A simplified Keller-Segel model: construction of exact solutions for the Cauchy and Neumann problems

A simplified Keller-Segel model is studied by means of Lie symmetry based approaches. It is shown that this (1+2)-dimensional nonlinear system is invariant with respect infinity-dimensional Lie algebra. The result is extended on the Cauchy and Neumann problems for this system. The Lie symmetries obtained are used for reduction of the problems in question to two-dimensional and, as a result, exact solutions of some two-dimensional problems are constructed. In particular, we have proved that the Cauchy problem for the (1+1)-dimensional Keller-Segel type system can be linearized and solved in an explicit form. Moreover, additional biologically motivated restrictions were established in order to obtain uniqueness of solution. An analogous result is also derived for the (1+1)-dimensional Neumann problem with the same governing system. This research is a natural continuation of the paper "Exact solutions of the simplified Keller-Segel model" published in Commun Nonlinear Sci Numer Simulat 2013; 18: 2960-2971. by Cherniha R. and Didovych M.