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

摘要

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>0under homogeneous Neumann boundary conditions and initial conditions, whereΩ?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 caseq>m+2N(). In the parabolic–elliptic case Sugiyama 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.