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 $\mathrm{\Omega }\subset {\mathbb{R}}^2$: \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 $\chi ,r,\mu ,\alpha ,\beta$ 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.