Conservation laws, symmetry reductions, and exact solutions of some Keller–Segel models

In this paper, three Keller–Segel models are considered from the point of Lie symmetry analysis, conservation laws, symmetry reduction, and exact solutions. By means of Lie symmetry analysis, we first obtain all the symmetries for the three models. Based on the obtained symmetries, many non-trivial and explicit conservation laws for the three models are obtained with the help of Ibragimov’s new conservation theorem. Applying the characteristic equations of the obtained symmetries, symmetry reductions and exact solutions are obtained, including solutions expressed by rational functions and Bessel functions.