Details
Original language | English |
---|---|
Article number | 16 |
Journal | Nonlinear Differential Equations and Applications NoDEA |
Volume | 28 |
Issue number | 2 |
Publication status | Published - Mar 2021 |
Externally published | Yes |
Abstract
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.
Keywords
- Chemotaxis, Finite-time blow-up, Logistic source
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
- Mathematics(all)
- Applied Mathematics
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In: Nonlinear Differential Equations and Applications NoDEA, Vol. 28, No. 2, 16, 03.2021.
Research output: Contribution to journal › Article › Research › peer review
}
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 -