## Abstract

We discuss the large time behavior of a weak solution of the Keller-Segel system of degenerate type: { ∂_{t}u- △u^{α} + div(u▽ψ) = 0, t > 0, x ε ℝ^{n},-△ψ + ψ = u, t> 0, x ε ℝ^{n}, u(0,x) = u_{0}(x) ≥ 0, x ε ℝ^{n}, where α > 1. It is known when the exponent a = 2-2/n then the problem shows the critical situation. In this case, we show that the small data global solution decays and its asymptotic profile converges to the Barenblatt-Pattle solution U(t) - (1 + t) ^{-n/σ}(A- |x|^{2}/(1 + t)^{1/(σ-1)}^ in L^{1} such as ||u(t)-u(t)||_{1} ≤C(1 + t)^{-v}, where v > 0 is depending on n and the regularity of the solution. To show this, we employ the forward self-similar transform and use the entropy dissipation term to derive the asymptotic profile due to Carrillo-Toscani [12] and Ogawa [47]. The Hölder continuity of the weak solution for the forward self-similar equation plays a crucial role. We derive the uniform Hölder continuity by using the rescaled alternative selection originated by DiBenedetto-Friedman [18, 19].

Original language | English |
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Pages (from-to) | 375-433 |

Number of pages | 59 |

Journal | Journal of Mathematical Sciences (Japan) |

Volume | 20 |

Issue number | 3 |

Publication status | Published - 2013 |

## Keywords

- Degenerate keller-segel system
- Hölder regularity
- Uniform asymptotic estimates