Blow-Up and Boundedness in Quasilinear Parabolic-Elliptic Chemotaxis System with Nonlinear Signal Production (Journal of Partial Differential Equations, 2023)
In this paper, we consider the quasilinear chemotaxis system of parabolic-elliptic type $$\begin{cases} u_t= abla\cdot(D(u) abla u)- abla\cdot(f(u) abla v), & x\in \Omega,\ t>0, \\ 0=\Delta v-\mu(t)+g(u), &  x\in \Omega, \ t>0 \end{cases}$$under homogeneous Neumann boundary conditions in a smooth bounded domain  $\Omega\subset\mathbb{R}^n, \ n\geq1$. The nonlinear diffusivity $D(\xi)$ and chemosensitivity $f(\xi)$ as well as nonlinear signal production $g(\xi)$ are supposed to extend the prototypes $$D(\xi)=C_{0}(1+\xi)^{-m}, \ \ f(\xi)=K(1+\xi)^{k}, \ \ g(\xi)=L(1+\xi)^{l}, \ \ C_{0}>0,\xi\geq 0,K,k,L,l>0,m\in\mathbb{R}.$$ We proved that if $m+k+l>1+\frac{2}{n}$, then there exists nonnegative radially symmetric initial data $u_{0}$ such that the corresponding solutions blow up in finite time. However, the system admits a global bounded classical solution for arbitrary initial datum when $m+k+l
Keywords: Chemotaxis; nonlinear diffusion; blow-up; boundedness; nonlinear signal production.