On parametric multisummable formal solutions to some nonlinear initial value Cauchy problems
Identifiers
Permanent link (URI): http://hdl.handle.net/10017/41409DOI: 10.1186/s13662-015-0541-4
ISSN: 1687-1839
Publisher
SpringerOpen
Date
2015-01-01Funders
Ministerio de Economía y Competitividad
Bibliographic citation
Lastra, A. & Malek, S. 2015, “On parametric multisummable formal solutions to some nonlinear initial value Cauchy problems”, Advances in Difference Equations, 2015, v. 2015, n. 200, pp. 1-78.
Keywords
Asymptotic expansion
Borel-Laplace transform
Fourier transform
Cauchy problem
Formal power series
Nonlinear integro-differential equation
Nonlinear partial differential equation
Singular perturbation
Project
info:eu-repo/grantAgreement/MINECO//MTM2012-31439/ES/ANALISIS DE PERTURBACIONES SINGULARES: ESTUDIO ASINTOTICO, CAPAS LIMITE Y FENOMENOS MULTIESCALA/
Document type
info:eu-repo/semantics/article
Version
info:eu-repo/semantics/publishedVersion
Publisher's version
http://dx.doi.org/10.1186/s13662-015-0541-4Rights
Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)
Access rights
info:eu-repo/semantics/openAccess
Abstract
We study a nonlinear initial value Cauchy problem depending upon a complex perturbation parameter whose coefficients depend holomorphically on (,t) near the origin in C2 and are bounded holomorphic on some horizontal strip in C w.r.t. the
space variable. In our previous contribution (Lastra and Malek in Parametric Gevrey
asymptotics for some nonlinear initial value Cauchy problems, arXiv:1403.2350), we
assumed the forcing term of the Cauchy problem to be analytic near 0. Presently, we
consider a family of forcing terms that are holomorphic on a common sector in time t
and on sectors w.r.t. the parameter whose union form a covering of some
neighborhood of 0 in C*, which are asked to share a common formal power series
asymptotic expansion of some Gevrey order as tends to 0. We construct a family of
actual holomorphic solutions to our Cauchy problem defined on the sector in time
and on the sectors in mentioned above. These solutions are achieved by means of a
version of the so-called accelero-summation method in the time variable and by
Fourier inverse transform in space. It appears that these functions share a common
formal asymptotic expansion in the perturbation parameter. Furthermore, this formal
series expansion can be written as a sum of two formal series with a corresponding
decomposition for the actual solutions which possess two different asymptotic
Gevrey orders, one stemming from the shape of the equation and the other
originating from the forcing terms. The special case of multisummability in is also
analyzed thoroughly. The proof leans on a version of the so-called Ramis-Sibuya
theorem which entails two distinct Gevrey orders. Finally, we give an application to
the study of parametric multi-level Gevrey solutions for some nonlinear initial value
Cauchy problems with holomorphic coefficients and forcing term in (,t) near 0 and
bounded holomorphic on a strip in the complex space variable.
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