Multi-level Gevrey solutions of singularly perturbed linear partial differential equations
aa r X i v : . [ m a t h . C V ] J u l Multi-level Gevrey solutions of singularly perturbed linear partialdifferential equations
A. Lastra ∗ , S. Malek † University of Alcal´a, Departamento de F´ısica y Matem´aticas,Ap. de Correos 20, E-28871 Alcal´a de Henares (Madrid), Spain,University of Lille 1, Laboratoire Paul Painlev´e,59655 Villeneuve d’Ascq cedex, France, [email protected]@math.univ-lille1.fr
September 6, 2018
Abstract
We study the asymptotic behavior of the solutions related to a family of singularly perturbed linearpartial differential equations in the complex domain. The analytic solutions obtained by means of a Borel-Laplace summation procedure are represented by a formal power series in the perturbation parameter.Indeed, the geometry of the problem gives rise to a decomposition of the formal and analytic solutions sothat a multi-level Gevrey order phenomenon appears. This result leans on a Malgrange-Sibuya theoremin several Gevrey levels.Key words: Linear partial differential equations, singular perturbations, formal power series, Borel-Laplace transform, Borel summability, Gevrey asymptotic expansions.2000 MSC: 35C10, 35C20
We study a family of singularly perturbed linear partial differential equations of the followingform(1)( ǫ r ( t k +1 ∂ t ) s + a )( ǫ r ( t k +1 ∂ t ) s + a ) ∂ Sz X ( t, z, ǫ ) = X ( s,κ ,κ ) ∈S b κ κ ( z, ǫ ) t s ( ∂ κ t ∂ κ z X )( t, z, ǫ ) , for given initial conditions(2) ( ∂ jz X )( t, , ǫ ) = φ i,j ( t, ǫ ) , ≤ j ≤ S − , where r and r stand for nonnegative integers (i. e. they belong to N = { , , ... } ), and s , s arepositive integers. We also fix a , a ∈ C ⋆ . S consists of a finite subset of elements ( s, κ , κ ) ∈ N . ∗ The author is partially supported by the project MTM2012-31439 of Ministerio de Ciencia e Innovacion,Spain † The author is partially supported by the french ANR-10-JCJC 0105 project and the PHC Polonium 2013project No. 28217SG.
We assume that
S > κ for every ( s, κ , κ ) ∈ S , and also that b s,κ ,κ ( z, ǫ ) belongs to the spaceof holomorphic functions in a neighborhood of the origin in C , O{ z, ǫ } .The initial data consist of holomorphic functions defined in a product of finite sectors withvertex at the origin.The framework of our study is the asymptotic study of singularly perturbed Cauchy problemsof the form(3) L ( t, z, ∂ t , ∂ z , ǫ )[ u ( t, z, ǫ )] = 0 , where L is a linear differential operator, for some given initial conditions ( ∂ jz u )( t, , ǫ ) = h j ( t, ǫ ),0 ≤ j ≤ ν − ǫ plays the role of a perturbationparameter near the origin and it turns out to be the variable in which asymptotic solutionsare being obtained. There is a wide literature dealing with the case where ǫ is real, L = ǫ m L ( t, z, ∂ t , ∂ z ) is acting on C ∞ ( R d ) functions or Sovolev spaces H s ( R d ). For a survey on thistopic, we refer to [5].On the other hand, the case for complex perturbation parameter ǫ has also been studied whensolving partial differential equations; in particular, when dealing with solutions belonging tospaces of analytic functions for singularly perturbed partial differential equations which exhibitseveral singularities of different nature. On this direction, one can cite the work by M. Canalis-Durand, J. Mozo-Fern´andez and R. Sch¨afke [3], S. Kamimoto [6], the second author [10, 11],and the first and the second author and J. Sanz [7]. In this last work, the appearance of both,irregular and fuchsian singularities in the problem causes that the Gevrey type concerning theasymptotic representation of the formal solution varies with respect to a problem in which onlyone type of such singularities appears.The asymptotic behavior of the solution in the problem under study (1), (2) differs from theprevious ones for the singularities are of different nature. Indeed, the appearance of two irregularsingularities t k +1 ∂ t perturbed by a certain power of ǫ enriches the accuracy of the informationprovided in the sense that different Gevrey orders can be distinguished.The main aim in this work is to construct actual holomorphic solutions X ( t, z, ǫ ) of (1), (2)which are represented by the formal solution(4) ˆ X ( t, z, ǫ ) = X β ≥ H β ( t, z ) ǫ β β ! , where H β belongs to an adecquate space of functions. The solution is holomorphic in a domainof the form T × U × E , where T and E are sectors of finite radius and vertex at the origin, and U is a neighborhood of the origin. In the asymptotic representation several Gevrey orders willappear.The strategy followed is to study, for every fixed ǫ ∈ E , a singular Cauchy problem (see (22),(23)) where Y ( t, z, ǫ ) := X ( ǫ − r t, z, ǫ ) turns out to be its solution. Of course, the domain ofdefinition of such a solution depends on the choice of ǫ ∈ E . More precisely, for every ǫ ∈ E onefinds a function ( T, z ) Y ( T, z, ǫ ) = X β ≥ Y β ( T, ǫ ) z β β !defined in a sector of radius depending on ǫ and wide enough opening in the variable T times aneighborhood of the origin (see Theorem 1). Indeed, the function T Y β ( T, ǫ ) is constructedas the m k − Laplace transform of τ W β ( τ, ǫ ) belonging to a well chosen Banach space (seeDefinition 1).At this point, we have handled a slightly modified version of the classical Laplace transformwhich better fits our needs, and has already been used in other works in the framework ofsingularly perturbed Cauchy problems with vanishing initial data, such as [8].It is worth noticing that some assumptions on the elements appearing on the equation ofthe singular Cauchy problem are made (see Assumption (D)) in order to be able to write theoperators involved of some form. This idea is reproduced from [13].The coefficients of the formal power series W ( τ, z, ǫ ) = X β ≥ W β ( τ, ǫ ) z β β !belong to some appropriate Banach space which depend on ǫ ∈ E ; and W ( τ, z, ǫ ) is constructedas the formal solution to the auxiliary Cauchy problem (13), (14) (see Proposition 2).The solution X ( t, z, ǫ ) is written in the form(5) X ( t, z, ǫ ) = X β ≥ Z L γ W β ( u, ǫ ) e − ( uǫrt ) k duu z β β ! , where L γ = [0 , ∞ ) e iγ for some γ ∈ [0 , π ).Regarding the singularities appearing, one realizes that the geometry of the problem iscrucial when approaching the auxiliary and the initial problems. Indeed, the singularities inequation (13) come from the zeroes in the variable τ of the equations ( kτ k ) s + a = 0 and ǫ r − s rk ( kτ k ) s + a = 0. The first equation provides fixed singularities which do not dependon ǫ whilst the second equation provides singularities that converge to the origin with ǫ . Thegeometry associated to this phenomenon is described in Section 2 and also in Assumption (B)in more detail. As a matter of fact, for every β ≥ τ W β ( τ, ǫ ) is a holomorphic functiondefined in a neighborhood of the origin which can be extended along an infinite sector (commonfor every ǫ ∈ E ). However, this initial neighborhood of 0 varies with ǫ ; all its complex numberswithin a certain range of directions and modulus larger than a function of ǫ which tends to 0with | ǫ | → E i , i = 1 , ..., ν −
1, where ( E i ) ≤ i ≤ ν − provides a good covering at 0 (see Definition 4). By means of a Ramis-Sibuya type theoremwith two levels we were able to estimate the difference of two consecutive solutions by deformingthe integration path of the m k − Laplace transform in (5). This deformation is made accordinglywith the geometry explained above so that, if some particular argument lies in between theintegration path of two consecutive solutions, then the Gevrey order within the asymptoticrepresentation is altered.We should mention that a similar phenomenon of parametric multilevel Gevrey asymptoticshas been observed recently by K. Suzuki and Y. Takei in [12] and Y. Takei in [14] for WKBsolutions of the Schr¨odinger equation ǫ ψ ′′ ( z ) = ( z − ǫ z ) ψ ( z )which possess 0 as fixed turning point and z ǫ = ǫ − as movable turning point. We stress thefact a resembling Ramis-Sibuya type theorem is used in this work.As a consequence, there exists a common ˆ X for every i = 1 , ..., ν − X ( t, z, ǫ ) = a ( t, z, ǫ ) + ˆ X ( t, z, ǫ ) + ˆ X ( t, z, ǫ ) , where a is a convergent series on some neighborhood of the origin, such that the solution X i ( t, z, ǫ ) is given by X i ( t, z, ǫ ) = a ( t, z, ǫ ) + X i ( t, z, ǫ ) + X i ( t, z, ǫ ) , where X ji admits ˆ X j as its Gevrey asymptotic expansion in E i of order ˆ r j for j = 1 , k − Borel-Laplace summability procedure in Section 4.1, we provide the solutions of asingular Cauchy problem (22), (23) which conform the support of the solution for the mainproblem in our work (32), (33). Finally, we estimate the difference of two solutions of the mainproblem in the intersection of their domain of definition in the perturbation parameter (seeTheorem 2) and obtain, by means of a Ramis-Sibuya theorem with two levels (see Section 6.1),a formal solution and a decomposition of both the analytic and the formal solution of the problemin two terms so that each term in the formal solution represents the corresponding term in theanalytic one under certain Gevrey type asymptotics (see Theorem 3).
Let ρ >
0. We denote D (0 , ρ ) the open disc in C , centered at 0 and with radius ρ . For d ∈ R ,we consider an unbounded sector { z ∈ C : | arg( z ) − d | < δ } , for some δ >
0, which is denotedby S d .Let E be an open and bounded sector with vertex at the origin. We put E = { ǫ ∈ C : | ǫ | < r E , θ , E < arg( ǫ ) < θ , E } , for some r E > ≤ θ , E < θ , E < π .Let δ >
0. For every ǫ ∈ E , we consider the open domain Ω( ǫ ) := ( S d ∪ D (0 , ρ )) \ Ω ( ǫ ),where Ω ( ǫ ) turns out to be a finite collection of sets of the form { τ ∈ C : | τ | > ρ ( | ǫ | ) , | arg( τ ) − d E | < δ } , where 0 ≤ d E < π is a real number depending on E , and x ∈ (0 , r E ) ρ ( x ) is amonotone increasing function with ρ ( x ) → x →
0. We give more technical details on theconstruction of this set afterwards, in Assumption (B.2), not to interrupt the reasonings. Weonly remark now that S d and Ω ( ǫ ) are such that S d ∩ Ω ( ǫ ) = ∅ for every ǫ ∈ E .Throughout this work, b and σ are fixed positive real numbers with b >
1, whilst k ≥ Definition 1
Let ǫ ∈ E and r ∈ Q , r > .For every β ≥ , we consider the vector space F β,ǫ, Ω( ǫ ) of holomorphic functions τ h ( τ, ǫ ) defined in Ω( ǫ ) such that k h ( τ, ǫ ) k β,ǫ, Ω( ǫ ) := sup τ ∈ Ω( ǫ ) ( (cid:12)(cid:12) τǫ r (cid:12)(cid:12) k (cid:12)(cid:12) τǫ r (cid:12)(cid:12) exp (cid:18) − σr b ( β ) (cid:12)(cid:12)(cid:12) τǫ r (cid:12)(cid:12)(cid:12) k (cid:19) | h ( τ, ǫ ) | ) < ∞ , where r b ( β ) = P βn =0 1( n +1) b . One can check that the pair ( F β,ǫ, Ω( ǫ ) , k·k β,ǫ, Ω( ǫ ) ) is a Banach space. Assumption (A):
Let a ∈ C with a = 0, and let s be a positive integer. We assume:( A.
1) arg( τ ) = π (2 j + 1) + arg( a ) ks , j = 0 , ..., ks − , for every τ ∈ S d \ { } .( A. ρ < | a | / ( ks k /k .The aim of the previous assumption is to avoid the roots of the function τ ( kτ k ) s + a when τ lies among the elements in Ω( ǫ ) for every ǫ ∈ E . This statement is clarified in thefollowing Lemma 1
Under Assumption (A), there exists a constant C > (which only depends on k , s , a ) such that (cid:12)(cid:12)(cid:12)(cid:12) kτ k ) s + a (cid:12)(cid:12)(cid:12)(cid:12) ≤ C , for every ǫ ∈ E and every τ ∈ Ω( ǫ ) . Proof
This proof follows analogous steps as the one of Lemma 1 in [7]. Let ǫ ∈ E .On the one hand, it is direct to check from Assumption (A.2) that any root of τ ( kτ k ) s + a keeps positive distance to D (0 , ρ ). This entails this distance provides an upper bound whensubstituting D (0 , ρ ) by D (0 , ρ ) \ Ω ( ǫ ) for every ǫ ∈ E .On the other hand, one has that1( kτ k ) s + a = ks − X j =0 A j τ − a / ( ks e iπ (cid:18) j +1 ks (cid:19) k /k , where A j = 1 a ks e − iπ (cid:16) ks − ks (cid:17) (2 j +1) a / ( ks )2 k /k , for every j = 0 , ..., ks −
1. Taking into account Assumption (A.1), there exists a constant C >
0, which does not depend on ǫ ∈ E , satisfying (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ − a / ( ks )2 e iπ (cid:16) j +1 ks (cid:17) k /k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C , for every τ ∈ S d and all j = 0 , ..., ks − ✷ We now give more detail on the construction of the set Ω( ǫ ) for each ǫ ∈ E . Assumption (B):
Let a ∈ C with a = 0 . Let r be a nonnegative integer, and r , s positive integers. We assume:( B. s r − s r > B.
2) For every ǫ ∈ E , the set Ω ( ǫ ) is constructed as follows:Ω ( ǫ ) := ks − [ j =0 { τ ∈ C : | τ | ≥ ρ ( | ǫ | ) , | arg( τ ) − d E ,j | < δ } , where ρ ( x ) = | a | / ( ks ) x s r − s r s s k k /k , x ≥ , (6) d E ,j = 1 ks (cid:18) π (2 j + 1) + arg( a ) + s r − s r s (cid:18) θ , E + θ , E (cid:19)(cid:19) , for every j = 0 , ..., ks −
1, and(7) δ > s r − s r ks s ( θ , E − θ , E ) . Assumption (B) is concerned with the nature of the roots of the function(8) τ ǫ r − s rk ( kτ k ) s + a , with(9) r := r s k . The dynamics of the singularities involved in the equation to study is related to the first itemin the previous assumption. More precisely, these tend to 0 with the perturbation parameter ǫ .The second enunciate in Assumption (B) is concerned with the distance of Ω ( ǫ ) to the roots of(8). Indeed, one can choose a positive lower bound for this distance which does not depend on ǫ ∈ E . Lemma 2
Let ǫ ∈ E . Under Assumption (B), there exists a constant C > (which onlydepends on k, s , s , r , r , a and which is independent of ǫ ∈ E ) such that (cid:12)(cid:12)(cid:12)(cid:12) ǫ r − s rk ( kτ k ) s + a (cid:12)(cid:12)(cid:12)(cid:12) ≤ C , for every τ ∈ Ω( ǫ ) . Proof
The proof of this result follows analogous steps as the corresponding one of Lemma 1.Let ǫ ∈ E . One can write 1 ǫ r − s rk ( kτ k ) s + a = ks − X j =0 B j ( ǫ ) τ − e iπ (cid:18) j +1 ks (cid:19) a / ( ks k /k ǫ r − s rkks , where B j ( ǫ ) = 1 a ks e − iπ (cid:16) ks − ks (cid:17) (2 j +1) a / ( ks )1 k /k ǫ s r − s r ks s , for every j = 0 , ..., ks −
1. Indeed, for all j = 0 , ..., ks − B j ( ǫ ) τ − e iπ (cid:18) j +1 ks (cid:19) a / ( ks k /k ǫ r − s rkks = a / ( ks )1 e − iπ (cid:16) ks − ks (cid:17) (2 j +1) k /k ks a ( τ ǫ − s r − s r ks s − k − /k e iπ (cid:16) j +1 ks (cid:17) a / ( ks )1 ) . At this point, it is sufficient to prove that the distance from τ ǫ − s r − s r ks s to k − /k e iπ (cid:16) j +1 ks (cid:17) a / ( ks )1 is upper bounded by a constant for every τ ∈ Ω( ǫ ), which does not depend on ǫ . Let τ ( ǫ ) ∈ C be satisfying(10) τ ( ǫ ) ǫ − s r − s r ks s − k − /k e iπ (cid:16) j +1 ks (cid:17) a / ( ks )1 = 0 . Regarding the construction of Ω ( ǫ ), the distance from τ ( ǫ ) to Ω ( ǫ ) might be attained at thecomplex points in Ω ( ǫ ) with arguments given by d E ,j ± δ or at the points in Ω ( ǫ ) of modulusequal to ρ ( | ǫ | ). In the first case, this distance is positive and does not depend on ǫ as it canbe deduced from (6) and (7). In the second case, the minimum distance is attained at τ ( ǫ ) / (cid:12)(cid:12)(cid:12)(cid:12) τ ( ǫ )2 ǫ − s r − s r ks s − k − /k e iπ (cid:16) j +1 ks (cid:17) a / ( ks )1 (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − a / ( ks )1 e iπ (cid:16) j +1 ks (cid:17) k /k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | a | / ( ks ) k /k > , which does not depend on ǫ . The conclusion is achieved from this point. ✷ Assumption (B.1) is substituted by the incoming Assumption (B.1)’. It deals with theexistence of attainable directions d ∈ R in such a way that S d ∩ ( ∪ ǫ ∈E Ω ( ǫ )) = ∅ . Indeed, forthis purpose one aims thatarg( τ ) = 1 ks (cid:20) π (2 j + 1) + arg( a ) + s r − s r s arg( ǫ ) (cid:21) , for any j = 0 , ..., ks − ǫ ∈ E and all τ ∈ S d \ { } .This entails that(11) ks arg( τ ) / ∈ (cid:18) π (2 j + 1) + arg( a ) + s r − s r s θ , E , π (2 j + 1) + arg( a ) + s r − s r s θ , E (cid:19) , for any j = 0 , ..., ks −
1. The overlapping of two consecutive sectors in Ω ( ǫ ) for some ǫ ∈ E would imply such d could not exist. Regarding (11), the existence of possible choices for direction d implies undertaking the following Assumption (C): θ , E − θ , E < πs s r − s r . which implies Assumption (B.1)’: s r − s r > s ¿From now on, we substitute Assumption (B.1) by Assumption (B.1)’, which is more restric-tive.The next lemmas are devoted to the behavior of the elements in the latter Banach spaceintroduced in Definition 1 under some operators, and its continuity. Lemma 3
Let ǫ ∈ E and β be a nonnegative integer. For every bounded continuous function g ( τ ) on Ω( ǫ ) such that M g := sup τ ∈ Ω( ǫ ) | g ( τ ) | does not depend on ǫ ∈ E , then k g ( τ ) h ( τ, ǫ ) k β,ǫ, Ω( ǫ ) ≤ M g k h ( τ, ǫ ) k β,ǫ, Ω( ǫ ) , for every h ∈ F β,ǫ, Ω( ǫ ) . Proof
It s a direct consecuence of the definition of the space F β,ǫ, Ω( ǫ ) . ✷ Proposition 1
Let ǫ ∈ E and r ∈ Q , r > . We consider real numbers ν ≥ and ξ ≥ − . Let S ≥ be a positive integer, r a positive rational number, and let α < β be nonnegative integers.Then, there exists a constant C > (depending on α , S , β , ξ , ν and which does not depend on ǫ ) with (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ k Z τ k ( τ k − s ) ν s ξ f ( s /k , ǫ ) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) β,ǫ, Ω( ǫ ) ≤ C | ǫ | rk (2+ ν + ξ ) (cid:18) ( β + 1) b β − α (cid:19) ν + ξ +3 k f ( τ, ǫ ) k α,ǫ, Ω( ǫ ) , for every f ∈ F α,ǫ, Ω( ǫ ) . Proof
Let f ∈ F α,ǫ, Ω( ǫ ) . For every τ ∈ Ω( ǫ ), the segment [0 , τ k ] is contained in Ω( ǫ ) for it is astar domain with respect to 0. By definition, we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ k Z τ k ( τ k − s ) ν s ξ f ( s /k , ǫ ) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) β,ǫ, Ω( ǫ ) = sup τ ∈ Ω( ǫ ) ( (cid:12)(cid:12) τǫ r (cid:12)(cid:12) k (cid:12)(cid:12) τǫ r (cid:12)(cid:12) exp (cid:18) − σr b ( β ) (cid:12)(cid:12)(cid:12) τǫ r (cid:12)(cid:12)(cid:12) k (cid:19) | τ | k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z τ k ( τ k − s ) ν s ξ f ( s /k , ǫ ) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)) ≤ sup τ ∈ Ω( ǫ ) (cid:12)(cid:12) τǫ r (cid:12)(cid:12) k (cid:12)(cid:12) τǫ r (cid:12)(cid:12) e − σr b ( β ) | τǫr | k | τ | k Z | τ | k s | ǫ r | k s /k | ǫ | r e − σr b ( α ) s | ǫr | k | f ( s /k e √− k arg( τ ) , ǫ ) | ( | τ | k − s ) ν s ξ s /k | ǫ | r s | ǫ r | k exp (cid:18) σr b ( α ) s | ǫ r | k (cid:19) ds . Taking into account that for every s ∈ [0 , | τ | k ] one hasexp (cid:18) − σr b ( β ) (cid:12)(cid:12)(cid:12) τǫ r (cid:12)(cid:12)(cid:12) k (cid:19) exp (cid:18) σr b ( α ) s | ǫ r | k (cid:19) ≤ exp (cid:18) − σ ( r b ( β ) − r b ( α )) (cid:12)(cid:12)(cid:12) τǫ r (cid:12)(cid:12)(cid:12) k (cid:19) =: e ( (cid:12)(cid:12)(cid:12) τǫ r (cid:12)(cid:12)(cid:12) k ) , and by the change of variable s = | ǫ r | k h , the last expression can be upper bounded by k f ( τ, ǫ ) k α,ǫ, Ω( ǫ ) sup τ ∈ Ω( ǫ ) (cid:12)(cid:12) τǫ r (cid:12)(cid:12) k (cid:12)(cid:12) τǫ r (cid:12)(cid:12) e ( (cid:12)(cid:12)(cid:12) τǫ r (cid:12)(cid:12)(cid:12) k ) | τ | k Z | τ | k | ǫr | k ( | τ | k − | ǫ r | k h ) ν | ǫ r | kξ h ξ h /k h | ǫ r | k dh ≤ | ǫ | rk (2+ ν + ξ ) k f ( τ, ǫ ) k α,ǫ, Ω( ǫ ) sup x ≥ B ( x ) , where B ( x ) = 1 + x x /k e ( x ) x Z x h /k h ( x − h ) ν h ξ dh. It only rests to provide a constant upper bound for B ( x ) in order to conclude the proof. Onecan estimate B ( x ) ≤ (1 + x ) e ( x ) x ν +1 Z x h ξ h dh = B ( x ) . ¿From standard calculations one arrives at B ( x ) ≤ C x ν + ξ +3 exp ( − σ ( r b ( β ) − r b ( α )) x )for some C >
0. The standard estimates x m e − m x ≤ (cid:18) m m (cid:19) m e − m , x ≥ m , m > r b , one concludes that B ( x ) ≤ C ( ν, ξ, σ ) (cid:18) ( β + 1) b β − α (cid:19) ν + ξ +3 . The result follows directly from here. ✷ In this section we study the existence of a formal solution for the forthcoming auxiliary Cauchyproblem (13), (14). After assuring the existence of a formal solution to this problem as a formalpower series in z , we provide estimates on its coefficients in terms of the norms in Definition 1.We keep the notations of Section 2, the construction of Ω( ǫ ) for every ǫ ∈ E and also thevalues of the constants r , r , s , s , r, k, b, σ, a and a hold.Let S be a positive integer and S be a finite subset of N . For every ( s, κ , κ ) ∈ S , b κ κ ( z, ǫ )is a holomorphic and bounded function in a product of discs centered at the origin. We put(12) b κ κ ( z, ǫ ) = X β ≥ b κ κ β ( ǫ ) z β β ! , for some holomorphic and bounded functions b κ κ β ( ǫ ) defined on some neighborhood of theorigin, which is common for every β ≥
0. We assume that b κ κ ( ǫ ) ≡ κ , κ , s ) ∈ S .We now make the following assumption on the elements of S . Assumption (D):
For every ( s, κ , κ ) ∈ S we have that S > κ , S > κ , κ ≥
1. Moreover,there exists a nonnegative integer δ κ ≥ k such that s = κ ( k + 1) + δ κ , and that S > j b (cid:16) δ κ k + κ (cid:17)k + 1.We also consider A κ ,p ∈ C for every ( s, κ , κ ) ∈ S and 1 ≤ p ≤ κ .It is worth mentioning that ǫ ∈ E remains fixed through the whole section, so that thesolution of the auxiliary Cauchy problem depends on ǫ .0For every fixed ǫ ∈ E we consider the following Cauchy problem(13) (( kτ k ) s + a )( ǫ r − s rk ( kτ k ) s + a ) ∂ Sz W ( τ, z, ǫ )= X ( s,κ ,κ ) ∈S b κ κ ( z, ǫ ) ǫ − r ( s − κ ) τ k Γ (cid:16) δ κ k (cid:17) Z τ k ( τ k − s ) δκ k − ( ks ) κ ∂ κ z W ( s /k , z, ǫ ) dss + X ≤ p ≤ κ − A κ ,p τ k Γ (cid:16) δ κ k ( κ − p ) k (cid:17) Z τ k ( τ k − s ) δκ k ( κ − p ) k − ( ks ) p ∂ κ z W ( s /k , z, ǫ ) dss , for given initial data(14) ( ∂ jz W )( τ, , ǫ ) = W j ( τ, ǫ ) ∈ F j,ǫ, Ω( ǫ ) , ≤ j ≤ S − . Proposition 2
Under Assumptions (A), (B), (C) on the geometric configuration of our frame-work, and under Assumption (D), there exists a formal power series solution of (13),(14), (15) W ( τ, z, ǫ ) = X β ≥ W β ( τ, ǫ ) z β β ! ∈ F β,ǫ, Ω( ǫ ) [[ z ]] , such that W β ( τ, ǫ ) ∈ F β,ǫ, Ω( ǫ ) for every β ≥ . Moreover, these coefficients satisfy the recursionformula (16) W β + S ( τ, ǫ ) β ! = 1(( kτ k ) s + a )( ǫ r − s rk ( kτ k ) s + a ) X ( s,κ ,κ ) ∈S X α + α = β b κ κ α ( ǫ ) α ! ǫ − r ( s − κ ) ×× τ k Γ (cid:16) δ κ k (cid:17) Z τ k ( τ k − s ) δκ k − ( ks ) κ W α + κ ( s /k , ǫ ) α ! dss + X ≤ p ≤ κ − A κ ,p τ k Γ (cid:16) δ κ + k ( κ − p ) k (cid:17) Z τ k ( τ k − s ) δκ k ( κ − p ) k − ( ks ) p W α + κ ( s /k , ǫ ) α ! dss , for every β ≥ , τ ∈ Ω( ǫ ) . Proof
Let β ≥ ǫ ∈ E and τ ∈ Ω( ǫ ). The recursion formula in (16) is directly obtained aftersubstitution of (15) in the equation (13). It is worth remarking that from the construction ofΩ( ǫ ) leading to Lemma 1 and Lemma 2, the function W β + S ( τ, ǫ ) is well defined and holomorphicin Ω( ǫ ) for every β ≥
0. We now prove that W β ( τ, ǫ ) ∈ F β,ǫ, Ω( ǫ ) for every β ≥ ≤ β ≤ S − w β ( ǫ ) := k W β ( τ, ǫ ) k β,ǫ, Ω( ǫ ) . Taking k·k β + S,ǫ, Ω( ǫ ) on both sides of the recursion formula (16), one obtains that w β + S ( ǫ ) β ! ≤ | ( kτ k ) s + a || ǫ r − s rk ( kτ k ) s + a | X ( s,κ ,κ ) ∈S X α + α = β | b κ κ α ( ǫ ) | α ! | ǫ | − r ( s − κ ) × × (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ k Γ (cid:16) δ κ k (cid:17) Z τ k ( τ k − s ) δκ k − ( ks ) κ W α + κ ( s /k , ǫ ) α ! dss (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) β + S,ǫ, Ω( ǫ ) + X ≤ p ≤ κ − | A κ ,p | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ k Γ (cid:16) δ κ + k ( κ − p ) k (cid:17) Z τ k ( τ k − s ) δκ k ( κ − p ) k − ( ks ) p W α + κ ( s /k , ǫ ) α ! dss (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) β + S,ǫ, Ω( ǫ ) . ¿From Lemma 1, Lemma 2, and Proposition 1, the right-hand side of the previous inequalitycan be upper bounded so that(17) w β + S ( ǫ ) β ! ≤ C X ( s,κ ,κ ) ∈S X α + α = β | b κ κ α ( ǫ ) | α ! | ǫ | − r ( s − κ ) ×× k κ | ǫ | rk ( δκ k + κ ) Γ (cid:16) δ κ k (cid:17) (cid:18) ( β + S + 1) b β + S − α − κ (cid:19) δκ k + κ +1 w α + κ ( ǫ ) α !+ X ≤ p ≤ κ − | A κ ,p | k p | ǫ | rk ( δκ k + κ ) Γ (cid:16) δ κ + k ( κ − p ) k (cid:17) (cid:18) ( β + S + 1) b β + S − α − κ (cid:19) δκ k ( κ − p ) k + p +1 w α + κ ( ǫ ) α ! , for some C >
0. Observe that k g ( τ, ǫ ) k α,ǫ, Ω( ǫ ) ≥ k g ( τ, ǫ ) k γ,ǫ, Ω( ǫ ) whenever α ≤ γ . FromAssumption (D), one has | ǫ | − r ( s − κ )+ rk ( δκ k + κ ) = 1 . Let M κ κ β > | b κ κ β ( ǫ ) | ≤ M κ κ β for all β ≥ s, κ , κ ) ∈S . We define B κ κ ( z ) = P β ≥ M κ κ β z β β ! . From the assumptions made on b κ κ there exist D , D > M κ κ β ≤ D D β β ! for every β ≥
0. The function B κ κ ( z ) turns out tobe a holomorphic and bounded function on some neighborhood of the origin.The terms (cid:16) ( β + S +1) b β + S − α − κ (cid:17) δκ k ( κ − p ) k + p +1 appearing in (17) can be upper bounded by C β ( β − · · · ( β − (cid:22) b ( δ κ k + κ ) (cid:23) )( β − (cid:22) b ( δ κ k + κ ) (cid:23) + 1)for some C > ∂ Sx u ( x, ǫ ) = C C X ( s,κ ,κ ) ∈S B κ κ ( x ) k κ Γ (cid:16) δ κ k (cid:17) + X ≤ p ≤ κ − | A κ ,p | k p Γ (cid:16) δ κ + k ( κ − p ) k (cid:17) (18) ∂ κ x x j b ( δκ k + κ ) k +1 ∂ j b ( δκ k + κ ) k +1 x u ( x, ǫ ) , with initial conditions(19) ( ∂ jx u )(0 , ǫ ) = w j ( ǫ ) , ≤ j ≤ S − . u ( x, ǫ ) = X β ≥ u β ( ǫ ) x β β ! ∈ R [[ x ]] . Moreover, its coefficients satisfy the recursion formula(20) u β + S ( ǫ ) β ! = C C X ( s,κ ,κ ) ∈S X α + α = β M κ κ α α ! | ǫ | − r ( s − κ ) β ! (cid:16) β − j b ( δ κ k + κ ) k(cid:17) ! × k κ | ǫ | rk ( δκ k + κ ) Γ (cid:16) δ κ k (cid:17) u α + κ ( ǫ ) α ! + X ≤ p ≤ κ − | A κ ,p | k p | ǫ | rk ( δκ k + κ ) Γ (cid:16) δ κ + k ( κ − p ) k (cid:17) u α + κ ( ǫ ) α ! . ¿From the initial conditions of the problem (18), (19), one gets that u j ( ǫ ) = w j ( ǫ ) for0 ≤ j ≤ S −
1. Regarding (17) and (20) one has w β ( ǫ ) ≤ u β ( ǫ ) , for every β ≥ ρ > w j ( ǫ ) < ρ for every 0 ≤ j ≤ S −
1, one has that the unique formal solution of (18),(19), u ( x, ǫ ) = P β ≥ u β ( ǫ ) x β β ! belongs to C { x } , with a radius of convergence Z >
0. Regardingthe previous steps one can affirm that this radius of convergence does not depend on the choiceof ǫ ∈ E .This yields the existence of M > X β ≥ u β ( ǫ ) Z β β ! < M, for every ǫ ∈ E which entails 0 < u β ( ǫ ) < M Z β β ! for every β ≥
0. The result is attained for(21) k W β ( τ, ǫ ) k β,ǫ, Ω( ǫ ) = w β ( ǫ ) ≤ u β ( ǫ ) ≤ M Z β β ! < ∞ , for every β ≥ ✷ In the present section we give some details on the k -Borel summability procedure of formal powerseries with coefficients belonging to a complex Banach space. This is a slightly modified version ofthe more classical one, which can be found in detail in [2], Section 3.2. This novel version entailsa different behavior of Borel and Laplace transforms with respect to the operators involved,which has already been used in the previous work [8] procuring fruitful results in the frameworkof Cauchy problems depending upon a complex perturbation parameter, with vanishing initialdata. We refer to [8] for further details.3 Definition 2
Let k ≥ be an integer. Let ( m k ( n )) n ≥ be the sequence m k ( n ) = Γ (cid:16) nk (cid:17) = Z ∞ t nk − e − t dt, n ≥ . Let ( E , k·k E ) be a complex Banach space. We say a formal power series ˆ X ( T ) = ∞ X n =1 a n T n ∈ T E [[ T ]] is m k -summable with respect to T in the direction d ∈ [0 , π ) if the following assertions hold:1. There exists ρ > such that the m k -Borel transform of ˆ X , B m k ( ˆ X ) , is absolutely conver-gent for | τ | < ρ , where B m k ( ˆ X )( τ ) = ∞ X n =1 a n Γ (cid:0) nk (cid:1) τ n ∈ τ E [[ τ ]] .
2. The series B m k ( ˆ X ) can be analytically continued in a sector S = { τ ∈ C ⋆ : | d − arg( τ ) | < δ } for some δ > . In addition to this, the extension is of exponential growth of order k in S , meaning that there exist C, K > such that (cid:13)(cid:13)(cid:13) B m k ( ˆ X )( τ ) (cid:13)(cid:13)(cid:13) E ≤ Ce K | τ | k , τ ∈ S. Under these assumptions, the vector valued Laplace transform of B m k ( ˆ X ) along direction d isdefined by L dm k (cid:16) B m k ( ˆ X ) (cid:17) ( T ) = k Z L γ B m k ( ˆ X )( u ) e − ( u/T ) k duu , where L γ is the path parametrized by u ∈ [0 , ∞ ) ue iγ ,for some appropriate direction γ de-pending on T , such that L γ ⊆ S and cos( k ( γ − arg( T ))) ≥ ∆ > for some ∆ > .The function L dm k ( B m k ( ˆ X ) is well defined and turns out to be a holomorphic and boundedfunction in any sector of the form S d,θ,R /k = { T ∈ C ⋆ : | T | < R /k , | d − arg( T ) | < θ/ } , forsome πk < θ < πk + 2 δ and < R < ∆ /K . This function is known as the m k -sum of the formalpower series ˆ X ( T ) in the direction d . The main aim in the present work is to study the asymptotic behavior of the solutionsof equation (1), (2) and relate them to its formal solution by means of Gevrey asymptoticexpansions. The following are some elementary properties concerning the m k -sums of formalpower series which will be crucial in our procedure.1) The function L dm k ( B m k ( ˆ X ))( T ) admits ˆ X ( T ) as its Gevrey asymptotic expansion of order1 /k with respect to t in S d,θ,R /k . More precisely, for every πk < θ < θ , there exist C, M > (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L dm k ( B m k ( ˆ X ))( T ) − n − X p =1 a p T p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E ≤ CM n Γ(1 + nk ) | T | n , for every n ≥ T ∈ S d,θ,R /k . Watson’s lemma (see Proposition 11 p.75 in [1]) allows us toaffirm that L dm k ( B m k ( ˆ X )( T ) is unique provided that the opening θ is larger than πk .42) The set of holomorphic functions having Gevrey asymptotic expansion of order 1 /k on asector with values in E turns out to be a differential algebra (see Theorem 18, 19 and 20 in [1]).This, and the uniqueness provided by Watson’s lemma provide some properties on m k -summableformal power series in direction d .We now assume E to be a Banach algebra for the product ⋆ . Let ˆ X , ˆ X ∈ T E [[ T ]] be m k -summable formal power series in direction d . Let q ≥ q ≥ X + ˆ X , ˆ X ⋆ ˆ X and T q ∂ q T ˆ X , which are elements of T E [[ T ]], are m k -summable in direction d . Then, one has L dm k ( B m k ( ˆ X ))( T ) + L dm k ( B m k ( ˆ X ))( T ) = L dm k ( B m k ( ˆ X + ˆ X ))( T ) , L dm k ( B m k ( ˆ X ))( T ) ⋆ L dm k ( B m k ( ˆ X ))( T ) = L dm k ( B m k ( ˆ X ⋆ ˆ X ))( T ) ,T q ∂ q T L dm k ( B m k ( ˆ X ))( T ) = L dm k ( B m k ( T q ∂ q T ˆ X ))( T ) , for every T ∈ S d,θ,R /k .The next proposition is written without proof for it can be found in [8], Proposition 6. Proposition 3
Let ˆ f ( t ) = P n ≥ f n t n ∈ E [[ t ]] , where ( E , k·k E ) is a Banach algebra. Let k, m ≥ be integers. The following formal identities hold. B m k ( t k +1 ∂ t ˆ f ( t ))( τ ) = kτ k B m k ( ˆ f ( t ))( τ ) , B m k ( t m ˆ f ( t ))( τ ) = τ k Γ (cid:0) mk (cid:1) Z τ k ( τ k − s ) mk − B m k ( ˆ f ( t ))( s /k ) dss . Let S ≥ r and positive integers r , s , s , k . The positive real number r is defined by (9). Let a , a ∈ C ⋆ and assume E , S d (and with it δ ) and D (0 , ρ ) are constructed in the shape of Section 2, for some d ∈ [0 , π ),and some ρ > γ ∈ [0 , π ) such that R + e iγ ⊆ S d ∪ { } .Let S be as in Section 3, which satisfies Assumption (D). For every ( s, κ , κ ) ∈ S we consideran holomorphic and bounded function b κ κ ( z, ǫ ) defined in a product of discs with center at theorigin which can be written as in (12), and A κ ,p ∈ C for every 1 ≤ p ≤ κ − ǫ ∈ E we consider the following Cauchy problem(22) (( T k +1 ∂ T ) s + a )( ǫ r − s rk ( T k +1 ∂ T ) s + a ) ∂ Sz Y ( T, z, ǫ )= X ( s,κ ,κ ) ∈S b κ κ ( z, ǫ ) ǫ − r ( s − κ ) T s ( ∂ κ T ∂ κ z Y )( T, z, ǫ ) , for given initial conditions(23) ( ∂ jz Y )( T, , ǫ ) = Y j ( T, ǫ ) , ≤ j ≤ S − . The initial conditions ( Y j ( T, ǫ )) ≤ j ≤ S − are constructed as follows: for every 0 ≤ j ≤ S −
1, let τ W j ( τ, ǫ ) be a holomorphic function defined in Ω( ǫ ). Moreover, assume there exists M > ǫ ∈E k W j ( τ, ǫ ) k j,ǫ, Ω( ǫ ) < M , ≤ j ≤ S − . Y j ( T, ǫ ) := L dm k ( W j ( τ, ǫ ))( T ) , where the Laplace transform is taken with respect to the variable τ , along the direction d .Observe from Definition 1 and Definition 2 that for every fixed ǫ ∈ E , the definition in (25)makes sense, providing a function T Y j ( T, ǫ ) which is well defined and holomorphic for all T = | T | e iθ such that cos( k ( γ − θ )) ≥ ∆, for some ∆ >
0, and | T | ≤ | ǫ | r ∆ /k ( σξ ( b )) /k , where ξ ( b ) = P n ≥ n +1) b .In the incoming result, we provide the solution of (22), (23) by means of the properties ofLaplace transform and the solution of the auxiliary Cauchy problem studied in Section 3. Theorem 1
Let ǫ ∈ E . Under the assumptions made at the beginning of the present section theproblem (22), (23) admits a holomorphic solution ( T, z ) Y ( T, z, ǫ ) defined in S d,θ, | ǫ | r (cid:16) ∆ σξ ( b ) (cid:17) /k × D (0 , /Z ) , for some Z > and some θ > π/k , where (26) S d,θ, | ǫ | r (cid:16) ∆ σξ ( b ) (cid:17) /k = ( T ∈ C ⋆ : | T | ≤ | ǫ | r (cid:18) ∆ σξ ( b ) (cid:19) /k , | arg( T ) − d | < θ ) . Proof
Taking into account Assumption (D), one can write T s ∂ κ T in the form T δ κ T κ ( k +1) ∂ κ T , for every ( s, κ , κ ) ∈ S , for some nonnegative integers δ κ . By means of the formula appearingin page 40 of [13], one can expand the previous operators in the form(27) T δ κ T κ ( k +1) ∂ κ T = T δ κ ( T k +1 ∂ T ) κ + X ≤ p ≤ κ − A κ ,p T k ( κ − p ) ( T k +1 ∂ T ) p , for some complex numbers A κ ,p ∈ C . Regarding (27), equation (22) is transformed into(28) (( T k +1 ∂ T ) s + a )( ǫ r − s rk ( T k +1 ∂ T ) s + a ) ∂ Sz Y ( T, z, ǫ )= X ( s,κ ,κ ) ∈S b κ κ ( z, ǫ ) ǫ − r ( s − κ ) T δ κ ( T k +1 ∂ T ) κ + X ≤ p ≤ κ − A κ ,p T k ( κ − p ) ( T k +1 ∂ T ) p ∂ κ z Y ( T, z, ǫ ) . One can apply the formal Borel transform B m k with respect to the variable T at both sidesof equation (27). The properties of this formal operator shown in Proposition 3 turn equation(27) into (13), with W ( τ, z, ǫ ) = B m k ( Y ( T, z, ǫ ))( τ ).Regarding (24), one has W j ∈ F j,ǫ, Ω( ǫ ) for 0 ≤ j ≤ S −
1. One can apply Proposition 2 tothe Cauchy problem with equation (13) and initial data given by(29) ( ∂ jz W )( τ, , ǫ ) = W j ( τ, ǫ ) , ≤ j ≤ S − X β ≥ W β ( τ, ǫ ) z β β ! ∈ F β,ǫ, Ω( ǫ ) [[ z ]] . Z , M > | W β ( τ, ǫ ) | ≤ M Z β β ! (cid:12)(cid:12) τǫ r (cid:12)(cid:12) (cid:12)(cid:12) τǫ r (cid:12)(cid:12) k exp (cid:18) σr b ( β ) (cid:12)(cid:12)(cid:12) τǫ r (cid:12)(cid:12)(cid:12) k (cid:19) , β ≥ , for every τ ∈ Ω( ǫ ).If we write T = | T | e iθ , we deduce that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k Z L γ W β ( u, ǫ ) e ( uT ) k duu (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k Z ∞ | W β ( se iγ , ǫ ) | e − sk | T | k cos( k ( γ − arg( T ))) ds ≤ kM Z β β ! Z ∞ exp( (cid:20) σξ ( b ) | ǫ | rk − ∆ | T | k (cid:21) s k ) ds, for every β ≥ L dm k ( W β ( τ, ǫ ))( T ) is well defined for T ∈ S d,θ, | ǫ | r (cid:16) ∆ σξ ( b ) (cid:17) /k , for every πk < θ < πk + 2 δ .Moreover, ( T, z ) Y ( T, z, ǫ ) := X β ≥ L dm k ( W β ( τ, ǫ ))( T ) z β β !defines a holomorphic function on S d,θ, | ǫ | r (cid:16) ∆ σξ ( b ) (cid:17) /k × D (0 , Z ), and it turns out to be a solutionof the problem (22), (23) from the properties of Laplace transform in 2), Section 4.1 and thefact that (30) is a formal solution of (13), (29). ✷ This section is devoted to the study of the formal and analytic solutions of the main problem inthe present work. The analytic solution is approximated by the formal solution in the perturba-tion parameter near the origin following different Gevrey levels which depend on the nature andlocation of the singular points involved. One may find two different situations depending on thegeometry of the problem: that in which only the singularities not depending on the perturbationparameter are involved, and other situation in which a moving singularity makes appearance.This last one depends on the perturbation parameter and makes the singularity tend to theorigin when the parameter vanishes.Let r be a nonnegative integer, and r , s , s , k be positive integers. We also fix a , a ∈ C ⋆ .We define r as in (9).We first recall the notion of a good covering and justify the geometric choices involved inthe framework of our problem. Definition 3
Let ( E i ) ≤ i ≤ ν − be a finite family of open sectors such that E i has its vertex at theorigin and finite radius r E i > for every ≤ i ≤ ν − . We say this family conforms a goodcovering in C ⋆ if E i ∩ E i +1 = ∅ for ≤ i ≤ ν − (we put E ν := E ) and ∪ ≤ i ≤ ν − E i = U \ { } for some neighborhood of the origin U . r E i := r E for every 0 ≤ i ≤ ν −
1, for somepositive real number r E , for our study is local at 0. Definition 4
Let ( E i ) ≤ i ≤ ν − be a good covering in C ⋆ . For every ≤ i ≤ ν − , we assume E i = { ǫ ∈ C ⋆ : | ǫ | < r E , θ , E i < arg( ǫ ) < θ , E i } , for some r E > and ≤ θ , E i < θ , E i < π . We write d E i for the bisecting direction of E i , ( θ , E i + θ , E i ) / . Let T be an open sector with vertex at 0 and finite radius, say r T > . We alsofix a family of open sectors S d i ,θ,r r E r T = (cid:26) t ∈ C ⋆ : | t | ≤ r r E r T , | d i − arg( t ) | < θ (cid:27) , with d i ∈ [0 , π ) for ≤ i ≤ ν − , and π/k < θ < π/k + δ , for some small enough δ > , underthe following properties:1. one has arg ( d i ) = π (2 j +1)+arg( a ) ks , for every j = 0 , ..., ks − .2. one has | arg( d i ) − d E i ,j | > δ i , for j = 0 , ..., ks − , where δ i := s r − s r ks s ( θ , E i − θ , E i ) ,and d E i ,j = ks (cid:16) π (2 j + 1) + arg( a ) + s r − s r s (cid:16) θ , E i + θ , E i (cid:17)(cid:17) .3. for every ≤ i ≤ ν − , t ∈ T and ǫ ∈ E i , one has ǫ r t ∈ S d i ,θ,r r E r T .Under the previous settings, we say the family { ( S d i ,θ,r r E r T ) ≤ i ≤ ν − , T } is associated to the goodcovering ( E i ) ≤ i ≤ ν − . Remark:
The previous construction is feasible under suitable choices for the elements in-volved. For example, if T is bisected by the positive real line and has a small enough opening,one can choose the constants in the definition of δ i such that δ i allows the third condition inthe previous definition to be satisfied for every 0 ≤ i ≤ ν − , π ).Let us consider a good covering in C ⋆ , ( E i ) ≤ i ≤ ν − . In the following, we identify the firstelement E with E ν .Let S ≥ S of N , and for every ( s, κ , κ ) ∈ S ,let b κ κ ( z, ǫ ) be as stated in Section 3, under the form (12).For each 0 ≤ i ≤ ν −
1, we study the Cauchy problem(32) ( ǫ r ( t k +1 ∂ t ) s + a )( ǫ r ( t k +1 ∂ t ) s + a ) ∂ Sz X i ( t, z, ǫ )= X ( s,κ ,κ ) ∈S b κ κ ( z, ǫ ) t s ( ∂ κ t ∂ κ z X i )( t, z, ǫ ) , for given initial conditions(33) ( ∂ jz ( X i ))( t, , ǫ ) = φ i,j ( t, ǫ ) , ≤ j ≤ S − , where the functions φ i,j ( t, ǫ ) are constructed in the following way:Let { ( S d i ,θ,r r E r T ) , T } be a family associated to the good covering ( E i ) ≤ i ≤ ν − . For the sake ofsimplicity in the notation, we will denote S d i ,θ,r E rr T by S d i from now on, for every 0 ≤ i ≤ ν − j ∈ { , ..., S − } and i ∈ { , ..., ν − } . We consider the construction in Section 2 forthe sets Ω( ǫ ), for a common sector S d i for every ǫ ∈ E i and define W ij ( τ, ǫ ) such that:8a) For every ǫ ∈ E i , the function τ W i,j ( τ, ǫ ) is an element in F j,ǫ, Ω( ǫ ) , with(34) k W i,j ( τ, ǫ ) k j,ǫ, Ω( ǫ ) < M , for some M > τ, ǫ ) W i,j ( τ, ǫ ) is a holomorphic function in ∪ ǫ ∈E i Ω( ǫ ) × E i .c) The function W i,j ( τ, ǫ ) coincides with W i +1 ,j ( τ, ǫ ) in the domain ∪ ǫ ∈ ( E i ∩E i +1 ) Ω( ǫ ) × ( E i ∩E i +1 ).Let γ i ∈ [0 , π ) be chosen in such a way that the set L γ i := R + e γ i √− ⊆ S d ∪ { } . Then, wedefine(35) φ i,j ( t, ǫ ) = Y i,j ( ǫ r t, ǫ ) := k Z L γi W i,j ( u, ǫ ) e − ( uǫrt ) k duu , for every ( t, ǫ ) ∈ T × E i . Regarding a), φ i,j is well defined and from b) one has φ i,j ( tǫ, ǫ ) turnsout to be a holomorphic function in T × E i .The next assumption is more restrictive than Assumption (B.1)’. We adopt it and substitute(B.1)’ for it in Assumption (B), for reasons that will be explained in the proof of Theorem 2.We are in conditions to construct the analytic solutions for the problem (32), (33). Theorem 2
Let the initial data (33) be constructed as above. Under Assumptions (A), (B)and (C) on the geometry of the problem, and under Assumption (D) on the constants involved,the problem (32), (33) has a holomorphic and bounded solution X i ( t, z, ǫ ) on ( T ∪ D (0 , h ′ )) × D (0 , R ) × E i , for every ≤ i ≤ ν − , for some R , h ′ > . Moreover, there exist < h ′′ < h ′ , K, M > (not depending on ǫ ), such that (36) sup t ∈T ∩ D (0 ,h ′′ ) z ∈ D (0 ,ρ / | X i +1 ( t, z, ǫ ) − X i ( t, z, ǫ ) | ≤ K exp (cid:18) − M | ǫ | ˆ r i (cid:19) , for every ǫ ∈ E i ∩ E i +1 , and some positive real number ˆ r i which depends on i . Proof
Let 0 ≤ i ≤ ν − ǫ ∈ E i . From Theorem 1, the Cauchy problem (22), with initialconditions given by ( ∂ jz Y j )( T, , ǫ ) = Y i,j ( T, ǫ ) , ≤ j ≤ S − , for the functions Y i,j defined in (35) admits a holomorphic solution ( T, z ) Y ( T, z, ǫ ) definedin S d i ,θ i , ∆ i | ǫ | r × D (0 , ∆ i ) , for some ∆ i , ∆ i > X i ( t, z, ǫ ) = Y ( ǫ r t, z, ǫ ), then X i turns out to be a holomorphic function defined in( T ∩ D (0 , h ′ )) × D (0 , R ) × E i , for some R , h ′ >
0, which turns out to be a solution of (32),(33)from its construction.We now give proof for the estimates in (36).For every ( t, z, ǫ ) ∈ ( T ∪ D (0 , h ′ )) × D (0 , R ) × ( E i ∩ E i +1 ), the difference of two solutionsrelated to two consecutive sectors of the good covering in the perturbation parameter can bewritten in the form(37) X i +1 ( t, z, ǫ ) − X i ( t, z, ǫ ) = X β ≥ ( X i +1 ,β ( t, ǫ ) − X i,β ( t, ǫ )) z β β ! , X i,β ( t, ǫ ) := k Z L γi W β,i ( u, ǫ ) e − ( utǫr ) k duu , with ( W i,β ( τ, ǫ )) β ≥ given by the recurrence (16), and with initial terms given by W i,j determinedin the construction of the present Cauchy problem.Before entering into details, it is worth mentioning the nature of the different values of ˆ r i ,depending on 0 ≤ i ≤ ν −
1. Indeed,(38) ˆ r i ∈ (cid:26) r s , r s (cid:27) . There are three different geometric situations one can find for each 0 ≤ i ≤ ν − π (2 j +1)+arg( a ) ks for j = 0 , ..., ks − d with | ˜ d i − arg( d E i ,j ) | ≤ δ i for j = 0 , ..., ks (we will say these are singular directions of second kind) in between γ i and γ i +1 , then one can deform the path L γ i +1 − L γ i to a point by means of Cauchy theoremso that the difference X i +1 − X i is null. In this case, one can reformulate the problem byconsidering a new good covering combining E i and E i +1 in a unique sector.2. If there exists at least a singular direction of first kind but no singular directions of secondkind in between γ i and γ i +1 , then the movable singularities depending on ǫ do not affectthe geometry of the problem, whereas the path can only be deformed taking into accountthose singularities which do not depend on ǫ . In this case ˆ r i := r /s .3. If there is at least a singular direction of second kind in between γ i and γ i +1 , then themovable singularities depend on ǫ , and tend to zero. As a consequence, this affects thegeometry of the problem, and the path deformation has to be made accordingly. In thiscase, ˆ r i := r /s .Observe that Assumption (B.1) leads to r /s < r /s so that the Gevrey order in the secondscenary is always greater than in the third one, i.e. ˆ r ≥ ˆ r .We first consider the situation in which only singular directions of first kind appear. Fromc) in the construction of the initial conditions of the Cauchy problem, one can deform theintegration path for the integrals in (37). For every ǫ ∈ E i ∩ E i +1 and t ∈ T ∩ D (0 , h ′ ) one has X i +1 ,β ( t, ǫ ) − X i,β ( t, ǫ ) = k Z L ρ / ,γi +1 W i +1 ,β ( u, ǫ ) e − ( utǫr ) k duu − k Z L ρ / ,γi W i,β ( u, ǫ ) e − ( utǫr ) k duu + k Z C ( ρ / ,γ i ,γ i +1 ) W i,i +1 ,β ( u, ǫ ) e − ( utǫr ) k duu . Here, L ρ / ,γ i +1 := [ ρ , + ∞ ) e √− γ i +1 , L ρ / ,γ i := [ ρ , + ∞ ) e √− γ i and C ( ρ / , γ i , γ i +1 ) is an arcof circle with radius ρ / ρ / e √− γ i +1 and ρ / e √− γ i with a well chosen orientation.Moreover, W i,i +1 ,β denotes the function W i,β in an open domain which contains the closed path( L γ i +1 \ L ρ / ,γ i +1 ) − C ( ρ / , γ i , γ i +1 ) − ( L γ i \ L ρ / ,γ i ), in which W i,β and W i +1 ,β coincide. Thisis a consequence of c) in the construction of the initial data for our problem.We first give estimates for I := k (cid:12)(cid:12)(cid:12)(cid:12)R L ρ / ,γi W i,β ( u, ǫ ) e − ( utǫr ) k duu (cid:12)(cid:12)(cid:12)(cid:12) . The corresponding ones for I := k (cid:12)(cid:12)(cid:12)(cid:12)R L ρ / ,γi +1 W i +1 ,β ( u, ǫ ) e − ( utǫr ) k duu (cid:12)(cid:12)(cid:12)(cid:12) follow the same argument, so we omit them.0 I ≤ k Z ∞ ρ / | W i,β ( se √− γ i , ǫ ) | exp (cid:18) − s k | t | k | ǫ | rk cos( k ( γ i − arg( t ) − r arg( ǫ ))) (cid:19) ds. Direction γ i was chosen depending on ǫ r t , in order that a positive real number ∆ exists withcos( k ( γ i − t − r arg( ǫ ))) ≥ ∆ >
0, for every ǫ ∈ E i ∩ E i +1 and t ∈ T ∩ D (0 , h ′ ). Bearing in mindthat a) in the construction of the initial conditions holds, there exist M , Z > I ≤ kM Z β β ! Z ∞ ρ / s | ǫ | r s k | ǫ r | k exp (cid:18) σξ ( b ) s k | ǫ | rk (cid:19) exp (cid:18) − s k ∆ | t | k | ǫ | rk (cid:19) ds. Indeed, if h ′ < (cid:16) ∆ σξ ( b )+∆ (cid:17) /k for some ∆ >
0, the previous expression is upper bounded by kM Z β β ! Z ∞ ρ / s | ǫ | r exp( − ∆ s k | ǫ | rk ) ds. Taking into account that k ≥ s ≥ ρ / s − k ≤ ( ρ / − k . The previous expressionequals kM Z β β ! Z ∞ ρ / s − k ( − k ) s k − | ǫ | r exp( − ∆ s k | ǫ | rk ) ds = kM Z β β ! | ǫ | r ( k − ( − k Z ∞ ρ / s − k ( − k ) s k − ∆ | ǫ | rk exp( − ∆ s k | ǫ | rk ) ds ≤ kM Z β β ! | ǫ | r ( k − ( − k ( ρ / − k exp( − ∆ s k | ǫ | rk ) | s →∞ s = ρ / = kM Z β β ! | ǫ | r ( k − k ( ρ / − k exp( − ∆ ( ρ / k | ǫ | rk ) ≤ M Z β β ! exp (cid:18) − K | ǫ | rk (cid:19) , (39)for some M , K > I = k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z C ( ρ / ,γ i ,γ i +1 ) W i,i +1 ,β ( u, ǫ ) e − ( utǫr ) k duu (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) yield(40) I ≤ M Z β β ! exp (cid:18) − K | ǫ | rk (cid:19) , whenever t ∈ T ∩ D (0 , h ′ ) for some M , K >
0. ¿From (39) and (40) one concludes there exist
M, K > | X i +1 ,β ( t, ǫ ) − X i,β ( , ǫ ) | ≤ M Z β β ! exp (cid:18) − K | ǫ | ˆ r i (cid:19) , for every β ≥ t ∈ T ∩ D (0 , h ′ ), ǫ ∈ E i ∩ E i +1 , and with ˆ r i = r /s .We now study the third situation which can occur, it is to say, that in which at least asingular direction of second kind lies in between the directions γ i and γ i +1 . Now, the coefficients1appearing in the series in (37) are such that the integration path under consideration in thedefinition of the Laplace transforms is deformed in a different way. Indeed, one can write forevery ǫ ∈ E i ∩ E i +1 , t ∈ T ∩ D (0 , h ′ ), that X i +1 ,β ( t, ǫ ) − X i,β ( t, ǫ ) = k Z L ρ ( | ǫ | ) / ,γi +1 W i +1 ,β ( u, ǫ ) e − ( utǫr ) k duu − k Z L ρ ( | ǫ | ) / ,γi W i,β ( u, ǫ ) e − ( utǫr ) k duu + k Z C ( ρ ( | ǫ | ) / ,γ i ,γ i +1 ) W i,i +1 ,β ( u, ǫ ) e − ( utǫr ) k duu . Here, the paths are L ρ ( | ǫ | ) / ,γ i +1 := [ ρ ( | ǫ | ) / , + ∞ ) e √− γ i +1 , L ρ ( | ǫ | ) / ,γ i := [ ρ ( | ǫ | )2 , + ∞ ) e √− γ i and C ( ρ ( | ǫ | ) / , γ i , γ i +1 ) is an arc of circle with radius ρ ( | ǫ | ) / ρ ( | ǫ | ) / e √− γ i +1 and ρ ( | ǫ | ) / e √− γ i with a well chosen orientation.We omit most of the calculs to estimate I := k (cid:12)(cid:12)(cid:12)(cid:12)R L ρ ( | ǫ | ) / ,γi +1 W i +1 ,β ( u, ǫ ) e − ( utǫr ) k duu (cid:12)(cid:12)(cid:12)(cid:12) , I := k (cid:12)(cid:12)(cid:12)(cid:12)R L ρ ( | ǫ | ) / ,γi W i,β ( u, ǫ ) e − ( utǫr ) k duu (cid:12)(cid:12)(cid:12)(cid:12) and I := k (cid:12)(cid:12)(cid:12)(cid:12)R C ( ρ ( | ǫ | ) / ,γ i ,γ i +1 ) W i,β ( u, ǫ ) e − ( utǫr ) k duu (cid:12)(cid:12)(cid:12)(cid:12) for they fol-low analogous steps as in the first case under study. Indeed, bounds for I and I can be obtainedunder the same arguments. For the study of I , one can follow the first same steps as in theestimates for I to get that I ≤ kM Z β β ! exp (cid:18) − ∆ ρ ( | ǫ | ) k | ǫ | rk (cid:19) , for some M , ∆ > ǫ . One has ρ ( | ǫ | ) k | ǫ | rk = | a | s | ǫ | s r − s r s s k | ǫ | r s = | a | s k | ǫ | − r s , which yields the existence of positive constants M , K such that I ≤ M Z β β ! exp − K | ǫ | r s ! , for t ∈ T ∩ D (0 , h ′ ). We also omit the study of I for the previous study can be reproduced.In view of these results, one can conclude that, in the case of a movable singularity betweenthe arguments γ i and γ i +1 , it is to say in the third case considered, one concludes there exist M, K > | X i +1 ,β ( t, ǫ ) − X i,β ( t, ǫ ) | ≤ M Z β β ! exp (cid:18) − K | ǫ | ˆ r i (cid:19) , for every β ≥
0, for t ∈ T ∩ D (0 , h ′ ), ǫ ∈ E i ∩ E i +1 , for ˆ r i := r s .In view of (41) and (42), one can plug this information into (37) to conclude there exist M, K > | X i +1 ( t, z, ǫ ) − X i ( t, z, ǫ ) | ≤ M X β ≥ Z β | z | β exp (cid:18) − K | ǫ | ˆ r i (cid:19) < M X β ≥ (1 / β exp (cid:18) − K | ǫ | ˆ r i (cid:19) , for every t ∈ T ∩ D (0 , ρ / z ∈ D (0 , / (2 Z )) and all ǫ ∈ E i ∩ E i +1 , for every 0 ≤ i ≤ ν − ✷ The different behavior of the difference of two solutions with respect to the perturbation param-eter in the intersection of adjacent sectors of the good covering studied in Theorem 2 providestwo different levels in the asymptotic approximation of the analytic solution in the variable ǫ .This behavior has also appeared in the previous work by the second author [9] when studyinga family of singularly perturbed difference-differential nonlinear partial differential equations,where small delays depending on the perturbation parameter occur in the time variable. Definition 5
Let ( E , k·k E ) be a complex Banach space and E be an open and bounded sectorwith vertex at 0. We also consider a positive real number α .We say that a function f : E → E , holomorphic on E , admits a formal power series ˆ f ( ǫ ) = P k ≥ a k ǫ k ∈ E [[ ǫ ]] as its α − Gevrey asymptotic expansion if, for any closed proper subsector
W ⊆ E with vertex at the origin, there exist
C, M > such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f ( ǫ ) − N − X k =0 a k ǫ k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E ≤ CM N N ! /α | ǫ | N , for every N ≥ , and all ǫ ∈ W . In this section, we state a new version of the classical Ramis-Sibuya theorem (see [4], TheoremXI-2-3) in two different Gevrey levels. We have decided to include the proof, which followsanalogous steps as the one in [9] for a Gevrey level and the 1 + level, for the sake of clarity anda self-contained argumentation. In addition to this, the enunciate is written in terms of justtwo different Gevrey levels in order to fit our necessities, but there is no additional difficulty onconsidering any finite number of different levels. Theorem (RS)
Let ( E , k·k E ) be a complex Banach space, and let ( E i ) ≤ i ≤ ν − be a goodcovering in C ⋆ . We assume G i : E i → E is a holomorphic function for every 0 ≤ i ≤ ν − i ( ǫ ) = G i +1 ( ǫ ) − G i ( ǫ ) for every ǫ ∈ Z i := E i ∩ E i +1 .Here we have made the identification of the elements with index ν with the correspondingones under index 0.Moreover, we assume The functions G i ( ǫ ) are bounded as ǫ ∈ E i tends to the origin, for every 0 ≤ i ≤ ν − We consider ˆ r > r >
0, and two nonempty subsets of { , ..., ν − } , say I and I ,such that I ∩ I = ∅ and I ∪ I = { , ..., ν − } . For every j = 1 , , and every i ∈ I j there exist K i , M i > k ∆ i ( ǫ ) k E ≤ K i e − Mi | ǫ | ˆ rj , for every ǫ ∈ Z i .Then, there exists a convergent power series a ( ǫ ) ∈ E { ǫ } defined on some neighborhood ofthe origin and ˆ G ( ǫ ) , ˆ G ( ǫ ) ∈ E [[ ǫ ]] such that G i can be written in the form(43) G i ( ǫ ) = a ( ǫ ) + G i ( ǫ ) + G i ( ǫ ) , where G ji ( ǫ ) is holomorphic on E i and has ˆ G j ( ǫ ) as its ˆ r j -Gevrey asymptotic expansion on E i ,for j = 1 , , and i ∈ { , ..., ν − } .3 Proof
For every 0 ≤ i ≤ ν − ji ( ǫ ) on the sectors Z i by∆ ji ( ǫ ) = ∆ i ( ǫ ) δ ij , j = 1 , . Here, δ ij is a Kronecker type function with value 1 if i ∈ I j and 0 otherwise.A direct consequence of Lemma XI-2-6 from [4] provided by the classical Ramis-Sibuyatheorem in Gevrey classes is that for every 0 ≤ i ≤ ν − j = 1 ,
2, there exist holomorphicfunctions Ψ ji : E i → C such that ∆ ji ( ǫ ) = Ψ ji +1 ( ǫ ) − Ψ ji ( ǫ )for every ǫ ∈ Z i , where by convention Ψ jν ( ǫ ) = Ψ j ( ǫ ). Moreover, there exist formal power series P m ≥ φ m,j ǫ m ∈ E [[ ǫ ]] such that for each 0 ≤ ℓ ≤ ν − W ⊆ E l with vertex at 0, there exist ˘ K ℓ , ˘ M ℓ > (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Ψ jℓ ( ǫ ) − M − X m =0 φ m,j ǫ m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E ≤ ˘ K ℓ ( ˘ M ℓ ) M M ! / ˆ r j | ǫ | M , for every ǫ ∈ W , and all positive integer M .We consider the bounded holomorphic functions a i ( ǫ ) = G i ( ǫ ) − Ψ i ( ǫ ) − Ψ i ( ǫ ) , for every0 ≤ i ≤ ν −
1, and ǫ ∈ E i . For every 0 ≤ i ≤ ν − a i +1 ( ǫ ) − a i ( ǫ ) = G i +1 ( ǫ ) − G i ( ǫ ) − ∆ i ( ǫ ) − ∆ i ( ǫ ) = G i +1 ( ǫ ) − G i ( ǫ ) − ∆ i ( ǫ ) = 0 , for ǫ ∈ Z i . Therefore, there exists a holomorphic function a ( ǫ ) defined on U \ { } , for someneighborhood of the origin U such that a i ( ǫ ) = a ( ǫ ) for every 0 ≤ i ≤ ν −
1. Since a ( ǫ ) is boundedon this domain, 0 turns out to be a removable singularity, and a ( ǫ ) defines a holomorphic functionon U .Finally, one can write G i ( ǫ ) = a ( ǫ ) + Ψ i ( ǫ ) + Ψ i ( ǫ ) , for ǫ ∈ E i , and every 0 ≤ i ≤ ν −
1. Moreover, Ψ ji ( ǫ ) admits ˆ G j ( ǫ ) = P m ≥ φ m,j ǫ m as itsˆ r j -Gevrey asymptotic expansion on E i , for j = 1 , ✷ Remark:
We put ˆ r := r /s and ˆ r := r /s and recall that ˆ r ≤ ˆ r . Assume that a sector E i has opening a bit larger than π/ ˆ r and if i ∈ I is such that I δ ,i,δ = { i − δ , . . . , i, . . . , i + δ } ⊂ I for some integers δ , δ ≥ E i ⊂ S π/ ˆ r ⊂ [ h ∈ I δ ,i,δ E h where S π/ ˆ r is a sector centered at 0 with aperture a bit larger than π/ ˆ r . Then, from the proofof Theorem (RS), we see that in the decomposition (43), the function G i ( ǫ ) can be analyticallycontinued on the sector S π/ ˆ r and has the formal series ˆ G ( ǫ ) as Gevrey asymptotic expansionof order ˆ r on S π/ ˆ r . Hence, G i ( ǫ ) is the ˆ r − sum of ˆ G ( ǫ ) on S π/ ˆ r in the sense of the definitiongiven in [1], Section 3.2. Moreover, the function G i ( ǫ ) has ˆ G ( ǫ ) as ˆ r − Gevrey asymptoticexpansion on E i , meaning that G i is the ˆ r − sum of ˆ G ( ǫ ) on E i .In other words, using the characterisation of multisummability given in [1], Theorem 1 p.57, the formal series ˆ G ( ǫ ) is (ˆ r , ˆ r ) − summable on E i and its (ˆ r , ˆ r ) − sum is the function G i ( ǫ )on E i .4The question that naturally arises is whether such situation can hold for certain practicalsituation. The answer is positive.Let us assume that s = 1 and s is much larger than 1. We denote ap ( E i ) the apertureof E i . We assume that T is a small and thin sector that is bisected by the positive real axis.Then, from the third property in Definition 4, we can assume that ap ( E i ) is slightly larger than π/ ( r /s ) for some element in the good covering ( E i ) ≤ i ≤ ν − . Taking into account Assumption(C), we take 2 πs s r − s r > ap ( E i ) > π/ ( r /s )Hence, r > s r − s r s r − s r > s , which means under these settingsthat(45) r /s > r + 1Now, the consecutive “movable” roots of P ǫ, ( τ ) = ǫ r − s rk ( kτ k ) s + a are separated by anangle of 2 π/ks = 2 π/k . The consecutive “fixed” roots of P ( τ ) are separated by an angle of2 π/ ( ks ).If s is much larger than 1, in between two consecutive roots of P ǫ, ( τ ) one can find at leastmore than two consecutive roots of P ( τ ) (the number of roots of P ( τ ) is far larger than thenumber of roots of P ǫ, ( τ ))We observe that the difference of any two neighboring solutions X i , X i +1 obtained as Laplacetransform along directions d i , d i +1 lies between these ”fixed” roots is of exponential decay oforder r /s . Hence, such consecutive integers i , i + 1 belong to the subset I (with the notationat the beginning of the remark) and the aperture of the sectors E i , E i +1 are larger than π/ ( r /s ).In addition, we observe that if r /s is not too large compared to r in (45), then the union ofthe E i over these aformentioned indices i can contain a sector S π/r of aperture π/r .In other words, we are in the configuration (44). The main result of this work states the existence of a formal power series in ǫ which can besplitted in two formal power series, each one linked to one of the different types of singularitiesappearing in the problem. In addition to this, the analytic solution is written as the sum of twofunctions which are represented by the forementioned formal power series under some Gevreytype asymptotics. Theorem 3
Under Assumptions (A), (B) and (C) on the geometric configuration of our prob-lem under study, and under Assumption (D), there exists a formal power series (46) ˆ X ( t, z, ǫ ) = X β ≥ H β ( t, z ) ǫ β β ! ∈ E [[ ǫ ]] , where E stands for the Banach space of holomorphic and bounded functions on the set ( T ∩ D (0 , h ′′ )) × D (0 , R ) equipped with the supremum norm, for some h ′′ , R > provided by Theo-rem 2, which formally solves the equation (47) ( ǫ r ( t k +1 ∂ t ) s + a )( ǫ r ( t k +1 ∂ t ) s + a ) ∂ Sz ˆ X ( t, z, ǫ )5= X ( s,κ ,κ ) ∈S b κ κ ( z, ǫ ) t s ( ∂ κ t ∂ κ z ˆ X )( t, z, ǫ ) . Moreover, ˆ X can be written in the form ˆ X ( t, z, ǫ ) = a ( t, z, ǫ ) + ˆ X ( t, z, ǫ ) + ˆ X ( t, z, ǫ ) , where a ( t, z, ǫ ) ∈ E { ǫ } is a convergent series on some neighborhood of ǫ = 0 and ˆ X ( t, z, ǫ ) , ˆ X ( t, z, ǫ ) are elements in E [[ ǫ ]] . Moreover, for every ≤ i ≤ ν − , the E -valued function ǫ X i ( t, z, ǫ ) constructed in Theorem 2 is of the form (48) X i ( t, z, ǫ ) = a ( t, z, ǫ ) + X i ( t, z, ǫ ) + X i ( t, z, ǫ ) , where ǫ X ji ( t, z, ǫ ) is a E -valued function which admits ˆ X j ( t, z, ǫ ) as its ˆ r j -Gevrey asymptoticexpansion on E i , for j = 1 , . Proof
We consider the family of functions ( X i ( t, z, ǫ )) ≤ i ≤ ν − constructed in Theorem 2. Forevery 0 ≤ i ≤ ν −
1, we define G i ( ǫ ) := ( t, z ) X i ( t, z, ǫ ), which turns out to be a holo-morphic and bounded function from E i into the Banach space E of holomorphic and boundedfunctions defined in ( T ∩ D (0 , h ′′ )) × D (0 , R ), for certain positive constants R and h ′′ definedin Theorem 2.The estimates (36) yield that the cocycle ∆ i ( ǫ ) = G i +1 ( ǫ ) − G i ( ǫ ) satisfies exponentially flatbounds of certain Gevrey order ˆ r i , depending on 0 ≤ i ≤ ν −
1. Theorem (RS) guarantees theexistence of formal power series ˆ G ( ǫ ) , ˆ G ( ǫ ) , ˆ G ( ǫ ) ∈ E [[ ǫ ]] such that one has the decomposition G i ( ǫ ) = a ( ǫ ) + ˆ G i ( ǫ ) + ˆ G i ( ǫ )for ǫ ∈ E i , where G ji ( ǫ ) is a holomorphic function on E i and admits ˆ G ji ( ǫ ) as its Gevrey asymptoticexpansion of order ˆ r j for all j = 1 , G ( ǫ ) =: ˆ X ( t, z, ǫ ) = X β ≥ H k ( t, z ) ǫ k k ! . The proof is concluded if we show that ˆ X ( t, z, ǫ ) satisfies (47). For any 0 ≤ i ≤ ν − j = 1 ,
2, the fact that G ji ( ǫ ) admits ˆ G ji ( ǫ ) as its Gevrey expansion of some order ˆ r j in E i impliesthat(49) lim ǫ → ,ǫ ∈E i sup ( t,z ) ∈ ( T ∩{| t |
0. From (49) we get a recursion formula for the coefficients in (46) given by a a ∂ Sz (cid:18) H ℓ ( t, z ) ℓ ! (cid:19) = X ( s,κ ,κ ) ∈S ℓ X m =1 ℓ ! m !( ℓ − m )! b κ κ m ( z ) m ! ∂ κ t ∂ κ z H ℓ − m ( t, z )( ℓ − m )! − a ( t k +1 ∂ t ) s ∂ Sz (cid:18) H ℓ − r ( t, z )( ℓ − r )! (cid:19) − a ( t k +1 ∂ t ) s ∂ Sz (cid:18) H ℓ − r ( t, z )( ℓ − r )! (cid:19) − ( t k +1 ∂ t ) s + s ∂ Sz H ℓ − ( r + r ) ( t, z )( ℓ − ( r + r ))! . G ( ǫ ) and the coefficients ofthe analytic solution, written as a power in the perturbation parameter, coincide. This yieldsˆ X ( t, z, ǫ ) is a formal solution of (32), (33). ✷ Remark:
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