Finite-time blow-up in hyperbolic Keller–Segel system of consumption type with logarithmic sensitivity

This paper deals with finite-time blow-up of a hyperbolic Keller–Segel system of consumption type with the logarithmic sensitivity <?CDATA ∂ₜ ρ = - χ∇ · (ρ ∇ log c), ∂ₜ c = - μ cρ (χ,\,μ>0)?> ∂ t ρ = − χ ∇ ⋅ ρ ∇ log c , ∂ t c = − μ c ρ χ , μ > 0 in <?CDATA Rᵈ\; (d x2A7E1)?> R d ( d ⩾ 1 ) for nonvanishing initial data. This system is closely related to tumor angiogenesis, an important example of chemotaxis. Our singularity formation is not because c touches zero (which makes <?CDATA log c?> log c diverge) but due to the blowup of <?CDATA C¹× C²?> C 1 × C 2 -norm of <?CDATA (ρ,c)?> ( ρ , c ) . As a corollary, we also construct initial data near any constant equilibrium state which blows up in finite time for any <?CDATA dx2A7E1?> d ⩾ 1 .