Uniform boundedness for the two-dimensional Keller–Segel system with Gompertz growth

It is well known that in two dimensions the classical Keller–Segel model may exhibit cell aggregation phenomena, which are automatically controlled when a logistic source with quadratic degradation is introduced. Considerable effort has been devoted to identifying weaker damping mechanisms that still reproduce the stabilizing effect of the quadratic term. In Xiang (2018), it was shown that, under suitable assumptions on the initial distribution of the cell density, even sub-logistic sources of the form f ( s ) = a s − b s 2 ln ( ln ( s + e ) ) , a , b > 0 , and s > 0 , can achieve this effect. In the present paper, we introduce the Gompertz function f ( s ) = α s ln K s , α , K > 0 , and s > 0 , as an external source in minimal 2D chemotaxis model. This term exhibits a weaker damping effect than the prototype considered in Xiang (2018). We investigate its influence on the system and establish conditions ensuring global existence and uniform boundedness of solutions.