Global existence of classical solutions to a system modeling propagation of urban crime with quadratic logistic damping

Abstract

We are concerned with the following partial differential equations arising from urban crime modeling: \begin{equation} \label{main} \begin{cases} u_t = Δu - χ\nabla \cdot \left( u \dfrac{\nabla v}{v} \right) - uv + B_1 + ru - μu^2, \\[4pt] v_t = Δv - v + uv + B_2, \end{cases} \end{equation} under no-flux boundary conditions in a smoothly bounded domain $\Omega \subset {\mathbb{R}}^n$ with $n\ge 2$, where $\chi$, $r$, and $\mu$ are positive constants. It is shown in this paper that if the nonnegative source terms $B_1$ and $B_2$ are sufficiently regular and \begin{equation*} μ> \frac{3}{n} + \frac{1}{n}\left( χn + \frac{n(χ(n-1)-2)^2}{4(n-1)} \right)^{\frac{n+1}{n}} \cdot \left( \frac{(2n+\sqrt{n})^2}{2n-1} \right)^{\frac{1}{n}} + n^{12n+3}, \end{equation*} then the system \eqref{main} with suitably regular initial data possesses a global classical solution. Moreover, under the assumption that \begin{equation*} \label{cond.B2} \inf_{t>0} \int_ΩB_2(\cdot,t) > 0, \end{equation*} the solutions are uniformly bounded in time.