On the Borel summability of divergent solutions of the heat equation
On the Borel summability of divergent solutions of the heat equation
复制标题
热方程发散解的 Borel 可求和性
DOI：
10.1017/s0027763000025289
复制
发表时间：
1999
影响因子：
0.8
通讯作者：
R. Schäfke
中科院分区：
文献类型：
--
作者：
D. Lutz;M. Miyake;R. Schäfke
摘要 近年来，建立了单变量发散幂级数的Borel可求和性或多重可和性理论，并证明了具有不规则奇点的常微分方程的每一个形式解都是多重可和的。对于偏微分方程，发散解的可求和问题还没有得到很好的研究，本文将尝试发展复杂热方程柯西问题的发散解的Borel可求和性，因为热方程是一个典型且重要的满足发散解的方程。总之，形式解的 Borel 可求和性具有解析连续性及其柯西数据沿条带域向无穷大增长的条件，而 Borel 和只不过是热核积分表达式给出的解。我们还提供了通过特殊的柯西数据从 Borel 和中获取热核的新方法。
Abstract In recent years, the theory of Borel summability or multisummability of divergent power series of one variable has been established and it has been proved that every formal solution of an ordinary differential equation with irregular singular point is multisummable. For partial differential equations the summability problem for divergent solutions has not been studied so well, and in this paper we shall try to develop the Borel summability of divergent solutions of the Cauchy problem of the complex heat equation, since the heat equation is a typical and an important equation where we meet diveregent solutions. In conclusion, the Borel summability of a formal solution is characterized by an analytic continuation property together with its growth condition of Cauchy data to infinity along a stripe domain, and the Borel sum is nothing but the solution given by the integral expression by the heat kernel. We also give new ways to get the heat kernel from the Borel sum by taking a special Cauchy data.