We consider a class of macroscopic models for the spatio-temporal evolution of urban crime, as originally going back to Ref. 29 [M. B. Short, M. R. D’Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl. Sci. 18 (2008) 1249–1267]. The focus here is on the question of how far a certain porous medium enhancement in the random diffusion of criminal agents may exert visible relaxation effects. It is shown that sufficient regularity of the non-negative source terms in the system and a sufficiently strong nonlinear enhancement 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 globally bounded under mild additional conditions on the source terms. These results are supplemented by numerical evidence which illustrates smoothing effects in solutions with sharply structured initial data in the presence of such porous medium-type diffusion and support the existence of singular structures in the linear diffusion case, which is the type of diffusion proposed in Ref. 29.

中文翻译：

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城市犯罪传播二维交叉扩散模型中非线性扩散增强的弛豫

我们考虑了一类城市犯罪时空演变的宏观模型，最初可以追溯到参考文献。29 [MB Short、MR D'Orsogna、VB Pasour、GE Tita、PJ Brantingham、AL Bertozzi 和 LB Chayes，犯罪行为统计模型，数学。模型方法应用程序。科学。18 (2008) 1249–1267]。这里的重点是犯罪分子随机扩散中的某种多孔介质增强作用在多大程度上可以发挥明显的松弛效应的问题。结果表明，系统中非负源项的足够规律性和足够强的非线性增强确保了在平滑有界平面凸域中提出的相应的 Neumann 型初边界值问题，对于宽类任意初始数据。此外，这个解在源条件的温和附加条件下是全局有界的。这些结果得到了数值证据的补充，这些证据说明了在存在这种多孔介质型扩散的情况下具有清晰结构化初始数据的解决方案中的平滑效应，并支持线性扩散情况下奇异结构的存在，这是参考文献中提出的扩散类型. 29.