城市犯罪模型中的无界现象
Unboundedness phenomenon in a model of urban crime
AI总结：
本文研究城市犯罪反应扩散系统的热点形成，通过 ε→0 的非局部标量极限及爆破构造，反证大初始质量下解可在任意给定时间内无界。
摘要：
我们证明空间模式（“热点”）可能在如下犯罪模型中形成：u_t = (1/ε)Δu − (χ/ε)∇·((u/v)∇v) − εuv，v_t = Δv − v + uv。我们在 Ω=B_R(0)⊂R^n 中考虑该系统，其中 R>0、n≥3，且 ε>0、χ>0；初始数据 u_0、v_0 具有足够大的初始质量 m:=∫_Ωu_0。更确切地说，对于每个 T>0 以及固定的 Ω、χ 和（较大的）m，我们构造初始数据 v_0，使其呈现以下无界现象：给定任意 M>0，都能找到 ε>0，使得相应最大解的第一分量在时间 T 之前于 Ω 中某点变得大于 M。由于 u 的 L^1 范数递减，这意味着必然形成某种异质结构。为此，我们首先在某区间 (0,T) 上明确不发生无界现象的假设下，通过令 ε↘0 取极限，从犯罪模型的解构造非局部标量问题 w_t=Δw+m w^{χ+1}/∫_Ωw^χ 的经典解。随后，我们为该标量问题构造可在时间 T 前导致爆破的初始数据。由于该标量问题的解唯一，这便通过反证法证明了我们的核心结果。
英文摘要：
We show that spatial patterns ("hotspots") may form in the crime model \begin{equation} \left\{\; \begin{aligned} u_{t} &= \tfrac{1}{\varepsilon}Δu - \tfracχ{\varepsilon} \nabla \cdot \left(\tfrac{u}{v} \nabla v \right) - \varepsilon uv, \\ v_{t} &= Δv - v + u v, \end{aligned} \right. \end{equation} which we consider in $Ω= B_R(0) \subset \mathbb R^n$, $R > 0$, $n \geq 3$ with $\varepsilon > 0$, $χ> 0$ and initial data $u_0$, $v_0$ with sufficiently large initial mass $m := \int_Ωu_0$. More precisely, for each $T > 0$ and fixed $Ω$, $χ$ and (large) $m$, we construct initial data $v_0$ exhibiting the following unboundedness phenomenon: Given any $M>0$, we can find $\varepsilon > 0$ such that the first component of the associated maximal solution becomes larger than $M$ at some point in $Ω$ before the time $T$. Since the $L^1$ norm of $u$ is decreasing, this implies that some heterogeneous structure must form. We do this by first constructing classical solutions to the nonlocal scalar problem \[ w_t = Δw + m \frac{w^{χ+1}}{\int_Ωw^χ} \] from the solutions to the crime model by taking the limit $\varepsilon \searrow 0$ under the assumption that the unboundedness phenomenon explicitly does not occur on some interval $(0,T)$. We then construct initial data for this scalar problem leading to blow-up before time $T$. As solutions to the scalar problem are unique, this proves our central result by contradiction.
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。
