This paper investigates the classical solutions of a chemotaxis system with weakly singular sensitivity, \( - \nabla \left( \frac{u}{v^k} \nabla v \right) \) for \( k \in (0,1) \) , and logistic sources under general nonlinear Neumann boundary conditions. Previous studies have shown that quadratic logistic damping can prevent blowup under homogeneous Neumann boundary conditions. Our work extends these results by proving that the logistic term, or even a sub-logistic term, is sufficiently strong to ensure global existence and boundedness of solutions, even under general boundary conditions that allow population influx into the domain.
