Chemotaxis models with signal-dependent sensitivity and a logistic-type source, I: Boundedness and global existence

TL;DR

This work analyzes boundedness and global existence for a parabolic-elliptic chemotaxis system with signal-dependent sensitivity $\chi(v)=\dfrac{\chi_0}{(1+v)^\beta}$, nonlinear cross-diffusion $u^m/(1+v)^\beta$, and logistic damping $a u - b u^{1+\alpha}$ on a bounded domain. It develops three complementary analytical angles to ensure bounded, globally existing solutions: (i) negative chemotaxis ($\chi_0\le 0$) yields immediate control and global existence for $m\ge 1$; (ii) weak nonlinear cross diffusion ($0<m\le 1$, $\beta\ge 1$) gives $L^p$-based controls and global existence under a smallness constraint on $\chi_0$ (with the $m=1$ case requiring $\chi_0<\tfrac{2(2\beta-1)}{\max\{2,\gamma N\}}$); and (iii) relatively strong logistic sources (several regimes on $\alpha$, $m$, $\gamma$, $\beta$) yield boundedness and global existence, with explicit interplays among the parameters. Part II extends the analysis to asymptotic behavior, including persistence, stability, and bifurcation of equilibria. The results unify and extend known bounds for special cases and provide explicit, dimension-dependent threshold criteria for global well-posedness in the presence of signal-dependent sensitivity.

Abstract

We study, in Part I of this series, boundedness and global existence of positive classical solutions to a parabolic-elliptic chemotaxis system with signal-dependent sensitivity and a logistic-type source on a bounded smooth domain $Ω\subset\mathbb{R}^N$: \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*} Here, $u$ denotes the population density and $v$ the chemical concentration. The parameters $α,γ,m,μ,ν$ are positive, $χ_0$ is real, and $a,b,β$ are nonnegative. We analyze boundedness from three viewpoints: negative chemotaxis ($χ_0<0$), the strength of the nonlinear cross diffusion rate $\frac{u^m}{(1+v)^β}$, and the strength of the logistic-type damping $u(a-bu^α)$. Under explicit conditions reflecting these mechanisms, all positive classical solutions remain bounded. Moreover, when $m\ge 1$, boundedness implies global existence. Although the decay of $χ(v) = \dfrac{χ_0}{(1+v)^β}$ for large $v$ has a damping effect, it also introduces new analytical difficulties; our techniques yield, for example, global existence for $m=1$ provided that \begin{equation*} β>\max\left\{1,\frac12+\frac{χ_0}{4}\max\{2,γN\}\right\}. \end{equation*} Several known results for special cases are recovered. Part II is devoted to the asymptotic behavior of globally defined solutions, including uniform persistence as well as stability and bifurcation of positive constant equilibria.

Figures (2)

• Figure 1.1: The constants $\Psi_\beta$, $\Theta_\beta$, and $\Theta_{2\beta-1}$ viewed as functions of $\beta$.
• Figure 1.2: Overlapping parameter regimes guaranteeing global classical solutions of the parabolic--elliptic system \ref{['E:main-PE']} in the $(m,\alpha)$-plane (see Theorems \ref{['T:weak-cross-diffusion']} and \ref{['T:strong-logistic-source']}). Left: open regions determined by the three lines $\alpha = m + \gamma - 1$, $\alpha = 2m + \gamma - 2$, and $m = 1$; the red wedge strictly above $\alpha = m + \gamma - 1$ requires $\beta \ge 0$, the crosshatched region strictly above $\alpha = 2m + \gamma - 2$ requires $\beta \ge \tfrac{1}{2}$, and the blue strip with $0 < m < 1$ requires $\beta \ge 1$. As we enlarge the admissible region to include smaller values of $\alpha$ and $m$, the lower bound on $\beta$ is strengthened from $\beta \ge 0$ to $\beta \ge \tfrac{1}{2}$ and then to $\beta \ge 1$, which shows that one needs larger $\beta$ to help stabilize the system. Right: the same three boundary lines, on which we have the corresponding critical cases; along each line global existence holds provided $\chi_0$ satisfies the indicated smallness condition.

Theorems & Definitions (48)

• Definition 1.1
• Proposition 1.1: Local existence
• Theorem 1.1: Boundedness and global existence with negative chemotaxis sensitivity
• Theorem 1.2: Boundedness and global existence with weak nonlinear cross diffusion
• Theorem 1.3: Boundedness and global existence with relatively strong logistic source
• Remark 1.1: Local existence
• Remark 1.2: Boundedness and global existence
• Remark 1.3: Boundedness and global existence with negative sensitivity
• Remark 1.4: Boundedness and global existence with weak nonlinear cross diffusion rate
• Remark 1.5: Boundedness and global existence with relatively strong logistic source