Properties of blow-up solutions and their initial data for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type

This paper is concerned with blow-up solutions to the quasilinear degenerate Keller-Segel systems of parabolic-parabolic type{ut=∇⋅(∇um−uq−1∇v),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0 under homogeneous Neumann boundary conditions and initial conditions, where Ω⊂RN (N≥3), m≥1, q≥2. As the basis on this study, it was recently shown that there exist radial initial data such that the corresponding solutions blow up in the case q>m+2N ([5]). In the parabolic-elliptic case Sugiyama [27] established behavior of blow-up solutions; however, behavior in the parabolic-parabolic case has not been studied. The purpose of this paper is to give many finite-time blow-up solutions and behavior of blow-up solutions in a neighborhood of blow-up time in the parabolic-parabolic case.