On e-regularity theorem and asymptotic behaviors of solutions for keller-se gel systems

We deal with the equation (KS)_m for the critic al case of q = m+ 2/N with N ≥ 3, m > 1, q ≥ 2: _tu = △_um - ▽ (uq-1▽v), x ∈ ℝ^N, t > 0; 0 = △v - γv + u, x ∈ ℝ^N, t > 0; u(x, 0) = u₀(x), tv(x, 0) = tv₀(x), x ∈ ℝN Based on a e-regularity theorem in [Y. Sugiyama, Partial regularity and blow-up asymptotics of weak solutions to degenerate parabolic systems of porous medium type, submitted], we first show that the set S_u of blow-up points of the weak solution μ has at most the zero- Hausdorff dimension if u ∈ C_w ([0, T]); L ¹ ℝ^N. Next, we give various conditions on the weak solution u so that the set S_u consists of finitely many points. Furthermore, we obtain an explicit constant for e in such a way that if the local concentratio n of mass around some point x ∈ S_u is less than e,then u is in fact locally bounded around x, which may be regarded as a removable singularity theorem. Simultaneously, we shall show that the solution u in C([0, T]; L¹(R^N)) can be continued beyond t = T, which gives an extension criterion in t he scaling invariant class associated with (KS)_m. Copyright by SIAM.