On Gevrey solutions of threefold singular nonlinear partial differential equations

Lastra, Alberto; Malek, Stéphane; Sanz, Javier

doi:10.1016/j.jde.2013.07.031

Cited by 32 publications

(55 citation statements)

References 16 publications

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“…The case of a complex perturbation parameter has led to results in which the nature of the singularities arising from the problem describe different types of singularities. We refer to the work by M. Canalis-Durand, J. Mozo-Fernández and R. Schäfke [3], the second author [18], the authors [7] and the authors and J. Sanz [14]. In [8], the authors study the parametric multi-level Gevrey solutions coming from a splitting of the equation which generates two Gevrey levels.…”

Section: Introductionmentioning

confidence: 99%

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

Lastra

Malek

2017

Journal of Difference Equations and Applications

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We study the asymptotic behavior of the solutions related to a singularly perturbed q-differencedifferential problem in the complex domain. The analytic solution can be splitted according to the nature of the equation and its geometry so that both, Gevrey and q−Gevrey asymptotic phenomena are observed and can be distinguished, relating the analytic and the formal solution.The proof leans on a two level novel version of Ramis-Sibuya theorem under Gevrey and q-Gevrey orders.

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Section: Introductionmentioning

confidence: 99%

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

Lastra

Malek

2017

Journal of Difference Equations and Applications

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“…In the case κ 1 = k, we remark in particular that F d (T, z, ) is a convergent series on D(0, T 0 /2) w.r.t T , and defines a bounded holomorphic function w.r.t z on H β and w.r.t on D(0, 0 ). On the other hand, when 0 < κ 1 < k, we apply the inequality (17) in the particular case α = n/k and β = n( 1…”

Section: Outline Of the Main Initial Value Problem And Related Auxilimentioning

confidence: 99%

On Singularly Perturbed Linear Initial Value Problems with Mixed Irregular and Fuchsian Time Singularities

Lastra

Malek

2019

J Geom Anal

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We consider a family of linear singularly perturbed PDE relying on a complex perturbation parameter . As in the former study [14] of the authors, our problem possesses an irregular singularity in time located at the origin but, in the present work, it entangles also differential operators of Fuchsian type acting on the time variable. As a new feature, a set of sectorial holomorphic solutions are built up through iterated Laplace transforms and Fourier inverse integrals following a classical multisummability procedure introduced by W. Balser. This construction has a direct issue on the Gevrey bounds of their asymptotic expansions w.r.t which are shown to bank on the order of the leading term which combines both irregular and Fuchsian types operators.

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“…After that, asymptotic expansion theory in a monomial is frequently used in studying formal power series solutions of certain class of differential equations (or systems) [3][4] [5].…”

Section: Introductionmentioning

confidence: 99%

On the Existence and Uniqueness of the Formal Solution to a Singular Partial Differential Equation

Li¹,

Li²

2016

Proceedings of the 2016 4th International Conference on Machinery, Materials and Information Technology Applications

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Abstract. According to the theory used in researching certain type of doubly singular differential equations, by computing directly, we find out that a singular partial differential equation in two complex variables has a unique formal power series solution which is in a special form and whose coefficients can be worked out explicitly. By estimating coefficients of the solution, we prove this formal solution is 1-Gevrey in a monomial. This gives a way to study Gevrey order in a monomial for formal solutions to certain classes of differential equations (or systems).

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On Gevrey solutions of threefold singular nonlinear partial differential equations

Cited by 32 publications

References 16 publications

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

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

On Singularly Perturbed Linear Initial Value Problems with Mixed Irregular and Fuchsian Time Singularities

On the Existence and Uniqueness of the Formal Solution to a Singular Partial Differential Equation

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