Abstract
We consider classical solutions to a parabolic-elliptic system. The system is a simplified version of a chemotaxis model. It is well known that solutions to the system have many kinds of behaviors. If the initial function is sufficiently small in the sense of a suitable functional space, the solution exists globally in time. If the initial function is sufficiently large in the sense of a suitable functional space, the solution blows up in finite time. In addition, there exist solutions blowing up in infinite time. In this paper, we show a stability of radial stationary solutions, and construct oscillating solutions in high dimensional case.
