Exclusion of finite-time singularities under weak logistic damping

Prove that finite-time singularities cannot occur for generalized solutions of the singular chemotaxis urban crime system when the logistic-damping coefficient \(\mu>0\) is sufficiently small, including in dimensions two and three.

Background

Theorem \ref{th-ggs} establishes global generalized solutions for every $\mu>0$, together with eventual smoothness and convergence under additional hypotheses in dimensions at most three. However, the theorem does not exclude the possibility that generalized solutions may develop singularities before the eventual regularity regime begins when $\mu$ is small. Remark (R8) explicitly identifies eliminating such finite-time singularities as an open problem.