一类城市犯罪模型一维解的整体存在性和定性性态
On the global existence and qualitative behavior of one-dimensional solutions to a model for urban crime
摘要：
我们考虑了交叉扩散演化系统\begin{eqnarray*} \left\{ \begin{array}{ll} u_t = u_{xx} - χ\big(\frac{u}{v} \partial_x u \big)_x - uv +B_1(x,t), \qquad & x\in Ω, \ t>0, \\[1mm] v_t = v_{xx} +uv - v + B_2(x,t), \qquad & x\in Ω, \ t>0, \end{array} \right. \end{eqnarray*}的无通量初边值问题，该系统由Short等人在[Short2008]中引入，其中$χ=2$描述了有界区间$Ω\subset\mathbb{R}$中的城市犯罪动力学，并规定了适当的正则非负函数$B_1$和$B_2$，我们首先证明了对于任何选择的$χ>0$和所有合理正则的非负初始数据，整体经典解的存在性。接下来，我们解决的问题，确定的定性行为的解决方案在适当的假设下的渐近性质的$B_1$和$B_2$。事实上，对于任意的$χ>0$，我们得到严格的正性的平均$B_2$的域的解决方案的有界性，此外，可以看出，施加一个温和的衰减假设$B_1$意味着$u$必须衰减到零的长期限制。我们的最终结果对所有包含相关值$χ=2$的$χ\in\left(0,\frac{\sqrt{6\sqrt{3}+9}}{2}\right),$都有效，表明在上述关于$B_1$的衰减假设下，如果进一步$B_2$适当地稳定到非平凡函数$B_{2,\infty}$，则$(u,v)$接近极限$(0,v_\infty)$，其中$v_\infty$表示\begin{eqnarray*} \left\{ \begin{array}{l} -\partial_{xx}v_\infty + v_\infty = B_{2,\infty}, \qquad x\in Ω, \\[1mm] \partial_x v_{\infty}=0, \qquad x\in\partialΩ. \end{array} \right. \end{eqnarray*}的解
英文摘要：
We consider the no-flux initial-boundary value problem for the cross-diffusive evolution system \begin{eqnarray*} \left\{ \begin{array}{ll} u_t = u_{xx} - χ\big(\frac{u}{v} \partial_x u \big)_x - uv +B_1(x,t), \qquad & x\in Ω, \ t>0, \\[1mm] v_t = v_{xx} +uv - v + B_2(x,t), \qquad & x\in Ω, \ t>0, \end{array} \right. \end{eqnarray*} which was introduced by Short et al. in [Short2008] with $χ=2$ to describe the dynamics of urban crime In bounded intervals $Ω\subset\mathbb{R}$ and with prescribed suitably regular nonnegative functions $B_1$ and $B_2$, we first prove the existence of global classical solutions for any choice of $χ>0$ and all reasonably regular nonnegative initial data. We next address the issue of determining the qualitative behavior of solutions under appropriate assumptions on the asymptotic properties of $B_1$ and $B_2$. Indeed, for arbitrary $χ>0$ we obtain boundedness of the solutions given strict positivity of the average of $B_2$ over the domain; moreover, it is seen that imposing a mild decay assumption on $B_1$ implies that $u$ must decay to zero in the long-term limit. Our final result, valid for all $χ\in\left(0,\frac{\sqrt{6\sqrt{3}+9}}{2}\right),$ which contains the relevant value $χ=2$, states that under the above decay assumption on $B_1$, if furthermore $B_2$ appropriately stabilizes to a nontrivial function $B_{2,\infty}$, then $(u,v)$ approaches the limit $(0,v_\infty)$, where $v_\infty$ denotes the solution of \begin{eqnarray*} \left\{ \begin{array}{l} -\partial_{xx}v_\infty + v_\infty = B_{2,\infty}, \qquad x\in Ω, \\[1mm] \partial_x v_{\infty}=0, \qquad x\in\partialΩ. \end{array} \right. \end{eqnarray*}
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