Uniqueness and convergence on equilibria of the Keller–Segel system with subcritical mass

This paper is concerned with the uniqueness of solutions to the following nonlocal semi-linear elliptic equation (Formula presented.) where Ω is a bounded domain in R ² and ß,λ are positive parameters. The above equation arises as the stationary problem of the well-known classical Keller–Segel model describing chemotaxis. For Eq. (*) with Neumann boundary condition, we establish an integral inequality and prove that the solution of Eq. (*) is unique if 0<λ≤8π and u satisfies some symmetric properties. While for Eq. (*) with Dirichlet boundary condition, the same uniqueness result is obtained without symmetric condition by a different approach inspired by some recent works (Gui and Moradifam, 2018, Invent. Math. 214(3):1169–1204; Gui and Moradifam, Proc. Am. Math. Soc. 146(3):1231–1124). As an application of the uniqueness results, we prove that the radially symmetric solution of the classical Keller–Segel system with subcritical mass subject to Neumann boundary conditions will converge to the unique constant equilibrium as time tends to infinity if Ω is a disc in two dimensions. As far as we know, this is the first result that asserts the exact asymptotic behavior of solutions to the classical Keller–Segel system with subcritical mass in two dimensions.