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

Abstract

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 \(Ω\subset \mathbb{R}^2\): \begin{equation*} \begin{cases} u_t = Δu - χ\nabla \cdot \left( \frac{u}{v^k} \nabla v \right) + ru - μu^2, \qquad &\text{in } Ω\times (0,T_{\rm max}), v_t = Δv - αv + βu, \qquad &\text{in } Ω\times (0,T_{\rm max}), \end{cases} \end{equation*} where \( k \in (0,1) \), 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.