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

• Published in: Communications in partial differential equations Vol. 44; no. 7; pp. 545 - 572
• Main Authors: Wang, Jun, Wang, Zhian, Yang, Wen
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 03.07.2019; Taylor & Francis Ltd
• Subjects: Asymptotic properties; Boundary conditions; Convergence; Dirichlet problem; Formulas (mathematics); Graphical user interface; Keller-Segel system; mean field equation; subcritical-mass; uniqueness; Subcritical mass; Uniqueness
• ISSN: 0360-5302; 1532-4133
• DOI: 10.1080/03605302.2019.1581804

This paper is concerned with the uniqueness of solutions to the following nonlocal semi-linear elliptic equation where Ω is a bounded domain in and 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 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.