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, $-\mu u^2$, can prevent blow-up under the largeness assumption on $\mu$. In this paper, we improve the result by replacing the quadratic degradation by sub-logistic one, $-\frac{\mu u^2}{\ln^\beta(u+e)}$ with $\beta \in (0,1)$, and removing the largeness assumption on $\mu$.