Finite-time blow-up in a class of chemotaxis systems with spatially heterogeneous diffusion sensitivity

Abstract

\indent In this paper, we study a class of parabolic-elliptic Keller-Segel systems with diffusion sensitivity dependent on spatial position, given by type \begin{equation} \left\{ \begin{array}{ll} u_{t} = \bigtriangledown\cdot(|x|^β \bigtriangledown u)-\bigtriangledown\cdot(u^α \bigtriangledown v), 0=\bigtriangleup v-μ+u, \qquad μ:=\frac{1}{|Ω|}\int_Ωudx,\end{array}\right. \end{equation} under homogeneous Neumann conditions in a ball $\Omega =B_R(0)\subset {\mathbb{R}}^n$ with $\alpha \ge 1$, $\beta >0$ and $n\ge 2$.\par \indent It is proved that any nonconstant nonnegative radial initial data $u_0\in C^{\theta }(\overline{\Omega })$, where $\theta \in (0,1)$, there exists a radially symmetric classical solution of the system (0.1) in $(\Omega \setminus \{0\})\times (0,T)$ for some $T>0$; moreover, if the initial values $u_0\in C^{1+\theta }(\overline{\Omega })$ for some $\theta \in (0,1)$ and satisfy a certain compatibility criterion and are radially decreasing, then this solution is bounded and unique in $(\Omega \setminus \{0\})\times (0,T^{\ast })$ with $T^{\ast }<T$.\par Finally, it is found that the initial mass corresponding to this parabolic-elliptic problem (0.1) is sufficiently concentrated to allow the solution to blow up in finite time.