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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

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Abstract

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 (Remy in J Dyn Control Syst 22(4):693–711, 2016; J Dyn Control Syst 23(4):853–878, 2017) and for the semilinear heat equation (Remy in J Math Anal Appl 494(2):124619, 2021). In addition, we present some technical results on the generalized binomial and multinomial coefficients, which are needed for the proof our main theorem.

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Notes

  1. We denote \({\widetilde{f}}\) with a tilde to emphasize the possible divergence of the series \({\widetilde{f}}\).

  2. A subsector \(\varSigma \) of a sector \(\varSigma '\) is said to be a proper subsector and one denotes \(\varSigma \Subset \varSigma '\) if its closure in \({\mathbb {C}}\) is contained in \(\varSigma '\cup \{0\}\).

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  1. Laboratoire de Mathématiques de Versailles, Université de Versailles Saint-Quentin, 45 avenue des Etats-Unis, Versailles Cedex, 78035, France

    Pascal Remy

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Remy, P. Summability of the Formal Power Series Solutions of a Certain Class of Inhomogeneous Partial Differential Equations with a Polynomial Semilinearity and Variable Coefficients. Results Math 76, 118 (2021). https://doi.org/10.1007/s00025-021-01428-z

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