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Large densities in a competitive two-species chemotaxis system in the non-symmetric case

Shohei Kohatsu, Johannes Lankeit

Abstract

This paper deals with the two-species chemotaxis system with Lotka-Volterra competitive kinetics, \begin{align*} \begin{cases} u_t = d_1 Δu - χ_1 \nabla \cdot (u \nabla w) + μ_1 u (1 - u - a_1 v), & x\inΩ,\ t>0,\\ v_t = d_2 Δv - χ_2 \nabla \cdot (v \nabla w) + μ_2 v (1 - a_2 u - v), & x\inΩ,\ t>0,\\ 0 = d_3 Δw + αu + βv - γw, & x\inΩ,\ t>0, \end{cases} \end{align*} under homogeneous Neumann boundary conditions and suitable initial conditions, where $Ω\subset \mathbb{R}^n$ $(n \in \mathbb{N})$ is a bounded domain with smooth boundary, $d_1, d_2, d_3, χ_1, χ_2, μ_1, μ_2 > 0$, $a_1, a_2 \ge 0$ and $α, β, γ> 0$. Under largeness conditions on $χ_1$ and $χ_2$, we show that for suitably regular initial data, any thresholds of the population density can be surpassed, which extends the previous results to the non-symmetric case. The paper contains a well-posedness result for the hyperbolic-elliptic limit system with $d_1=d_2=0$.

Large densities in a competitive two-species chemotaxis system in the non-symmetric case

Abstract

This paper deals with the two-species chemotaxis system with Lotka-Volterra competitive kinetics, \begin{align*} \begin{cases} u_t = d_1 Δu - χ_1 \nabla \cdot (u \nabla w) + μ_1 u (1 - u - a_1 v), & x\inΩ,\ t>0,\\ v_t = d_2 Δv - χ_2 \nabla \cdot (v \nabla w) + μ_2 v (1 - a_2 u - v), & x\inΩ,\ t>0,\\ 0 = d_3 Δw + αu + βv - γw, & x\inΩ,\ t>0, \end{cases} \end{align*} under homogeneous Neumann boundary conditions and suitable initial conditions, where is a bounded domain with smooth boundary, , and . Under largeness conditions on and , we show that for suitably regular initial data, any thresholds of the population density can be surpassed, which extends the previous results to the non-symmetric case. The paper contains a well-posedness result for the hyperbolic-elliptic limit system with .
Paper Structure (14 sections, 22 theorems, 143 equations)

This paper contains 14 sections, 22 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Parabolic--parabolic--elliptic case
  4. Some basic results of solutions to (1.1)
  5. Controlling the W1q-norm of a solution by its L infty-norm
  6. Solutions to (1.1) exist long enough
  7. Boundedness of ut and vt
  8. Hyperbolic--hyperbolic--elliptic case
  9. Definition and uniqueness of the strong W1q-solution
  10. Local existence and approximation
  11. Existence on maximal time intervals
  12. Proof of the main theorem
  13. Blow-up of solutions to the HHE system
  14. Proof of main theorem

Key Result

Theorem 1.1

Let $n\in ℕ$, and let $\Omega \subset \mathbb{R}^n$ be a bounded domain with smooth boundary. Let $d_3, \chi_1, \chi_2, \mu_1, \mu_2 > 0$, $a_1, a_2 \ge 0$ and $\alpha, \beta, \gamma > 0$. Then for all $p > 1$ which satisfy one of the following there is $C(p) > 0$ with the following property : Whenever $q > \max\{n, 2\}$ and $u_0, v_0 \in W^{1,q}(\Omega)$ are nonnegative such that there is $T_{\

Theorems & Definitions (51)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.2
  • Remark 1.8
  • ...and 41 more