Chemotaxis effect vs logistic damping on boundedness in the 2-D minimal Keller-Segel model

In this paper, we study chemotaxis effect vs logistic dampening on boundedness for the two-dimensional minimal Keller-Segel model with logistic source in a 2-D smooth and bounded domain. It is well-known that this model allows only for global and uniform-in-time bounded solutions for any chemotactic strength and logistic dampening. Here, we carefully employ a simple and new method to regain its boundedness and, with particular attention to how boundedness depends qualitatively on the coefficient of chemotactic strength and logistic dampening rate. Up to a scaling constant depending only on initial data and the domain, we provide explicit upper bounds for the the solution components of the corresponding initial-boundary value problem. This qualitative boundedness results seems to be the first result in the regard.