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Stability of constant equilibria in a Keller--Segel system with gradient dependent chemotactic sensitivity and sublinear signal production

Shohei Kohatsu

TL;DR

The paper addresses the Keller–Segel system with gradient-dependent chemotactic sensitivity and sublinear signal production under Neumann boundary conditions on a bounded domain. It proves the spatially homogeneous equilibrium is asymptotically stable when $p$ and $\theta$ satisfy the stated bounds, generalizing the $\theta=1$ case of Kohatsu–Yokota. The approach combines an $\varepsilon$-regularization, uniform $L^r$ and $L^q$ estimates, and semigroup/Duhamel techniques to pass to a global weak solution and establish decay toward the mean state. These results extend stability theory for flux-limited chemotaxis with nonlinear production to sublinear production, providing insight into long-time behavior of such systems.

Abstract

This paper deals with the homogeneous Neumann boundary-value problem for the Keller--Segel system \begin{align*} \begin{cases} u_t=Δu - χ\nabla \cdot (u|\nabla v|^{p-2}\nabla v),\\[] v_t=Δv - v + u^θ \end{cases} \end{align*} in $n$-dimensional bounded smooth domains for suitably regular nonnegative initial data, where $χ> 0$, $p \in (1, \infty)$ and $θ\in (0,1]$. Under smallness conditions on $p$ and $θ$, we prove that the spatially homogeneous equilibrium solution is stable. This generalizes the result in Kohatsu--Yokota (Le Matematiche, 2023; 78; 213--237) from the case $θ= 1$ to more general values of $θ$.

Stability of constant equilibria in a Keller--Segel system with gradient dependent chemotactic sensitivity and sublinear signal production

TL;DR

The paper addresses the Keller–Segel system with gradient-dependent chemotactic sensitivity and sublinear signal production under Neumann boundary conditions on a bounded domain. It proves the spatially homogeneous equilibrium is asymptotically stable when and satisfy the stated bounds, generalizing the case of Kohatsu–Yokota. The approach combines an -regularization, uniform and estimates, and semigroup/Duhamel techniques to pass to a global weak solution and establish decay toward the mean state. These results extend stability theory for flux-limited chemotaxis with nonlinear production to sublinear production, providing insight into long-time behavior of such systems.

Abstract

This paper deals with the homogeneous Neumann boundary-value problem for the Keller--Segel system \begin{align*} \begin{cases} u_t=Δu - χ\nabla \cdot (u|\nabla v|^{p-2}\nabla v),\\[] v_t=Δv - v + u^θ \end{cases} \end{align*} in -dimensional bounded smooth domains for suitably regular nonnegative initial data, where , and . Under smallness conditions on and , we prove that the spatially homogeneous equilibrium solution is stable. This generalizes the result in Kohatsu--Yokota (Le Matematiche, 2023; 78; 213--237) from the case to more general values of .
Paper Structure (3 sections, 6 theorems, 71 equations)

This paper contains 3 sections, 6 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Global existence of approximate solutions
  3. Proof of Theorem \ref{['thm1']}

Key Result

Theorem 1.1

Let $n \in \mathbb{N}$. Assume that $u_0$ and $v_0$ satisfy initial. Let $\overline{u_0} := \frac{1}{|\Omega|} \int_{\Omega} u_0$ and $m := \| u_0 \|_{L^1(\Omega)}$. Suppose that $\theta \in (0,1]$, and that Then there exist a global weak solution $(u, v)$ of KSPP and $t_1 > 0$ such that where $C > 0$, $\alpha > 0$ and $\beta > 0$ are constants. In particular, one can find $\eta_0 > 0$ such that

Theorems & Definitions (15)

  • Definition 1.1
  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 5 more