Global stabilization of the full attraction-repulsion Keller-Segel system

Hai Yang Jin, Zhi An Wang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

7 Citations (Scopus)

Abstract

We are concerned with the following full Attraction-Repulsion Keller-Segel (ARKS) system x ∈ Ω, t > 0, x ∈ Ω, t > 0u w u vt ( t t x = = = 0) D D 1 u = 2 u vw0 + (+ x · αu ) (γu χu v ( x βv, v δw, 0) ) + = v 0 · ( ( x ξu ) w w (x ) 0) = w0(x) x ∈ Ω (∗) x ∈ Ω, t > 0, in a bounded domain Ω ⊂ R2 with smooth boundary subject to homogeneous Neumann boundary conditions. By constructing an appropriate Lyapunov functions, we establish the boundedness and asymptotical behavior of solutions to the system (∗) with large initial data (u0, v0, w0) ∈ [W1,∞(Ω)]3. Precisely, we show that if the parameters satisfy χα ξγ ≥ max n D D 1 2 D 1 , δ, β δ o for D2 β all positive parameters D1, D2, χ, ξ, α, β, γ and δ, the system (∗) has a unique global classical solution (u, v, w), which converges to the constant steady state (ū0 α β ū0 γ δ ū0) as t → +∞, where ū0 = | Ω 1 | R Ω u0dx. Furthermore, the decay rate is exponential if χα ξγ > max n β δ β δ o . This paper provides the first results on the full ARKS system with unequal chemical diffusion rates (i.e. D1 6= D2) in multi-dimensions.

Original languageEnglish
Pages (from-to)3509-3527
Number of pages19
JournalDiscrete and Continuous Dynamical Systems- Series A
Volume40
Issue number6
DOIs
Publication statusPublished - Jun 2020

Keywords

  • Attraction-repulsion
  • Chemotaxis
  • Exponential decay rate
  • Global stability

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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