Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion model for urban crime propagation

Abstract

We consider a class of macroscopic models for the spatio-temporal evolution of urban crime, as originally going back to Short et al. (Math. Mod. Meth. Appl. Sci. 18, 2008). The focus here is on the question how far a certain nonlinear enhancement in the random diffusion of criminal agents may exert visible relaxation effects. Specifically, in the context of the system \begin{eqnarray*} \left\{ \begin{array}{l} u_t = \nabla \cdot (u^{m-1} \nabla u) - \chi \nabla \cdot \Big(\frac{u}{v} \nabla v \Big) - uv + B_1(x,t), \\[1mm] v_t = \Delta v +uv - v + B_2(x,t), \end{array} \right. \end{eqnarray*} it is shown that whenever $\chi>0$ and the given nonnegative source terms $B_1$ and $B_2$ are sufficiently regular, the assumption \begin{eqnarray*} m>\frac{3}{2} \end{eqnarray*} is sufficient to ensure that a corresponding Neumann-type initial-boundary value problem, posed in a smoothly bounded planar convex domain, admits locally bounded solutions for a wide class of arbitrary initial data. Furthermore, this solution is seen to be globally bounded if both $B_1$ and $B_2$ are bounded and $\liminf_{t\to\infty} \int_\Omega B_2(\cdot,t)$ is positive. This is supplemented by numerical evidence which, besides illustrating associated smoothing effects in particular situations of sharply structured initial data in the presence of such porous medium type diffusion mechanisms, indicates a significant tendency toward support of singular structures in the linear diffusion case $m=1$.