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

Abstract

Abstract This paper deals with finite-time blow-up of a hyperbolic Keller–Segel system of consumption type with the logarithmic sensitivity 0\right)$?> ∂ t ρ = − χ ∇ ⋅ ρ ∇ log c , ∂ t c = − μ c ρ χ , μ > 0 in 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 log c diverge) but due to the blowup of C 1 × C 2 -norm of ( ρ , c ) . As a corollary, we also construct initial data near any constant equilibrium state which blows up in finite time for any d ⩾ 1 .