Absence of blow-up in a fully parabolic chemotaxis system with weak singular sensitivity and logistic damping in dimension two

It is shown in this paper that blow-up does not occur in the following chemotaxis system under homogeneous Neumann boundary conditions in a smooth, open, bounded domain \ (R²\): equation* cases uₜ = u - (uvᵏ v) + ru - u², &in (0, T ₌₀ₗ), vₜ = v - v + u, &in (0, T ₌₀ₗ), cases equation* where \ (k (0, 1) \), and \ (, r, , , \) are positive parameters. Known results have already established the same conclusion for the parabolic-elliptic case. Here, we complement these findings by extending the result to the fully parabolic case.