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

Abstract

This paper deals with finite-time blow-up of a hyperbolic Keller–Segel system of consumption type with the logarithmic sensitivity 0\\right)$?> ∂tρ=−χ∇⋅ρ∇logc,∂tc=−μcρχ,μ>0 in Rd(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 logc diverge) but due to the blowup of C1×C2 -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 .