Global existence of solutions to a parabolic–parabolic system for chemotaxis with weak degradation

We study the global existence of solutions to a parabolic–parabolic system for chemotaxis with a logistic source in a two-dimensional domain, where the degradation order of the logistic source is weaker than quadratic. We introduce nonlinear production of a chemoattractant, and show the global existence of solutions under certain relations between the degradation and production orders.