Global Solvability and Stabilization in a Three-Dimensional Cross-Diffusion System Modeling Urban Crime Propagation

In this paper, we will focus on the initial-boundary value problem with no-flux boundary conditions for the three-dimensional cross-diffusion system {u(t) = Delta u - chi del . (u/v del v) - uv + B-1(x, t), x is an element of Omega, t > 0, v(t) = Delta v + uv - v + B-2(x, t), x is an element of Omega, t > 0. Under some basic assumptions on the functions B-1 and B-2, we will show that for each chi is an element of (0, root 3), the system has at least one global renormalized solution in case that each ingredient of the system is radially symmetric with regard to the center of Omega. Moreover, if chi is an element of (0, root 6/2), the solution will approach the solution of an elliptic boundary value problem as t -> infinity under some extra hypotheses.