Details
| Original language | English |
|---|---|
| Article number | 103868 |
| Journal | Nonlinear Analysis: Real World Applications |
| Volume | 73 |
| Early online date | 23 Mar 2023 |
| Publication status | Published - Oct 2023 |
Abstract
We discuss the influence of possible spatial inhomogeneities in the coefficients of logistic source terms in parabolic–elliptic chemotaxis-growth systems of the form ut=Δu−∇⋅(u∇v)+κ(x)u−μ(x)u2,0=Δv−v+u in smoothly bounded domains Ω⊂R2. Assuming that the coefficient functions satisfy κ,μ∈C0(Ω¯) with μ≥0 we prove that finite-time blow-up of the classical solution can only occur in points where μ is zero, i.e. that the blow-up set B is contained in {x∈Ω¯∣μ(x)=0}.Moreover, we show that whenever μ(x0)>0 for some x0∈Ω¯, then one can find an open neighbourhood U of x0 in Ω¯ such that u remains bounded in U throughout evolution.
Keywords
- Blow-up set, Chemotaxis, Logistic source, Spatial heterogeneity, Spatially local bounds
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
- Engineering(all)
- General Engineering
- Economics, Econometrics and Finance(all)
- Mathematics(all)
- Computational Mathematics
- Mathematics(all)
- Applied Mathematics
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In: Nonlinear Analysis: Real World Applications, Vol. 73, 103868, 10.2023.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Possible points of blow-up in chemotaxis systems with spatially heterogeneous logistic source
AU - Black, Tobias
AU - Fuest, Mario
AU - Lankeit, Johannes
AU - Mizukami, Masaaki
N1 - Funding Information: T.B. acknowledges support of the Deutsche Forschungsgemeinschaft, Germany in the context of the project Emergence of structures and advantages in cross-diffusion systems (No. 411007140 , GZ: WI 3707/5-1).
PY - 2023/10
Y1 - 2023/10
N2 - We discuss the influence of possible spatial inhomogeneities in the coefficients of logistic source terms in parabolic–elliptic chemotaxis-growth systems of the form ut=Δu−∇⋅(u∇v)+κ(x)u−μ(x)u2,0=Δv−v+u in smoothly bounded domains Ω⊂R2. Assuming that the coefficient functions satisfy κ,μ∈C0(Ω¯) with μ≥0 we prove that finite-time blow-up of the classical solution can only occur in points where μ is zero, i.e. that the blow-up set B is contained in {x∈Ω¯∣μ(x)=0}.Moreover, we show that whenever μ(x0)>0 for some x0∈Ω¯, then one can find an open neighbourhood U of x0 in Ω¯ such that u remains bounded in U throughout evolution.
AB - We discuss the influence of possible spatial inhomogeneities in the coefficients of logistic source terms in parabolic–elliptic chemotaxis-growth systems of the form ut=Δu−∇⋅(u∇v)+κ(x)u−μ(x)u2,0=Δv−v+u in smoothly bounded domains Ω⊂R2. Assuming that the coefficient functions satisfy κ,μ∈C0(Ω¯) with μ≥0 we prove that finite-time blow-up of the classical solution can only occur in points where μ is zero, i.e. that the blow-up set B is contained in {x∈Ω¯∣μ(x)=0}.Moreover, we show that whenever μ(x0)>0 for some x0∈Ω¯, then one can find an open neighbourhood U of x0 in Ω¯ such that u remains bounded in U throughout evolution.
KW - Blow-up set
KW - Chemotaxis
KW - Logistic source
KW - Spatial heterogeneity
KW - Spatially local bounds
UR - http://www.scopus.com/inward/record.url?scp=85150765119&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2209.14184
DO - 10.48550/arXiv.2209.14184
M3 - Article
AN - SCOPUS:85150765119
VL - 73
JO - Nonlinear Analysis: Real World Applications
JF - Nonlinear Analysis: Real World Applications
SN - 1468-1218
M1 - 103868
ER -