Mathematical analysis of a continuous version of statistical model for criminal behavior

Abstract

This paper is concerned with a reaction‐advection‐diffusion system, which is a continuous version of statistical model for criminal activity proposed by Short et al (Math. Models Methods Appl. Sci., 8 (2008), 1249‐1267). From very recently results, it is known that the classical solution of the corresponding initial‐boundary value problem exists globally. On the other hand, Rodriguez (Phys. D, 260 (2013), 191‐200) asserted the uniform‐in‐time boundedness of classical solution; however, there is a gap in the proof. In this work, we first solve this open problem by establishing some crucial a priori estimates. Then, invoking the uniform boundedness, we present that the classical solution converges exponentially to the steady state. Our results are consistent with the experiment observation and numerical simulation in the existing literatures.