Blow-up prevention by sub-logistic sources in 2D Keller–Segel chemotaxis systems with superlinear signal production

This paper focuses on studying blow-up prevention by sub-logistic sources in 2D Keller–Segel chemotaxis systems with superlinear signal production. An application of a result on parabolic gradient regularity for parabolic equations in Orlicz spaces shows that the presence of sub-logistic sources is indeed sufficiently strong to ensure the global existence and boundedness of solutions. Our proof also relies on several techniques, including parabolic regularity in Sobolev spaces, variational arguments, interpolation inequalities in Sobolev spaces, and Moser iteration method.