TY - JOUR
T1 - A blow-up result for a quasilinear chemotaxis system with logistic source in higher dimensions
AU - Lin, Ke
AU - Mu, Chunlai
AU - Zhong, Hua
PY - 2018/8/1
Y1 - 2018/8/1
N2 - In this paper we consider the quasilinear chemotaxis system {ut = ∇ ⋅ (D(u)∇u) − χ∇ ⋅ (u∇v) + f(u), x ∈ Ω, t > 0, 0 = Δv − μ(t) + u, x ∈ Ω, t > 0, with homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ Rn with n ≥ 2, where χ > 0, μ(t) := [Formula presented]∫Ωu(x,t)dx and f ∈ C ([0, ∞)) ∩ C1((0, ∞)) is a logistic source of the form f(s) = as − bsκ with a ≥ 0, b > 0, κ > 1 and s ≥ 0, and the diffusion D ∈ C2([0, ∞)) is supposed to satisfy D(s) ≥ D0s−m for all s > 0 with some D0 > 0 and m ∈ R. Given any b > 0, when the logistic source is strong enough in the sense that κ > m + 3 −[Formula presented] and κ > 2, it is shown that for any initial data u0 ∈ C0 (Ω¯) and n ≥ 2 the problem possesses a unique global bounded classical solution. However, when D(s)=D0s−m for all s > 0 with [Formula presented] −1 < m ≤ 0 in the sense that n ≥ 5, and the effect of logistic source is weaker in the sense that κ ∈ (1, [Formula presented]), it is shown that for arbitrary prescribed M0 > 0 there exists initial data u0 ∈ C∞(Ω¯) satisfying ∫Ωu0 = M0 such that the corresponding solution (u, v) of the system blows up in finite time in a ball Ω = B0 (R) ⊂ Rn with some R > 0. This result extends the blow-up arguments of the Keller–Segel chemotaxis model with logistic cell kinetics in Winkler [39] to more general quasilinear case. Moreover, since there is a gap in the proof of Zheng et al. [46], it also presents modified results for the mistake.
AB - In this paper we consider the quasilinear chemotaxis system {ut = ∇ ⋅ (D(u)∇u) − χ∇ ⋅ (u∇v) + f(u), x ∈ Ω, t > 0, 0 = Δv − μ(t) + u, x ∈ Ω, t > 0, with homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ Rn with n ≥ 2, where χ > 0, μ(t) := [Formula presented]∫Ωu(x,t)dx and f ∈ C ([0, ∞)) ∩ C1((0, ∞)) is a logistic source of the form f(s) = as − bsκ with a ≥ 0, b > 0, κ > 1 and s ≥ 0, and the diffusion D ∈ C2([0, ∞)) is supposed to satisfy D(s) ≥ D0s−m for all s > 0 with some D0 > 0 and m ∈ R. Given any b > 0, when the logistic source is strong enough in the sense that κ > m + 3 −[Formula presented] and κ > 2, it is shown that for any initial data u0 ∈ C0 (Ω¯) and n ≥ 2 the problem possesses a unique global bounded classical solution. However, when D(s)=D0s−m for all s > 0 with [Formula presented] −1 < m ≤ 0 in the sense that n ≥ 5, and the effect of logistic source is weaker in the sense that κ ∈ (1, [Formula presented]), it is shown that for arbitrary prescribed M0 > 0 there exists initial data u0 ∈ C∞(Ω¯) satisfying ∫Ωu0 = M0 such that the corresponding solution (u, v) of the system blows up in finite time in a ball Ω = B0 (R) ⊂ Rn with some R > 0. This result extends the blow-up arguments of the Keller–Segel chemotaxis model with logistic cell kinetics in Winkler [39] to more general quasilinear case. Moreover, since there is a gap in the proof of Zheng et al. [46], it also presents modified results for the mistake.
KW - Boundedness
KW - Chemotaxis
KW - Finite-time blow up
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U2 - 10.1016/j.jmaa.2018.04.015
DO - 10.1016/j.jmaa.2018.04.015
M3 - 21_Publication in refereed journal
VL - 464
SP - 435
EP - 455
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
SN - 0022-247X
IS - 1
ER -