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How strong a logistic damping can prevent blow-up for the minimal Keller–Segel chemotaxis system'

Research paper by Tian Xiang

Indexed on: 17 Dec '17Published on: 12 Dec '17Published in: Journal of Mathematical Analysis and Applications



Abstract

Publication date: 15 March 2018 Source:Journal of Mathematical Analysis and Applications, Volume 459, Issue 2 Author(s): Tian Xiang We study nonnegative solutions of parabolic–parabolic Keller–Segel minimal-chemotaxis-growth systems with prototype given by { u t = ∇ ⋅ ( d 1 ∇ u − χ u ∇ v ) + κ u − μ u 2 , x ∈ Ω , t > 0 , v t = d 2 Δ v − β v + α u , x ∈ Ω , t > 0 in a smooth bounded smooth but not necessarily convex domain Ω ⊂ R n ( n ≥ 3 ) with nonnegative initial data u 0 , v 0 and homogeneous Neumann boundary data, where d 1 , d 2 , α , β , μ > 0 , χ , κ ∈ R . We provide quantitative and qualitative descriptions of the competition between logistic damping and other ingredients, especially, chemotactic aggregation to guarantee boundedness and convergence. Specifically, we first obtain an explicit formula μ 0 = μ 0 ( n , d 1 , d 2 , α , χ ) for the logistic damping rate μ such that the system has no blow-ups whenever μ > μ 0 . In particular, for Ω ⊂ R 3 , we get a clean formula for μ 0 : μ 0 ( 3 , d 1 , d 2 , α , χ ) = { 3 4 d 1 α χ , if  d 1 = d 2 , χ > 0  and  Ω  is convex , 3 10 − 2 ( 1 d 1 + 2 d 2 ) α χ ,  otherwise . This offers a quantized effect of the logistic source on the prevention of blow-ups. Our result extends the fundamental boundedness principle by Winkler [42] with d 1 = 1 , d 2 = α = β : = 1 / τ , Ω being convex and sufficiently large values of μ beyond a certain number not explicitly known (except the simple case τ = 1 and χ > 0 ) and quantizes the qualitative result of Yang et al. [52]. Besides, in non-convex domains, since μ 0 ( 3 , 1 , 1 , 1 , χ ) = ( 7.743416 ⋯ ) χ , the recent boundedness result, μ > 20 χ , of Mu and Lin [25] is greatly improved. Then we derive another explicit formula: μ 1 = μ 1 ( d 1 , d 2 , α , β , κ , χ ) = α χ 4 κ + d 1 d 2 β for the logistic damping rate so that convergence of bounded solutions is ensured and the respective convergence rates are explicitly calculated out whenever μ > μ 1 . Recent convergence results of He and Zheng [9] are therefore complemented and refined.