Summability of the formal power series solutions of a certain class of inhomogeneous nonlinear partial differential equations with a single level
type: Item
instance of: scholarly article
title: Summability of the formal power series solutions of a certain class of inhomogeneous nonlinear partial differential equations with a single level
zbMATH Open document ID: 1487.35183
author: Pascal Remy
published in: Journal of Differential Equations
publication date: 2022-02-10T00:00:00Z
review text: The work is devoted to the study of a family of inhomogeneous nonlinear Cauchy problems of the form \[\partial_{t}^{\kappa}u-\sum_{i\in\mathcal{K}}\sum_{q\in Q_i}P_{i,q}(t,x,u)\partial_t^i\partial_x^qu=\tilde{f}(t,x),\] under initial conditions \(\partial_t^ju(0,x)=\varphi_j(x)\), for \(0\le j\le \kappa-1\), for some \(\kappa\ge1\), and where \(Q_i\) (resp. \(\mathcal{K}\)) is a nonempty subset of \(\mathbb{N}\) for all \(i\) (resp. \(\{0,\ldots,\kappa-1\}\)). \(P_{i,q}\) stands for a polynomial in its last variable with analytic coefficients in a product of discs at the origin. The inhomogeneity \(\tilde{f}\) is a formal power series in \(t\) with coefficients in a common neighborhood of the origin, which coincides with the domain of holomorphy of the initial condition. Under the assumption of the existence of a single slope of the Newton polygon associated to the problem, the author proves summability of the unique formal solution of the main problem under study, say \(\tilde{u}(t,x)\), iff the inhomogeneity and the formal series \(\partial_x^n\tilde{u}(t,0)\) are so along the same direction. The sum satisfies the same equation with the inhomogeneity substituted by the corresponding sum (Theorem 3.8). The proof of the result is based on tight upper estimates and technical results applied to an auxiliary equation leading to the conclusion.
reviewed by: Alberto Lastra
Mathematics Subject Classification ID: 35C10
Mathematics Subject Classification ID: 35C20
Mathematics Subject Classification ID: 40B05
zbMATH DE Number: 7471739
zbMATH Keywords: summability
zbMATH Keywords: inhomogeneous partial differential equation
zbMATH Keywords: nonlinear partial differential equation
zbMATH Keywords: formal power series
zbMATH Keywords: divergent power series
zbMATH Keywords: Newton polygon
MaRDI profile type: Publication
full work available at URL: j.jde.2022.01.006
OpenAlex ID: W4205689133
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DOI: 10.1016/J.JDE.2022.01.006
description: scientific article; zbMATH DE number 7471739