Solutions to a chemotaxis system with spatially heterogeneous diffusion sensitivity

TL;DR

This paper investigates a chemotaxis model with spatially varying diffusion sensitivity, proving local existence, boundedness under certain conditions, and demonstrating nonexistence of globally bounded solutions for specific initial data.

Contribution

It establishes the existence, uniqueness, and boundedness of solutions to a chemotaxis system with spatially dependent diffusion sensitivity, and shows conditions leading to nonexistence of global solutions.

Findings

Existence of classical solutions for radially symmetric initial data.

Boundedness and uniqueness under certain initial conditions.

Nonexistence of globally bounded solutions for large initial data.

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.