Blow-up behavior of solutions to a degenerate parabolic–parabolic Keller–Segel system

Abstract

Qualitative properties of solutions blowing up in finite time are obtained for a degenerate parabolic–parabolic Keller–Segel system, the nonlinear diffusion being of porous medium type with an exponent smaller or equal to the critical one $$m_c:=2(N-1)/N$$ . In both cases, it is shown that only type II blow-up is possible, that is, blow-up at the same rate as backward self-similar solutions never occurs. Further information on the generation of singularities induced by mass concentration are given in the critical case.