The logistic chemotaxis system with singular sensitivity and signal absorption in dimension two. (December 2019) - Wang, Wei
Abstract: In this paper we study the non-flux chemotaxis system: u t = Δ u − χ ∇ ⋅ ( u v ∇ v ) + b u − μ u 2, v t = Δ v − u v, in a bounded and smooth domain Ω ⊂ R 2, with b ∈ R and χ, μ > 0 . The existing literature, on the premise of relatively weak cross-diffusion with 0 < χ < 1, has established the global existence of classical solutions for arbitrary μ > 0 . In the present paper, we extend to prove that for any χ ≥ 1, there exists μ ∗ ( χ ) > 0 such that whenever μ > μ ∗ ( χ ), any reasonably smooth initial data emanates a unique global classical solution. Moreover, it is shown that there exist α ( Ω ) > 0 and β ( Ω ) > 0 such that when b > 0, if μ > α ( Ω ) b + β ( Ω ) for 0 < χ < 1, or μ > max { μ ∗ ( χ ), α ( Ω ) b + β ( Ω ) } for χ ≥ 1, then any classical solution must be globally bounded with the convergence that ( u ( ⋅, t ), | ∇ v ( ⋅, t ) | v ( ⋅, t ), v ( ⋅, t ) ) → ( b μ, 0, 0 ) in [ L ∞ ( Ω ) ] 3 as t → ∞ . Instead, when b ≤ 0, all classical solutions remain uniformly bounded in time as long as μ > 0 for 0 < χ < 1, or μ > μ ∗ ( χ ) for χ ≥ 1, and it also holds that ( u ( ⋅, t ), | ∇ v ( ⋅, t ) | v ( ⋅, t ), v ( ⋅, t ) ) → ( 0, 0, λ ) in [ L ∞ ( Ω ) ] 3 as t → ∞ with 0 ≤ λ < 1 | Ω | ∫ Ω v ( ⋅, 0 ) .