This research is a natural continuation of the recent paper "Exact solutions of the simplified Keller-Segel model" (Commun Nonlinear Sci Numer Simulat 2013, 18, 2960-2971. It is shown that a (1+2)-dimensional Keller-Segel type system is invariant with respect infinite-dimensional Lie algebra. All possible maximal algebras of invariance of the Neumann boundary value problems based on the Keller-Segel system in question were found. Lie symmetry operators are used for constructing exact solutions of some boundary value problems. Moreover, it is proved that the boundary value problem for the (1+1)-dimensional Keller-Segel system with specific boundary conditions can be linearized and solved in an explicit form.

No abstract

A simplified Keller-Segel model is studied by means of Lie symmetry based approaches. It is shown that a (1 + 2)-dimensional Keller-Segel type system, together with the correctly-specified boundary and/or initial conditions, is invariant with respect to infinite-dimensional Lie algebras. A Lie symmetry classification of the Cauchy problem depending on the initial profile form is presented. The Lie symmetries obtained are used for reduction of the Cauchy problem to that of (1 + 1)-dimensional. Exact solutions of some (1 + 1)-dimensional problems are constructed. In particular, we have proved that the Cauchy problem for the (1 + 1)-dimensional simplified Keller-Segel system can be linearized and solved in an explicit form. Moreover, additional biologically motivated restrictions were established in order to obtain a unique solution. The Lie symmetry classification of the (1 + 2)-dimensional Neumann problem for the simplified Keller-Segel system is derived. Because Lie symmetry of boundary-value problems depends essentially on geometry of the domain, which the problem is formulated for, all realistic (from applicability point of view) domains were examined. Reduction of the the Neumann problem on a strip is derived using the symmetries obtained. As a result, an exact solution of a nonlinear two-dimensional Neumann problem on a finite interval was found.

No abstract