Boundedness in a chemotaxis system with weakly singular sensitivity in dimension two with arbitrary sub-quadratic degradation sources

We study the global existence and boundedness of solutions to a chemotaxis system with weakly singular sensitivity and sub-logistic sources in a two dimensional domain. X. Zhao (Nonlinearity; 2023; 36; 3909-3938 ) showed that the logistic degradation, -μ u², can prevent blow-up under the largeness assumption on μ. In this paper, we improve the result by replacing the quadratic degradation by sub-logistic one, -μ u²/ln^β(u+e) with β ∈ (0,1), and removing the largeness assumption on μ.