Prevention of infinite-time blowup by slightly super-linear degradation in a Keller–Segel system with density-suppressed motility

An initial-Neumann boundary value problem for a Keller–Segel system with density-suppressed motility and source terms is considered. Infinite-time blowup of the classical solution was previously observed for its source-free version when dimension N ⩾ 2 . In this work, we prove that with any source term involving a slightly super-linear degradation effect on the density, of a growth order of s log ⁡ s at most, the classical solution is uniformly-in-time bounded when N ⩽ 3 , thus preventing the infinite-time explosion detected in the source-free counter-part. The cornerstone of our proof lies in an improved comparison argument and a construction of an entropy inequality.