GLOBAL LARGE-DATA GENERALIZED SOLUTIONS TO A TWO-DIMENSIONAL CHEMOTAXIS SYSTEM STEMMING FROM CRIME MODELLING

We consider a chemotaxis system over a bounded domain Omega subset of R-2 of the following form {ut = triangle u - chi del. middot (u/v del u) - u + a, vt = triangle v - v + uv(1 - v) + b, (*) with chi > 0, a >= 0 and b > 0. The particular version of (*) with chi = 2 was proposed by Pitcher to describe the evolution of population dynamics of criminal agents. Recent results reveal that the system (*) associated with Neumann boundary conditions admits a global classical solution, under appropriate smallness conditions on both the initial data and the parameter chi. The present study indicates that nevertheless, for all reasonably regular initial data and any chi > 0, the corresponding Neumann initial-boundary value problem possesses a global generalized solution. Furthermore, it also demonstrates that, whenever a = 0, such global generalized solution becomes bounded and smooth at least eventually. In particular, it approaches the spatial equilibria at exponential rate in the large time limit.