Divergent solutions of the heat equation: On an article of Lutz, Miyake and Schäfke
Divergent solutions of the heat equation: On an article of Lutz, Miyake and Schäfke
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热方程的发散解：关于 Lutz、Miyake 和 Schäfke 的文章
DOI：
10.2140/pjm.1999.188.53
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发表时间：
1999
影响因子：
0.6
通讯作者：
W. Balser
中科院分区：
文献类型：
--
作者：
W. Balser
D.A. Lutz，M. Miyake和R. Schäfke [5]研究了上述问题，假设φ ∈（z）收敛（对于|z| < r，假设r > 0）到解析函数φ（z）（可以获得这篇文章的副本，例如，（作者为今）。他们证明了（0.2）（用φ（z）代替φ（z））在方向d上是一可和的，当且仅当φ（z）可以在方向d/2和π + d/2上解析地继续到无穷大，并且当在这些方向上无穷大时，它的指数大小至多为2;这意味着[3]对于足够大的常数cj，我们有|φ（z）|≤ c1 exp（c2| z|).在这里，我们得到类似的结果更一般的初始数据。特别地，我们将证明形式解（0.2）的可和性类型随给定的初始条件φ（z）而变化，它甚至不需要在原点解析，但可以是具有某些可和性性质的形式级数。然而，对于不同的初始数据，结果仍然不完全令人满意，因此本文第二节在性质上略显粗略。
D.A. Lutz, M. Miyake and R. Schäfke [5] have studied the above problem, assuming φ̂(z) convergent (for |z| < r, say, with r > 0) to the analytic function φ(z) (a copy of this article may be obtained, e.g., from the author of the present one). They showed that (0.2) (with φ(z) instead of φ̂(z)) is one–summable in a direction d if and only if φ(z) can be analytically continued to infinity in directions d/2 and π + d/2, and is of exponential size at most two when going to infinity in these directions; by this we mean [3] that for sufficiently large constants cj we have |φ(z)| ≤ c1 exp(c2|z|). Here, we obtain analogous results for more general initial data. In particular, we shall show that the type of summability for the formal solution (0.2) varies with the given initial condition φ̂(z), which need not even be analytic at the origin, but may be a formal series with certain summability properties. The results for divergent initial data, however, are still not fully satisfactory, so Section 2 of this paper is slightly sketchy in nature.