Divergent solutions of the heat equation: On an article of Lutz, Miyake and Schäfke
type: Item
instance of: scholarly article
title: Divergent solutions of the heat equation: On an article of Lutz, Miyake and Schäfke
zbMATH Open document ID: 0960.35045
author: Werner Balser
published in: Pacific Journal of Mathematics
publication date: 2000-08-23T00:00:00Z
review text: The author considers an extension of a paper by \textit{D. A. Lutz, M. Miyake} and \textit{R. Schäfke} [On the Borel summability of divergent solutions of the heat equations, Nagoya Math. J. 154, 1-29 (1999; Zbl 0958.35061)] dealing with the summability of formal solutions of the Cauchy problem for the complex heat equation \[ u_t(t,z) = u_{zz}(t,z), \quad u(0,z) = \widehat{\varphi}(z)= \sum \varphi_n z^n. \] He establishes the result that the type of summability for the formal solution \(\sum \frac {\widehat{\varphi}^{(2n)}(z)} {n!} t^n\) varies with the initial condition \(\widehat{\varphi}(z)\). He notes that \(\widehat{\varphi}(z)\) need not even be analytic at the origin; it may be a formal series with certain summability properties. On the other hand, the question of divergent initial data has not yet been satisfactorily dealt with and must be studied further.
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 35C10
zbMATH DE Number: 1344285
zbMATH Keywords: summability of formal solutions
zbMATH Keywords: Cauchy problem
zbMATH Keywords: complex heat equation
reviewed by: Deborah Tepper Haimo
MaRDI profile type: Publication
full work available at URL: pjm.1999.188.53
OpenAlex ID: W2059518258
DOI: 10.2140/PJM.1999.188.53
description: scientific article; zbMATH DE number 1344285