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 equationwhere Ω is a bounded domain in R 2 and β, λ are positive parameters. The above equation arises as the stationary problem of the well-known classical Keller-Segel model describing chemotaxis. For equation ( * ) with Neumann boundary condition, we establish an integral inequality and prove that the solution of ( * ) is unique if 0 < λ ≤ 8π and u satisfies some symmetric properties. While for ( * ) 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 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.

On the parabolic‐elliptic Keller–Segel system with signal‐dependent motilities: A paradigm for global boundedness and steady states

“…), which has been extensively studied in the past few decades and a vast number of results have been obtained (cf. previous works 3‐7 and references therein). In contrast, the results of () with non‐constant γ ( v ) and ϕ ( v ) are very few and to the best of our knowledge the existing results are available only for the special case , that is, in (), which simplifies the Keller–Segel system () into ().…”

“…We remark for the radially symmetric domain , if the solution is only required to be constant on the boundary (not necessarily radially symmetric), it was shown in previous work 7 that () only admits a unique constant solution.…”

On the parabolic-elliptic Keller-Segel system with signal-dependent motilities: a paradigm for global boundedness and steady states

“…We remark for the radially symmetric domain Ω ⊂ R 2 , if the solution is only required to be constant on the boundary (not necessarily radially symmetric), it was shown in [40] that (4.10) only admits a unique constant solution.…”

“…[29]), which has been extensively studied in the past few decades and a vast number of results have been obtained (cf. [7,10,16,17,34,40] and references therein). In contrast, the results of (1.1) with non-constant γ(v) and φ(v) are very few and to the best of our knowledge the existing results are available only for the special case φ(v) = −γ ′ (v), i.e.…”

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

“…Precisely, the global bounded solutions exists in one dimension [28]. In space of two dimensions (n = 2), there exists a critical mass m * = 4π χ such that the solution is bounded and asymptotically converges to its unique constant equilibrium if Ω u 0 dx < m * [27,35] and blows up if Ω u 0 dx > m * [18], where Ω u 0 dx denotes the initial cell mass. In the higher dimensions (n ≥ 3), for any Ω u 0 dx > 0, the solution may blow up in finite time [39].…”