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

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.