Gevrey Asymptotics for Logarithmic-Type Solutions to Singularly Perturbed Problems with Nonlocal Nonlinearities

We investigate a family of nonlinear partial differential equations which are singularly perturbed in a complex parameter ϵ and singular in a complex time variable t at the origin. These equations combine differential operators of Fuchsian type in time t and space derivatives on horizontal strips in...

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Bibliographic Details
Published inAbstract and applied analysis Vol. 2023; pp. 1 - 42
Main Author Malek, Stéphane
Format Journal Article
LanguageEnglish
Published New York Hindawi 2023
John Wiley & Sons, Inc
Hindawi Limited
Subjects
Asymptotic series
Differential equations, Partial
Functions (mathematics)
Logarithms
Mathematical research
Nonlinear differential equations
Nonlinear theories
Nonlinearity
Numbers
Operators (mathematics)
Ordinary differential equations
Parameters
Partial differential equations
Polynomials
France
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Summary:We investigate a family of nonlinear partial differential equations which are singularly perturbed in a complex parameter ϵ and singular in a complex time variable t at the origin. These equations combine differential operators of Fuchsian type in time t and space derivatives on horizontal strips in the complex plane with a nonlocal operator acting on the parameter ϵ known as the formal monodromy around 0. Their coefficients and forcing terms comprise polynomial and logarithmic-type functions in time and are bounded holomorphic in space. A set of logarithmic-type solutions are shaped by means of Laplace transforms relatively to t and ϵ and Fourier integrals in space. Furthermore, a formal logarithmic-type solution is modeled which represents the common asymptotic expansion of the Gevrey type of the genuine solutions with respect to ϵ on bounded sectors at the origin.
ISSN:1085-3375
1687-0409
DOI:10.1155/2023/3025513