Vanishing logistic-damping threshold at zero chemotactic sensitivity

Determine whether the logistic-damping threshold \(\mu_0(\chi,n)\) for global existence and boundedness of classical solutions to the singular chemotaxis urban crime system can be chosen so that \(\mu_0(0,n)=0\).

Background

Theorem \ref{th-global} introduces an explicit threshold $\mu_0(\chi,n)>0$ such that sufficiently strong logistic damping yields global bounded classical solutions for the singular chemotaxis urban crime system. Remark (R2) gives rough upper bounds for this threshold in dimensions two and three and notes that these bounds attain their minimum at the physically relevant value $\chi=2$. The unresolved issue is whether the threshold itself vanishes when chemotactic sensitivity is absent, as one might expect from the absence of chemotactic aggregation.