European Journal of Applied Mathematics ( IF 1.9 ) Pub Date : 2023-09-25 , DOI: 10.1017/s0956792523000268 Bin Li , Li Xie
We consider a parabolic-parabolic chemotaxis system with singular chemotactic sensitivity and source functions, which is originally introduced by Short et al to model the spatio-temporal behaviour of urban criminal activities with the particular value of the chemotactic sensitivity parameter $\chi =2$. The available analytical findings for this urban crime model including $\chi =2$ are restricted either to one-dimensional setting, or to initial data and source functions with appropriate smallness, or to initial data and source functions with some radial symmetry. In the present work, our first result asserts that for any $\chi \gt 0$ the initial-boundary value problem of this urban crime model possesses a global generalised solution in the two-dimensional setting, without imposing any small or radial conditions on initial data and source functions. Our second result presents the asymptotic behaviour of such solution, under some additional assumptions on source functions.
中文翻译:
城市犯罪二维抛物线-抛物线趋化系统的广义解:全局存在和大时间行为
我们考虑具有奇异趋化敏感性和源函数的抛物线-抛物线趋化系统,该系统最初由 Short 等人引入,用于模拟城市犯罪活动的时空行为,并具有趋化敏感性参数的特定值