Boundedness of solutions to a quasilinear parabolic–parabolic chemotaxis model with variable logistic source

On an attraction-repulsion chemotaxis model involving logistic source

“…Let Ω ⊂ R N (N ≥ 3) be a bounded domain with smooth boundary, the parameters χ, ξ, α 1 , α 2 , β 1 , β 2 > 0. Assume that the nonnegative initial data satisfy (u 0 , υ 0 , ω 0 ) ∈ W 1,∞ (Ω) 3 and f (u) ≤ au − µu 2 with a ≥ 0, µ > 0. Then there is a maximal existence time T max ∈ (0, ∞] and a unique triple (u, υ, ω) of nonnegative bounded functions belong to…”

“…(x, t) ∈ Ω × (0, T ), ∂u ∂ν = ∂υ ∂ν = ∂ω ∂ν = 0, (x, t) ∈ ∂Ω × (0, T ), u(x, 0) = u 0 (x) , ρυ(x, 0) = ρυ 0 (x) , ρω(x, 0) = ρω 0 (x) , x ∈ Ω, (1.1) where Ω ⊂ R N (N ≥ 3) is a bounded domain with smooth boundary, ρ ≥ 0, ∂ ∂ν is the derivative with respect to the outer normal of ∂Ω and nonnegative initial data (u 0 , υ 0 , ω 0 ) satisfying suitable regularity, the parameters χ, ξ, α 1 , α 2 , β 1 , β 2 > 0, where χ and ξ are respectively measure the strength of the attraction and repulsion and u(x, t), υ(x, t), ω(x, t) denote the cell density, the chemoattractant concentration, the chemorepellent concentration, respectively and the logistic source f ∈ C ∞ ([0, ∞)) fulfills f (u) ≤ au − µu 2 with f (0) ≥ 0, a ≥ 0, µ > 0.…”

“…The correlation study can be divided into two types: parabolic-elliptic chemotaxis model i.e. ρ = 0 (see [3,39,42,47]) and parabolic-parabolic chemotaxis model i.e. ρ ̸ = 0 (see [1,2,17,38,41,43,46]).…”

“…ρ = 0 (see [3,39,42,47]) and parabolic-parabolic chemotaxis model i.e. ρ ̸ = 0 (see [1,2,17,38,41,43,46]). We refer the reader to Horstmann's paper [16] to have general knowledge about the Keller-Segel model.…”

Global attractors in a two-species chemotaxis system with two chemicals and variable logistic sources

“…The mathematical study of such models has some difficulties with respect to "classical" models. We refer the interested reader to the recently published articles [3][4][5] with logistic source involving the exponents depending on the spatial variables.…”