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 \(Ω ⊂ R^2\): {equation*} {cases} u_t = Δ u - χ ∇ · ( u/v^k ∇ v ) + ru - μ u^2, &in Ω × (0,Tₘₐₓ), v_t = Δ v - α v + β u, &in Ω × (0,Tₘₐₓ), {cases} {equation*} where \( k ∈ (0,1) \), and \(χ, r, μ, α, β \) 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.