We consider a class of logarithmic Keller-Segel type systems modeling the spatio-temporal behavior of either chemotactic cells or criminal activities in spatial dimensions two and higher. Under certain assumptions on parameter values and given functions, the existence of classical solutions is established globally in time, provided that initial data are sufficiently regular. In particular, we enlarge the range of chemotactic sensitivity χ, compared to known results, in case that spatial dimensions are between two and eight. In addition, we provide new type of small initial data to obtain global classical solution, which is also applicable to the urban crime model. We discuss long-time asymptotic behaviors of solutions as well.
Bibliographical noteFunding Information:
J. Ahn is supported by the Dongguk University Research Fund of 2020. K. Kang is partially supported by NRF-2019R1A2C1084685 and NRF-2015R1A5A1009350. J. Lee is supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1701-05.
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All Science Journal Classification (ASJC) codes
- Applied Mathematics