Finite-time blowup in Cauchy problem of parabolic-parabolic chemotaxis system

This paper is concerned with blowup in a parabolic-parabolic system describing chemotactic aggregation. In a disk, radial solutions blow up in finite time if their initial energy is less than some value. In the whole plane, the energy diverges to −∞ as time goes to +∞ for any forward selfsimilar solution. This implies that one cannot expect to get a sufficient condition for finite-time blowup using energy as in a disk. For a solution ( u , v ) , u and v denote density of cells and of chemical substance, respectively. Let τ be the coefficient of time derivative of v. We first prove that for τ > 0 there exists M ( τ ) > 0 with M ( τ ) → ∞ as τ → ∞ such that all radial solutions ( u , v ) with initial mass of u larger than M ( τ ) blow up in finite time. On the other hand, it was shown in [22] that any blowup in the system with τ = 1 is type II (not necessarily in radial case). Removing the restriction on τ, we get the conclusion for all τ > 0 .