Boundary layer expansions for initial value problems with two complex time variables

We study a family of partial differential equations in the complex domain, under the action of a complex perturbation parameter $\epsilon$. We construct inner and outer solutions of the problem and relate them to asymptotic representations via Gevrey asymptotic expansions with respect to $\epsilon$, in adequate domains. The construction of such analytic solutions is closely related to the procedure of summation with respect to an analytic germ, put forward in[J. Mozo-Fern\'andez, R. Sch\"afke, Asymptotic expansions and summability with respect to an analytic germ, Publ. Math. 63 (2019), no. 1, 3--79.], whilst the asymptotic representation leans on the cohomological approach determined by Ramis-Sibuya Theorem.


Introduction
The main aim in this work is to describe the analytic solutions and asymptotic behavior of the solutions of a family of initial value problems in the complex domain. Such family consists of partial differential equations in two complex time variables of the form (1) Q(∂ z )u(t 1 , t 2 , z, ǫ) = P (t k 1 +1 1 ∂ t 1 , t k 2 +1 2 ∂ t 2 , ∂ z , z, ǫ)u(t 1 , t 2 , z, ǫ) + f (t 1 , t 2 , z, ǫ), under given initial data u(0, t 2 , z, ǫ) ≡ u(t 1 , 0, z, ǫ) ≡ 0. Here, Q(X) ∈ C[X] and P (T 1 , T 2 , Z, z, ǫ) stands for a polynomial in (T 1 , T 2 , Z) with holomorphic coefficients w.r.t. (z, ǫ) on H β ×D(0, ǫ 0 ), where H β stands for the horizontal strip in the complex plane H β := {z ∈ C : |Im(z)| < β}, for some β > 0, and D(0, ǫ 0 ) ⊆ C stands for the open disc centered at the origin with radius ǫ 0 , for some small ǫ 0 > 0. The symbol ǫ acts as a small complex perturbation parameter in the equation. Moreover, k 1 , k 2 are positive ingers with 1 ≤ k 1 < k 2 . The forcing term, constructed in detail in Section 3, turns out to be a holomorphic function in C 2 × H β ′ × D(0, ǫ 0 ). In this paper, we also adopt the notation D(0, r) for the closed disc centered at 0 ∈ C and radius r > 0. The precise constrains involving the parameters involved in each of the equations determining the family of PDEs under study is described in Section 3. This is the continuation of a series of works devoted to the study of PDEs in the complex domain under the action of two complex time variables. In [13], the authors have studied a family of nonlinear initial value Cauchy problems of the form (2) Q(∂ z )∂ t 1 ∂ t 2 u(t 1 , t 2 , z, ǫ) = (P 1 (∂ z , ǫ)u(t 1 , t 2 , z, ǫ))(P 2 (∂ z , ǫ)u(t 1 , t 2 , z, ǫ)) + P (t 1 , t 2 , ∂ t 1 , ∂ t 2 , ∂ z , ǫ)u(t 1 , t 2 , z, ǫ) + f (t 1 , t 2 , z, ǫ), where the terms in Q, P, P 1 , P 2 are such that the action of t 1 and t 2 is symmetric. Moreover, we assume that the polynomial where L 1 involves leading terms of the differential operator P , can be factorized in such a way that each of the factors only depend on one of the times variables, i.e. P(t 1 , t 2 , ∂ t 1 , ∂ t 2 , ∂ z , ǫ) = P 1 (t 1 , ∂ t 1 , ∂ z , ǫ)P 2 (t 2 , ∂ t 2 , ∂ z , ǫ).
From this symmetric configuration one is able to construct families of analytic bounded solutions for open finite sectors with vertex at the origin in C, β ′′ > 0, and (E h ) 0≤h≤ι−1 is a good covering of C ⋆ (see Definition 4). Moreover, an asymptotic behavior of such solutions can be observed with respect to the perturbation parameter ǫ. Indeed, there exists a formal power series ǫ →û(t 1 , t 2 , z, ǫ) ∈ E[[ǫ]], where E stands for the Banach space of holomorphic and bounded functions defined in T 1 × T 2 × H β ′′ with the norm of the supremum, which turns out to be a formal solution of (2). In addition to this, a multisummability result joins both, analytic and formal solutions (see [13], Theorem 2).
In the second study [14], the property of symmetry of the equations drops, and P in (3) is no longer factorizable into two terms which only present dependence on one of the time variables. This asymmetry causes that the procedure followed in [13] is no longer valid in that second framework and the procedure followed differs from that in [13].
In both studies, a Borel-Laplace method is applied. In the symmetric case, the analytic solution is constructed as the Laplace transform with respect to τ 1 and τ 2 of an auxiliary function ω(τ 1 , τ 2 ), which is well defined in a domain of the form (S 1 ∪ D(0, ρ)) × (S 2 ∪ D(0, ρ)), for some ρ > 0 and certain sectors S 1 , S 2 with vertex at the origin. Moreover, such function admits an exponential growth at infinity with respect to τ 1 ∈ S 1 and τ 2 ∈ S 2 . This is the suitable configuration in order to apply Borel-Laplace techniques on each of the variables involved and achieve summability results (see Section 6.1). On the other hand the asymmetric settings in the problem considered in [14] causes the function ω(τ 1 , τ 2 ) only be defined in sets of the form S 1 × (S 2 ∪D(0, ρ)), and a small divisor phenomena is observed. Therefore, the summability conditions are not satisfied and a different approach has to be followed, focused on studying the natural domains and asymptotic behavior of ω(τ 1 , τ 2 ), and apply summability results asymmetrically.
In this sense, this work is concerned with a family of equations in which none of the previous strategies is satisfactory. On the one hand, the symmetric situation does not hold in the present work, so the strategy followed in [13] is not available. On the other hand, the strategy considered in [14] does not apply because the auxiliary function ω(τ 1 , τ 2 ) requires that, at least for one of the variables, a neighborhood of the origin is contained in its domain of definition. This is not the case, so a summability procedure can not be followed. The reason for failure is that the deformation path accomplished when computing the difference of two consecutive solutions of the main problem, written as the Laplace transform ω, is no longer applicable. A small divisor phenomena occurs which does not allow to determine the Gevrey orders involved in the relationship between the analytic and the formal solution. The precise reasoning on the failure of this procedure is detailed in Section 3.1.
A second novelty in the present work is the appearance of two different kinds of families of analytic solutions of the main problem for which one can give a picture of their asymptotic behavior with respect to the perturbation parameter. Following the terminology in the study of boundary layer solutions of equations, we distinguish the inner solutions (see Section 5.1) and the outer solutions (see Section 5.2) of the main problem, and describe their asymptotic representation with respect to the perturbation parameter near the origin. A recent work by the authors [18] constructs boundary layer expansions for certain initial value problem with merging turning points, regarding inner and outer solutions, which only considers the action of one time variable in the equation. In that previous work and also in the present work, the Gevrey orders of the asymptotic representation of the inner and outer solutions are different in general. As mentioned, we observe a comparable phenomenon in the present situation. However, in our context the inner solutions might not be λ 1 k 1 −summable in general, for some λ 1 > 0 to be precised.
The so-called inner and outer expansions are of great interest in mathematics, under the theory of matched asymptotic expansions. For a detailed theory on this subject we refer to classical textbooks such as [3,7,11,20,21,22]. For the general aspects on Gevrey asymptotic expansions in this context, we refer to the book [9].
The study of singularly perturbed PDEs in the complex domain is a topic of increasing interest. In 2015, H. Yamazawa and M. Yoshino [23] studied parametric Borel summability in semilinear systems of PDEs of fuchsian type, and of combined irregular and fuchsian type by M. Yoshino [24].
The theory of monomial summability was put forward by M. Canalis-Durand, J. Mozo-Fernández and R. Schäfke in [4]. Recently, S.A. Carrillo and J. Mozo-Fernández have studied further properties on monomial summability and Borel-Laplace methods on this theory in [5,6], and this technique has been successfully applied to families of singularly perturbed ODEs and PDEs. A step further is given by J. Mozo-Fernández and R. Schäfke in [19], where the authors put forward novel Gevrey asymptotic expansions and summability with respect to an analytic germ, and apply their technique to different families of PDEs and ODEs. In the present work, we make use of a similar approach to search for solutions of our main problem in the form of a Laplace-like transform with a meromorphic kernel at 0 (see Remark at page 9 and (19)).
We now give a general overview of the sections in which the present study is divided, and the main results obtained. The statement of the main problem under consideration is settled in Section 3, where we give arguments on the reason of failure of the methods used in [13,14] in this family of PDEs (see Section 3.1) and determine the shape of the analytic solution as a Laplace-like transform of a function related to the meromorphic kernel Ω provided in (19) (see Section 3.2): given two good coverings of C ⋆ , (E 0 h 1 ) 0≤h 1 ≤ι 1 −1 and (E ∞ h 2 ) 0≤h 2 ≤ι 2 −1 , with the first good covering consisting of sectors with wide enough opening (see Definition 4), we construct two sets of analytic solutions of (1) in the form The elements of the first family are constructed on a domain of the form The elements in the second family are constructed on domains of the form Here, T 1 is a finite sector, T 2 is an infinite sector and the direction ξ h ∈ R is an appropriate argument, the set T 2,ǫ is a bounded sector which depends on ǫ ∈ E ∞ h 2 and tends to infinity with ǫ → 0; λ 1 , λ 2 ∈ N. The function ω(u, m, ǫ) comes as a result of a fixed point argument in the Borel plane, in certain Banach spaces of functions (see Section 3.3 and Section 3.4). We finally relate the analytic solutions to an asymptotic representation in different subdomains achieving the construction and asymptotic results on the inner solutions (see Section 5.1) and on the outer solutions (see Section 5.2). In both situations, we provide differences of consecutive solutions (in the sense that they are associated to consecutive sectors in the fixed good covering), and apply Ramis-Sibuya theorem (see Theorem (RS)) to arrive at the existence of a common asymptotic representation of all inner solutions and an asymptotic representation of all the outer solutions. These asymptotic behavior appears in the form of Gevrey asymptotic expansions of order 1/(λ 1 k 1 ) with respect to the perturbation parameter ǫ on E ∞ h 2 regarding the inner solutions, whilst λ 2 k 2 −Gevrey summability can be observed regarding the outer solutions, with respect to the perturbation parameter (see Theorem 4).
The paper is organized as follows. In Section 2, we recall some definitions and results on Laplace transform, asymptotic expansions and Fourier transform. In Section 3, we state the main problem under study and analyze different ways to approach the problem. The section ends with the construction of the solution to an auxiliary problem in the Borel plane within a Banach space of functions with exponential growth and decay. Section 5 is devoted to the construction of the analytic solution of the main problem in addition to the inner and outer solutions. The work ends in Section 6 with the study of the parametric Gevrey asymptotic expansions of both types of solutions in appropriate domains, with respect to the perturbation parameter. The last section is focused on the technical proof of Lemma 4, left at the end of the paper for the sake of clarity in the ongoing argumentation.

Laplace transform, asymptotic expansions and Fourier transform
In this section, we recall the main definitions and results involved in the theory of Borel summable formal power series with coefficients in a fixed complex Banach space (E, · E ). For a detailed description on the classical theory, we refer to [1], Section 3.2. For the sake of simplicity, we have decided to make use of a slightly modified version of the classical theory. This slightly modified version of the k−Borel transform has been used in previous works by the authors such as [15,16,17], and in the study of singularly perturbed families of equations such as [13].
Definition 1 Let k ≥ 1 be an integer. For every n ≥ 1, we define m k (n) := Γ( n k ). A formal power seriesf (t) = ∞ n=1 f n t n ∈ tE [[t]] is said to be m k −summable with respect to t in the direction d ∈ [0, 2π) if i) there exists ρ > 0 such that the formal power series, known as the formal m k −Borel transform off , is absolutely convergent in D(0, ρ).
ii) there exists δ > 0 such that the function B m k (f )(τ ), which in principal is only defined on some neighborhood of the origin, can be analytically continued with respect to τ in a sector S d,δ = {τ ∈ C * : |d − arg(τ )| < δ}. Moreover, there exist C, K > 0 such that Under the previous hypotheses, the vector valued Laplace transform of B m k (f )(τ ) in the direction d is defined by The integration path consists of the half-line where γ depends on T and is chosen in such a way that cos(k(γ − arg(t))) ≥ δ 1 > 0, for some fixed δ 1 . The function L d m k (B m k (f ))(t) is well defined, holomorphic and bounded in any sector where π k < θ < π k + 2δ and 0 < R < δ 1 /K. This function is called the m k −sum of the formal seriesf (t) in the direction d.
We use the notation B m k ,t (resp. L d m k ,t ) to emphasize that t is the variable with respect to which Borel (resp. Laplace) transform is applied, if necessary.
The following result holds regarding the properties of the m k −Borel transform.
. Let k ≥ 1 be an integer number. Then, the following formal identity holds: We also recall some classical properties of inverse Fourier transform, which are used in our construction.
The inverse Fourier transform of f , given by can be extended to an analytic function on the horizontal strip Let φ(m) := imf (m) ∈ E (β,µ−1) . Then, the following statements hold: , and consider the convolution product of f and g,

Statement of the main problem and solution of an auxiliary problem
Our main aim in this work is to provide analytic and formal solutions to the main problem under study (1), and give information about the asymptotic behavior relating both. In this section, we detail the elements involved in the main problem under study, and provide different approaches which might be followed in order to search analytic and asymptotically related formal solutions. Let 1 ≤ k 1 < k 2 , and D 1 , D 2 ≥ 2 be integer numbers. We also fix λ 1 , λ 2 ∈ N. For 1 ≤ ℓ 1 ≤ D 1 and 1 ≤ ℓ 2 ≤ D 2 , we consider non negative integers δ ℓ 1 , δ ℓ 2 and ∆ ℓ 1 ℓ 2 .
We assume that We consider polynomials with complex coefficients Q, R D 1 D 2 , and R ℓ 1 ℓ 2 , for every 1 ≤ ℓ 1 ≤ D 1 − 1 and 1 ≤ ℓ 2 ≤ D 2 − 1. We assume that where A Q,R D 1 ,R D 2 stands for the sectorial annulus In addition to that, we assume We consider the main initial value problem under study under given initial conditions u(0, t 2 , z, ǫ) ≡ 0, and u(t 1 , 0, z, ǫ) ≡ 0.
where m → C ℓ 1 ℓ 2 (m, ǫ) ∈ E (β,µ) and satisfies uniform bounds with respect to ǫ ∈ D(0, ǫ 0 ). More precisely, there exists C ℓ 1 ℓ 2 > 0 such that (11) sup The function f is constructed as follows. Assume that ψ : C × R × D(0, ǫ 0 ) → C represents an entire function with respect to the first variable, continuous in R with respect to the second one, and holomorphic with respect to the third variable on the disc D(0, ǫ 0 ). Moreover, there exist C ψ , β, µ, ν ∈ R, with C ψ, β, ν > 0 and such that the next bound holds (12) |ψ In this situation, it is straight to check that the function where L γ = [0, ∞)e γ √ −1 can spin around the origin in order to guarantee that F is a holomorphic function on C 2 with respect to (T 1 , T 2 ) by analytic continuation. We define (14) f Therefore, f is a holomorphic function in C 2 × H β ′ × D(0, ǫ 0 ), for every 0 < β ′ < β. The reason why the forcing term is built in such a restrictive manner will be put into light later on in Section 3.2.
In this framework, we search for solutions of (10) of the form In view of (6), the properties of Fourier inverse transform, and the definition of f and c ℓ 1 ℓ 2 , we get that the expression U (T 1 , T 2 , m, ǫ) turns out to be a solution of

A first approach
As a first approach, one is tempted to follow techniques used in the previous works of the authors dealing with singularly perturbed partial differential equations in two complex time variables. On the one hand, the family of equations studied in [13] shows a symmetric role of the time variables in the equation. Although this is the case for (10), in that previous study it holds that the principal part of any of the equations in the family is factorizable as a product of two operators which split the dependence on the time variables. For this reason, that procedure is no longer valid in the present framework. On the other hand, the study made in [14] does not fit the main problem under study, as it can be deduced from the forthcoming argument.
Departing from the auxiliary equation (16), we proceed to apply m k 1 −Borel transformation with respect to T 1 and m k 2 −Borel transformation with respect to T 2 . Then, it holds that solves an equation of the form is a holomorphic function on C 2 w.r.t. the first variables, on D(0, ǫ 0 ) w.r.t. the fourth, and continuous on R w.r.t. the second variable, in view of (13). Let Proposition 3 Under the previous construction leading to (17), it holds that the possible actual holomorphic solutions ω(τ 1 , τ 2 , m, ǫ) of (17) can not be defined on any set of the form (S 1 ∪ D(0, ρ)) × S 2 (resp. S 1 × (S 2 ∪ D(0, ρ))) with respect to (τ 1 , τ 2 ), for any ρ > 0 and any infinite sectors S 1 , S 2 with vertex at the origin in C.
Proof Due to the equation (17) exhibits a symmetric behavior with respect to τ 1 and τ 2 , we only give details on the first of the previous statements, whilst the second follows from a symmetric argument. Fix m ∈ R and let ρ 0 > 0. We consider τ 2 ∈ S 2 such that We derive that any τ 1 ∈ C such that P m (τ 1 , τ 2 ) = 0 would satisfy that It follows that all the k 1 δ D 1 roots of τ 1 → P m (τ 1 , τ 2 ) for such choice of τ 2 belong to the disc D(0, ρ 0 ). The limit ρ 0 → 0 concludes the result. ✷ As a matter of fact, a small divisor phenomena is observed, which does not allow a summability procedure. Moreover, the possible actual holomorphic solutions ω(τ 1 , τ 2 , m, ǫ) of (17) are only expected to be well defined and holomorphic on products of sectors with infinite radius, say S 1 × S 2 . This construction does not allow to use the procedure applied neither in [13] nor [14], in order to analyze the asymptotic properties of the solutions with respect to the small perturbation parameter ǫ (see the introduction of this work for further details).

Second approach
In view of the failure of the approaches in [13,14] (see Section 3.1), we need to adopt another perspective.
We search for solutions of (16) of the special form where L d stands for a half line departing from the origin and with bisecting direction of argument given by d ∈ R, for some d ∈ R to be determined. If one pursues the construction of the first approach, one can construct genuine solutions of (16) as a double Laplace transform along well chosen halflines L d j ⊂ S j , j = 1, 2, as it was the case in our previous studies [13,14]. Our idea consists on merging this double integral along the product L d 1 × L d 2 into a simple integral along a halfline L d ⊂ C. Geometrically, it consists in a projection on the diagonal part (u, u) ∈ C 2 , u ∈ C of the space C 2 . The advantage is that in the related problem associated to the Borel map w(u, m, ǫ) (see (20)) involves now P m (τ, τ ) as a denominator which is this time well defined on a full neighborhood of 0 w.r.t τ and analytically continuable along unbounded sectors S d with suitable directions d ∈ R. As a result, the asymptotic analysis in ǫ becomes a tractable task. The drawback of this approach concerns the class of equations we are able to handle which is reduced compared to our previous studies and contains linear PDEs with special time reliance.
Remark: Observe that in the case k 1 = k 2 , the function U d turns out to be a Laplace transform of order k 1 in the meromorphic function near 0 ∈ C 2 . That situation is directly linked to a summability procedure with respect to a germ of function in C 2 , as described in [19]. However, in our situation, the function Ω is meromorphic near 0, not analytic.
Under the hypothesis that the solution of (16) is of the form (19), then ω in (19) solves the problem We substitute (17) by (20), as an auxiliary problem in order to solve the main equation. In the next section, we study some spaces of functions which are involved in the construction of the solution of (20).

Banach spaces of functions with exponential growth and decay
The Banach spaces described in this section are modified versions of those appearing in [15]. We omit the details which can be derived directly from that work, and the variations of that norm, stated in [13,14].
We fix positive real numbers β, µ, ν, with µ > 1, and an integer k ′ > 0. Let ρ > 0 and S d be an open and unbounded sector with bisecting direction d ∈ R. We denote D(0, ρ) the closed disc centered at 0 and positive radius ρ.
The next result is straighforward from the definition of the norm · (ν,β,µ,k ′ ) .
The proof of the next result follows analogous arguments as those in Proposition 2 [12], and refer to that work for a complete proof.

Solution of an auxiliary equation
At this point, we provide a brief summary on the strategy to trace. We continue with the approach described in Section 3.2, searching for solutions of (16) in the form (19). In this section, we guarantee the existence of ω(τ, m, ǫ) by means of a fixed point argument in the Banach space of functions introduced in Section 3.3. At this point, we follow a similar guideline as the one initiated in our former study [15]. We consider the next polynomial In the following, we need lower bounds of the expression P m (τ ) with respect to m and τ . In order to achieve this goal, we can factorize the polynomial w.r.t τ , namely where its roots q l (m) can be displayed explicitely as We set an unbounded sector S d centered at 0, a small disc D(0, ρ) and we adjust the sector A Q,R D 1 D 2 in a way that the next condition hold : a constant m > 0 can be chosen with D(0, ρ). Indeed, the inclusion (8) implies in particular that all the roots q l (m), 0 ≤ l ≤ k 1 δ D 1 + k 2 δ D 2 − 1 remain apart of some neighborhood of the origin, i.e satisfy |q l (m)| ≥ 2ρ for an appropriate choice of ρ > 0. Furthermore, when the aperture of A Q,R D 1 D 2 is taken close enough to 0, all these roots q l (m) stay inside a union U of unbounded sectors centered at 0 that do not cover a full neighborhood of 0 in C * . We assign a sector S d with By construction, the quotients q l (m)/τ fall apart some small disc centered at 1 in C for all τ ∈ S d , m ∈ R, 0 ≤ l ≤ k 1 δ D 1 + k 2 δ D 2 − 1. Then, (23) follows.
We are now ready to supply lower bounds for P m (τ ).

Actual solutions of the auxiliar problem (20)
Let E be a finite sector of C ⋆ with vertex at the origin. Let d ∈ R the bisecting direction of an infinite sector S d satisfying the hypotheses of Proposition 5. Let ω d be the solution of (20) constructed in Proposition 5. LetT 1 be a bounded sector with vertex at the origin, and letT 2 be an unbounded sector with vertex at the origin such that for all T j ∈T j , some ξ ∈ R (which might depend on T 1 and T 2 ) such that e √ −1ξ ∈ S d , all ǫ ∈ E and some δ > 0; for j = 1, 2.
The function U ξ defined in (19) turns out to be an actual solution of the auxiliary problem (20) in the domainT 1 ×T 2 × R × E. Moreover, the next estimates hold Due to k ′ ∈ (k 1 , k 2 ), the function L(x, y) is well defined in {(x, y) ∈ R 2 : x ≥ 0, y ≥ 0}. We write L(|T 1 |, |T 2 |) = L 1 (|T 1 |, |T 2 |) + L 2 (|T 1 |, |T 2 |), with for some ρ > 0. The proof of the following technical lemma is left to the Appendix at the end of the work, in order not to interfere with the ongoing arguments.
Following our new approach, we are able to construct global solutions U ξ (T 1 , T 2 , m, ǫ) in the time variable T 2 on an unbounded sectorT 2 . This was not possible in our two previous studies [13,14], where only local in time solutions were built up. This feature allows us to study the asymptotic expansions w.r.t ǫ in two different situations : when T 2 remains in a prescribed bounded domain (which is related to the forthcoming outer solution, constructed for the main equation 10) and when T 2 tends to ∞ in a related manner with ǫ (linked to the inner solutions of (10) that we plan to build in Section 5.1).

Inner and outer solutions of the main problem
In this section, we preserve the conditions established in the statement of the main problem under study in Section 3. More precisely, we assume the conditions (6)-(9) hold on the elements involved in the problem (10), with forcing term f determined by the construction and conditions in (12)- (14).
Our main aim is to construct solutions of the main problem (10) together with their asymptotic behavior in different situations. This will be done via (16) and the shape (19), as stated in our second approach in Section 3.2.
We first specify some geometric constructions on the domain of definition of the solutions. First, we recall the definition of a good covering in C ⋆ , and that of a good covering of prescribed opening.
Definition 4 Let ι 1 , ι 2 ≥ 2 be integer numbers. We consider two sets (E 0 are open sectors with vertex at the origin which satisfy the following assumptions: ii) The intersection of three different elements of each family is empty.
iii) The union of the elements of each family covers a punctured disc centered at 0 in C.
Definition 5 Let T 1 be a bounded sector with vertex at the origin and let T 2 be an unbounded sector with vertex at the origin. Let (E 0 h 1 ) 0≤h 1 ≤ι 1 −1 be a good covering of C ⋆ of opening π/(λ 2 k 2 ), be an infinite sector of bisecting direction d h 1 and let S ∞ d h 2 be an infinite sector of bisecting direction d h 2 , for all 0 ≤ h 2 ≤ ι 2 − 1. We say that the set for all t j ∈ T j , some ξ h 1 ∈ R (which might depend on t j and ǫ) such that e √ −1ξ h 1 ∈ S d h 1 , all ǫ ∈ E 0 h 1 and some δ > 0; for j = 1, 2.
for all t j ∈ T j , some ξ h 2 ∈ R (which might depend on t j and ǫ) such that e √ −1ξ h 2 ∈ S d h 2 , all ǫ ∈ E ∞ h 2 and some δ > 0; for j = 1, 2.
For every two consecutive outer solutions associated to (10), which are jointly defined in for two positive constantsĈ,D > 0.

Parametric Gevrey asymptotic expansions of the solutions
This section is devoted to the study of the asymptotic expansions associated to the outer and inner solutions of (10), with respect to the perturbation parameter. We make use of the cohomological criterion for k−summability of formal power series with coefficients in a Banach space (see [2], p. 121, or [10], Lemma XI-2-6), known as Ramis-Sibuya theorem. We first recall the main definition of this summability theory.

k−summable formal power series and Ramis-Sibuya Theorem
Let (E, · E ) be a complex Banach space.
Definition 7 Let k ≥ 1 be an integer number. A formal power serieŝ is said to be k−summable with respect to ǫ in the direction d ∈ R if there exists a bounded holomorphic function f defined in a finite sector V d of bisecting direction d and opening larger than π/k and values in E, such that it admitsf (ǫ) as its Gevrey asymptotic expansion of order 1/k on V d , i.e. for every proper subsector for every N ≥ 1 and ǫ ∈ V 1 . Such function is unique and it is called the k−sum of the formal power series. Furthermore, we can reconstruct the function f by means of a similar Borel-Laplace procedure as that stated in Section 2.
Then, for all 0 ≤ h ≤ ι−1, the functions G h (ǫ) have a common formal power seriesĜ(ǫ) ∈ E[[ǫ]] as Gevrey asymptotic expansion of order 1/k on E h . Moreover, if the aperture of one sector E h 0 is barely larger than π/k, then G h 0 (ǫ) is promoted as the k−sum ofĜ(ǫ) on E h 0 .

Parametric Gevrey asymptotic expansions of the inner and outer solutions of the main problem
In this section, we display the main results of the present work, namely the asymptotic behavior of the inner and outer solutions of (10), constructed in the previous section. In the present section, we assume the conditions (6)-(9) hold on the elements involved in the main equation under study (10), with forcing term f determined by the construction and conditions in (12)- (14). We depart from two good coverings (E ∞ h 2 ) 0≤h 2 ≤ι 2 −1 of C ⋆ , and (E 0 h 1 ) 0≤h 1 ≤ι 1 −1 of C ⋆ , this second good covering with opening π/(λ 2 k 2 ); and choose T 1 , T 2 satisfying (31), (32). We also fix χ ∞ 2 as in Definition 6, which allow us to construct the family of inner solutions associated to the first good covering. In addition to this, we choose ρ 2 > 0 and the corresponding outer solutions associated to the second good covering.
Let E 1 denote the Banach space of holomorphic and bounded functions defined in T 1 × χ ∞ 2 × H β ′ endowed with the sup. norm, and E 2 the Banach space of holomorphic and bounded functions in T 1 × (T 2 ∩ D(0, ρ 2 )) × H β ′ , with sup. norm.

Proof of Lemma 4
It is straight to check statement 1) in Lemma 4, due to k ′ < k 2 . We proceed to give proof to 2.a): let ρ ∞ 2 , ρ 1 > 0. One has that We focus on determining upper bounds for L(x), as x → ∞. Such bounds have already been observed in the proof of Proposition 4 of our recent contribution [12]. We provide a complete set of arguments for the sake of better readability. By dominated convergence, and the series representation of exp(νr k ′ ) we derive that (νr k ′ ) n n! e − r k 2 x dr = n≥0 ν n n! ∞ 0 (r k ′ ) n e − r k 2 x dr, for all x > 0. Let L n (x) = ∞ 0 (r k ′ ) n e − r k 2 x dr.
The change of variable r k 2 /x =r yields We have being the last integral upper bounded by a positive constant, for k 2 > k ′ . In conclusion, there exists some C 2.1 > 0 such that the statement in 2.b) holds.