A chemotaxis system with singular sensitivity for burglaries in the higher-dimensional settings: generalized solvability and long-time behavior

We study the no-flux initial-boundary value problem of a chemotaxis system with singular sensitivity of the following form ⋆ \(\begin{aligned} \left\{ \begin{aligned}&u_t= \Delta u-\chi \nabla \cdot \left( u\nabla \ln v\right) - u+ g_1, \\&v_t=\Delta v- v+ uv(1-v)+g_2, \end{aligned} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}& u_t=\mathrm{\Delta }u-\chi \mathrm{\nabla }·\left(u,\mathrm{\nabla },ln,v\right)-u+g_1,\hfill \\ \hfill & v_t=\mathrm{\Delta }v-v+uv(1-v)+g_2,\hfill \end{array}\right)\end{array}$$ over a bounded domain \(\Omega \subset {\mathbb {R}}^n\) $\mathrm{\Omega }\subset {\mathbb{R}}^n$ , with chemotaxis coefficient \(\chi >0\) $\chi >0$ and nonnegative source functions \(g_1\) $g_1$ and \(g_2\) $g_2$ , which was proposed by Pitcher to describe the dynamics of burglaries. From the recent results it is known that, if \(n=2\) $n=2$ , then such problem admits a global generalized solution for any initial data and any \(\chi >0\) $\chi >0$ , and possesses a global classical solution for small initial data and small \(\chi \) $\chi$ . This paper presents that for all reasonably regular initial data and any \(\chi >0\) $\chi >0$ the corresponding homogeneous Neumann initial-boundary value problem possesses a generalized solution in higher-dimensional settings (i.e., \(n\ge 3\) $n\ge 3$ ). The asymptotic behavior of generalized solutions is explored as well under some additional assumptions on the source functions.