Global existence, boundedness and asymptotic behavior to a logistic chemotaxis model with density-signal governed sensitivity and signal absorption

TL;DR

This paper investigates a chemotaxis model with density-dependent sensitivity and logistic growth, establishing conditions for global existence, boundedness, and long-term behavior of solutions in various dimensions.

Contribution

It proves the existence of global classical solutions for certain sensitivity parameters and analyzes their boundedness and asymptotic behavior in two dimensions.

Findings

Global solutions exist for eta<1 in any dimension.

Solutions are uniformly bounded and converge to steady states in 2D when etand ig.

Large nd ig ensure stability and convergence of solutions.

Abstract

In present paper, we consider a chemotaxis consumption system with density-signal governed sensitivity and logistic source: $u_t=\Delta u-\nabla\cdot(\frac{S(u)}{v}\nabla v)+ru-\mu u^2$, $v_t=\Delta v-uv$ in a smooth bounded domain $\Omega\subset\mathbb{R}^n$ $(n\ge2)$, where parameters $r,\mu>0$ and density governed sensitivity fulfills $ S(u) \simeq u(u+1)^{\beta-1}$ for all $u\ge0$ with $\beta\in \mathbb{R}$. It is proved that for any $r,\mu>0$, there exists a global classical solution if $\beta<1$ and $n\ge2$. Moreover, the global boundedness and the asymptotic behavior of the classical solution are determined for the case $\beta\in[0,1)$ in two dimensional setting, that is, the global solution $(u,v)$ is uniformly bounded in time and $\Big(u,v,\frac{|\nabla v|}{v}\Big)\longrightarrow\Big(\frac{r}{\mu},0,0\Big)~in~L^{\infty}(\Omega)~as~t\rightarrow\infty$, provided $\mu$ sufficiently large.