发表机构
Westlake Institute for Advanced Study, Westlake University（西湖高等研究院，西湖大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究城市犯罪传播模型（含二次逻辑阻尼的偏微分方程组）在无通量边界条件下经典解的全局存在性，给出关于阻尼系数μ的充分条件，并证明在源项正性假设下解时间一致有界。
AI 中文摘要
本文关注以下源于城市犯罪建模的偏微分方程组：\begin{equation} \label{main} \begin{cases} u_t = \Delta u - \chi \nabla \cdot \left( u \dfrac{\nabla v}{v} \right) - uv + B_1 + ru - \mu u^2, \\\\[4pt] v_t = \Delta v - v + uv + B_2, \end{cases} \end{equation} 在光滑有界区域 $\Omega \subset \mathbb{R}^n$（$n \geq 2$）上满足无通量边界条件，其中 $\chi$、$r$ 和 $\mu$ 为正常数。本文证明，若非负源项 $B_1$ 和 $B_2$ 足够正则，且 \begin{equation*} \mu > \frac{3}{n} + \frac{1}{n}\left( \chi n + \frac{n(\chi(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*} 则系统 \eqref{main} 在适当正则的初始数据下具有全局经典解。此外，在假设 \begin{equation*} \label{cond.B2} \inf_{t>0} \int_\Omega B_2(\cdot,t) > 0 \end{equation*} 下，解在时间上一致有界。
英文摘要
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 $Ω\subset \mathbb{R}^n$ with $n \geq 2$, where $χ$, $r$, and $μ$ 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.
