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 Rodriguez for crime modeling as follows: {u(t)= Delta u - chi del center dot(u del ln nu) - kappa u nu + h(1), v(t)= Delta v - v + u + h(2), in a bounded and smooth spatial domain Omega subset of R-n withn >= 3, with the parameters chi >0 and kappa > 0, andwith the nonnegative functions h(1) and h(2). For the case that kappa = 0, h(1) equivalent to 0 and h(2) equivalent to 0, recent resultsshowed that the corresponding initial-boundary value problem admits a global generalized solutionprovided 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 globalgeneralized solutions provided that chi < chi(1) with some chi(1) > chi(0), which seems to confirm that the mixed-type damping -kappa u nu has a regularization effect on solutions. Besides the existence result for generalizedsolutions, a statement on the large-time behavior of such solutions is derived as well.