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

Abstract

We consider the effects 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 $\gamma, \alpha, \chi$. In particular, for $\gamma=0,$ 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 $L^{\frac{N}{2}}$-norm plays an important role in construction of the time global and blow-up solutions of (KS). Furthermore, we give essential examples of small-$L^1$ initial data which yield blow-up solutions. Consequently, we give the answer to the conjecture by Childress-Percus [2] for $N \ge 3$; i.e., that even though the $L^1$-norm of the initial data is small, the blow-up solutions of (KS) exist in the case of $N \ge 3$. This implies that the smallness of the $L^1$-norm of the initial data does not give us any criterion on the existence of global solutions except when $N=2$.