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

Abstract

This paper is concerned with the uniqueness of solutions to the following nonlocal semi-linear elliptic equation \begin{equation}\label{ellip}\tag{$\ast$} Δu-βu+λ\frac{e^u}{\int_Ωe^u}=0~\mathrm{in}~Ω, \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 $0<\lambda \le 8\pi$ 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.

24 pages, 1 figure

References in corpus (1)

• The Sphere Covering Inequality and Its Applications

Cited by in corpus (1)

• Critical mass on the Keller-Segel system with signal-dependent motility