Double-scale Gevrey Asymptotics for Logarithmic Type Solutions to Singularly Perturbed Linear Initial Value Problems

We examine a family of linear partial differential equations both singularly perturbed in a complex parameter ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document} and singular in complex time t at the origin. These equations entail forcing terms which combine polynomial and logarithmic type functions in time and that are bounded holomorphic on horizontal strips in one complex space variable. A set of sectorial holomorphic solutions are built up by means of complete and truncated Laplace transforms w.r.t t and ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document} and Fourier inverse integral in space. Asymptotic expansions of these solutions relatively to t and ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document} are investigated and two distinguished Gevrey type expansions in monomial and logarithmic scales are exhibited.


Introduction
In this paper, we focus our attention on a family of singularly perturbed linear partial differential equations with the shape Q(∂ z )u(t, z, ) = ( t) dD (t∂ t ) δD R D (∂ z )u(t, z, ) +P (t, z, , t∂ t , ∂ z )u(t, z, ) + f (t, z, ) for vanishing initial data u(0, z, ) ≡ 0, where d D , δ D ≥ 1 are integers, Q(X), R D (X) stand for polynomials with complex coefficients and P (t, z, , V 1 , V 2 ) represents a polynomial in the arguments t, V 1 , V 2 with holomorphic coefficients The forcing term f (t, z, ) involves coefficients that depend polynomially on the time variable t, analytically on on D 0 and holomorphically in z on H β . This term also combines logarithmic type functions expressed as truncated Laplace transforms along a fixed segment [−a, 0] for some radius a > 0 that count in the function 1/ log( t). Its expression is chosen in a way that when a > 0 is taken large, it becomes proximate to a general logarithmic type map f ∞ in t, displayed as polynomials in both t and 1/ log( t) with coefficients that are entire functions on the strip H β , see (24).
Observe that the main equation (1) involves powers of the basic differential operator of Fuchsian type t∂ t . For a conspicuous textbook about Fuchsian ordinary and partial differential equations, we refer to [7]. However, under the sufficient conditions set on (1) listed in Sect. 2.2, it turns out that (1) will be reduced throughout the work to an auxiliary prominent equation, stated in (41), that brings in only powers of basic differential operators of so-called irregular type u k1+1 1 ∂ u1 and u 2 2 ∂ u2 in two independent complex variables u 1 and u 2 . The definition of irregular type differential operators can be found in the classical textbook [1] in the framework of ordinary differential equations and in the work [15] in the context of partial differential equations.
Remark that the limit map f ∞ displayed in (24) (obtained from the forcing term as its truncated Laplace radius a tends to +∞) is made up with pieces that separately solve explicit nonlinear ordinary differential equations that are singularly perturbed and comprise first order differential operators of irregular type, see (25).
In the present study, our objective is the construction of a set of holomorphic solutions to (1) and the description of their asymptotic expansions as tends to 0 (stated in Theorem 1 of Sect. 4.3). We model these solutions as functions representable as double Laplace transforms and Fourier integral. Such an approach has already been successfully applied in the recent works [11,12] by A. Lastra and the author and in [3] by G. Chen, A. Lastra and the author in the analysis of singularly perturbed initial value problems in two complex time variables. Under the list of conditions applied to the shape of (1), detailed in Sect. 2.2, one can single out Fourier integral in the space variable z, where the so-called Borel/Fourier map ω p (τ 1 , τ 2 , m, ) stands for a function -which is analytic near τ 1 = 0 and for τ 2 ∈ D a , -with (at most) of exponential growth of order k 1 on an infinite sector containing the half line L dp = [0, +∞)e √ −1dp w.r.t τ 1 , -continuous and under exponential decay w.r.t m on R, -with analytic reliance on in the punctured disc D 0 \ {0}.
It is worth noticing that these solutions u p (t, z, ) cannot be constructed as a complete Laplace transform in the map 1/ log( t), but only as a truncated one for some fixed radius a > 0. The reason is that the radius a > 0 of the disc, on which ω p (τ 1 , τ 2 , m, ) is holomorphic relatively to τ 2 and built up in Proposition 4, can be freely chosen as large as desired but cannot be set as +∞ since a and 0 are related by a rule of the form n0 0 a n1 ≤ M for some suitable small constant M > 0 and positive integers n 0 , n 1 ≥ 1. The explicit constraints relating these two quantities are given through the technical bounds (79) and (89).
According to the very structure of these solutions, the family {u p } p∈I1 owns asymptotic expansions of Gevrey type in two particular scales of functions.
All the functions → u p (t, z, ), p ∈ I 1 , share a common asymptotic formal expansionû 1 (t, z, ) =  (134). Furthermore, these asymptotic expansions turn out to be of Gevrey order 1/k 1 on every sectors E p , meaning that constants K 1 p , M 1 p > 0 can be chosen for which the error bounds hold for all integers N ≥ 0, all ∈ E p , uniformly in t ∈ T and z ∈ H β . In Proposition 9, we show that the coefficients G 1 n are subjected to an explicit 198 Page 4 of 61 S. Malek Results Math differential recursion relation w.r.t n ≥ 0 that may be useful for their effective computations. For each p ∈ I 1 , the function (t, ) → u p (t, z, ) possesses a generalized asymptotic formal expansion (in the sense defined in the classical textbooks [6,18] (1/ log( t)) n n! on the domain T ×E p , in the scale of logarithmic functions {(1/ log( t)) n } n≥0 , for bounded holomorphic coefficients G 2 n,p on T × H β × E p . These asymptotic expansions share the common feature to be of Gevrey order 1 on the sectors E p , giving rise to constants K 2 , M 2 > 0 for which the error estimates occur for all integers N ≥ 0, all ∈ E p and t ∈ T , uniformly in z ∈ H β . In addition, the coefficients G 2 n,p are proved to fulfill some partial differential recursion relation in regard to n ≥ 0 that may be helpful for their practical reckoning, see Proposition 10.
Each map U dp,dq is modeled as a rescaled version of a bounded holomorphic map (u 1 , u 2 , z) → U dp,dq (u 1 , u 2 , z, ) by means of U dp,dq (t, u 2 , z, ) = U dp,dq ( t, u 2 , z, ) on domains U 1,dp × U 2,dq × H β for all ∈ D 0 \ {0} where U 1,dp are bounded sectors bisected by the directions d p , described in Definition 6 of the work. These maps U dp,dq (u 1 , u 2 , z, ) are shown to solve a set of auxiliary linear partial differential equations given by (104) which combine powers of the basic differential operators u k1+1 implies in particular that the functions u p , p ∈ I 1 , solve our main problem (1) on T × H β × E p . The two asymptotic properties (2) and (3) for u p stem from sharp exponential bound estimates for the differences of neighboring maps U dp,dq reached in Proposition 7, for which a well known criterion for the existence of asymptotic expansions of Gevrey type established by J-P. Ramis and Y. Sibuya can be applied, see Sect. 4.2.
In the framework of linear partial differential equations of so-called Fuchsian type, the construction of logarithmic type solutions is a well established subject. In the papers [19,20], H. Tahara considers so-called linear Fuchsian partial differential equations (introduced by M. Baouendi and C. Goulaouic in [2]) with the shape for linear differential operators P j , with order less than j, with holomorphic coefficients near t = 0 and x = (x 1 , . . . , x n ) = 0. Under conditions of non resonance of the characteristic exponents at x = 0, he has characterized the holomorphic solutions to (4) on , which can be expressed as where u l,j (t, x) = r l,j k=1 u l,j,k (x)(log(t)) k−1 , for positive integers r l,j ≥ 1 and holomorphic maps u l,j,k on D R , where λ l (x) represent the characteristic exponents of the equation (4) at x. Later on in the year 2000, T. Mandai was able to extend this important result to the general situation without any assumption on the characteristic exponents by following a similar approach to the method of Frobenius for ordinary differential equations with regular singularity at a point, see [16].
In the context of linear partial differential equations of so-called irregular type in which our present contribution falls, much less results are known and represents a promising trend for upcoming research. Nevertheless, in that direction, we can mention the striking paper [23] by H. Yamazawa published in 2017. Therein, the author examines linear partial differential equations of the form for x = (x 1 , . . . , x n ) ∈ C n , with holomorphic coefficients a j,α and forcing term f near (t, x) = (0, 0) ∈ C n+1 , for some monic polynomial ρ → C l (ρ, x) with  (6), the author constructs formal solutions with the shapeû(t, x) =û 1 (t, x) +û 2 (t, log(t), x) witĥ where ρ p (x) are the roots of ρ → C l (ρ, x) with positive real parts at x = 0 and u i (x), ϕ i,k,p (x) are holomorphic coefficients on some small disc D R , R > 0. The formal series in t,û 1 (t, x) (resp.û 2 (t, y, x)) with holomorphic coefficients on the domain D R (resp. D R/τ × D R for any fixed small parameter τ > 0) are divergent in general but are shown to be (multi)-summable in several levels of Gevrey orders on suitable sectors (in the sense defined in the book [1]). At last, we quote two compelling recent works that are somehow related to the result of the present study.
In [21], H. Tahara investigates higher order analogs of nonlinear singular partial differential equations of first order with so-called Briot-Bouquet type (see the textbook [7] by R. Gérard and H. Tahara for the origin of this terminology). These equations are written in the form for x = (x 1 , . . . , x n ) ∈ C n , under some restrictions of the analytic map F (t, x, Z) near the origin. The author studies spaces of solutions u(t, x) to (7), that are holomorphic on a product S × D R for some sector S centered at 0 and given disc D R with radius R > 0, restricted to upper bounds of the form for some constants C, a > 0 provided that t ∈ S. Sufficient conditions on the characteristic exponents at x = 0 of (7) are given for which the bounds (8) imply the stronger bounds for some constants K, b > 0, whenever t ∈ S. As a consequence, the structure of all the solutions u(t, x) of (7) subjected to (8) (and ressembling the one given by (5)) can be completely described by a former result by R. Gérard and H. Tahara stated in [7], Chap. 8.
In [17], the authors study families of formal power series f (x) with real coefficients in double scales of power and logarithmic functions with the shape where λ, α > 0 are real numbers and S is contained in a finitely generated additive semi-group in (0, +∞). Given such an f , normal forms for the conjugacy class ϕ −1 • f • ϕ are completely classified in that paper and a so-called embedding theorem is reached. These sets of formal expansions extend the classical formal Dulac series for c 0 > 0, increasing sequences {λ i } i≥0 of positive real numbers and polynomials P i with real coefficients, which appear to represent asymptotic expansions at the origin of Poincaré maps P stemming from analytic planar vector fields.

Layout of the Main Initial Value Problem and Tied up
Auxiliary Problems

Laplace Transforms of Order k and Fourier Inverse Maps
In this short subsection, we include a prefatory material about Laplace transforms and Fourier inverse maps that will be used in the upcoming sections. Let k ≥ 1 be an integer. We recall the definition of the Laplace transform of order k as introduced in [9]. Definition 1. We set S d,δ = {τ ∈ C * : |d − arg(τ )| < δ} as some unbounded sector with bisecting direction d ∈ R and aperture 2δ > 0 and D ρ as a disc centered at 0 with radius ρ > 0. Consider a holomorphic function w : S d,δ ∪ D ρ → C that vanishes at 0 and withstands the bounds : there exist C > 0 and K > 0 such that for all τ ∈ S d,δ . We define the Laplace transform of w of order k in the direction d as the integral transform where γ depends on T and is chosen in such a way that cos(k(γ − arg(T ))) ≥ δ 1 , for some fixed real number δ 1 > 0. The function L d k (w)(T ) is well defined, holomorphic and bounded on any sector where 0 < θ < π k + 2δ and 0 < R < δ 1 /K. We remind some useful property : if w(τ ) represents an entire function w.r.t τ ∈ C with the bounds (9), its Laplace transform L d k (w)(T ) does not depend on the direction d in R and represents a bounded holomorphic function on D R 1/k whose Taylor expansion is represented by the convergent series Results Math X(T ) = n≥1 w n Γ( n k )T n on D R 1/k , where Γ(x) stands for the Gamma function.
We remind the reader the definition of some family of Banach spaces used for the first time by the author in [14] and introduced in [5].
Finally, we restate the definition of the inverse Fourier transform acting on the latter Banach spaces and some of its close at hand formulas relative to derivation and convolution product as expounded in [9].
The inverse Fourier transform of f is given by for all x ∈ R. The function F −1 (f ) extends to an analytic bounded function on the strips for all given 0 < β < β. a) Define the function m → φ(m) = imf (m) which belongs to the space E (β,μ−1) . Then, the next identity as the convolution product of f and g. Then, ψ belongs to E (β,μ) and moreover,

The Main Problem Outlined
In this subsection, we disclose the principal linear initial value problem under study in this paper. It is stated as follows, where D ≥ 2 is some integer, for vanishing initial data u(0, z, ) ≡ 0. The constants d D ,δ D ,Δ l , d l and δ l for 1 ≤ l ≤ D − 1 are positive integers that are submitted to the next list of technical constraints: 1. There exists an integer k 1 ≥ 1 such that for all 1 ≤ l ≤ D − 1.

The next three inequalities
hold for all 1 ≤ l ≤ D − 1.
The maps Q(X), R D (X) and R l (X) are polynomials with complex coefficients that are subjected to the next two restrictions: 3. The next bounds hold, for all 1 ≤ l ≤ D − 1, where deg(P ) denotes the degree of a polynomial P (X). 4. One can select an unbounded sectorial annulus with bisecting direction d Q,RD ∈ R, aperture η Q,RD > 0 and inner radius r Q,RD > 0 (prescribed later on in the work), for which the next inclusion holds true. The coefficients a l (z, ), 1 ≤ l ≤ D − 1, are built up in the following way. For 1 ≤ l ≤ D − 1, let m → A l (m, ) be maps • that belong to the Banach space E (β,μ) , for some given real numbers β > 0 and μ > 1 that fulfills the restriction for all 1 ≤ l ≤ D − 1. • that depend analytically on on a disc D 0 centered at 0 in C with radius 0 > 0 and for which a constant A l, 0 > 0 can be be singled out with sup We set for all 1 ≤ l ≤ D − 1. According to Definition 3, the maps (z, ) → a l (z, ) represent bounded holomorphic maps on the product H β × D 0 , for any prescribed 0 < β < β.
The forcing term f (t, z, ) is constructed in the next manner. Let J 1 , J 2 be given finite subsets of the positive integers N * . For j 1 ∈ J 1 , j 2 ∈ J 2 , we denote by m → F j1,j2 (m, ) a function which 1. appertains to the Banach space E (β,μ) , for β > 0, μ > 1 given above. 2. relies analytically on on the disc D 0 , with a constant F j1,j2, 0 such that We introduce the next polynomial in the variables τ 1 , τ 2 , with coefficients in E (β,μ) that depends analytically in on D 0 . We consider some given real number a > 0 and we set where L d1 = [0, ∞)e √ −1d1 stands for a halfline in direction d 1 ∈ R, which depends on τ 1 in a way that cos(k 1 (d 1 − arg(u 1 ))) remains strictly positive and where L π,a = [0, a]e √ −1π stands for the segment [−a, 0]. Owing to Definition 1, the map F π,a can be written in the form of a polynomial in u 1 , whose coefficients (u 2 , z, ) → F π,a,j1 (u 2 , z, ) are holomorphic on C * × H β × D 0 . Notice that the expression F π,a does not depend on the choice of the direction d 1 . An explicit expression of the maps F π,a,j1 will be disclosed later on in the work (see Sect. 5, Proposition 10, Lemma 6). Due to Definition 1, we observe that when the radius a > 0 tends to infinity, each partial map u 2 → F π,a,j1 (u 2 , z, ) becomes close to the polynomial in u 2 , with bounded holomorphic coefficients on H β × D 0 , for any 0 ≤ β < β, provided that the variable u 2 satisfies for some small δ > 0. The forcing term f (t, z, ) is defined by the logarithmic type function Here log(z) stands for the principal value of the logarithm, namely log(z) = ln |z| + √ −1arg(z) with arg(z) ∈ (−π, π). By construction, we notice that provided that |z| is small enough with z / ∈ (−∞, 0], for any given small δ > 0. As a result, from the expansion (21), one checks that f (t, z, ) represents a holomorphic function in z ∈ H β and t, for t ∈ D (some small disc centered at 0) and ∈ D 0 \ {0}, provided that t / ∈ (−∞, 0] and as long as 0 > 0 is taken small enough. Furthermore, from the discussion above, when the radius a > 0 is taken large enough, the forcing term f (t, z, ) becomes proximate to the explicit logarithmic type function in t, Notice that each single piece ψ j1,j2 (t, ) = ( t) j1 /(log( t)) j2 , for j 1 ∈ J 1 , j 2 ∈ J 2 solves an explicit nonlinear ordinary differential equation which is both singularly perturbed in the parameter and possesses a differential operator of first order with irregular type, namely for all t, ∈ C such that t / ∈ (−∞, 0]. The reason for which we need to restrict ourselves to a truncated Laplace transform in the variable u 2 for the expression (20) instead of a complete Laplace integral (a = +∞) will be expounded later on in the work.

A Family of Related Initial Value Problems
In this subsection, we reduce the study of the main problem (11) to the analysis of a set of auxiliary problems which involves four independent complex variables. We plan to seek for solutions u(t, z, ) to the equation (11) with vanishing initial data at t = 0 of the form for some expression U π (u 1 , u 2 , z, ) in the four independent variables u 1 , u 2 , z and .
Following the usual chain rule (applied at a formal level at this stage of the work), the next computation holds As a result, the expression u(t, z, ) (formally) solves the equation (11) with vanishing data at t = 0 if the expression U π (u 1 , u 2 , z, ) solves the next equation for given vanishing initial data U π (0, 0, z, ) ≡ 0. Here the symbol (u 1 ∂ u1 − u 2 2 ∂ u2 ) h stands for the h−iterate of the differential operator u 1 ∂ u1 − u 2 2 ∂ u2 for any given integer h ≥ 1.
In order to be able to build genuine solutions to (11) and furthermore to study their asymptotic properties as tends to 0, we need to examine a more general family of related problems stated as follows.
For any given direction where the direction d 1 ∈ R depends on τ 1 in a way that cos(k 1 (d 1 − arg(u 1 ))) remains strictly positive and where L d2,a = [0, a]e √ −1d2 stands for a segment of length a in direction d 2 . Owing to Definition 1, we notice that the expression F d2,a does not rely on the direction d 1 .

Construction of Analytic Solutions to the Set of Related Initial Value Problems
In this section, we plan to construct a family of analytic solutions U d1,d2 (u 1 , u 2 , z, ) to the auxiliary problem (30) obtained for well chosen directions d 1 ∈ R, for any given direction d 2 ∈ R.

The Shape of the Analytic Solutions and Associated Convolution Equation
For any given direction d 2 ∈ R, we search for a family of solutions to (30) in the form of a double Laplace transform and inverse Fourier integral Here, we assume that the so-called Borel-Fourier map ω d1 belongs to a Banach space of functions denoted F d1 (ν,β,μ,k1,ρ,a, ) described in the next definition.
Remark. Similar Banach spaces that involve functions with two complex and one real variables have been recently introduced in the works [11,12] by A. Lastra and the author.
Our main task within this subsection is to derive some convolution equation fulfilled by the Borel-Fourier map ω d1 .
We recall some features of the Laplace transform under the action of multiplication by a monomial and differential operators already stated in our foregoing work [9]. A detailed proof of the formulas stated in the forthcoming lemma can be found in the work [10], Lemma 2.
1. The action of the differential operator u k1+1 2. Let m ≥ 1 be an integer. The multiplication by u m 1 acting on (31) is expressed through 3. The differentiel operator u 2 ∂ u2 applies on (31) by means of In the next steps, we plan to express all the differential operators acting on the variables u 1 , u 2 that appear in the related initial value problem (30) by the agency of the basic operator listed in the above lemma.
At first, since the operators u 1 ∂ u1 and u 2 2 ∂ u2 commute to each other, one can rewrite (30) by way of the classical binomial formula as for given initial data U d2 (0, 0, z, ) ≡ 0.
In a second step, we apply a useful lemma already stated in the previous work of A. Lastra and the author, [13] which provides expansions for the iterations of the basic fuchsian operator u 1 ∂ u1 .
By dint of this lemma, we can recast the last equation (35) in the form In a last undertaking, we bring into play a helpful formula introduced in the work [22] and which appears in many papers of the author and his colleagues, going back to its earliest occurrence in [9], that is stated as follows.
On the basis of the identities displayed in Definition 3 and Lemma 1, this last way (41) of rephrasing (30) allows us to reach the following statement.

Action of Linear Convolution Operators
In this subsection, we investigate continuity properties of two useful linear convolutions operators acting on the Banach spaces given in Definition 4 and appearing in the above equation (42).
We consider the map for all x ≥ 0. Notice that G(x) is well defined for all x ≥ 0, according to (44), since we can recast its expression by means of the change of variable g 1 = xg 2 for 0 ≤ g 2 ≤ 1 in the form for all x ≥ 0. According to the sharp bounds reached in Proposition 1 of the paper [10], we can single out a constant K 1 > 0 (depending on the constants for all x ≥ 1. Two cases arise. A) Assume first that τ 1 ∈ S d1 ∪ D ρ is chosen such that Then, under the condition (44), one finds a constant C 1.1 > 0 (relying on γ 1 , γ 3 , k 1 , K 1 ) such that for all τ 1 ∈ S d1 ∪ D ρ , under the constraint (50), all τ 2 ∈ D a and m ∈ R. B) Take for granted that Then, owing to (48), there exists a constant C 1.2 > 0 (leaning on γ 2 , γ 3 , k 1 ) with for all τ 1 ∈ S d1 ∪ D ρ submitted to (52), all τ 2 ∈ D a and m ∈ R. At last, the collection of the upper bound (47) coupled with (51) and (53) spawns the awaited estimates (45).

Solving the Associated Convolution Equation
In this subsection we uniquely solve the auxiliary convolution equation reached in (42) (91)). In the process, we need to perform a division by the next parameter depending polynomial provided that τ 1 ∈ S d1 ∪ D ρ . Crucial lower bounds are stated in the next lemma.

Lemma 4.
For a suitable choice of the inner radius r Q,RD > 0 and aperture η Q,RD > 0 of the sector S Q,RD set up in (15), there exist unbounded sectors S d1 centered at 0 with bisecting direction d 1 ∈ R and a small enough radius ρ > 0 for which the next lower estimates hold. One can select a constant C P > 0 with Proof. Since the complex roots q l (m), 0 ≤ l ≤ k 1 δ D − 1 of τ 1 → P m (τ 1 ) are explicit, we can factorize the polynomial as follows where for all 0 ≤ l ≤ k 1 δ D − 1, for any τ 1 ∈ C and m ∈ R. We single out an unbounded sector S d1 centered at 0, a small disc D ρ and we arrange the sector S Q,RD given in (15) in a way that the next two features hold: 1) A constant M 1 > 0 can be found such that 2) There exists a constant M 2 > 0 with We now provide some explanations for the two above bounds. • Concerning the first point 1), we observe that under the hypothesis (15), the roots q l (m) are bounded from below and satisfy |q l (m)| ≥ 2ρ for all m ∈ R, all 0 ≤ l ≤ δ D k 1 − 1 for a suitable choice of the radii r Q,RD , ρ > 0. Furthermore, for all m ∈ R, all 0 ≤ l ≤ δ D k 1 −1, these roots remain inside an union Q of unbounded sectors centered at 0 that do not cover a full neighborhood of 0 in C * whenever the aperture η Q,RD > 0 of S Q,RD is taken small enough. Therefore, we may choose a sector S d1 such that Such sector satisfies that for all 0 ≤ l ≤ δ D k 1 − 1, the quotients q l (m)/τ 1 lay outside some small disc centered at 1 in C for all τ 1 ∈ S d1 , all m ∈ R. As a consequence, (63) follows. • We select the sector S d1 and disc D ρ as above. The second point 2) follows from the fact that for any fixed 0 ≤ l 0 ≤ δ D k 1 − 1, the quotient τ 1 /q l0 (m) stays apart a small disc centered at 1 in C for all τ 1 ∈ S d1 ∪D ρ , all m ∈ R. Departing from the factorization (62) and taking benefit from the two lower bounds (63), (64) reached overhead, we arrive at as long as τ 1 ∈ S d1 ∪ D ρ , for all m ∈ R.
We introduce the next linear map In the next proposition, we show that H represents a 1/2−Lipschitz map on some suitable ball of the Banach space introduced in Definition 4.

Proposition 3.
We select an appropriate inner radius r Q,RD > 0 and aperture η Q,RD > 0 of the sector S Q,RD defined in (15), together with an unbounded sector S d1 and a radius ρ that fulfill the requirements of Lemma 4. Then, one occurs whenever ω 1 , ω 2 ∈B .
In the sequel, we provide bounds estimates for each piece of the map H .
In the next six estimates, we make use of the bounds obtained in Proposition 1 and Lemma 4. Namely, owing to the equality in (12) we get Besides, we check that Furthermore, we observe that , τ 2 , m)ds 1 || (ν,β,μ,k1,ρ,a, ) In the upcoming next three upper bounds, we apply both Propositions 1, 2 and Lemma 4. Indeed, on the basis of the inequality in (12)  inequality in (13) combined with (14), (16) and (17), we get Moreover, assumed from the second and third inequalities of (13) together with (14), (16) and (17), we reach provided that 1 ≤ q ≤ δ l along with whenever 1 ≤ p ≤ q − 1 and 1 ≤ q ≤ δ l . At last, we need to control the norm of the last term of the map H . Indeed, according to the upper and lower bounds (18), (61), we achieve || F(τ 1 , τ 2 , m, ) P m (τ 1 ) || (ν,β,μ,k1,ρ,a, ) = sup Now, we select 0 > 0 close enough to 0 and take suitably > 0 in a way that the next inequality C1 CP (rQ,R D ) Eventually, the collection of the ten above bounds (69) We turn to the second item addressing the 1/2−Lipschitz feature (68). Let ω 1 , ω 2 be elements of the closed ballB in F d1 (ν,β,μ,k1,ρ,a, ) where the radius has been fixed in the first item.
In the ensuing proposition, we come up with analytic solutions to the auxiliary problem (30).

Proposition 5.
For all unbounded sectors S d1 with bisecting directions d 1 ∈ R and radius ρ > 0 that obey the requirements of Lemma 4, for all directions d 2 ∈ R, we define the partial map (0, 0, z, ) ≡ 0.
Furthermore, the sectors U 1,d1 and U 2,d2 are submitted to the next technical constraints: • There exists a positive real number Δ 1 > 0 such that for all u 1 ∈ U 1,d1 , one can select a direction d 1,u1 ∈ R (relying on u 1 ) such that • The radius r U 1,d 1 > 0 of U 1,d1 suffers the next upper bounds for some positive real numberΔ 1 > 0 and where Δ 1 > 0 is defined in the above item. • There exists some positive real number Δ 2 > 0 such that Proof. We focus on the first item of the proposition. We consider the map ω d1 (τ 1 , τ 2 , m, ) built up in Proposition 4 and we select two bounded sectors U 1,d1 and U 2,d2 fulfilling the above prerequisite (93), (94) and (95). We set u j ∈ U j,dj , for j = 1, 2 and take for given real numbers r 1 ≥ 0, r 2 ∈ [0, a]. Then, a constant > 0 can be found for which the next bounds hold for all r 1 ≥ 0, r 2 ∈ [0, a] and all m ∈ R, where r U 2,d 2 > 0 stands for the radius of U 2,d2 . As a result, we reach the next upper bounds provided that u j ∈ U j,dj , j = 1, 2, z ∈ H β and all ∈ D 0 \ {0}, by the change of variable r 2 = r 1 /| | in the integral. The right handside of (97) turns out to be a constant unconstrained to on D 0 \ {0}. The first item follows.

Construction of a Finite Set of Genuine Solutions to a Properly Chosen Finite Set of Related Initial Value Problems
We first need to refresh the reader's memory of the definition of a good covering in C * as stated in the reference text book [8], Section XI-2.

Definition 5.
Let ς ≥ 2 be an integer. For all 0 ≤ p ≤ ς − 1, we set U p as open bounded sectors centered at 0 that share the next three qualities 1. The intersection U p ∩ U p+1 of two consecutive sectors U p and U p+1 is not empty for any 0 ≤ p ≤ ς − 1, where U ς = U 0 by convention. 2. The intersection of any three sectors U p ∩ U q ∩ U r is empty for distinct integers p, q, r ∈ {0, . . . , ς − 1}. 3. The union of all the sectors U p , 0 ≤ p ≤ ς − 1, covers some punctured neighborhood of 0 in C * , that is for some neighborhood U of 0 in C. Such a set U = {U p } 0≤p≤ς−1 of sectors is labelled a good covering in C * .
Furthermore, we introduce a notion of fitting finite sets of sectors.

Definition 6.
We consider three families of bounded open sectors centered at 0, for integers ς j ≥ 2, j = 1, 2 and a bounded sector T centered at 0 that are submitted to the next list of constraints: 1. For all 0 ≤ p ≤ ς 1 − 1 and any fixed ∈ D 0 \ {0}, for some given radius 0 > 0, the sectors U 1,dp with bisecting direction d p ∈ R is subjected to the next three conditions • For each direction d p ∈ R, 0 ≤ p ≤ ς 1 − 1, one can single out an unbounded sector S dp centered at 0 with bisecting direction d p that satisfies the requirements of Lemma 4 (namely for which the lower bounds (63) and (64) hold) • For each 0 ≤ p ≤ ς 1 − 1, there exists a positive real number Δ p > 0 such that for all u 1 ∈ U 1,dp , one can choose a direction d p,u1 ∈ R (depending on u 1 ) such that • The radius r U 1,dp > 0 of U 1,dp is submitted to the next upper bounds for some positive real numberΔ p > 0 and where Δ p > 0 is defined in the above item. 2. The radius r T > 0 of the sector T is subjected to the bounds where Δ p ,Δ p > 0 are defined in 1., for all 0 ≤ p ≤ ς 1 − 1. Besides, the sectors E p share a common radius given by 0 , for 0 ≤ p ≤ ς 1 − 1. 3. For all 0 ≤ p ≤ ς 1 − 1, the sectors E p and T satisfy the next feature t ∈ U 1,dp for all ∈ E p , all t ∈ T . 4. The set E represents a good covering in C * . Furthermore, the aperture of the sector T is taken close enough to 0 in a way that the set is not empty. 5. For all 0 ≤ q ≤ ς 2 − 1, the sectors U 2,dq with bisecting direction d q ∈ R obey the next constraint : there exists some positive real numberΔ q > 0 such that for all u 2 ∈ U 2,dq . Furthermore, we assume the existence of an index q 1 ∈ {0, . . . , ς 2 − 1} such that d q1 = π. 6. The set U 2 forms a good covering in C * . We say that the sets sectors U 1 , U 2 , E and T are fitting.
In the coming proposition, we build up a finite family of analytic solutions to auxiliary problems (30) with well chosen forcing terms. Proposition 6. Consider sets of sectors U 1 , U 2 , E and a sector T that are fitting (in the sense of Definition 6). For each 0 ≤ q ≤ ς 2 − 1, the equation where the forcing term F dq,a is given by the expression (29), possesses a finite set of holomorphic solutions (u 1 , u 2 , z) → U dp,dq (u 1 , u 2 , z, ), for 0 ≤ p ≤ ς 1 −1, on the domain U 1,dp × U 2,dq × H β , for all ∈ D 0 \ {0}, where 0 > 0 is taken small enough, for any 0 < β < β, that fulfills the constraint U dp,dq (0, 0, z, ) ≡ 0. These maps own the next two important features.
Proof. This proposition is a straight consequence of Proposition 5 and the definition of the fitting sectors chosen overhead in the proposition.
In the next proposition we study a finite set of maps related to the analytic solutions (105) to the problems (104) stated in Proposition 6. In particular we are interested in the control of their consecutive differences which turns out to be a crucial information for reaching their asymptotic features (see Sect. 4.2).
we introduce the map U dp,dq (t, u 2 , z, ) := U dp,dq ( t, u 2 , z, ) where U dp,dq is built up in Proposition 6. The next properties hold.
• For all 0 ≤ p ≤ ς 1 − 1, 0 ≤ q ≤ ς 2 − 1, the maps U dp,dq (t, u 2 , z, ) are bounded holomorphic on the product one can find two constants M p,1 , K p,1 > 0 such that two constants M q,2 , K q,2 > 0 can be singled out for which where the convention d ς2 = d 0 is taken.
Proof. The first item follows directly from the properties of the maps U dp,dq stated in Proposition 6 and from the features of the sectors E p and T listed in the points 2. and 3. of Definition 6. The second item needs more effort and uses a path deformation argument. Let us fix q = q 0 ∈ {0, . . . , ς 2 − 1} and take p ∈ {0, . . . , ς 1 − 1}. For any given τ 2 ∈ D a , m ∈ R, ∈ D 0 \ {0}, the partial maps τ 1 → ω dj (τ 1 , τ 2 , m, ), j = p, p + 1, are analytic continuation on the sector S dj of a common analytic map denoted τ 1 → ω(τ 1 , τ 2 , m, ) on the disc D ρ .
For any fixed ∈ E p+1 ∩ E p and t ∈ T , we deform the oriented path L dp+1, t − L dp, t into the union of three oriented pieces • An arc of circle centered at 0 with radius ρ/2 that rely the above two halflines By means of the classical Cauchy's theorem, we recast the next difference as a sum of three terms U dp+1,dq 0 (t, u 2 , z, ) − U dp,dq 0 (t, u 2 , z, ) Upper bounds are provided for the first building block of (110), Based on the bounds (96) and (106) together with the requirements described in Definition 6, we arrive at where C k1,ρ,Δp,p+1 = sup x≥0 provided that ∈ E p+1 ∩ E p , t ∈ T , u 2 ∈ U 2,dq 0 and z ∈ H β . In summary, the splitting (110) along with the bounds (111), (112), (117) breed the awaited estimates (108).
The third item leads to comparable bounds as the ones reached in the second item and leans again on a path deformation argument. Indeed, we set p 0 ∈ {0, . . . , ς 1 − 1} and take q ∈ {0, . . . , ς 2 − 1}. For any prescribed τ 1 ∈ S dp 0 , m ∈ R and ∈ E p0 , the partial map τ 2 → ω dp 0 (τ 1 , τ 2 , m, ) is analytic on the disc D a . As a result, we may bend the oriented path L dq+1,a − L dq,a into the union of three oriented basic geometrical loci • two segments • An arc of circle centered at 0 with radius a/2, joining the above two segments.
By dint of the classical Cauchy's theorem, we can reorganize the following difference as a sum of three contributions We plan to upper bound the first part of (118), namely Drew on the bounds (96) and (106) along with the requirements described in Definition 6 and by means of the change of variable s 1 = r 1 /| | in the integral, we deduce that provided that ∈ E p0 , t ∈ T , u 2 ∈ U 2,dq+1 ∩ U 2,dq and z ∈ H β . Much the same as above, the second piece of the decomposition (118), can be upper controled as as long as ∈ E p0 , t ∈ T , u 2 ∈ U 2,dq+1 ∩ U 2,dq and z ∈ H β . The closing block of (118), can be measured as follows. Owing to the lower bounds (103) set up in Definition 6, we check that cos(θ − arg(u 2 )) >Δ q,q+1 := min(Δ q ,Δ q+1 ) provided that u 2 ∈ U 2,dq+1 ∩ U 2,dq , whenever the angle θ belongs to (d q , d q+1 ) or (d q+1 , d q ). On the basis of the bounds (96), (106) and (124) together with the requirements stemming from Definition 6 and by means of the change of variable s 1 = r 1 /| | in the integral, we reach for all ∈ E p0 , t ∈ T , u 2 ∈ U 2,dq+1 ∩ U 2,dq and z ∈ H β . Eventually, the bounds (120), (122) and (125) reached above for the quantities J 1 , J 2 and J 3 applied to the splitting of the difference (118) beget the forecast bounds (109).

Gevrey Asymptotic Expansions for the Related Maps to the Analytic
Solutions of (104) We first remind the reader a result known as the Ramis-Sibuya theorem in the literature, see Lemma XI-2-6 in [8]. It represents a prominent tool in the proof of our main result stated in the next subsection.
Theorem (R.S.). Let (F, ||.|| F ) be a Banach space over C and we consider a good covering {U p } 0≤p≤ς−1 in C * as described in Definition 5. For all 0 ≤ p ≤ ς − 1, we set G p : U p → F as holomorphic functions that are subjected to the next two constraints 1. The maps G p are bounded on U p for all 0 ≤ p ≤ ς − 1.
2. The difference Θ p (u) = G p+1 (u) − G p (u) defines a holomorphic map on the intersection Z p = U p+1 ∩ U p which is exponentially flat of order k, for some integer k ≥ 1, meaning that one can select two constants C p , A p > 0 for which holds provided that u ∈ Z p , for all 0 ≤ p ≤ ς − 1. By convention, we set G ς = G 0 and U ς = U 0 .
Then, one can single out a formal power seriesĜ(u) = n≥0 G n u n with coefficients G n belonging to F, which is the common Gevrey asymptotic expansion of order 1/k relatively to u on U p for all the maps G p , for 0 ≤ p ≤ ς − 1. It means that two constants K p , M p > 0 can be pinpointed with the error bounds for all integers N ≥ 0, all u ∈ U p , all 0 ≤ p ≤ ς − 1.
In the next proposition, we come up with asymptotic expansions of Gevrey type for the maps U dp,dq (t, u 2 , z, ), built up in Proposition 7, relatively to each variable and u 2 . +∂ m u2 F dq,a ( t, u 2 , z, ) for all m ≥ 0, provided that t ∈ T , u 2 ∈ U 2,dq , z ∈ H β and ∈ E p0 . In order to allow the variable u 2 become close to the origin on the sector U 2,dq in the above relation, the next lemma is needed with F j1,j2 (z, ) = 1 (2π) 1/2 +∞ −∞ F j1,j2 (m, )e izm dm for all integers j 1 ∈ J 1 , j 2 ∈ J 2 , where t ∈ T , u 2 ∈ U 2,dq , z ∈ H β and ∈ E p0 . Using the parametrization τ 2 = ρ 2 exp( √ −1d q ) for 0 ≤ ρ 2 ≤ a, we can rewrite At this stage, we observe that this last integral can be explicitely computed. Indeed, the next recursion relation x m e −x/A dx, for any given positive real number a > 0, non vanishing complex number A ∈ C * and all integers m ≥ 1. We deduce the existence of polynomials P 1 a,j2−1 (X) and P 2 a,j2−1 (X) with real coefficients relying on a, j 2 such that F d q ,a,j 1 (u 2 , z, ) = j 2 ∈J 2 F j 1 ,j 2 (z, )(exp( √ −1d q )) j 2 P 2 a,j 2 −1 u 2 exp − provided that u 2 ∈ U 2,dq , z ∈ H β and ∈ E p0 . For this reason, for any prescribed integer m ≥ 0, all j 2 ∈ J 2 and given a > 0, a polynomial P 2 a,j2−1,m (X) ∈ R[X] and a rational function Q 1 a,j2−1,m (X) ∈ R(X) with one single pole at X = 0 can be singled out such that