{"title":"Gevrey Asymptotics for Logarithmic-Type Solutions to Singularly Perturbed Problems with Nonlocal Nonlinearities","authors":"S. Malek","doi":"10.1155/2023/3025513","DOIUrl":null,"url":null,"abstract":"We investigate a family of nonlinear partial differential equations which are singularly perturbed in a complex parameter \n \n ϵ\n \n and singular in a complex time variable \n \n t\n \n at the origin. These equations combine differential operators of Fuchsian type in time \n \n t\n \n and space derivatives on horizontal strips in the complex plane with a nonlocal operator acting on the parameter \n \n ϵ\n \n 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 \n \n t\n \n and \n \n ϵ\n \n 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 \n \n ϵ\n \n on bounded sectors at the origin.","PeriodicalId":7061,"journal":{"name":"Abstract and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2023-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Abstract and Applied Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1155/2023/3025513","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 1
Abstract
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.
期刊介绍：
Abstract and Applied Analysis is a mathematical journal devoted exclusively to the publication of high-quality research papers in the fields of abstract and applied analysis. Emphasis is placed on important developments in classical analysis, linear and nonlinear functional analysis, ordinary and partial differential equations, optimization theory, and control theory. Abstract and Applied Analysis supports the publication of original material involving the complete solution of significant problems in the above disciplines. Abstract and Applied Analysis also encourages the publication of timely and thorough survey articles on current trends in the theory and applications of analysis.
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