (u(⋅, 0), V(⋅, 0)) ∈ (C0(Ω)) and w(⋅, 0) ∈ W1,r(Ω) for some r > n, the system possesses a unique global solution that is uniformly bounded under some appropriate conditions on the coefficients μ1, μ2 and the chemotaxis sensitivity functions χ1(w), χ2(w). Under some appropriate conditions, we prove that the solutions are uniformly bounded in time using an iterative method

Preliminaries and Main Results

Local Existence of Solutions