On multiscale Gevrey and Gevrey asymptotics for some linear difference differential initial value Cauchy problems

Lastra, Alberto; Malek, Stéphane

doi:10.1080/10236198.2017.1337104

Cited by 23 publications

(35 citation statements)

References 29 publications

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“…Therefore, a small divisor phenomena is observed , which does not allow a summability procedure. This occurrence has already been noticed in another context in previous works: in the framework of q−differencedifferential equations [16]; in the context of multilevel Gevrey solutions of PDEs in the complex domain in [15], etc.…”

Section: Introductionsupporting

confidence: 56%

On Parametric Gevrey Asymptotics for Some Initial Value Problems in Two Asymmetric Complex Time Variables

Lastra

Malek

2018

Results Math

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We study a family of nonlinear initial value partial differential equations in the complex domain under the action of two asymmetric time variables. Different Gevrey bounds and multisummability results are obtain depending on each element of the family, providing a more complete picture on the asymptotic behavior of the solutions of PDEs in the complex domain in several complex variables.The main results lean on a fixed point argument in certain Banach space in the Borel plane, together with a Borel summability procedure and the action of different Ramis-Sibuya type theorems. variable coefficients (see [5,6,21]). The second author [22], both authors [13] and the two authors and J. Sanz [17] have also contributed in this theory. Recently, multisummability of formal solutions of PDEs has also been put forward in different works. W. Balser [4] described a multisummability phenomenon in certain PDEs with constant coefficients. S. Ouchi [23] constructed multisummable formal solutions of nonlinear PDEs, coming from perturbation of ordinary differential equations. H. Tahara and H. Yamazawa [24] have made progresses on general linear PDEs with non constant coefficients under entire initial data. In [20], G. Lysik constructs summable formal solutions of the one dimensional Burgers equation by means of the Cole-Hopf transform. O. Costin and S. Tanveer [8] construct summable formal power series in time variable to 3D Navier Stokes equations. The authors have obtained results in this direction [14, 15].A recent overview on summability and multisummability techniques under different points of view is displayed in [18].The purpose of the present work is to study the solutions of a family of singularly perturbed partial differential equations from the asymptotic point of view. More precisely, we consider a problem of the formThe elements which conform the nonlinear part P 1 , P 2 are polynomials in their second variable with coefficients being holomorphic functions defined on some neighborhood of the origin, say D(0, 0 ), continuous up to their boundary.Here, D(0, 0 ) stands for the open disc in the complex plane centered at 0, and with positive radius 0 > 0. We write D(0, 0 ) for its closure. Moreover, P stands for some polynomial of six variables, with complex coefficients, and the forcing term f (t 1 , t 2 , z, ) is a holomorphic and bounded function in D(0, ρ) 2 × H β × D(0, 0 ), for some ρ > 0, and where H β stands for the horizontal stripfor some β > 0.The precise configuration of the elements involved in the problem is stated and described in Section 2.2.This paper provides a step beyond in the study of the asymptotic behavior of the solutions of a subfamily of singularly perturbed partial differential equations of the form (1). We first recall some previous advances made in this respect, which motivate the present framework.In [13], we studied under the asymptotic point of view the solutions of certain family of PDEs of the formwhere the elements involved in the problem depend only on one time variable t. Our next aim w...

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Section: Introductionsupporting

confidence: 56%

On Parametric Gevrey Asymptotics for Some Initial Value Problems in Two Asymmetric Complex Time Variables

Lastra

Malek

2018

Results Math

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“…In this section we preserve the statements and constructions concerning the main problem under study (17), and the geometric and algebraic conditions held on the elements involved in the main equation.…”

Section: Study Of Auxiliary Equationsmentioning

confidence: 99%

“…Assume that t 2 ∈ T 2 is such that |t 2 | < ρ 2 for some fixed ρ 2 > 0 which is independent of , then we say that u h 2 (t, z, ) represents an outer solution of (17).…”

Section: Outer Solutions Of the Main Problemmentioning

confidence: 99%

“…In this section, we preserve all the assumptions made on the elements involved in the main problem (17), detailed in Section 3. Moreover, we depart from the geometric construction of the elements used to construct the analytic solutions of (17), described in Sections 4 and 5, and under Assumption (31).…”

Section: Asymptotic Expansions For the Analytic Solutions Of The Mainmentioning

confidence: 99%

“…Afterwards, we provide the definition and related properties of some Banach spaces involved in the construction of the solution. In Section 3 we state the main problem under study (17) and two auxiliary equations. The elements involved in them in addition to the domains of existence and upper bounds of the solutions of such equations are detailed.…”

Section: Introductionmentioning

confidence: 99%

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On a q—Analog of a Singularly Perturbed Problem of Irregular Type with Two Complex Time Variables

Lastra

Malek

2019

Mathematics

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Analytic solutions and their formal asymptotic expansions for a family of the singularly perturbed q−difference-differential equations in the complex domain are constructed. They stand for a q−analog of the singularly perturbed partial differential equations considered in [15]. In the present work, we construct outer and inner analytic solutions of the main equation, each of them showing asymptotic expansions of essentially different nature with respect to the perturbation parameter. The appearance of the −1-branch of Lambert W function will be crucial in this respect.

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Summability of Formal Solutions for Some Generalized Moment Partial Differential Equations

2021

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On multiscale Gevrey and Gevrey asymptotics for some linear difference differential initial value Cauchy problems

Cited by 23 publications

References 29 publications

On Parametric Gevrey Asymptotics for Some Initial Value Problems in Two Asymmetric Complex Time Variables

On Parametric Gevrey Asymptotics for Some Initial Value Problems in Two Asymmetric Complex Time Variables

On a q—Analog of a Singularly Perturbed Problem of Irregular Type with Two Complex Time Variables

Summability of Formal Solutions for Some Generalized Moment Partial Differential Equations

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