Solutions to a chemotaxis system with spatially heterogeneous diffusion sensitivity

Abstract

We consider a parabolic-elliptic Keller-Segel system with spatially dependent diffusion sensitivity \begin{eqnarray*} \left\{ \begin{array}{l} u_t = \nabla \cdot (|x|^\beta \nabla u) - \nabla \cdot (u\nabla v), \\[1mm] 0 = \Delta v - \mu + u, \qquad \mu:=\frac{1}{|\Omega|} \int\limits_\Omega u, \end{array} \right. \qquad \qquad (\star) \end{eqnarray*} under homogeneous Neumann boundary conditions in the ball $\Omega=B_R(0)\subset \mathbb R^n$. For $\beta>0$ and radially symmetric H\"older continuous initial data, we prove that there exists a pointwise classical solution to $(\star)$ in $(\Omega\setminus \{0\})\times (0,T)$ for some $T>0$. For radially decreasing initial data satisfying certain compatibility criteria, this solution is bounded and unique in $(\Omega\setminus \{0\})\times (0,T^*)$ for some $T^*>0$. Moreover, for $n \geq 2$ and sufficiently accumulated initial data, there exists no solution $(u,v)$ to $(\star)$ in the sense specified above which is globally bounded in time.