Approaching optimality in blow-up results for Keller–Segel systems with logistic-type dampening

• Published in: Nonlinear differential equations and applications Vol. 28; no. 2
• Main Author: Fuest, Mario
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.03.2021; Springer Nature B.V
• Subjects: Analysis; Boundary value problems; Mathematics; Mathematics and Statistics; Logistic source; Chemotaxis; Finite-time blow-up
• ISSN: 1021-9722; 1420-9004
• DOI: 10.1007/s00030-021-00677-9

Nonnegative solutions of the Neumann initial-boundary value problem for the chemotaxis system in smooth bounded domains Ω ⊂ R n , n ≥ 1 , are known to be global in time if λ ≥ 0 , μ > 0 and κ > 2 . In the present work, we show that the exponent κ = 2 is actually critical in the four- and higher dimensional setting. More precisely, if n ≥ 4 , κ ∈ ( 1 , 2 ) and μ > 0 or n ≥ 5 , κ = 2 and μ ∈ 0 , n - 4 n , for balls Ω ⊂ R n and parameters λ ≥ 0 , m 0 > 0 , we construct a nonnegative initial datum u 0 ∈ C 0 ( Ω ¯ ) with ∫ Ω u 0 = m 0 for which the corresponding solution ( u ,  v ) of ( ⋆ ) blows up in finite time. Moreover, in 3D, we obtain finite-time blow-up for κ ∈ ( 1 , 3 2 ) (and λ ≥ 0 , μ > 0 ). As the corner stone of our analysis, for certain initial data, we prove that the mass accumulation function w ( s , t ) = ∫ 0 s n ρ n - 1 u ( ρ , t ) d ρ fulfills the estimate w s ≤ w s . Using this information, we then obtain finite-time blow-up of u by showing that for suitably chosen initial data, s 0 and γ , the function ϕ ( t ) = ∫ 0 s 0 s - γ ( s 0 - s ) w ( s , t ) cannot exist globally.