A Maillet Type Theorem for First Order Singular Nonlinear Partial Differential Equations

Abstract

We shall study the first order singular nonlinear

Gevrey index theorem for the inhomogeneous n-dimensional heat equation with a power-law nonlinearity and variable coefficients

“…The existing results concern mainly Gevrey properties, especially the convergence (e.g. [10,16,18,20,25,[36][37][38][48][49][50][51][52][53][54]), and there are very few results about the summation (see [17,22,26,40,43]).…”

Formal Gevrey solutions -- in analytic germs -- for higher order holomorphic PDEs

“…The Gevrey class of partial differential equations has been extensively studied by several authors, see, e.g., [13,18,22,23,27] and also [8], Chapters 6 and 7, together with the references therein. In particular, equation (3.1) is contained in a family of equations treated in Chapter 6 of [8] for the case d = k = 1 where the Gevrey class is computed as the maximum of the values…”

“…The increasing interest on this type of results has provided novel advances in different frameworks. For instance, on generalized power series solutions of ordinary differential equations [9], partial differential equations [13,22,23], in singularly perturbed problems [3], integro-differential equations [21], moment partial differential equations [2,12,17,25], difference and q-difference equations [11,26,28], among others. We can also mention results in dynamical systems, such as the Gevrey character of parametrizations of invariant curves associated to fixed points of analytic local diffeomorphism [1,14].…”

Convergence or Divergence of Formal Solutions of First Order Singular Nonlinear Ordinary Differential Equations in Complex Domain

“…For a nonlinear partial di¤erential equation which is written in a normal form was studied in a book by R. Gérard and T. Tahara [1], and the Maillet type theorem was proved in a general form. Their studies were continued by A. Shirai [4,5] and he determined the Gevrey index for divergent solution for more general equation.…”