Finite-time blowup in Cauchy problem of parabolic-parabolic chemotaxis system (Q2310804)

The main results of this paper are: radially symmetric solutions of the doubly parabolic Keller-Segel system \[u_t=\Delta u-\nabla\cdot(u\nabla v),\] \[\tau v_t=\Delta v+u,\] in the whole plane \({\mathbb R}^2\) blowup in a finite \(T\) whenever the total mass of the initial condition is sufficiently large: \(\|u_0\|_1>M(\tau)\) for some \(M(\tau)\ge \max\{8\pi,C\tau\}\) with some \(C>0\). Moreover, even for nonsymmetric solutions, the blowup is of type II, i.e., it is not true that \(\|u(t)\|_\infty\le \frac{c}{T-t}\) with the blowup time \(T\) and some \(c<\infty\). The proofs involve modified energy estimates and the second moment of solutions.