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Stability of constant steady states of an attraction-repulsion chemotaxis system

Hiroshi Wakui, Tetsuya Yamada

TL;DR

This paper studies the stability of constant steady states for the attraction-repulsion chemotaxis system on $\mathbb{R}^n$. By solving the elliptic equations for $\psi_1$ and $\psi_2$, it reduces the system to a nonlocal scalar equation for $u$ with kernel $K=\beta_1 B_{\lambda_1}-\beta_2 B_{\lambda_2}$ and analyzes the linearized operator around $u\equiv A$ via its symbol $h_A(\xi)$. The stability of the constant state is governed by $M(A)=\max_{\xi} h_A(\xi)$, with a stable regime $0<A\le A_*$ (where $M(A)\le 0$) and an unstable regime $A>A_*$; a further threshold $A^*$ marks Lyapunov instability for large $A$. The paper proves local and global well-posedness for small perturbations in uniformly local spaces, derives $L^p$-$L^q$ semigroup estimates, and establishes instability in the large-$A$ regime, thereby clarifying when attraction-repulsion steady states are stable. These results extend the understanding of constant steady-state stability beyond purely attracting or repelling limiting cases.

Abstract

The Cauchy problem for the attraction-repulsion chemotaxis system in the whole $n$-dimensional space has uncountable constant steady states. In the attraction chemotaxis system, each positive constant steady state is stable if it is in a certain region. On the other hand, in the repulsion chemotaxis system, every positive constant steady state is stable. Our main purpose of this paper is to give a suitable condition under which the attraction-repulsion chemotaxis system has also stable constant steady states.

Stability of constant steady states of an attraction-repulsion chemotaxis system

TL;DR

This paper studies the stability of constant steady states for the attraction-repulsion chemotaxis system on . By solving the elliptic equations for and , it reduces the system to a nonlocal scalar equation for with kernel and analyzes the linearized operator around via its symbol . The stability of the constant state is governed by , with a stable regime (where ) and an unstable regime ; a further threshold marks Lyapunov instability for large . The paper proves local and global well-posedness for small perturbations in uniformly local spaces, derives - semigroup estimates, and establishes instability in the large- regime, thereby clarifying when attraction-repulsion steady states are stable. These results extend the understanding of constant steady-state stability beyond purely attracting or repelling limiting cases.

Abstract

The Cauchy problem for the attraction-repulsion chemotaxis system in the whole -dimensional space has uncountable constant steady states. In the attraction chemotaxis system, each positive constant steady state is stable if it is in a certain region. On the other hand, in the repulsion chemotaxis system, every positive constant steady state is stable. Our main purpose of this paper is to give a suitable condition under which the attraction-repulsion chemotaxis system has also stable constant steady states.

Paper Structure

This paper contains 19 sections, 24 theorems, 243 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Main result
  4. Preliminaries
  5. Local-in-time solutions to problem \ref{['eq;simplified-ar-ks']} in ${\mathcal{L}}_{\mathrm{uloc}}^p(\mathbb{R}^n)$
  6. Useful tools
  7. Linearized problem of equation \ref{['eq;simplified-ar-ks']}
  8. Properties of the linearized operator $L_A$
  9. $L^p$-$L^q$ estimate of $\mathrm{e}^{t\mathcal{L}_A}$
  10. Decay estimates of $\mathrm{e}^{t\mathcal{L}_A}$
  11. Exponential growth of $\mathrm{e}^{t\mathcal{L}_A}$
  12. Perturbed problem near constant steady states
  13. Existence of local-in-time solutions to problem \ref{['eq;lin']}
  14. Uniqueness of solution to problem \ref{['eq;lin']}
  15. Global-in-time solutions to problem \ref{['eq;lin']}
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $\beta_1, \beta_2, \lambda_1, \lambda_2 > 0, \lambda_1 \not = \lambda_2$, and $p$ be and $n < q \le 2p$. Assume that where Then there exists $\varepsilon_0 > 0$ such that for every $v_0 \in L^p(\mathbb{R}^n)$ with $\| v_0 \|_p < \varepsilon_0$, problem eq;simplified-ar-ks corresponding to the initial data $u_0 = A + v_0$ has a unique global-in-time mild solution $u = u(t,x)$ satisfying $u -

Figures (2)

  • Figure 1: Positivity of $f_A(\tau)$
  • Figure 2: Monotonicity of $g_A(\tau)$

Theorems & Definitions (46)

  • Theorem 2.1
  • Proposition 3.1
  • Lemma 3.2
  • Lemma 3.3: Multivariable Faà di Bruno's formula
  • proof : Proof of Lemma \ref{['lem;1']}
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • Definition 4.1
  • Lemma 4.2
  • ...and 36 more