The Gevrey asymptotics in the initial value problem for singularly perturbed nonlinear differential equations

In this paper, the initial value problem for singularly perturbed nonlinear holomorphic differential equations with a small parameter is considered in a complex domain, and the asymptotic behavior of solutions with respect to the parameter is studied. Under suitable conditions, a formal power series...

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Bibliographic Details
Published in:Journal of Differential Equations 2023-11, Vol.373, p.283-326
Main Author: Tahara, Hidetoshi
Format: Article
Language:English
Subjects:
Gevrey asymptotics
Initial value problem
Nonlinear differential equation
Singular perturbation
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Summary:In this paper, the initial value problem for singularly perturbed nonlinear holomorphic differential equations with a small parameter is considered in a complex domain, and the asymptotic behavior of solutions with respect to the parameter is studied. Under suitable conditions, a formal power series solution is constructed, and then it is proved that this formal solution is summable in a suitable direction. This guarantees the existence of a holomorphic solution that admits the formal power series solution as an asymptotic expansion of Gevrey type. It is then proved that this solution also has a Gevrey type asymptotic expansion with respect to the parameter.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2023.07.020