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

TL;DR

This paper investigates finite-time blow-up phenomena in a class of chemotaxis systems with spatially varying diffusion sensitivity, establishing conditions for existence, boundedness, and blow-up of solutions.

Contribution

It introduces a novel analysis of chemotaxis systems with spatially dependent diffusion, demonstrating finite-time blow-up under certain initial conditions.

Findings

Existence of classical solutions for nonconstant radial initial data.

Boundedness and uniqueness of solutions under specific initial conditions.

Finite-time blow-up occurs when initial mass is sufficiently concentrated.

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|^{\beta} \bigtriangledown u)-\bigtriangledown\cdot(u^{\alpha} \bigtriangledown v), 0=\bigtriangleup v-\mu +u, \qquad \mu:=\frac{1}{|\Omega|}\int_{\Omega}udx,\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 $(\Omega \setminus \{ 0 \})\times (0,T^{*})$ with $T^{*}<T$.\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.