  We consider a parabolic-elliptic type of Keller-Segel equations with
generalized diffusion and logistic source under homogeneous Neumann-Neumann
boundary conditions. We construct bounded weak solutions globally in time in an
unbalanced optimal transport framework, provided that the magnitude of the
chemotactic sensitivity can be restricted depending on parameters. In the case
of subquadratic degradation of the logistic source, we quantify the chemotactic
sensitivity, in particular, in terms of the power of degradation and the
pointwise bound of the initial density.
