Regularization effect of the mixed-type damping in a higher-dimensional logarithmic Keller-Segel system related to crime modeling

We study a logarithmic Keller-Segel system proposed by Rodríguez for crime modeling as follows: $ \begin{equation*} \left\{ \begin{split} &u_t = \Delta u-\chi\nabla\cdot\left(u\nabla\ln v\right)- \kappa uv+ h_1,\\ &v_t = \Delta v- v+ u+h_2, \end{split} \right. \end{equation*} $ in a bounded and smooth spatial domain $ \Omega\subset \mathbb R^n $ with $ n\geq3 $, with the parameters $ \chi > 0 $ and $ \kappa > 0 $, and with the nonnegative functions $ h_1 $ and $ h_2 $. For the case that $ \kappa = 0 $, $ h_1\equiv0 $ and $ h_2\equiv0 $, recent results showed that the corresponding initial-boundary value problem admits a global generalized solution provided that $ \chi < \chi_0 $ with some $ \chi_0 > 0 $. In the present work, our first result shows that for the case of $ \kappa > 0 $ such problem possesses global generalized solutions provided that $ \chi < \chi_1 $ with some $ \chi_1 > \chi_0 $, which seems to confirm that the mixed-type damping $ -\kappa uv $ has a regularization effect on solutions. Besides the existence result for generalized solutions, a statement on the large-time behavior of such solutions is derived as well.