A blow-up result for a quasilinear chemotaxis system with logistic source in higher dimensions

In this paper we consider the quasilinear chemotaxis system {u_t = ∇ ⋅ (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 − bs^κ 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.