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• W2797549917 abstract "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+lambdafrac{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 $0<lambda leq 8pi$ 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." @default.
• W2797549917 created "2018-04-24" @default.
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• W2797549917 date "2018-04-09" @default.
• W2797549917 modified "2023-09-27" @default.
• W2797549917 title "Uniqueness and convergence on equilibria of the Keller-Segel system with subcritical mass" @default.
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• W2797549917 doi "https://doi.org/10.48550/arxiv.1804.03319" @default.
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