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 (\Omega \subset \mathbb{R}^2): \begin{equation*} \begin{cases} u_t = \Delta u - \chi \nabla \cdot \left( \frac{u}{v^k} \nabla v \right) + ru - \mu u^2, \qquad &\text{in } \Omega \times (0,T_{\rm max}), v_t = \Delta v - \alpha v + \beta u, \qquad &\text{in } \Omega \times (0,T_{\rm max}), \end{cases} \end{equation*} where ( k \in (0,1) ), and (\chi, r, \mu, \alpha, \beta ) 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.