Multi-species Patlak-Keller-Segel system

We study the regularity and large-time behavior of a crowd species driven by chemo-tactic interactions. What distinguishes different is way they interact with rest crowd: collective motion chemical reactions which end up in coupled system parabolic Patlak-Keller-Segel equations. show that densities diffuse to zero provided interactions between satisfy certain sub-critical condition; latter intimately related log-Hardy-Littlewood-Sobolev inequality for systems due Shafrir & Wolansky. Thus example, when two interact, one has mass less than $4\pi$, then 2-system stays smooth all time independent total system, sharp contrast well-known breakdown specie initial mass$> 8\pi$.