Classification of type I and type II behaviors for a supercritical nonlinear heat equation
Classification of type I and type II behaviors for a supercritical nonlinear heat equation
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DOI：
10.1016/j.jfa.2008.05.021
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发表时间：
2009-02
影响因子：
1.7
通讯作者：
H. Matano;F. Merle
中科院分区：
文献类型：
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作者：
H. Matano;F. Merle
在Dirichlet边界条件下，研究了非线性热方程ut=Δu+|u|p - 1u径向对称解在RNor上的爆破问题。我们假设p>pS:=N+2N−2，并且初始数据是有界的，可能会改变符号。我们的第一个目标是建立I型和II型爆炸的各种特征。在许多其他条件中，我们证明下列条件是等价的：(a)爆炸是II型的；(b)当s→∞时，重标解w（y,s）收敛于φ∗(y)或- φ∗(y)，其中φ∗表示奇异平稳解；(c) u(x，T)/φ∗(x)在x→0时趋于±1，其中T为爆破时间。我们的第二个目标是研究爆炸之外的延续。除其他外，我们证明了如果一个膨胀是I型且不完全的，那么它的极限l1延拓在膨胀后立即变得平滑，并且I型膨胀意味着“I型正则化”，也就是说，[公式：见文本]有界为t。我们还给出了完全爆炸和不完全爆炸的各种标准。
We study blow-up of radially symmetric solutions of the nonlinear heat equation ut=Δu+|u|p−1u either on RNor on a finite ball under the Dirichlet boundary conditions. We assume p>pS:=N+2N−2 and that the initial data is bounded, possibly sign-changing. Our first goal is to establish various characterizations of type I and type II blow-ups. Among many other things we show that the following conditions are equivalent: (a) the blow-up is of type II; (b) the rescaled solution w(y,s) converges to either φ∗(y) or −φ∗(y) as s→∞, where φ∗denotes the singular stationary solution; (c) u(x,T)/φ∗(x) tends to ±1 as x→0, where T is the blow-up time. Our second goal is to study continuation beyond blow-up. Among other things we show that if a blow-up is of type I and incomplete, then its limit L1continuation becomes smooth immediately after blow-up, and that type I blow-up implies “type I regularization,” that is, [Formula: see text] is bounded as t↘T. We also give various criteria for complete and incomplete blow-ups.