| Abstract: |
| We study a parabolic-elliptic Keller-Segel system with spatially dependent chemotactic sensitivity of the form
\begin{eqnarray*}
\left\{ \begin{array}{l}
u_t = \Delta u - \nabla \cdot (|x|^\alpha u\nabla v), \[1mm]
0 = \Delta v - \mu + u,
\qquad \mu:=\frac{1}{|\Omega|} \int_{\Omega} u,
\end{array} \right.
\qquad \qquad (\star)
\end{eqnarray*}
for $\alpha>0$ under homogeneous Neumann boundary conditions in the ball $\Omega=B_R(0)\subset \mathbb R^n$.
This model accounts for a reduction in chemotactic sensitivity in the vicinity of the origin.
For radially symmetric initial data, we discuss the solvability of $(\star)$ and analyze the behavior of solutions with respect to global boundedness and finite-time blow-up. |
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