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 \begin{equation}\label{ellip}\tag{$\ast$} \Delta u-\beta u+\lambda\frac{e^u}{\int_{\Omega}e^u}=0~\mathrm{in}~\Omega, \end{equation} where $\Omega$ is a bounded domain in $\mathbb{R}^2$ and $\beta, \lambda$ are positive parameters. The above equation arises as the stationary problem of the well-known classical Keller-Segel model describing chemotaxis. For equation \eqref{ellip} with Neumann boundary condition, we establish an integral inequality and prove that the solution of (\ref{ellip}) is unique if $0<\lambda \leq 8\pi$ and $u$ satisfies some symmetric properties. While for \eqref{ellip} with Dirichlet boundary condition, the same uniqueness result is obtained without symmetric condition by a different approach inspired by some recent works [19,21]. 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 $\Omega$ 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.