Eventual smoothness of weak solutions for the logarithmic Keller–Segel system

Determine whether global weak solutions of the logarithmic Keller–Segel system with quadratic logistic damping in dimensions n≥3 become smooth in finite or eventual time.

Background

The paper recalls prior results for the logarithmic Keller–Segel system obtained from the urban crime model by removing the −uv term from the first equation, replacing +uv by +u in the second equation, and taking quadratic logistic damping. Global weak solutions are known in dimensions n≥3, but the regularity of these weak solutions is unresolved. This question is distinct from the paper’s own theorem, which establishes eventual smoothness for generalized solutions of the urban crime system under additional assumptions and in dimensions n≤3.