Large time behavior of solution to a parabolic-elliptic chemotaxis system with weak singular sensitivity and logistic kinetics: Boundedness, persistence, stability

TL;DR

This paper investigates the long-term behavior of solutions to a chemotaxis system with weak singular sensitivity and logistic growth, establishing boundedness, persistence, and exponential stability under certain parameter conditions.

Contribution

It provides new thresholds for parameters ensuring boundedness, persistence, and stability of solutions in a chemotaxis model with weak singular sensitivity.

Findings

Existence of parameter thresholds for boundedness and persistence.

Global boundedness of classical solutions under certain conditions.

Exponential convergence to steady state for bounded solutions.

Abstract

This paper deals with the long-term behavior of positive solutions for the following parabolic-elliptic chemotaxis competition system with weak singular sensitivity and logistic source \begin{equation} \label{abstract-eq} \begin{cases} u_t=\Delta u-\chi \nabla\cdot (\frac{u}{v^{\lambda}} \nabla v) +ru- \mu u^2, \quad &x\in \Omega,\cr 0=\Delta v- \alpha v +\beta u,\quad &x\in \Omega, \cr \frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=0,\quad &x\in\partial\Omega, \end{cases}\, \end{equation} where $\Omega \subset \mathbb{R}^N (N \ge 2)$ is a smooth bounded domain, the parameters $\chi,\, r, \, \mu, \, \alpha,\,\beta$ are positive constants and $\lambda \in (0,1).$ In this article, for all suitably smooth initial data $u_0\in C^0(\bar\Omega)$ with $u_0 \not \equiv 0,$ it has been proven that: First, there exists $\mu > \mu_1^*(p,\lambda,\chi,\beta)$ such that any globally defined positive solution is $L^p(\Omega)$-bounded with $p \ge 2.$ Next, there exists $\mu > \mu_2^*(N,\lambda,\chi,\beta)$ such that any globally defined classical solutions is globally bounded. Third, there exists $\mu > \mu_3^*(N,\lambda,\chi,\beta)$ such that any globally defined positive solution is uniformly bounded above and below eventually by some positive constants that are independent of its initial function $u_0.$ Last, there exists $\mu > \mu_4^*(N,\lambda,\chi,\alpha,\beta,r,\Omega)$ such that any globally bounded classical solution to system (0.1) exponentially converges to the constant steady state $(\frac{r}{\mu},\frac{\beta}{\alpha}\frac{r}{\mu}).$