Chemotaxis models with signal-dependent sensitivity and a logistic-type source, II: Persistence and stabilization
Le Chen, Ian Ruau, Wenxian Shen
Abstract
This paper is Part II of a series on global existence and asymptotic behavior of positive solutions to \begin{equation*} \begin{cases} \displaystyle u_t=Δu-χ_0\nabla\cdot\left(\frac{u^m}{(1+v)^β}\nabla v\right)+au-bu^{1+α}, & x\inΩ, \cr \displaystyle 0=Δv-μv+νu^γ, & x\inΩ, \cr \displaystyle \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0, & x\in\partialΩ, \end{cases} \end{equation*} where $Ω\subset\mathbb{R}^N$ is a bounded and smooth domain. The parameters $α,γ,m,μ,ν$ are positive, $χ_0$ is real, and $a,b,β$ are nonnegative. In Part I, we established boundedness and global existence. Here, we study persistence and stabilization, quantifying how $β$ and $χ_0$ influence long-time dynamics. First, we prove uniform persistence if $m\ge 1$. Next, for $a,b>0$, the unique positive equilibrium is $(u^*,v^*) = \left((\tfrac{a}{b})^{1/α},(\tfracνμ)(\tfrac{a}{b})^{γ/α}\right)$. We identify a threshold $χ^*(u^*)$: $(u^*,v^*)$ is linearly stable if $χ_0<χ^*(u^*)$, with local exponential decay, unstable if $χ_0>χ^*(u^*)$. We also give conditions ensuring every bounded solution converges exponentially to $(u^*,v^*)$. For $a=b=0$, we study stability of the constant equilibria under mass constraint, obtaining a linear stability threshold and global stabilization. We extend the Lyapunov method from $m=1$ to $m>1$ and the rectangle/ODE method from $β=0$ to $β>0$. For $m\ge 1$, signal saturation (large $β$) or repulsion ($χ_0<0$) prevents aggregation and promotes relaxation. In Part III, we study bifurcation and pattern formation when $χ_0$ passes through critical thresholds.