On parametric Gevrey asymptotics for some nonlinear initial value problems in symmetric complex time variables

Lastra, Alberto; Malek, Stéphane

doi:10.3233/asy-191568

Cited by 7 publications

(47 citation statements)

References 17 publications

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“…As a first approach, one is tempted to follow techniques used in the previous works of the authors dealing with singularly perturbed partial differential equations in two complex time variables. On the one hand, the family of equations studied in [13] shows a symmetric role of the time variables in the equation. Although this is the case for (10), in that previous study it holds that the principal part of any of the equations in the family is factorizable as a product of two operators which split the dependence on the time variables.…”

Section: A First Approachmentioning

confidence: 99%

“…along well chosen halflines L d j ⊂ S j , j = 1, 2, as it was the case in our previous studies [13,14]. Our idea consists on merging this double integral along the product L d 1 × L d 2 into a simple integral along a halfline L d ⊂ C. Geometrically, it consists in a projection on the diagonal part (u, u) ∈ C 2 , u ∈ C of the space C 2 .…”

Section: Second Approachmentioning

confidence: 99%

“…The Banach spaces described in this section are modified versions of those appearing in [15]. We omit the details which can be derived directly from that work, and the variations of that norm, stated in [13,14].…”

Section: Banach Spaces Of Functions With Exponential Growth and Decaymentioning

confidence: 99%

“…This is the continuation of a series of works devoted to the study of PDEs in the complex domain under the action of two complex time variables. In [13], the authors have studied a family of nonlinear initial value Cauchy problems of the form (2) Q(∂ z )∂ t 1 ∂ t 2 u(t 1 , t 2 , z, ǫ) = (P 1 (∂ z , ǫ)u(t 1 , t 2 , z, ǫ))(P 2 (∂ z , ǫ)u(t 1 , t 2 , z, ǫ))…”

Section: Introductionmentioning

confidence: 99%

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Boundary layer expansions for initial value problems with two complex time variables

Lastra

Malek

2020

Adv Differ Equ

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We study a family of partial differential equations in the complex domain, under the action of a complex perturbation parameter ǫ. We construct inner and outer solutions of the problem and relate them to asymptotic representations via Gevrey asymptotic expansions with respect to ǫ, in adequate domains. The construction of such analytic solutions is closely related to the procedure of summation with respect to an analytic germ, put forward in [19], whilst the asymptotic representation leans on the cohomological approach determined by Ramis-Sibuya Theorem.under given initial data u(0, t 2 , z, ǫ) ≡ u(t 1 , 0, z, ǫ) ≡ 0. Here, Q(X) ∈ C[X] and P (T 1 , T 2 , Z, z, ǫ) stands for a polynomial in (T 1 , T 2 , Z) with holomorphic coefficients w.r.t. (z, ǫ) on H β ×D(0, ǫ 0 ), where H β stands for the horizontal strip in the complex planefor some β > 0, and D(0, ǫ 0 ) ⊆ C stands for the open disc centered at the origin with radius ǫ 0 , for some small ǫ 0 > 0. The symbol ǫ acts as a small complex perturbation parameter in the *

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Section: A First Approachmentioning

confidence: 99%

Section: Second Approachmentioning

confidence: 99%

Section: Banach Spaces Of Functions With Exponential Growth and Decaymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Boundary layer expansions for initial value problems with two complex time variables

Lastra

Malek

2020

Adv Differ Equ

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“…In [12], we described a study of a family of equations of the shape (1) which showed a symmetric behaviour with respect to the asymptotic properties of the analytic solutions with respect to both time variables, as initially expected from the generalization of the one-time variable case. More precisely, we proved the following result: given a good covering of C , {E p 1 ,p 2 } 0≤p 1 ≤ς 1 −1 coefficients which are holomorphic functions with respect the perturbation parameter on some neighborhood of the origin, under assumptions (8)- (10).…”

Section: Introductionmentioning

confidence: 85%

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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Summability of the Formal Power Series Solutions of a Certain Class of Inhomogeneous Partial Differential Equations with a Polynomial Semilinearity and Variable Coefficients

Remy

2021

Results Math

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In this article, we investigate the summability of the formal power series solutions in time of a certain class of inhomogeneous partial differential equations with a polynomial semilinearity, and with variable coefficients. In particular, we give necessary and sufficient conditions for the k-summability of the solutions in a given direction, where k is a positive rational number entirely determined by the linear part of the equation. These conditions generalize the ones given by the author for the linear case [?,?] and for the semilinear heat equation [?]. In addition, we present some technical results on the generalized binomial and multinomial coefficients, which are needed for the proof our main theorem.Summability, Inhomogeneous partial differential equation, Nonlinear partial differential equation, Formal power series, Divergent power series 35C10, 35C20, 40B05

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On parametric Gevrey asymptotics for some nonlinear initial value problems in symmetric complex time variables

Cited by 7 publications

References 17 publications

Boundary layer expansions for initial value problems with two complex time variables

Boundary layer expansions for initial value problems with two complex time variables

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

Summability of the Formal Power Series Solutions of a Certain Class of Inhomogeneous Partial Differential Equations with a Polynomial Semilinearity and Variable Coefficients

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