On parametric Gevrey asymptotics for initial value problems with infinite order irregular singularity and linear fractional transforms

This paper is a continuation a previous work of the authors where parametric Gevrey asymptotics for singularly perturbed nonlinear PDEs has been studied. Here, the partial differential operators are combined with particular Moebius transforms in the time variable. As a result, the leading term of the main problem needs to be regularized by means of a singularly perturbed infinite order formal irregular operator that allows us to construct a set of genuine solutions in the form of a Laplace transform in time and inverse Fourier transform in space. Furthermore, we obtain Gevrey asymptotic expansions for these solutions of some order $K>1$ in the perturbation parameter.


Introduction
Within this paper, we focus on a family of nonlinear singularly perturbed equations which combines linear fractional transforms, partial derivatives and differential operators of infinite order of the form (1) Q(∂ z )u(t, z, ) = exp(α k t k+1 ∂ t )R(∂ z )u(t, z, ) + P (t, , {m κ,t, } κ∈I , ∂ t , ∂ z )u(t, z, ) where α, k > 0 are real numbers, Q(X), R(X), Q 1 (X), Q 2 (X) stand for polynomials with complex coefficients and P (t, , {U κ } κ∈I , V 1 , V 2 ) represents a polynomial in t, V 1 , V 2 , linear in U κ , with holomorphic coefficients w.r.t near the origin in C, where the symbol m κ,t, denotes a Moebius operator acting on the time variable through m κ,t, u(t, z, ) = u( t 1 + κ t , z, ) arXiv:1807.07453v1 [math.CV] 19 Jul 2018 for κ belonging to some finite subset I of the positive real numbers R * + . The forcing term f (t, z, ) embodies an analytic function in the vicinity of the origin relatively to (t, ) and holomorphic w.r.t z on a horizontal strip in C of the form H β = {z ∈ C/|Im(z)| < β} for some β > 0.
This work is a continuation of our previous study [14] where we aimed attention at the next problem (2) Q(∂ z )∂ t y(t, z, ) = H(t, , ∂ t , ∂ z )y(t, z, ) + Q 1 (∂ z )y(t, z, )Q 2 (∂ z )y(t, z, ) + f (t, z, ) for given vanishing initial data y(0, z, ) ≡ 0, where Q 1 , Q 2 , H stand for polynomials and f (t, z, ) is built up as above. Under suitable constraints on the components of (2), by means of Laplace and inverse Fourier transforms, we constructed a set of genuine bounded holomorphic solutions y p (t, z, ), 0 ≤ p ≤ ς − 1, for some integer ς ≥ 2, defined on domains T × H β × E p , for some well selected bounded sector T with vertex at 0 and E = {E p } 0≤p≤ς−1 a set of bounded sectors whose union contains a full neighborhood of 0 in C * . On the sectors E p , the solutions y p are shown to share w.r.t a common asymptotic expansionŷ(t, z, ) = n≥0 y n (t, z) n that represents a formal power series with bounded holomorphic coefficients y n (t, z) on T ×H β . Furthermore, this asymptotic expansion turns out to be (at most) of Gevrey order 1/k , for some integer k ≥ 1 (see Definition 7 for an explanation of this terminology) that comes out in the highest order term of the differential operator H which is of irregular type in the sense of [20] and displayed as for some integer δ D ≥ 2 and a polynomial R D (X). In the case when the aperture of E p can be taken slightly larger than π/k , the function → y p (t, z, ) represents the k −sum ofŷ on E p as described in Definition 7. Through the present contribution, our purpose is to carry out a comparable statement namely the existence of sectorial holomorphic solutions and associated asymptotic expansions as tends to 0 with controlled Gevrey bounds. However, the appearance of the nonlocal Moebius operator m κ,t, changes drastically the whole picture in comparison with our previous investigation [14]. Namely, according to our approach, a leading term of finite order δ D ≥ 2 in time as above (3) is insufficient to ensure the construction of actual holomorphic solutions to our initial problem (1). We need to supplant it by an exponential formal differential operator of infinite order w.r.t t, where (t k+1 ∂ t ) (p) represents the p−th iterate of the irregular differential operator t k+1 ∂t. As a result, (1) becomes singularly perturbed of irregular type but of infinite order in time. The reason for the choice of such a new leading term will be put into light later on in the introduction. A similar regularization procedure has been introduced in a different context in the paper [4] in order to obtain entire solutions in space for hydrodynamical PDEs such as the 3D Navier Stokes equations ∂ t v(t, x) + v(t, x) · ∇v(t, x) = −∇p(t, x) − µ∆v(t, x) , ∇ · v(t, x) = 0 for given 2π−periodic initial data v(0, x) = v 0 (x 1 , x 2 , x 3 ) on R 3 , where the usual Laplacian ∆ = 3 j=1 ∂ 2 x j is asked to be replaced by a (pseudo differential) operator exp(λA 1/2 ), where λ > 0 and A stands for the differential operator −∇ 2 , whose Fourier symbol is exp(λ|k|) for k ∈ Z 3 \ {0}. The resulting problem is shown to possess a solution v(t, x) that is analytic w.r.t x in C 3 for all t > 0 whereas the solutions of the initial problem are expected to develop singularities in space.
Under appropriate restrictions on the shape of (1) listed in the statement of Theorem 1, we can select 1. a set E of bounded sectors as mentioned above, which forms a so-called good covering in C * (see Definition 5), 2. a bounded sector T with bisecting direction d = 0 3. and a set of directions d p ∈ (− π 2 , π 2 ), 0 ≤ p ≤ ς − 1 organized in a way that the halflines L dp = R + exp( √ −1d p ) avoid the infinite set of zeros of the map τ → Q(im) − exp(αkτ k )R(im) for all m ∈ R, for which we can exhibit a family of bounded holomorphic solutions u p (t, z, ) on the products T × H β × E p . Each solution u p can be expressed as a Laplace transform of some order k and Fourier inverse transform (4) u p (t, z, ) = k (2π) 1 where w dp (u, m, ) stands for a function with (at most) exponential growth of order k on a sector containing L dp w.r.t u, owning exponential decay w.r.t m on R and relying analytically on near 0 (Theorem 1). Moreover, we show that the functions → u p (t, z, ) admit a common asymptotic expansionû(t, z, ) = m≥0 h m (t, z) m on E p that defines a formal power series with bounded holomorphic coefficients on T × H β . Besides, it turns out that this asymptotic expansion is (at most) of Gevrey order 1/k and leads to k−summability on E p 0 provided that one sector E p 0 has opening larger than π/k (Theorem 2). Another substantial contrast between the problems (1) and (2) lies in the fact that the real number k is asked to be less than 1. The situation k = 1 is not covered by the technics developped in this work and is postponed for future inspection. However, the special case k = 1 has already been explored by the authors for some families of Cauchy problems and gives rise to double scale structures involving 1 and so-called 1 + Gevrey estimates, see [16], [19]. Observe that if one performs the change of variable t = 1/s through the change of function u(t, z, ) = X(s, z, ) then the equation (1) is mapped into a singularly perturbed PDE combined with small shifts T κ, X(s, z, ) = X(s + κ , z, ), for κ ∈ I. This restriction concerning the Gevrey order of formal expansions of the analytic solutions is rather natural in the context of difference equations as observed by B. Braaksma and B. Faber in [5]. Namely, if A(x) stands for an invertible matrix of dimension n ≥ 1 with meromorphic coefficients at ∞ and G(x, y) represents a holomorphic function in 1/x and y near (∞, 0), under suitable assumptions on the formal fundamental matrix Y (x) of the linear equation y(x + 1) = A(x)y(x), any formal solutionŷ(x) ∈ C n [[1/x]] of the nonlinear difference equation can be decomposed as a sum of formal seriesŷ(x) = q h=1ŷ h (x) where eachŷ h (x) turns out to be k h −summable on suitable sectors for some real numbers 0 < k h ≤ 1, for 1 ≤ h ≤ q.
In order to construct the family of solutions {u p } 0≤p≤ς−1 mentioned above, we follow an approach that has been successfully applied by B. Faber and M. van der Put, see [7], in the study of formal aspects of differential-difference operators such as the construction of Newton polygons, factorizations and the extraction of formal solutions and consists in considering the translation x → x + κ as a formal differential operator of infinite order through the Taylor expansion at x, see (25). In our framework, the action of the Moebius transform T → T 1+κT is seen as an irregular operator of infinite order that can be formally written in the exponential form If one seeks for genuine solutions in the form (4), then the so-called Borel map w dp (τ, m, ) is asked to solve a related convolution equation (31) that involves infinite order operators exp(−κC k (τ )) where C k (τ ) denotes the convolution map given by (28). It turns out that this operator exp(−κC k (τ )) acts on spaces of analytic functions f (τ ) with (at most) exponential growth of order k, i.e bounded by C exp(ν|τ | k ) for some C, ν > 0 but increases strictly the type ν by a quantity depending on κ, k and ν as shown in Proposition 2, (48). It is worthwhile mentioning that the use of precise bounds for the so-called Wiman special function E α,β (z) = n≥0 z n /Γ(β +αn) for α, β > 0 at infinity is crucial in the proof that the order k is preserved under the action of exp(−κC k (τ )). Notice that this function also played a central role in proving multisummability properties of formal solutions in a perturbation parameter to certain families of nonlinear PDEs as described in our previous work [15]. As a result, the presence of an exponential type term exp(αkτ k ) in front of the equation (31) and therefore the infinite order operator exp(α k t k+1 ∂ t ) as leading term of (1) seems unavoidable to compensate such an exponential growth. We mention that a similar strategy has been carried out by S.Ōuchi in [17] who considered functional equations where p ≥ 1 is an integer, a j ∈ C * and ϕ j (z),f (z) stand for holomorphic functions near z = 0. He established the existence of formal power series solutionsû(z) ∈ C[[z]] that are proved to be p−summable in suitable directions by solving an associated convolution equation of infinite order for the Borel transform of order p in analytic functions spaces with (at most) exponential growth of order p on convenient unbounded sectors. More recently, in a work in progress [9], S. Hirose, H. Yamazawa and H. Tahara are extending the above statement to more general functional PDEs such as for analytic coefficients a 1 , a 2 , f near 0 ∈ C 2 for which formal series solutionŝ can be built up that are shown to be multisummable in appropriate multidirections in the sense defined in [2].
In a wider framework, there exists a gigantic literature dealing with infinite order PDEs/ODEs both in mathematics and in theoretical physics. We just quote some recent references somehow related to our research interests. In the paper [1], the authors study formal solutions and their Borel transform of singularly perturbed differential equations of infinite order j≥0 j P j (x, ∂ x )ψ(x, ) = 0 where P j (x, ξ) = k≥0 a j,k (x)ξ k represent entire functions with appropriate growth features. For a nice introduction of the point of view introduced by M. Sato called algebraic microlocal analysis, we refer to [11]. Other important contributions on infinite order ODEs in this context of algebraic microlocal analysis can be singled out such as [12], [13]. The paper is arranged as follows. In Section 2, we remind the reader the definition of Laplace transform for an order k chosen among the positive real numbers and basic formulas for the Fourier inverse transform acting on exponentially flat functions. In Section 3, we display our main problem (11) and describe the strategy used to solve it. In a first step, we restrain our inquiry for the sets of solutions to time rescaled function spaces, see (12). Then, we elect as potential candidates for solutions Laplace transforms of order k and Fourier inverse transforms of Borel maps w with exponential growth on unbounded sectors and exponential decay on the real line. In the last step, we write down the convolution problem (31) which is asked to be solved by the map w.
In Section 4, we analyze bounds for linear/nonlinear convolution operators of finite/infinite orders acting on different spaces of analytic functions on sectors. In Section 5, we solve the principal convolution problem (31) within the Banach spaces described in Sections 3 and 4 by means of a fixed point argument. In Section 6, we provide a set of genuine holomorphic solutions (104) to our initial equation (11) by executing backwards the lines of argument described in Section 3. Furthermore, we show that the difference of any two neighboring solutions tends to 0, for in the vicinity of the origin, faster than a function with exponential decay of order k. Finally, in Section 7, we prove the existence of a common asymptotic expansion of Gevrey order 1/k > 1 for the solutions mentioned above leaning on the flatness estimates reached in Section 6, by means of a theorem by Ramis and Sibuya.

Laplace, Borel transforms of order k and Fourier inverse maps
We recall the definition of Laplace transform of order k as introduced in [14] but here the order k is assumed to be a real number less than 1 and larger than 1/2. If z ∈ C * denotes a non vanishing complex number, we set z k = exp(k log(z)) where log(z) stands for the principal value of the complex logarithm defined as log(z) = log |z| + iarg(z) with −π < arg(z) < π.
Consider a holomorphic function w : S d,δ → C that 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 > 0, for some fixed δ 1 . The function L d k (w)(T ) is well defined, holomorphic and bounded on any sector S d,θ,R 1/k = {T ∈ C * : |T | < R 1/k , |d − arg(T )| < θ/2}, where π k < θ < π k + 2δ and 0 < R < δ 1 /K.
We restate the definition of some family of Banach spaces introduced in [14].
Finally, we remind the reader the definition of the inverse Fourier transform acting on the latter Banach spaces and some of its handy formulas relative to derivation and convolution product as stated in [14].
Definition 3 Let f ∈ E (β,µ) with β > 0, µ > 1. 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 occurs. b) Take g ∈ E (β,µ) and set as the convolution product of f and g. Then, ψ belongs to E (β,µ) and moreover,

Outline of the main initial value problem and related auxiliary problems
We set k ∈ ( 1 2 , 1) as a real number. Let D ≥ 2 be an integer, α D > 0 be a positive real number and c 12 , c f be complex numbers in C * . For 1 ≤ l ≤ D − 1, we consider complex numbers c l ∈ C * and non negative integers d l , δ l , ∆ l , together with positive real numbers κ l > 0 submitted to the next constraints. We assume that for all 1 ≤ l ≤ D − 2. We also take for granted that We consider a sequence of functions m → F n (m, ), for n ≥ 1 that belong to the Banach space E (β,µ) for some β > 0 and µ > max(deg(Q 1 )+1, deg(Q 2 )+1) and that depend analytically on ∈ D(0, 0 ), where D(0, 0 ) denotes the open disc centered at 0 in C with radius 0 > 0. We assume that there exist constants K 0 , T 0 > 0 such that which represents a convergent series on D(0, T 0 /2) with holomorphic and bounded coefficients on H β for any given width 0 < β < β. For all 1 ≤ l ≤ D − 1, we set the polynomials A l (T, ) = n∈I l A l,n ( )T n where I l are finite subsets of N and A l,n ( ) represent bounded holomorphic functions on the disc D(0, 0 ). We put for all 1 ≤ l ≤ D − 1. By construction, f (t, z, ) (resp. a l (t, )) defines a bounded holomorphic function on D(0, r) × H β × D(0, 0 ) (resp. D(0, r) × D(0, 0 )) for any given 0 < β < β and radii r, 0 > 0 with r 0 ≤ T 0 /2.
Let us introduce the next differential operator of infinite order formally defined as where (t k+1 ∂ t ) (p) stands for the p−th iterate of the differential operator t k+1 ∂ t . We consider a family of nonlinear singularly perturbed initial value problems which involves this latter operator of infinite order as leading term and linear fractional transforms for vanishing initial data u(0, z, ) = 0. Within this work, we search for time rescaled solutions of (11) of the form (12) u(t, z, ) = U ( t, z, ) Then, through the change of variable T = t, the expression U (T, z, ) is subjected to solve the next nonlinear singular problem involving fractional transforms for given initial data U (0, z, ) = 0. According to the assumption (7), there exists a real number Besides, with the help of the formula (8.7) from [21] p. 3630, we can expand the next differential operators Hence, according to (14) together with (15), we can write down the next equation for U (T, z, ), namely We now provide the definition of a modified version of some Banach spaces introduced in the papers [14], [15] that takes into account a ramified variable τ k for k given as above.
Definition 4 Let S d be an unbounded sector centered at 0 with bisecting direction d ∈ R. Let ν, β, µ > 0 and ρ > 0 be positive real numbers. Let k ∈ ( 1 2 , 1) defined as above. We set F d 2) The norm Lemma 1 For β, µ given in (9), there exists ν > 0 such that the series define a function that belongs to the space F d (ν,β,µ,k,ρ) for all ∈ D(0, 0 ), for any radius ρ > 0, any sector S d for d ∈ R.
Proof By Definition of the norm ||.|| (ν,β,µ,k,ρ) , we get the next upper bounds Due to the classical estimates for any real numbers m 1 ≥ 0, m 2 > 0, together with the Stirling formula (see [3], Appendix B.3) as n tends to +∞, we get two constants A 1 > 0 depending on k, ν and A 2 > 0 depending on k such that for all n ≥ 1. Therefore, if ν 1/k > A 2 /T 0 then we obtain the bounds By construction, according to the very definition of the Gamma function, the function F (T, z, ) can be represented as a Laplace transform of order k in direction d and Fourier inverse transform where the integration path L γ = R + e √ −1γ stands for a halfline with direction γ ∈ R which belongs to the set S d ∪ {0}, whenever T belongs to a sector S d,θ, with bisecting direction d, aperture π k < θ < π k + Ap(S d ) and radius with Ap(S d ) the aperture of S d for some > 0 and z appertains to a strip H β for any 0 < β < β together with ∈ D(0, 0 ).
In the next step, we seek for solutions U (T, z, ) of (16) on the same domains as above that can be expressed similarly to F (T, z, ) as integral representations through Laplace transforms of order k and Fourier inverse transform Our goal is the statement of a related problem fulfilled by the expression w(τ, m, ) that is forecast to be solved in the next section among the Banach spaces introduced above in Definition 4. Overall this section, let us assume that the function w(τ, m, ) belongs to the Banach space F d (ν,β,µ,k,ρ) . We first display some formulas related to the action of the differential operators of irregular type and multiplication by monomials. A similar statement has been given in Section 3 of [14] for formal series expansions.
Lemma 2 1) The action of the differential operator T k+1 ∂ T on U γ is given by 2) Let m > 0 be a real number. The action of the multiplication by T m on U γ is described through 3) The action of the differential operators Q(∂ z ) and multiplication with the resulting functions Q(∂ z )U γ maps U γ into a Laplace and Fourier transform, Proof Here we present direct analytic proofs which avoids the use of summability arguments through the Watson's lemma. The first point 1) is obtained by a mere derivation under the symbol. We turn to the second point 2). By application of the Fubini theorem we get that On the other hand, by successive path defor- By the very definition of the Gamma function and a path deformation yields As a result, according to the path deformation s = u k , we finally get which implies the identity (23). We aim our attention to the point 3). Again the Fubini theorem yields Therefore, we obtain As a result, and according to the paths deformations s = u k and v = u k , we get at last from which the identity (24) follows. 2 At the next level, we describe the action of the Moebius transform T → T 1+κ l T on U γ . It needs some preliminaries.
We depart as in the work of B. Faber and M. Van der Put [7] which describes the translation x → x + κ l as a differential operator of infinite order through the Taylor expansion. Namely, for any holomorphic function f : U → C defined on an open convex set U ⊂ C containing x and x + κ l , the next Taylor formula holds If one performs the change of variable x = 1/T through the change of function f (x) = U (1/x), one obtains a corresponding formula for U (T ), where (T 2 ∂ T ) (p) represents the p−th iterate of the irregular operator T 2 ∂ T . According to our hypothesis k ∈ (1/2, 1), we can rely on Lemma 2 1)2) for the next expansions As a result, if one denotes C k the operator defined as then the expression U γ ( T 1+κ l T , z, ) can be written as Laplace transform of order k in direction d and Fourier inverse transform where the integrant is formally presented as a series of operators and C (p) k stands for the k−th order iterate of the operator C k described above. By virtue of the identities (22), (23) and (24) presented in Lemma 2 and according to the integral representation for the Moebius map acting on U γ as described above in (29), we are now in position to state the main equation that shall fulfill the expression w(τ, m, ) provided that U γ (T, z, ) solves the equation in prepared form (16), namely

Action of convolution operators on analytic and continuous function spaces
The principal goal of this section is to present bounds for convolution maps acting on function spaces that are analytic on sectors in C and continuous on R. As in Definition 4, S d denotes an unbounded sector centered at 0 with bisecting direction d in R and D(0, ρ) \ L − stands for a cut disc centered at 0 where L − = (−ρ, 0]. Proposition 1 Let k ∈ ( 1 2 , 1) be a real number. We set γ 2 , γ 3 as real numbers submitted to the next assumption for all τ ∈ S d , all m ∈ R. Assume moreover that for all m ∈ R, the map τ → f (τ, m) extends analytically on the cut disc D(0, ρ) \ L − and for which one can choose a constant C 1 > 0 such that whenever τ ∈ D(0, ρ) \ L − and m ∈ R. We set Then, 1) The map (τ, m) → C k,γ 2 ,γ 3 (f )(τ, m) is a continuous function on S d × R, holomorphic w.r.t τ on S d for which one can sort a constant K 1 > 0 (depending on γ 2 ,σ) such that 2) For all m ∈ R, the function τ → C k,γ 2 ,γ 3 (f )(τ, m) extends analytically on D(0, ρ) \ L − . Furthermore, the inequality Proof We first investigate the global behaviour of the convolution operator C k,γ 2 ,γ 3 w.r.t τ on the unbounded sector S d , namely the point 1). Owing to the assumed bounds (33), we get In the next part of the proof, we need to focus on sharp upper bounds for the function We move onward as in Proposition 1 of [15] but we need to keep track on the constants appearing in the bounds in order to provide accurate estimates regarding the dependence with respect to the constants γ 3 and N . In accordance with the uniform expansion e σh = n≥0 (σh) n /n! on every compact interval [0, x], x ≥ 0, we can write down the expansion According to the Beta integral formula (see Appendix B from [3]), we recall that holds for any real numbers x ≥ 0 and α > 0, β > 0. Therefore, since N + γ 3 ≥ 1 and γ 2 > −1, we can rewrite for all x ≥ 0. On the other hand, as a consequence of the Stirling formula Γ(x) ∼ (2π) 1/2 x x e −x x −1/2 as x → +∞, for any given a > 0, there exist two constants K 1.1 , K 1.2 > 0 (depending on a) such that for all x ≥ 1. As a result, we get a constants K 1.2 > 0 (depending on γ 2 ) for which for all n ≥ 0. Hence, we get a constant K 1.3 > 0 (depending on γ 2 ) for all x ≥ 0. A second application of (40), shows the existence of a constant K 1.1 > 0 (depending in γ 2 ) for which 1 (n + 1) γ 2 +1 ≤ Γ(n + 1) holds for all n ≥ 0. Subsequently, we obtain a constant K 1.4 > 0 (depending on γ 2 ) such that for all x ≥ 0.
Owing to the asymptotic property at infinity of the Wiman function E α,β (z) = n≥0 z n /Γ(β+ αn), for any α, β > 0 stated in [6] p. 210 we get a constant K 1.5 > 0 (depending on γ 2 ,σ) with for all x ≥ 0. In accordance with this last inequality, by going back to our departing inequality (38), we obtain the expected bounds stated in the inequality (36), namely In a second part of the proof, we study local properties near the origin w.r.t τ . First, we can rewrite C k,γ 2 ,γ 3 by using the parametrization s = τ k u for 0 ≤ u ≤ 1. Namely, holds for all τ ∈ D(0, ρ)\L − whenever m ∈ R. Under the fourth assumption of (32) and from the construction of f (τ, m), the representation (43) induces that for all m ∈ R, τ → C k,γ 2 ,γ 3 (f )(τ, m) extends analytically on D(0, ρ) \ L − . Furthermore, granting to (34), one can deduce the bounds With the help of (39), we deduce that Proposition 2 Let k ∈ ( 1 2 , 1) be a real number. Let (τ, m) → f (τ, m) be a continuous function on S d × R, holomorphic w.r.t τ on S d for which there exist constants C 2 > 0, ν > 0 and µ > 1, β > 0 fulfilling for all τ ∈ S d , all m ∈ R. Take for granted that for all m ∈ R, the map τ → f (τ, m) extends analytically on the cut disc D(0, ρ)\L − suffering the next bounds : there exists a constant C 2 > 0 with Let κ l > 0 be a real number. We consider the operator k denotes the iterate of order p ≥ 0 of the operator C k defined as with the convention that C ). Then, 1) The map (τ, m) → (exp(−κ l C k )f )(τ, m) represents a continuous function on S d × R, holomorphic w.r.t τ on S d for which there exists a constant K 1 > 0 (depending on k,ν) such that 2) For all m ∈ R, the function τ → (exp(−κ l C k )f )(τ, m) extends analytically on D(0, ρ) \ L − . Furthermore, for all τ ∈ D(0, ρ) \ L − , all m ∈ R.
Proof We proceed as in the proof of Proposition 3 of [14]. Namely, according to the norm's definition 4, we can rewrite According to the triangular inequality |m| ≤ |m − m 1 | + |m 1 | and bearing in mind the definition of the norms of f and g, we deduce

Now, we get bounds from above that can be broken up in two parts
In the last step of the proof, we show that C 3.2 and C 3.3 have finite values. By construction, three positive constants Q 1 , Q 2 and R can be found such that for all m, m 1 ∈ R. Hence, that is finite owing to µ > max(deg(Q 1 ) + 1, deg(Q 2 ) + 1) submitted to the constraints (52) as shown in Lemma 4 from [18]. On the other hand, which is also finite. 2

Manufacturing of solutions to an auxiliary integral equation relying on a complex parameter
The main objective of this section is the construction of a unique solution of the equation (31) for vanishing initial data within the Banach spaces given in Definition 4. The first disclose further analytic assumptions on the leading polynomials Q(X) and R D (X) in order to be able to transform our problem (31) into a fixed point equation as stated below, see (101).
Namely, we take for granted that there exists a bounded sectorial annulus , small aperture η Q,R D > 0 for some radii r Q,R D ,2 > r Q,R D ,1 > 1 such that for all m ∈ R. For any integer l ∈ Z, we set See Figure 1 for a configuration of the points a l (m), l ∈ Z, and the set S Q,R D related to their definition. By construction, we see that Indeed, by construction of τ k = exp(k log(τ )), this equation is equivalent to write for some h ∈ Z. According to the hypothesis r Q,R D ,1 > 1, we know that |arg(a l (m))| < π/2 and hence (65) | arg(a l (m)) k | < π 2k < π since we assume that 1 2 < k < 1. Owing to the fact that arg(τ ) belongs to (−π, π), it forces h = 0 and hence arg(τ ) = arg(a l (m))/k.
We consider the set of so-called forbidden directions. We choose the aperture η Q,R D > 0 small enough in a way that for all directions d ∈ (−π/2, π/2) \ Θ Q,R D , we can find some unbounded sector S d centered at 0 with small aperture δ S d > 0 and bisecting direction d such that τ l / ∈ S d ∪ D(0, ρ) for some fixed ρ > 0 small enough and for all l ∈ Z.
For all τ ∈ C \ R − , all m ∈ R, we consider the function Let d ∈ (−π/2, π/2) \ Θ Q,R D and take a sector S d and a disc D(0, ρ) as above. 1) Our first goal is to provide lower bounds for the function |H(τ, m)| when τ ∈ S d and m ∈ R. Let τ ∈ S d . Then, we can write for some well chosen l ∈ Z, where r ≥ 0 and where θ belong to some small interval I S d which is close to 0 but such that 0 / ∈ I S d . In particular, we choose I S d in a way that arg(τ l ) + θ belongs to (−π, π) for all θ ∈ I S d .
Hence, owing to the fact that τ l solves (62), we can rewrite In particular, if the radius r Q,R D ,2 > r Q,R D ,1 is chosen close enough to r Q,R D ,1 , we get a constant η 1,l > 0 (depending on l) for which  In a second step, we aim attention at lower bounds for large values of |τ | on S d . We first carry out some preliminary computations, namely we need to expand We assume that the segment I S d is close enough to 0 in a way that we can find a constant ∆ 1 > 0 submitted to the next inequality for all m ∈ R, all θ ∈ I S d . Besides, according to the inclusion (59), we notice that holds for all m ∈ R. As a result, collecting (71), (72) and (73) yields the lower bounds Departing from the factorization (69) we get the next estimates from below We select a real number r 1 > 0 large enough such that for all r ≥ r 1 . Under this last constraint (76), we deduce from (74) and (75) that for all r ≥ r 1 , all θ ∈ I S d , all m ∈ R. Now, in view of the decomposition (67), we get in particular that |τ | = r|τ l |. Consequently, we see that for all τ ∈ S d with |τ | ≥ r 1 |τ l |. As a result, gathering (70) and (77), together with the shape of a l (m) and τ l given in (60), (63), we obtain two constants A H,d , B H,d > 0 depending on k, S Q,R D , S d for which for all τ ∈ S d , all m ∈ R.

Proposition 4
We make the next additional assumptions for all 1 ≤ l ≤ D − 1, where K 1 is a constant depending on k, ν defined in Proposition 2 1) and B H,d is selected in (78). Under the condition that the moduli |c 12 |,|c f | and |c l | for 1 ≤ l ≤ D − 1 are chosen small enough, we can find a constant > 0 for which the equation (31) has a unique solution w d (τ, m, ) in the space F d (ν,β,µ,k,ρ) controlled in norm in a way that ||w d (τ, m, )|| (ν,β,µ,k,ρ) ≤ for all ∈ D(0, 0 ), where β, µ > 0 are chosen as in (9), ν > 1 is taken as in Lemma 1, the sector S d and the disc D(0, ρ) are suitably selected in a way that τ l / ∈ S d ∪ D(0, ρ) for all l ∈ Z where τ l is displayed by (63) as described above.
Proof We initiate the proof with a lemma that introduces a map related to (31) and describes some of its properties that will allow us to apply a fixed point theorem for it.
Lemma 3 One can sort the moduli |c 12 |,|c f | and |c l | for 1 ≤ l ≤ D − 1 tiny in size for which a constant > 0 can be picked up in a way that the map H defined as fulfills the next features: 1) The next inclusion holds whereB(0, ) represents the closed ball of radius > 0 centered at 0 in F d (ν,β,µ,k,ρ) for all ∈ D(0, 0 ).
Proof Foremost, we focus on the first property (83). Let w(τ, m) belonging to F d (ν,β,µ,k,ρ) . We take ∈ D(0, 0 ) and set > 0 such that ||w(τ, m)|| (ν,β,µ,k,ρ) ≤ . In particular, we notice that the next estimates As a consequence of Proposition 2, we get that (τ, m) → exp(−κ l C k )(w)(τ, m) defines a continuous function on S d × R, holomorphic w.r.t τ on S d and a constant K 1 > 0 (depending on k, ν) can be found such that for all τ ∈ S d , all m ∈ R. Furthermore, the application of Proposition 1 for γ 2 = n+d l,k k − 1, γ 3 = δ l − 1 with n ∈ I l grants a constant C 4 > 0 (depending on I l ,k,κ l ,d l ,δ l ,ν) with Looking back to the lower bounds (78) and having a glance at the constraints (81) allows us to reach the estimates On the other hand, Proposition 2 guarantees that for all m ∈ R, the function τ → (exp(−κ l C k )w)(τ, m) extends analytically on D(0, ρ) \ L − with the bounds for all τ ∈ D(0, ρ)\L − , all m ∈ R. As a consequence, Proposition 1 specialized for γ 2 = n+d l,k k −1, γ 3 = δ l − 1 with n ∈ I l gives raise to a constant C 4 > 0 (depending on I l ,k,κ l ,d l ,δ l ,ν,ρ) for which Keeping in mind the lower bounds (80) we notice that By clustering (87) and (89), we conclude that there exists a constant C 5 > 0 (depending on I l , k, κ l , d l , δ l , ν, ρ, S Q,R D , S d , R l , Q) with Keeping in view the bounds (86), an application of Proposition 1 for where n ∈ I l , with 1 ≤ p ≤ δ l − 1 yields a constant C 6 > 0 (depending on I l ,k,κ l ,d l ,δ l ,ν) with for all τ ∈ S d , m ∈ R and 1 ≤ p ≤ δ l − 1.
Owing to the lower bounds (78) under the restriction (81), we deduce that Using the bounds (88) and calling up Proposition 1 for the values where n ∈ I l , with 1 ≤ p ≤ δ l − 1, we obtain a constant C 6 > 0 (depending on I l ,k,κ l ,d l ,δ l ,ν,ρ) for which With the help of the lower bounds (80), we deduce provided that τ ∈ D(0, ρ) \ L − and m ∈ R.
By grouping (91) and (92), we deduce the existence of a constant C 7 > 0 (depending on I l , k, κ l , d l , δ l , ν, ρ, S Q,R D , S d , R l , Q) with On the other hand, taking into account the assumption (8) and the lower bounds (70) together with (80), the application of Proposition 3 induces a constant C 3 > 0 (depending on Q 1 , Q 2 , Q, µ, k, ν) and a constant η 2 > 0 (equal to η 2,l from (70)) for which Furthermore, owing to Lemma 1 and in view of the lower estimates (70), (80), we obtain a constant K f > 0 (depending on k, ν and K 0 , T 0 from (9)) and η 2 > 0 such that for all ∈ D(0, 0 ). Now, we select |c 12 |, |c f | with |c l |, 1 ≤ l ≤ D − 1 small enough in a way that one can find a constant > 0 with Finally, if one collects the norms estimates (90), (93) in a row with (94) and (95) under the restriction (96), on gets the inclusion (83).
In the next part of the proof, we turn to the second feature (84). Namely, let w 1 (τ, m), w 2 (τ, m) belonging toB(0, ) inside F d (ν,β,µ,k,ρ) . From the very definition, we get in particular that the next bounds Following exactly the same steps as the sequence of inequalities (85), (86), (87), (88), (89) and (90), we observe that for the constant C 5 > 0 appearing in (90). Similarly, tracking the progression (85), (86), (88), (91), (92) and (93) yields the next bounds for all 1 ≤ p ≤ δ l − 1, where the constant C 7 > 0 shows up in (93). In order to handle the nonlinear term, we need to present the next difference in prepared form Then, in view of the assumption (8) and the lower bounds (70), (80), Proposition 3 gives raise to constants C 3 > 0 and η 2 > 0 appearing in (94) for which Now, we restrict the constants |c 12 | and |c l |, 1 ≤ l ≤ D − 1 in a way that one arrives at the next bounds At last, by assembling the estimates (97), (98) with (99) submitted to the constraints (100), one achieves the forcast shrinking property (84). Ultimately, we select |c 12 |, |c f | and |c l |, 1 ≤ l ≤ D − 1 small enough in a way that (96) and (100) are simultaneously fulfilled. Lemma 3 follows. 2 We turn back again to the proof of Proposition 4. For > 0 chosen as in Lemma 3, we set the closed ballB(0, ) ⊂ F d (ν,β,µ,k,ρ) which represents a complete metric space for the distance d(x, y) = ||x − y|| (ν,β,µ,k,ρ) . Owing to the lemma above, we observe that H induces a contractive application from (B(0, ), d) into itself. Then, according to the classical contractive mapping theorem, the map H possesses a unique fixed point that we set as w d (τ, m, ), meaning that

Analytic solutions on sectors to the main initial value problem
We turn back to the formal constructions realized in Section 3 by taking into consideration the solution of the related problem (31) built up in Section 5 within the Banach spaces described in Definition 4. At the onset, we remind the reader the definition of a good covering in C * and we disclose a modified version of so-called associated sets of sectors as proposed in our previous work [14].
Definition 5 Let ς ≥ 2 be an integer. For all 0 ≤ p ≤ ς −1, we set E p as an open sector centered at 0, with radius 0 > 0 such that E p ∩ E p+1 = ∅ for all 0 ≤ p ≤ ς − 1 (with the convention that E ς = E 0 ). Furthermore, we take for granted that the intersection of any three different elements of {E p } 0≤p≤ς−1 is empty and that ∪ ς−1 p=0 E p = U \ {0}, where U stands for some neighborhood of 0 in C. A set of sector {E p } 0≤p≤ς−1 with the above properties is called a good covering in C * .

Definition 6
We consider a good covering E = {E p } 0≤p≤ς−1 in C * . We fix a real number ρ > 0 and an open sector T centered at 0 with bisecting direction d = 0 and radius r T > 0 and we set up a family of open sectors with aperture θ > π/k and d p ∈ [−π, π), 0 ≤ p ≤ ς − 1 represent their bisecting directions. We say that the data {{S dp,θ, 0 r T } 0≤p≤ς−1 , T , ρ} are associated to E if the next two constraints hold: 1) There exists a set of unbounded sectors S dp , 0 ≤ p ≤ ς − 1 centered at 0 with suitably chosen bisecting direction d p ∈ (−π/2, π/2) and small aperture satisfying the property that τ l / ∈ S dp ∪ D(0, ρ) for some fixed radius ρ > 0 and all l ∈ Z where τ l stand for the complex numbers defined through (63).
2) For all ∈ E p , all t ∈ T , (102) t ∈ S dp,θ, 0 r T Figure 3: Good covering in C for all 0 ≤ p ≤ ς − 1. Figure 3 shows a configuration of a good covering of three sectors, one of them of opening larger than π/k for some k close to 1. We illustrate in Figure 4 a configuration of associated sectors.
In the following first principal result of the work, we build up a set of actual holomorphic solutions to the main initial value problem (11) defined on the sectors E p w.r.t . We also provide an upper control for the difference between any two neighboring solutions on E p ∩ E p+1 that turn out to be at most exponentially flat of order k.
Theorem 1 Let us assume that the constraints (6), (7), (8), (9) and (59) hold. We consider a good covering E = {E p } 0≤p≤ς−1 for which a set of data {{S dp,θ, 0 r T } 0≤p≤ς−1 , T , ρ} associated to E can be singled out. We take for granted that the constants α D and κ l , 1 ≤ l ≤ D − 1 appearing in the problem (11) are submitted to the next inequalities for all 0 ≤ p ≤ ς − 1, where B H,dp is framed in the construction (78) and depends on k,S Q,R D , S dp and K 1 > 0 is a constant relying on k, ν defined in Proposition 2 1). Then, whenever the moduli |c 12 |,|c f | and |c l |, 1 ≤ l ≤ D − 1 are taken sufficiently small, a family {u p (t, z, )} 0≤p≤ς−1 of genuine solutions of (11) can be established. More precisely, each function u p (t, z, ) defines a bounded holomorphic function on the product (T ∩D(0, σ))×H β ×E p for any given 0 < β < β and suitably tiny σ > 0 (where β comes out in (9)) and can be expressed as a Laplace transform of order k and Fourier inverse transform along a halfline L γp = R + e √ −1γp ⊂ S dp ∪ {0} and where w dp (τ, m, ) stands for a function that belongs to the Banach space F dp (ν,β,µ,k,ρ) for all ∈ D(0, 0 ). Furthermore, one can choose Figure 4: A configuration associated to the good covering in Figure 3 constants K p , M p > 0 and 0 < σ < σ (independent of ) with for all ∈ E p+1 ∩ E p , all 0 ≤ p ≤ ς − 1 (owing to the convention that u ς = u 0 ).
As a consequence, the Laplace transform of order k and Fourier inverse transform along a halfline L γp ⊂ S dp ∪ {0} represents 1) A holomorphic bounded function w.r.t T on a sector S dp,θ, with bisecting direction d p , aperture π k < θ < π k + Ap(S dp ), radius , where Ap(S dp ) stands for the aperture of S dp , for some real number > 0.
As a result, the function u p (t, z, ) = U γp ( t, z, ) defines a bounded holomorphic function w.r.t t on T ∩ D(0, σ) for some σ > 0 small enough, ∈ E p , z ∈ H β for any given 0 < β < β, owing to the fact that the sectors E p and T from the associated data fulfill the crucial feature (102). Moreover, u p (t, z, ) solves the main initial value problem (11) on the domain described above (T ∩ D(0, σ)) × H β × E p , for all 0 ≤ p ≤ ς − 1.
In the final part of the proof, we are concerned with the bounds (105). The steps of verification are comparable to the arguments displayed in Theorem 1 of [14] but we still decide to present the details for the benefit of clarity.
If the conditions above are fulfilled, the vector valued Laplace transform of order k of B k (â)(τ ) in the direction d is defined by where γ depends on and is chosen in such a way that cos(k(γ − arg( ))) ≥ δ 1 > 0, for some fixed δ 1 , for all in a sector S d,θ,R 1/k = { ∈ C * : | | < R 1/k , |d − arg( )| < θ/2}, where the angle θ and radius R suffer the next restrictions, π k < θ < π k + 2δ and 0 < R < δ 1 /K. Notice that this Laplace transform of order k differs slightly from the one introduced in Definition 1 which turns out to be more suitable for the problems under study in this work.
The function L d k (B k (â))( ) is called the k−sum of the formal seriesâ( ) in the direction d. It represents a bounded and holomorphic function on the sector S d,θ,R 1/k and is the unique such function that possesses the formal seriesâ( ) as Gevrey asymptotic expansion of order 1/k with respect to on S d,θ,R 1/k which means that for all π k < θ 1 < θ, there exist C, M > 0 such that ||L d k (B k (â))( ) − n−1 p=0 a p p || F ≤ CM n Γ(1 + n k )| | n for all n ≥ 1, all ∈ S d,θ 1 ,R 1/k .
In the sequel, we present a cohomological criterion for the existence of Gevrey asymptotics of order 1/k for suitable families of sectorial holomorphic functions and k−summability of formal series with coefficients in Banach spaces (see [3], p. 121 or [10], Lemma XI-2-6) which is known as the Ramis-Sibuya theorem in the literature. This result is an essential tool in the proof of our second main statement (Theorem 2).
Theorem (RS) Let (F, ||.|| F ) be a Banach space over C and {E p } 0≤p≤ς−1 be a good covering in C * . For all 0 ≤ p ≤ ς − 1, let G p be a holomorphic function from E p into the Banach space (F, ||.|| F ) and let the cocycle Θ p ( ) = G p+1 ( ) − G p ( ) be a holomorphic function from the sector Z p = E p+1 ∩ E p into E (with the convention that E ς = E 0 and G ς = G 0 ). We make the following assumptions.
1) The functions G p ( ) are bounded as ∈ E p tends to the origin in C, for all 0 ≤ p ≤ ς − 1.
2) The functions Θ p ( ) are exponentially flat of order k on Z p , for all 0 ≤ p ≤ ς − 1. This means that there exist constants C p , A p > 0 such that ||Θ p ( )|| F ≤ C p e −Ap/| | k for all ∈ Z p , all 0 ≤ p ≤ ς − 1.
Then, for all 0 ≤ p ≤ ς − 1, the functions G p ( ) have a common formal power serieŝ G( ) ∈ F[[ ]] as Gevrey asymptotic expansion of order 1/k on E p . Moreover, if the aperture of one sector E p 0 is slightly larger than π/k, then G p 0 ( ) represents the k−sum ofĜ( ) on E p 0 . 7.2 Gevrey asymptotic expansion in the complex parameter for the analytic solutions to the initial value problem Within this subsection, we disclose the second central result of our work, namely we establish the existence of a formal power series in the parameter whose coefficients are bounded holomorphic functions on the product of a sector T with small radius centered at 0 and a strip H β in C 2 , which represent the common Gevrey asymptotic expansion of order 1/k of the actual solutions u p (t, z, ) of (11) constructed in Theorem 1.
The second main result of this work can be stated as follows. as Gevrey asymptotic expansion of order 1/k. Strictly speaking, for all 0 ≤ p ≤ ς − 1, we can pick up two constants C p , M p > 0 with sup t∈T ∩D(0,σ ),z∈H β |u p (t, z, ) − n−1 m=0 h m (t, z) m | ≤ C p M n p Γ(1 + n k )| | n for all n ≥ 1, whenever ∈ E p . Furthermore, if the aperture of one sector E p 0 can be taken slightly larger than π/k, then the map → u p 0 (t, z, ) is promoted as the k−sum ofû(t, z, ) on E p 0 .
Proof We focus on the family of functions u p (t, z, ), 0 ≤ p ≤ ς − 1 constructed in Theorem 1. For all 0 ≤ p ≤ ς − 1, we define G p ( ) := (t, z) → u p (t, z, ), which represents by construction a holomorphic and bounded function from E p into the Banach space F of bounded holomorphic functions on (T ∩ D(0, σ )) × H β equipped with the supremum norm, where T is a bounded sector selected in Theorem 1, the radius σ > 0 is taken small enough and H β is a horizontal strip of width 0 < β < β. In accordance with the bounds (105), we deduce that the cocycle Θ p ( ) = G p+1 ( ) − G p ( ) is exponentially flat of order k on Z p = E p ∩ E p+1 , for any 0 ≤ p ≤ ς − 1.
Owing to Theorem (RS) displayed overhead, we obtain a formal power seriesĜ( ) ∈ F[[ ]] which represents the Gevrey asymptotic expansion of order 1/k of each G p ( ) on E p , for 0 ≤ p ≤ ς − 1. Besides, when the aperture of one sector E p 0 is slightly larger than π/k, the function G p 0 ( ) defines the k−sum ofĜ( ) on E p 0 as described through Definition 7. 2