Borel summability of divergent solutions for singular first-order partial differential equations with variable coefficients. I.

*(English)*Zbl 1147.35016Summary: This first part I and the forthcoming second part [J. Differ. Equations 227, No. 2, 534–563 (2006; Zbl 1147.35017)] are concerned with the study of the Borel summability of divergent power series solutions for singular first-order linear partial differential equations of nilpotent type. Under one restriction on equations, we can divide them into two classes. In this part I, we deal with the one class and obtain the conditions under which divergent solutions are Borel summable. (The other class will be studied in part II.) In order to assure the Borel summability of divergent solutions, global analytic continuation properties for coefficients are required despite of the fact that the domain of the Borel sum is local.

##### MSC:

35C20 | Asymptotic expansions of solutions to PDEs |

35C10 | Series solutions to PDEs |

35C15 | Integral representations of solutions to PDEs |

35F05 | Linear first-order PDEs |

##### Keywords:

partial differential equations; divergent power series; Borel summability; asymptotic expansions; analytic continuation
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\textit{M. Hibino}, J. Differ. Equations 227, No. 2, 499--533 (2006; Zbl 1147.35016)

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