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Borel summability of divergent solutions of the Cauchy problem to non-Kowalewskian equations. (English) Zbl 0990.35005

Chen, Hua (ed.) et al., Partial differential equations and their applications. Proceedings of the conference, Wuhan, China, April 5-9, 1999. Singapore: World Scientific. 225-239 (1999).
The author considers the non-Kowalevskian Cauchy problem \[ D^p_t u(t,x)= D_x^qu(t,x), \quad 1\leq p<q, \]
\[ u(0,x)=f(x), \quad D^j_t u(0,x)=0\quad \text{if}\quad 1\leq j<p, \] where \((t,x)\in \mathbb{C}^2\) and the Cauchy data \(f(x)\) are assumed to be holomorphic in a neighborhood of the origin \(x=0\). The problem has a unique formal solution. The author discusses in detail convergence and Borel summability of the solution under different assumptions on the behaviour of \(f(x)\) at infinity. Relevant reference is given by S. Ōuchi [J. Math. Sci., Tokyo 2, 375-417 (1995; Zbl 0860.35018)], concerning the existence of the solutions in a sectorial domain for a similar problem.
For the entire collection see [Zbl 0969.00056].
Reviewer: L.Rodino (Torino)

MSC:

35A10 Cauchy-Kovalevskaya theorems
35C10 Series solutions to PDEs

Citations:

Zbl 0860.35018
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