Blow-up criterion via scaling invariant quantities with effect on coefficient growth in Keller-Segel system

We consider the e ects on a blow-up phenomena of the Keller-Segel system (KS) in terms of the mass and second moment of initial data in connection with three coefficients γ α,x. In particular, for γ = 0, our criterion on blow-up of solutions coincides with the quantity of the scaling invariant class associated with the Keller-Segel system. We also show that the size of the L N/2 -norm plays an important role in construction of the time global and blow-up solutions of (KS). Furthermore, we give essential examples of small-L¹ initial data which yield blow-up solutions. Consequently, we give the answer to the conjecture by Childress-Percus [2] for N ≥ 3; i.e., that even though the L¹-norm of the initial data is small, the blow-up solutions of (KS) exist in the case of N ≥ 3. This implies that the smallness of the L¹-norm of the initial data does not give us any criterion on the existence of global solutions except when N = 2.