Self-similar Solutions to a Parabolic System Modelling Chemotaxis

We study the forward self-similar solutions to a parabolic system modeling chemotaxis ut=∇·(∇u-u∇v), rvt=∇v+u in the whole space R2, where τ is a positive constant. Using the Liouville-type result and the method of moving planes, it is proved that self-similar solutions (u,v) must be radially symmetric about the origin. Then the structure of the set of self-similar solutions is investigated. As a consequence, it is shown that there exists a threshold in ∫R2u for the existence of self-similar solutions. In particular, for 0<r≤1/2, there exists a self-similar solution (u,v) if and only if ∫R2u