On parametric Gevrey asymptotics for singularly perturbed partial differential equations with delays. (English) Zbl 1470.35127

Lastra, Alberto; Malek, Stéphane

Abstr. Appl. Anal. 2013, Article ID 723040, 18 p. (2013).

Summary: We study a family of singularly perturbed \(q\)-difference-differential equations in the complex domain. We provide sectorial holomorphic solutions in the perturbation parameter \(\epsilon\). Moreover, we achieve the existence of a common formal power series in \(\epsilon\) which represents each actual solution and establish \(q\)-Gevrey estimates involved in this representation. The proof of the main result rests on a new version of the so-called Malgrange-Sibuya theorem regarding \(q\)-Gevrey asymptotics. A particular Dirichlet like series is studied on the way.

MSC:

35C10	Series solutions to PDEs
35C20	Asymptotic expansions of solutions to PDEs

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