This paper investigates an initial-boundary value problem for a chemotaxis system that models urban crime propagation with burglary fatigue. The system is formulated as follows: \(\begin{aligned} \left\{ \begin{aligned}&u_t=\Delta u^m -\chi \nabla \cdot (\frac{u}{v}\nabla v)-u+g_1(x, t), & x\in \Omega ,\,t>0,\\&v_t=\Delta v-v+uv(1-v)+g_2(x, t), & x\in \Omega ,\,t>0, \end{aligned} \right. \end{aligned}\) where \(\Omega \subset \mathbb {R}^n\) ( \(n\ge 2\) ) is a smoothly bounded domain, \(\chi >0\) is the chemotactic sensitivity coefficient, and \(g_1, g_2\) are non-negative source functions representing external crime stimuli and signal supplements, respectively. The core objective is to quantify the relaxation effects induced by enhanced nonlinear diffusion. Under no-flux boundary conditions on u and v, we prove that when \(m>\frac{3}{2}-\frac{1}{n}\) , the problem admits a globally bounded weak solution. Furthermore, the long-time behavior of the solution is analyzed under additional mild assumptions on the source functions.
