Large time behavior in a quasilinear chemotaxis model with indirect signal absorption. (September 2022) - Zhang, Wenji; Liu, Suying
Abstract: This paper considers the following chemotaxis system u t = ∇ ⋅ D ( u ) ∇ u − ∇ ⋅ S ( u ) ∇ v, x ∈ Ω, t > 0, v t = Δ v − v w, x ∈ Ω, t > 0, w t = − δ w + u, x ∈ Ω, t > 0, under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n ( n ⩾ 2 ), where the parameter δ > 0 and D, S are smooth functions satisfying D ( u ) ⩾ K 0 ( u + 1 ) α, 0 ⩽ S ( u ) ⩽ K 1 ( u + 1 ) β − 1 u with α, β ∈ R and K 0, K 1 > 0 . Suppose that β − α < 1 n + 1 2, this system admits a global bounded classical solution ( u, v, w ) fulfilling ‖ u ⋅ t − u ̄ 0 ‖ L ∞ Ω + ‖ v ⋅ t − 0 ‖ L ∞ Ω + ‖ w ⋅ t − u ̄ 0 δ ‖ L ∞ Ω → 0 as t → ∞, where u ̄ 0 ≔ 1 | Ω | ∫ Ω u 0 .