TY - JOUR

T1 - Approaching optimality in blow-up results for Keller–Segel systems with logistic-type dampening

AU - Fuest, Mario

N1 - Publisher Copyright: © 2021, The Author(s).

PY - 2021/3

Y1 - 2021/3

N2 - Nonnegative solutions of the Neumann initial-boundary value problem for the chemotaxis system [Figure not available: see fulltext.] 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)andμ>0orn≥5,κ=2andμ∈(0,n-4n),for balls Ω ⊂ R n and parameters λ≥ 0 , m> 0 , we construct a nonnegative initial datum u∈ C(Ω ¯) with ∫ Ωu= m for which the corresponding solution (u, v) of (⋆) blows up in finite time. Moreover, in 3D, we obtain finite-time blow-up for κ∈(1,32) (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)=∫0snρn-1u(ρ,t)dρ fulfills the estimate ws≤ws. Using this information, we then obtain finite-time blow-up of u by showing that for suitably chosen initial data, s and γ, the function ϕ(t)=∫0s0s-γ(s0-s)w(s,t) cannot exist globally.

AB - Nonnegative solutions of the Neumann initial-boundary value problem for the chemotaxis system [Figure not available: see fulltext.] 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)andμ>0orn≥5,κ=2andμ∈(0,n-4n),for balls Ω ⊂ R n and parameters λ≥ 0 , m> 0 , we construct a nonnegative initial datum u∈ C(Ω ¯) with ∫ Ωu= m for which the corresponding solution (u, v) of (⋆) blows up in finite time. Moreover, in 3D, we obtain finite-time blow-up for κ∈(1,32) (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)=∫0snρn-1u(ρ,t)dρ fulfills the estimate ws≤ws. Using this information, we then obtain finite-time blow-up of u by showing that for suitably chosen initial data, s and γ, the function ϕ(t)=∫0s0s-γ(s0-s)w(s,t) cannot exist globally.

KW - Chemotaxis

KW - Finite-time blow-up

KW - Logistic source

UR - http://www.scopus.com/inward/record.url?scp=85101903329&partnerID=8YFLogxK

U2 - 10.1007/s00030-021-00677-9

DO - 10.1007/s00030-021-00677-9

M3 - Article

VL - 28

JO - Nonlinear Differential Equations and Applications NoDEA

JF - Nonlinear Differential Equations and Applications NoDEA

SN - 1021-9722

IS - 2

M1 - 16

ER -