### Abstract

We deal with the equation (KS)_{m} for the critic al case of q = m+ 2/N with N ≥ 3, m > 1, q ≥ 2: _{t}u = △_{u}m - ▽ (uq-1▽v), x ∈ ℝ^{N}, t > 0; 0 = △v - γv + u, x ∈ ℝ^{N}, t > 0; u(x, 0) = u_{0}(x), tv(x, 0) = tv_{0}(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]; L^{1}(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.

Original language | English |
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Pages (from-to) | 1664-1692 |

Number of pages | 29 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 41 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2009 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Computational Mathematics
- Applied Mathematics

### Cite this

*SIAM Journal on Mathematical Analysis*,

*41*(4), 1664-1692. https://doi.org/10.1137/080721078