Details
| Original language | English |
|---|---|
| Article number | 103022 |
| Journal | Nonlinear Analysis: Real World Applications |
| Volume | 52 |
| Publication status | Published - Apr 2020 |
| Externally published | Yes |
Abstract
The Neumann initial–boundary problem for the chemotaxis system u t=Δu−∇⋅(u∇v)+κ(|x|)u−μ(|x|)u p,0=Δv− [Formula presented] +u,m(t)≔∫ Ωu(⋅,t)is studied in a ball Ω=B R(0)⊂R 2, R>0 for p≥1 and sufficiently smooth functions κ,μ:[0,R]→[0,∞). We prove that whenever μ ′,−κ ′≥0 as well as μ(s)≤μ 1s 2p−2 for all s∈[0,R] and some μ 1>0 then for all m 0>8π there exists u 0∈C 0(Ω¯) with ∫ Ωu 0=m 0 and a solution (u,v) to (⋆) with initial datum u 0 blowing up in finite time. If in addition κ≡0 then all solutions with initial mass smaller than 8π are global in time, displaying a certain critical mass phenomenon. On the other hand, if p>2, we show that for all μ satisfying μ(s)≥μ 1s p−2−ε for all s∈[0,R] and some μ 1,ε>0 the system (⋆) admits a global classical solution for each initial datum 0≤u 0∈C 0(Ω¯).
Keywords
- Chemotaxis, Critical mass, Finite-time blow-up, Logistic source
ASJC Scopus subject areas
- Engineering(all)
- Mathematics(all)
- Computational Mathematics
- Mathematics(all)
- Analysis
- Economics, Econometrics and Finance(all)
- Mathematics(all)
- Applied Mathematics
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In: Nonlinear Analysis: Real World Applications, Vol. 52, 103022, 04.2020.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Finite-time blow-up in a two-dimensional Keller–Segel system with an environmental dependent logistic source
AU - Fuest, Mario
N1 - Publisher Copyright: © 2019 Elsevier Ltd
PY - 2020/4
Y1 - 2020/4
N2 - The Neumann initial–boundary problem for the chemotaxis system u t=Δu−∇⋅(u∇v)+κ(|x|)u−μ(|x|)u p,0=Δv− [Formula presented] +u,m(t)≔∫ Ωu(⋅,t)is studied in a ball Ω=B R(0)⊂R 2, R>0 for p≥1 and sufficiently smooth functions κ,μ:[0,R]→[0,∞). We prove that whenever μ ′,−κ ′≥0 as well as μ(s)≤μ 1s 2p−2 for all s∈[0,R] and some μ 1>0 then for all m 0>8π there exists u 0∈C 0(Ω¯) with ∫ Ωu 0=m 0 and a solution (u,v) to (⋆) with initial datum u 0 blowing up in finite time. If in addition κ≡0 then all solutions with initial mass smaller than 8π are global in time, displaying a certain critical mass phenomenon. On the other hand, if p>2, we show that for all μ satisfying μ(s)≥μ 1s p−2−ε for all s∈[0,R] and some μ 1,ε>0 the system (⋆) admits a global classical solution for each initial datum 0≤u 0∈C 0(Ω¯).
AB - The Neumann initial–boundary problem for the chemotaxis system u t=Δu−∇⋅(u∇v)+κ(|x|)u−μ(|x|)u p,0=Δv− [Formula presented] +u,m(t)≔∫ Ωu(⋅,t)is studied in a ball Ω=B R(0)⊂R 2, R>0 for p≥1 and sufficiently smooth functions κ,μ:[0,R]→[0,∞). We prove that whenever μ ′,−κ ′≥0 as well as μ(s)≤μ 1s 2p−2 for all s∈[0,R] and some μ 1>0 then for all m 0>8π there exists u 0∈C 0(Ω¯) with ∫ Ωu 0=m 0 and a solution (u,v) to (⋆) with initial datum u 0 blowing up in finite time. If in addition κ≡0 then all solutions with initial mass smaller than 8π are global in time, displaying a certain critical mass phenomenon. On the other hand, if p>2, we show that for all μ satisfying μ(s)≥μ 1s p−2−ε for all s∈[0,R] and some μ 1,ε>0 the system (⋆) admits a global classical solution for each initial datum 0≤u 0∈C 0(Ω¯).
KW - Chemotaxis
KW - Critical mass
KW - Finite-time blow-up
KW - Logistic source
UR - http://www.scopus.com/inward/record.url?scp=85071624396&partnerID=8YFLogxK
U2 - 10.1016/j.nonrwa.2019.103022
DO - 10.1016/j.nonrwa.2019.103022
M3 - Article
VL - 52
JO - Nonlinear Analysis: Real World Applications
JF - Nonlinear Analysis: Real World Applications
SN - 1468-1218
M1 - 103022
ER -