TY - JOUR

T1 - Blow-up criterion via scaling invariant quantities with effect on coefficient growth in Keller-Segel system

AU - Sugiyama, Yoshie

PY - 2010/7/1

Y1 - 2010/7/1

N2 - 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.

AB - 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.

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M3 - Article

AN - SCOPUS:79961025582

SN - 0893-4983

VL - 23

SP - 619

EP - 634

JO - Differential and Integral Equations

JF - Differential and Integral Equations

IS - 7-8

ER -