论文标题
在城市犯罪传播的二维跨扩散模型中，非线性扩散增强放松
Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion model for urban crime propagation
论文作者
论文摘要
我们考虑了一类宏观模型，以实现城市犯罪的时空演化，因为最初可以追溯到Short等人。 （Math。Mod。Meth。Appl。Sci。18，2008）。这里的重点是一个问题，在犯罪分子的随机扩散中某种非线性增强的距离可能会产生可见的放松效果。具体来说，在系统的上下文中\ begin {eqnarray*} \ left \ {\ begin {array} {l} u_t = \ nabla \ cdot（u^{m -1} \ nabla u） - χ\ nabla \ cdot \ cdot \ big（\ frac {u} {v} {v} {v} \ nabla v \ big） - uv + b_1（x） 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凸形域，为广泛的任意初始数据提供了本地界限的解决方案。此外，如果$ b_1 $和$ b_2 $均受到界限，并且$ \ liminf_ {t \ to \ infty} \int_Ωb_2（\ cdot，t）$是正面的，则可以认为该解决方案在全球范围内。这补充了数值证据，除了在存在这种多孔培养基类型扩散机制的情况下，在特定情况下的特定情况下相关的平滑效果外，还表明在线性扩散案例$ M = 1 $中具有显着支持奇异结构的趋势。
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 $m=1$.
