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 partial derivative(t)rho = -chi del & sdot; (rho del logc), partial derivative(t)c = -mu c rho (chi,mu > 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 )x C-2 -norm of (rho,c) . As a corollary, we also construct initial data near any constant equilibrium state which blows up in finite time for any d >= 1 .