城市犯罪传播的二维交叉扩散模型中的非线性扩散增强松弛
Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion model for urban crime propagation
摘要：
我们考虑一类宏观模型的时空演变的城市犯罪，最初要回到短等。（数学。应用科学18，2008）。这里的重点是在多大程度上的犯罪代理人的随机扩散的某种非线性增强可能会产生明显的松弛效应的问题。具体而言，在系统\begin{eqnarray*} \left\{ \begin{array}{l} u_t = \nabla \cdot (u^{m-1} \nabla u) - χ\nabla \cdot \Big(\frac{u}{v} \nabla v \Big) - uv + B_1(x,t), \\[1mm] v_t = Δv +uv - v + B_2(x,t), \end{array} \right. \end{eqnarray*}的上下文中，它示出，只要$χ>0$和给定的非负源项$B_1$和$B_2$是足够经常的，假设\begin{eqnarray*} m>\frac{3}{2} \end{eqnarray*}是足以确保相应的Neumann型初边值问题，在一个光滑有界的平面凸区域，承认局部有界的解决方案，为广泛的一类任意初始数据。此外，如果$B_1$和$B_2$都是有界的，并且$\liminf_{t\to\infty} \int_ΩB_2(\cdot,t)$是正的，则该解是全局有界的。这是补充的数值证据，除了说明相关的平滑效果，在特定情况下，尖锐的结构化的初始数据，在这种多孔介质型扩散机制的存在下，表明一个显着的趋势，支持奇异结构的线性扩散情况下$m=1$。
英文摘要：
We consider a class of macroscopic models for the spatio-temporal evolution of urban crime, as originally going back to Short et al. (Math. Mod. Meth. Appl. Sci. 18, 2008). The focus here is on the question how far a certain nonlinear enhancement in the random diffusion of criminal agents may exert visible relaxation effects. Specifically, in the context of the system \begin{eqnarray*} \left\{ \begin{array}{l} u_t = \nabla \cdot (u^{m-1} \nabla u) - χ\nabla \cdot \Big(\frac{u}{v} \nabla v \Big) - uv + B_1(x,t), \\[1mm] v_t = Δv +uv - v + B_2(x,t), \end{array} \right. \end{eqnarray*} it is shown that whenever $χ>0$ and the given nonnegative source terms $B_1$ and $B_2$ are sufficiently regular, the assumption \begin{eqnarray*} m>\frac{3}{2} \end{eqnarray*} is sufficient to ensure that a corresponding Neumann-type initial-boundary value problem, posed in a smoothly bounded planar convex domain, admits locally bounded solutions for a wide class of arbitrary initial data. Furthermore, this solution is seen to be globally bounded if both $B_1$ and $B_2$ are bounded and $\liminf_{t\to\infty} \int_ΩB_2(\cdot,t)$ is positive. This is supplemented by numerical evidence which, besides illustrating associated smoothing effects in particular situations of sharply structured initial data in the presence of such porous medium type diffusion mechanisms, indicates a significant tendency toward support of singular structures in the linear diffusion case $m=1$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
