On the uniqueness of L1-continuation after blowup
On the uniqueness of L1-continuation after blowup
复制标题
论爆炸后L1-延续的唯一性
DOI：
10.1016/j.jfa.2008.01.014
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发表时间：
2008
影响因子：
1.7
通讯作者：
N. Mizoguchi
中科院分区：
文献类型：
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作者：
N. Mizoguchi
本文研究了一类半线性热方程Cauchy问题的L1-延拓在爆破后的唯一性，其中p>1，0<T ∞，u 0 ∈L∞（RN）.这里我们说u是（P）的一个L1-解，如果u∈C（[0，T <$）; Lloc 1（RN）），其中u∈Llocp（RN×（0，T <$））满足（P）在分布意义下的解.在pS<p<pJL的情形下，[M]给出了（P）爆破后径向L1-解的唯一性的反例. Fila，N. Mizoguchi，Multiple continuation beyond blow-up，Differential Integral Equations 20（2007）671-680]，其中pS和pL分别是Sobolev和Joseph和Lundgren的指数。在径向情形下，我们给出了p> pJ的L1-连续在爆破后唯一的一个充分条件。如果对于（P）的L1-解u，存在（P）的经典解序列{un}，使得对于初始数据序列{u 0，n}，当n→∞时，u 0，n→ u 0在L∞（RN）中，对于t∈（0，T ∈），当n→∞时，un（t）→u（t）在Llocp（RN）中，则u称为极限L1-解.在此充分条件的基础上，证明了当p>pJL时，爆破后具有径向对称性的极限L1-解的唯一性。
This paper is concerned with the uniqueness of L1-continuation beyond blowup for a Cauchy problem of a semilinear heat equation with p>1, 0<T˜⩽∞ and u0∈L∞(RN). Here we say that u is an L1-solution of (P) if u∈C([0,T˜);Lloc1(RN)) with u∈Llocp(RN×(0,T˜)) satisfies (P) in the distributional sense. In the case of pS<p<pJL, a counter example for the uniqueness of radial L1-solution of (P) after blowup was given in [M. Fila, N. Mizoguchi, Multiple continuation beyond blow-up, Differential Integral Equations 20 (2007) 671–680], where pSand pJLare the exponent of Sobolev and of Joseph and Lundgren, respectively. We give a sufficient condition for the uniqueness of L1-continuation beyond blowup for p>pJLin the radial case. If for an L1-solution u of (P) there exists a sequence {un} of classical solutions of (P) such that u0,n→u0in L∞(RN) as n→∞ for the sequence {u0,n} of initial data and that un(t)→u(t) in Llocp(RN) as n→∞ for t∈(0,T˜), then u is called a limit L1-solution. Based on the sufficient condition, we prove the uniqueness of limit L1-solution with radial symmetry after blowup for p>pJL.