Oscillating solutions to a parabolic-elliptic system related to a chemotaxis model

Abstract

In this paper, we consider solutions to a Cauchy problem for a parabolic-elliptic system in two dimensional space. This system is a simpli ed version of a chemotaxis model, and is also a model of self-interacting particles. The behavior of solutions to the problem closely depends on the $L^1$-norm of the solutions. If the quantity is larger than 8$\pi$, the solution blows up in nite time. If the quantity is smaller than the critical mass, the solution exists globally in time. In the critical case, in nite blowup solutions are found. In the present paper, we direct our attention to radial solutions to the problem whose $L^1$-norm is equal to 8$\pi$ and nd oscillating solutions.