Stabilization for small mass in a quasilinear parabolic-elliptic-elliptic attraction-repulsion chemotaxis system with density-dependent sensitivity: balanced case

Abstract

This paper deals with the problem of the quasilinear parabolic--elliptic--elliptic attraction-repulsion chemotaxis system with q = p and χα −ξγ = 0: 

1. u =∇·((u+1)m−1∇u−χu(u+1)p−2∇v+ξu(u+1)q−2∇w),

2. 0=∆v+αu−βv,

3. 0 = ∆w+γu−δw

in a bounded domain Ω ⊂ Rn (n ∈ N) with smooth boundary ∂ Ω, where m, p, q ∈ R, χ , ξ , α , β , γ , δ > 0 are constants. In the case that m ̸= 1, p ̸= 2 and q ̸= 2 boundedness and finite-time blow-up have been classified by the sizes of $p, q$ and the sign of χ α − ξ γ (Z. Angew.\ Math.\ Phys.; 2022; 73; 61), where the critical case χ α − ξ γ = 0 has been excluded. The purpose of this paper is to prove boundedness and stabilization in the case χα−ξγ =0.