Gevrey Properties and Summability of Formal Power Series Solutions of Some Inhomogeneous Linear Cauchy-Goursat Problems

Abstract

In this article, we investigate the Gevrey and summability properties of the formal power series solutions of some inhomogeneous linear Cauchy-Goursat problems with analytic coefficients in a neighborhood of p0, 0q P C 2 . In particular, we give necessary and sufficient conditions under which these solutions are convergent or are k-summable, for a convenient positive rational number k, in a given direction.

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

“…We must prove that the coefficients u j,˚p xq P OpD ρ1,...,ρn q of r upt, xq satisfy similar inequalities. The approach we present below is analoguous to the ones already developed in [5,[45][46][47] in the framework of linear partial and integro-differential equations and is based on the Nagumo norms [8,39,56] and on a technique of majorant series. However, our calculations appear to be much more complicated than in the linear case: the nonlinear term u m of equation eq.…”

“…We now shall bound the Nagumo norms }u j,˚} ps`1qj,ρ for any j. To do that, we shall proceed similarly as in [5,[45][46][47] by using a technique of majorant series. However, as we shall see, the calculations are much more complicated.…”

On the summability of the solutions of the inhomogeneous heat equation with a power-law nonlinearity and variable coefficients

“…To prove that the condition is sufficient, we shall proceed in a similar way as the proof of [5,Thm. 3.4] (see also [48][49][50]).…”

“…Then, r vpt, xq and r gpt, xq are both 1-summable in the direction θ (see Proposition 2.2) and identity (3.1) above tells us it suffices to prove that it is the same for r wpt, xq. To this end, we shall proceed similarly as [5,[48][49][50] through a fixed point method. Of course, as we shall see below, the nonlinear term pB´2 x r wq 2 induces much more complicated calculations.…”

Telescopic, Multiplicative, and Rational Extensions of Summations

“…a holomorphic function A(z) in a neighborhood of the origin, we write A(σ) for the formal series ∞ n=0 a n σ n . While formal power series are a fruitful object of study in their own right [1,3,12,15,19,23,26], mathematicians have a longstanding interest in assigning a definite "sum" to each series, or at least as many series as possible [4,5,6,7,8,13,14,17,18,25,28,29]. For a finitely supported series A, this is straightforward: if A ∈ R[σ], we set A(A) := A(1), and say A(A) is the sum of A.…”