Determination of blowup type in the parabolic-parabolic Keller-Segel system
Determination of blowup type in the parabolic-parabolic Keller-Segel system
复制标题
抛物线-抛物线 Keller-Segel 系统中爆破类型的确定
DOI：
10.1007/s00208-018-1772-y
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
N. Mizoguchi
中科院分区：
文献类型：
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作者：
E. Latos;T. Suzuki;and Y. Morita;N. Mizoguchi
本文涉及抛物线-抛物线 Keller-Segel 系统 {u_t= ∇ ⋅ (∇ u-u ∇ v) &\quad in\, Ω * (0, T),\v_t= Δ v-α v+ u &\quad in\, Ω * (0, T)。 ut=∇·(∇ u-u∇ v) in Ω×(0, T), vt= Δ v-α v+ u in Ω×(0, T) 常量 α ≥ 0 α≥ 0 且初始数据在光滑有界域 Ω ⊂ R^ 2 Ω⊂ R 2 中，在诺伊曼边界条件下或在 Ω= R^ 2 Ω= R 2 中。作为细菌聚集模型引入，在数学上可翻译为有限时间爆炸。如果\limsup _ t → T ‖ u (t) ‖ _ L^ ∞=+ ∞ lim性sup t→ T‖ u (t)‖ L∞=+∞，则称解 (u, v) 在 t= T<+ ∞ t= T<+∞ 处爆炸。当 (u, v) 在 t= T t= T 处爆炸时，如果 ‖ u (t) ‖ _ L^ ∞ ≤ C (Tt)^-1‖ u (t)‖ L∞≤ C (T-t)-1 对于 t ∈ 0, T) t∈ 0, T) 且某个常数 C> 0 C> 0，则该爆炸称为类型 I，否则为类型 II。 Mizoguchi (J Funct Anal 271: 3323–3347, 2016) 表明，每次爆炸在径向情况下都是 II 型。在本文中，我们在一般情况下得到了结论。
This paper is concerned with a parabolic–parabolic Keller–Segel system {u_t= ∇ ⋅ (∇ u-u ∇ v) &\quad in\, Ω * (0, T),\v_t= Δ v-α v+ u &\quad in\, Ω * (0, T). ut=∇·(∇ u-u∇ v) in Ω×(0, T), vt= Δ v-α v+ u in Ω×(0, T) with a constant α ≥ 0 α≥ 0 and nonnegative initial data in a smoothly bounded domain Ω ⊂ R^ 2 Ω⊂ R 2 under the Neumann boundary condition or in Ω= R^ 2 Ω= R 2. It was introduced as a model of aggregation of bacteria, which is mathematically translated as finite-time blowup. A solution (u, v) is said to blow up at t= T<+ ∞ t= T<+∞ if\limsup _ t → T ‖ u (t) ‖ _ L^ ∞=+ ∞ lim sup t→ T‖ u (t)‖ L∞=+∞. When (u, v) blows up at t= T t= T, the blowup is called type I if ‖ u (t) ‖ _ L^ ∞ ≤ C (Tt)^-1‖ u (t)‖ L∞≤ C (T-t)-1 for t ∈ 0, T) t∈ 0, T) with some constant C> 0 C> 0, and type II otherwise. It was shown in Mizoguchi (J Funct Anal 271: 3323–3347, 2016) that each blowup is type II in radial case. In this paper, we obtain the conclusion in general case.