Electron. J. Differential Equations, Vol. 2019 (2019), No. 55, pp. 1-75.

Parametric Borel summability for linear singularly perturbed Cauchy problems with linear fractional transforms

Alberto Lastra, Stephane Malek

We consider a family of linear singularly perturbed Cauchy problems which combines partial differential operators and linear fractional transforms. This work is the sequel of a study initiated in [17]. We construct a collection of holomorphic solutions on a full covering by sectors of a neighborhood of the origin in $\mathbb{C}$ with respect to the perturbation parameter $\epsilon$ . This set is built up through classical and special Laplace transforms along piecewise linear paths of functions which possess exponential or super exponential growth/decay on horizontal strips. A fine structure which entails two levels of Gevrey asymptotics of order 1 and so-called order $1^{+}$ is presented. Furthermore, unicity properties regarding the $1^{+}$ asymptotic layer are observed and follow from results on summability with respect to a particular strongly regular sequence recently obtained in [13].

Submitted April 20, 2018. Published April 29, 2019.
Math Subject Classifications: 35R10, 35C10, 35C15, 35C20.
Key Words: Asymptotic expansion; Borel-Laplace transform; Cauchy problem; difference equation; integro-differential equation; linear partial differential equation; singular perturbation.

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Alberto Lastra
Dpto. de Física y Matemáticas
Universidad de Alcalá, Ap. Correos 20
E-28871 Alcalá de Henares, Madrid, Spain
email: alberto.lastra@uah.es
Stephane Malek
University of Lille 1, Laboratoire Paul Painlevé
59655 Villeneuve d'Ascq cedex, France
email: Stephane.Malek@math.univ-lille1.fr

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