Electron. J. Differential Equations,
Vol. 2019 (2019), No. 55, pp. 175.
Parametric Borel summability for linear singularly perturbed
Cauchy problems with linear fractional transforms
Alberto Lastra, Stephane Malek
Abstract:
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
with respect to
the perturbation parameter
. 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 socalled order
is presented. Furthermore, unicity properties
regarding the
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; BorelLaplace transform; Cauchy problem;
difference equation; integrodifferential equation;
linear partial differential equation; singular perturbation.
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Alberto Lastra
Dpto. de Física y Matemáticas
Universidad de Alcalá, Ap. Correos 20
E28871 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.univlille1.fr

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