@article{Qiu2023, 
author = {Zixuan Qiu and Bin Li},
title = {Eventual smoothness of generalized solutions to a singular chemotaxis system for urban crime in space dimension 2},
year = {2023},
journal = {Electronic Research Archive},
volume = {31},
number = {6},
pages = {3218-3244},
keywords = {chemotaxis system, generalized solution, eventual smoothness, asymptotic behavior},
url = {https://www.sciopen.com/article/10.3934/era.2023163},
doi = {10.3934/era.2023163},
abstract = {This paper is concerned with a chemotaxis system in a two-dimensional setting as follows:               {                                                u            t                    =          Δ          u          −          χ          ∇          ⋅                      (                          u              ∇              ln              ⁡              v                        )                    −          κ          u          v          +          r          u          −          μ                      u            2                    +                      h            1                    ,                                                          v            t                    =          Δ          v          −          v          +          u          v          +                      h            2                    ,                                                                      (    ⋆    )  with the parameters    χ  ,  κ  ,  μ  >  0 and    r  ∈      R  , and with the given functions        h    1    ,      h    2    ≥  0. This model was originally introduced by Short et al for urban crime with the particular values    χ  =  2  ,  r  =  0 and    μ  =  0, and the logistic source term    r  u  −  μ      u    2   was incorporated into (   ⋆) by Heihoff to describe the fierce competition among criminals. Heihoff also proved that the initial-boundary value problem of (   ⋆) possesses a global generalized solution in the two-dimensional setting. The main purpose of this paper is to show that such a generalized solution becomes bounded and smooth at least eventually. In addition, the long-time asymptotic behavior of such a solution is discussed.}
}