The existence of generalised global supersolutions with a control upon the total muss is established for the parabolic–parabolic Keller–Segel system with logarithmic sensitivity for any space dimension. It is verified that smooth supersolutions of this sort are actually classical solutions. Unlike the previously existing constructions, neither is the chemotactic sensitivity coefficient required to be small, nor is it necessary for the initial data to be radially symmetric.
Zhigun, A. (2018). Generalised supersolutions with mass control for the Keller–Segel system with logarithmic sensitivity. Journal of Mathematical Analysis and Applications, 467(2), 1270–1286. https://doi.org/10.1016/j.jmaa.2018.08.001