Blow-up at space infinity for nonlinear equations
Blow-up at space infinity for nonlinear equations
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非线性方程在无限远空间的爆炸
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
Noriaki Umeda
中科院分区：
文献类型：
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作者：
Noriaki Umeda
。自Weissler和Friedman-McLeod的工作[15]和Friedman-McLeod[1]以来，已经有大量关于爆破点位置的文献。(我们不打算在本文中详尽列出参考文献。)然而，大多数结果要么考虑有界域，要么考虑解在有界域上衰减；这样的解不在有界域上爆破[2]。据作者所知，在[4]的结果之前，在有界域上讨论有界解的唯一文献是Lacey的工作[8]。他把狄利克雷特问题考虑得只剩下一半。他研究了各种非线性项，证明了解只有在空间中才会爆炸。他的方法是基于构造合适的下解和上解。然而，这个构造在很大程度上依赖于x=0的Dirichlet条件，即使在n=1的情况下也不适用于柯西问题。如前所述，Giga-Umeda[4]证明了定理1和2的陈述，假设Lim
. (It blows up only at space in nity.)There is a huge literature on location of blow-up points since the work ofWeissler [15] and Friedman-McLeod [1]. (We do not intend to list referencesexhaustively in this paper.) However, most results consider either boundeddomains or solutions decaying at space in nity; such a solution does not blowup at space in nity [2].As far as the authors know, before the result of [4] the only paper dis-cussing blow-up at space in nity is the work of Lacey [8]. He consideredthe Dirichlet problem in a half line. He studied various nonlinear terms andproved that a solution blows up only at space in nity. His method is basedon construction of suitable subsolutions and supersolutions. However, theconstruction heavily depends on the Dirichlet condition at x= 0 and doesnot apply to the Cauchy problem even for the case n= 1.As previously described, the Giga-Umeda [4] proved the statement ofTheorems 1 and 2 assuming that lim