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

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 $\Omega\subset\mathbb{R}^N$: \begin{equation*} \begin{cases} \displaystyle u_t=\Delta u-\chi_0\nabla\cdot\left(\frac{u^m}{(1+v)^\beta}\nabla v\right)+au-bu^{1+\alpha}, & x\in\Omega, \cr \displaystyle 0=\Delta v-\mu v+\nu u^\gamma, & x\in\Omega, \cr \displaystyle \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0, & x\in\partial\Omega. \end{cases} \end{equation*} Here, $u$ denotes the population density and $v$ the chemical concentration. The parameters $\alpha,\gamma,m,\mu,\nu$ are positive, $\chi_0$ is real, and $a,b,\beta$ are nonnegative. We analyze boundedness from three viewpoints: negative chemotaxis ($\chi_0<0$), the strength of the nonlinear cross diffusion rate $\frac{u^m}{(1+v)^\beta}$, and the strength of the logistic-type damping $u(a-bu^\alpha)$. 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 $\chi(v) = \dfrac{\chi_0}{(1+v)^\beta}$ 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*} \beta>\max\left\{1,\frac12+\frac{\chi_0}{4}\max\{2,\gamma N\}\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.