Divergent solutions of the heat equation: On an article of Lutz, Miyake and Schäfke
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.
scientific article; zbMATH DE number 1344285