On a crime model in higher-dimensional setting: global generalized solvability and eventual smoothness

Abstract

This paper is concerned with the system

$$\begin{aligned} \left\{ \begin{aligned}&u_t= \Delta u-\chi \nabla \cdot \left( \frac{u}{v}\nabla v\right) +g(u),&\ x\in \Omega ,\ t>0, \\&v_t=\Delta v-v+ uv(1-v)+b,&\ x\in \Omega ,\ t>0, \end{aligned} \right. \end{aligned}$$

(⋆)

in a smooth bounded domain $\Omega \subset {\mathbb {R}}^n(n\ge 2)$, where $\chi >0$, $b>0$, $g(s)\in C^1[0, \infty )$ satisfies

$$\begin{aligned} g(0)\ge 0\quad \textrm{and}\quad \frac{g(s)}{s}\rightarrow -\infty \quad \textrm{as}\quad s\rightarrow \infty . \end{aligned}$$

The first result demonstrates that for all properly regular initial data, the associated homogeneous Neumann initial boundary value problem admits global solutions within some generalized framework. Meanwhile, it reveals that when $n=2$ and $g(u)=-u^\alpha$ with $\alpha >1$, such generalized solutions eventually become smooth and converge to the homogeneous steady state (0, b) with respect to the norm in $(L^\infty (\Omega ))^2$ at least at algebraic rate as $t\rightarrow \infty$.