TY - JOUR
T1 - ε-Regularity theorem and its application to the blow-up solutions of Keller-Segel systems in higher dimensions
AU - Sugiyama, Yoshie
PY - 2010/4/1
Y1 - 2010/4/1
N2 - Let us consider (KS)m below for all N ≥ 2 and general exponents m and q. In particular, the 2-D semi-linear case such as N = 2, m = 1 and q = 2 is included. We establish an ε-regularity theorem for weak solutions. As an application, we give an extension criterion in C ([0, T] ; Lfrac(N (q - m), 2) (RN)) which coincides with a scaling invariant class of weak solutions associated with (KS)m. In addition, the Hausdorff dimension of its singular set is zero if u ∈ L∞ (0, T ; Lfrac(N (q - m), 2) (RN)) and ufrac(N (q - m), 2) ∈ Cw ([0, T] ; L1 (RN)).
AB - Let us consider (KS)m below for all N ≥ 2 and general exponents m and q. In particular, the 2-D semi-linear case such as N = 2, m = 1 and q = 2 is included. We establish an ε-regularity theorem for weak solutions. As an application, we give an extension criterion in C ([0, T] ; Lfrac(N (q - m), 2) (RN)) which coincides with a scaling invariant class of weak solutions associated with (KS)m. In addition, the Hausdorff dimension of its singular set is zero if u ∈ L∞ (0, T ; Lfrac(N (q - m), 2) (RN)) and ufrac(N (q - m), 2) ∈ Cw ([0, T] ; L1 (RN)).
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U2 - 10.1016/j.jmaa.2009.11.019
DO - 10.1016/j.jmaa.2009.11.019
M3 - Article
AN - SCOPUS:71249147604
VL - 364
SP - 51
EP - 70
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
SN - 0022-247X
IS - 1
ER -