Differential and Integral Equations

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

Yoshie Sugiyama

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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 $N=2$.

Article information

Differential Integral Equations, Volume 23, Number 7/8 (2010), 619-634.

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K57: Reaction-diffusion equations 35K45: Initial value problems for second-order parabolic systems


Sugiyama, Yoshie. Blow-up criterion via scaling invariant quantities with effect on coefficient growth in Keller-Segel system. Differential Integral Equations 23 (2010), no. 7/8, 619--634. https://projecteuclid.org/euclid.die/1356019187

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