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

Abstract

We are concerned with the following full Attraction-Repulsion Keller-Segel (ARKS) system       1. Introduction. To describe the aggregation of Microglia in the central nervous system in Alzhemer's disease due to the interaction of chemoattractant (i.e. βamyloid) and chemorepellent (i.e. TNF-α), Luca et al. [21] proposed the following

Global dynamics in a chemotaxis model describing tumor angiogenesis with/without mitosis in any dimensions

“…However, if Θ < 0 (i.e., attraction prevails over repulsion), based on the availability of Lyapunov functional, a critical mass phenomenon has been found (see [9,15,20] for more details). Recently, in the repulsion dominated case, i.e, Θ ≥ 0, the global stability of constant steady state has been studied in [17,22].…”

Global existence and boundedness in a fully parabolic attraction-repulsion chemotaxis system with signal-dependent sensitivities and logistic source

“…Also, when χ = ξ, Jin-Liu [4] studied the two-and three-dimensional cases. Recently, Jin-Wang [6] derived global existence and boundedness, and stabilization under the condition ξ χ ≥ C with some C > 0 in the two-dimensional setting. In this way, boundedness is well established in the system (1.1) with logistic term.…”

“…As to the system (1.1) with constant sensitivities (i.e., χ(v) ≡ χ, ξ(w) ≡ ξ, where χ, ξ > 0 are constants), global existence and boundedness were obtained in [3,4,5,6]. More precisely, Jin-Wang [5] investigated the one-dimensional case.…”

Boundedness in the higher‐dimensional attraction‐repulsion chemotaxis system involving tensor‐valued sensitivity with saturation

“…Lin et al 41 proved that the solution of () is globally bounded and the corresponding solution of small initial data can converge to steady states for the critical case and . Moreover, Jin and Wang 42 constructed a Lyapunov functional to establish the global boundedness and asymptotic behavior of solutions in some parameter regimes. Liu et al 43 studied the time‐periodic solutions and steady state solutions in the whole parameter regimes by employing local and global bifurcation theory. and D ( u ) ≡ 1.…”