Non-trivial ω -limit sets and oscillating solutions in a chemotaxis model in R2R2 with critical mass

This paper studies the Cauchy problem for a parabolic–elliptic system in R2R2 modeling chemotaxis as well as self-attracting particles. In the critical mass case the fine dynamics of the model is ascertained in terms of the structure of the underlying ω  -limit sets. According to the results of this paper, any nonnegative radially symmetric bounded solution either stabilizes to a steady-state as t↑∞t↑∞, or oscillates between two steady-states. Moreover, a rather general class of nonnegative initial data, not necessarily radially symmetric, for which the associated solutions exhibit a complex oscillatory behavior is constructed; their ω-limit sets consist of a nontrivial topological continuum of steady-states. Besides the technical difficulties inherent to the lack of compactness of the resolvent operators, one has to add the challenge that the problem is utterly non-local. Consequently, thought the basic ideas on the foundations of this paper might be considered classical, most of the proofs throughout are extremely sophisticated and absolutely new in their full generality.