Radially symmetric stationary solutions for certain chemotaxis systems in higher dimensions: A geometric approach

Abstract

This paper reports results on the existence, shapes, and asymptotic behavior of positive radially symmetric stationary solutions for several models of chemotaxis in higher dimensions. The systems treated in this paper are the simplest parabolic-elliptic Keller-Segel model and the simplest attraction-repulsion chemotaxis system, in which a positive-valued (resp. negative-valued) solution of one is a negative-valued (resp. positive-valued) solution of the other from symmetry. In particular, the construction of functions satisfying equations that diverge at the endpoints of finite intervals is an interesting result. The key to the discussion is to derive a scalar equation by using a transformation on the averaged mass for the equation satisfied by the radially symmetric stationary solution and to investigate the infinity dynamics as geometric information for the two-dimensional ordinary differential equations derived from it. To achieve this, we use a method that combines classical results from the continuous dynamical systems theory and Poincaré-Lyapunov compactification. In addition, the results for singular solutions are discussed in light of the results of previous studies.