On parametric Gevrey asymptotics for some initial value problems in two asymmetric complex time variables

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1 On parametric Gevrey asymptotics for some initial value problems in two asymmetric complex time variables A. Lastra S. Male University of Alcalá Departamento de Física y Matemáticas Ap. de Correos E-887 Alcalá de Henares Madrid Spain University of Lille Laboratoire Paul Painlevé Villeneuve d Ascq cedex France alberto.lastra@uah.es Stephane.Male@math.univ-lille.fr Abstract We study a family of nonlinear initial value partial differential equations in the complex domain under the action of two asymmetric time variables. Different Gevrey bounds and multisummability results are obtain depending on each element of the family providing a more complete picture on the asymptotic behavior of the solutions of PDEs in the complex domain in several complex variables. The main results lean on a fixed point argument in certain Banach space in the Borel plane together with a Borel summability procedure and the action of different Ramis-Sibuya type theorems. Key words: asymptotic expansion Borel-Laplace transform Fourier transform initial value problem formal power series nonlinear integro-differential equation nonlinear partial differential equation singular perturbation. MSC: 35C 35C. Introduction This wor is framed into the study of multisummable formal solutions of certain family of PDEs. Multisummability of formal solutions of functional equations is observed in recent studies made by some research groups in different directions and a growing interest has been observed in the scientific community. The present wor belongs to these trends of studies for which we provide a brief overview. Borel-Laplace summability procedures have been recently applied to solve partial differential equations. In the seminal wor [9] the authors obtain positive results on the linear complex heat equation with constant coefficients. This construction was extended to more general linear PDEs by W. Balser in [3] under the assumption of adequate extension of the initial data to an infinite sector. More recently M. Hibino [9] has made some advances in the study of linear first order PDEs. Subsequently several authors have studied complex heat lie equations with The author is partially supported by the project MTM C--P of Ministerio de Economía y Competitividad Spain The author is partially supported by the project MTM C--P of Ministerio de Economía y Competitividad Spain.

2 variable coefficients see [5 6 ]. The second author [] both authors [3] and the two authors and J. Sanz [7] have also contributed in this theory. Recently multisummability of formal solutions of PDEs has also been put forward in different wors. W. Balser [4] described a multisummability phenomenon in certain PDEs with constant coefficients. S. Ouchi [3] constructed multisummable formal solutions of nonlinear PDEs coming from perturbation of ordinary differential equations. H. Tahara and H. Yamazawa [4] have made progresses on general linear PDEs with non constant coefficients under entire initial data. In [] G. Lysi constructs summable formal solutions of the one dimensional Burgers equation by means of the Cole-Hopf transform. O. Costin and S. Tanveer [8] construct summable formal power series in time variable to 3D Navier Stoes equations. The authors have obtained results in this direction [4 5]. A recent overview on summability and multisummability techniques under different points of view is displayed in [8]. The purpose of the present wor is to study the solutions of a family of singularly perturbed partial differential equations from the asymptotic point of view. More precisely we consider a problem of the form Q z t ut t z = P z ut t z P z ut t z + P t t t t z + ft t z under initial conditions ut z u t z and where QX C[X]. The elements which conform the nonlinear part P P are polynomials in their second variable with coefficients being holomorphic functions defined on some neighborhood of the origin say D continuous up to their boundary. Here D stands for the open disc in the complex plane centered at and with positive radius >. We write D for its closure. Moreover P stands for some polynomial of six variables with complex coefficients and the forcing term ft t z is a holomorphic and bounded function in D ρ H β D for some ρ > and where H β stands for the horizontal strip for some β >. H β := {z C : Imz < β } The precise configuration of the elements involved in the problem is stated and described in Section.. This paper provides a step beyond in the study of the asymptotic behavior of the solutions of a subfamily of singularly perturbed partial differential equations of the form. We first recall some previous advances made in this respect which motivate the present framewor. In [3] we studied under the asymptotic point of view the solutions of certain family of PDEs of the form Q z t ut z = P z ut z P z ut z + P t t z ut z + ft z where the elements involved in the problem depend only on one time variable t. Our next aim was to chec whether the asymptotic properties of the solutions in this equation can be extended to functions of more number of time variables as stated in.

3 3 It is worth mentioning that in the previous wor [3] the linear part of the equation ruled by P t t z ut z was assumed to be more general than in the present configuration admitting an additional term of the form c t z R z ut z where c t z is given by a certain holomorphic function defined on a product D ρ H β D. We decided not to incorporate this term in the present study for the sae of simplicity. However the results can be written with no additional theoretical difficulties by adding the analog of such terms into the equation. As a matter of fact the decision of not considering this term in the present wor is due to emphasize other fact: an outstanding phenomena occurred when dealing with two complex variables arriving at substantially and qualitatively different asymptotic properties of the solutions attained. In [] we described a study of a family of equations of the shape which showed a symmetric behaviour with respect to the asymptotic properties of the analytic solutions with respect to both time variables as initially expected from the generalization of the one-time variable case. More precisely we proved the following result: given a good covering of C {E p p } p ς see Definition 3 involving sectors of opening larger than π/ there exist p ς sectors with vertex at the origin in C and finite radius say T and T such that a family of solutions {u p p t t z } p ς of is constructed. The function u p p t t z turns p ς out to be holomorphic in T T H β E p p for every p ς and p ς. In addition to this we obtain in this previous wor that the difference of two consecutive in the sense that they are related to consecutive sectors in the good covering solutions u p p and of can be classified into two categories: u p p. Those pairs p p p p U such that sup u p p t t z u p p t t z K p e Mp E p p E p p ; t t z T T H β. and those pairs p p p p U such that sup u p p t t z u p p t t z K p e Mp E p p E p p. t t z T T H β Here and are different positive integers involved in the definition of the polynomials appearing in the main equation and K p M p are positive constants. The application of a two-level Ramis-Sibuya type result entails the existence of a formal power series ût t z F[[]] where F stands for the Banach space of holomorphic and bounded functions in the domain T T H β with the supremum norm. Such formal power series is a formal solution of and can be split in the form ût t z = at t z + û t t z + û t t z where at t z belongs to F{} and û û F[[]]. Moreover for all p {... ς } and p {... ς } the function u p p t t z can be split analogously: u p p t t z = at t z + u p p t t z + u p p t t z where u j p p t t z is an F-valued function which admits û j t t z as its j -Gevrey asymptotic expansion on E p p for j = seeing û j as a formal power series in with

4 4 coefficients in F. In addition to this and under the assumption that < a multisummability result is also attained. Under the assumption that {p p p p p p p p... p y p y p y py } U for some y N := {...} and E p y py S π/ j y E p j pj for some sector S π/ with opening larger than π/ then it holds that ût m t z is indeed summable on E p y being its py -sum given by u p y on E py p y. py The role played by and in the previous framewor is completely symmetric. The assumption < is innocuous reaching symmetric results in the case that <. In that study the principal part of any of the equations in the family studied is factorisable as a product of two operators involving a single time variable yielding a multisummability phenomena in the perturbation parameter. On the other hand in the present study the sign of is crucial at the time of studying the asymptotic behavior of the analytic solution. In fact a negative sign provides less information on the asymptotic behavior which entails only Gevrey estimates whilst the positive one furnishes more precise information namely multisummability. Here is where the strength of the present results holds. More precisely we find a family of analytic solutions {u p p t t z } p ς p ς of the main problem under study which are holomorphic in T T H β E p p and such that one of the following hold:. In case > a formal power series ût t z F[[]] formal solution of exists such that for every p p {... ς } {... ς } the function u p p t t z admits ût t z as its asymptotic expansion of Gevrey order / in E p p see Theorem.. In case that > a formal power series ût t z F[[]] exists being formal solution of and such that ût t z shows analogous properties as those described in the family of equations in [] i.e. multisummability of the formal solution with Gevrey levels and see Theorem 3. The present study is based on the following approach: after establishing the main problem under study: Q z t + t d δ D t t d δ D t R D D z + 3 t d δ 3 D3 t R D3 z ut z = P z ut z P z ut z + l j D j j= l l t d l t d l δ l t δ l t R l l z ut z + ft z where D D d δ D d δ D 3 d 3 δ D3 are integer numbers and for all l D and l D we tae nonnegative integers d l d l δ l d l and l l under the assumptions 5-7. Moreover Q R D D R D3 and R l l are polynomials with complex coefficients for all l D and l D. The polynomials P P present

5 5 coefficients which are holomorphic functions with respect the perturbation parameter on some neighborhood of the origin under assumptions 8-. The forcing term ft t z is given by some holomorphic and bounded function on a neighborhood of the origin with respect to both variables and the perturbation parameter and some horizontal strip with respect to z variable. We search for analytic solutions of given as a Laplace and Fourier transform of certain function to be determined: 3 ut t z = π / ω d u u m e u u t du t du L d u u L d where L γj = R + e iγ j for some appropriate direction γ j R for j =. The problem of finding such a function is equivalent in view of Lemma to solve an auxiliary convolution equation in the Borel plane. More precisely there is a one-to-one correspondence between functions ut t z of the form 3 which solve and functions ωτ τ m admitting Laplace transform with respect to the first two variables along directions d and d resp. and Fourier transform with respect to m variable which turn out to be solutions of a convolution equation see 3. For every fixed value of the perturbation parameter τ τ m ω dτ τ m is obtained as the fixed point of the contractive operator H see 33 for its definition acting on some Banach space of functions owing exponential decay at infinity on the Fourier variable and defined on some neighborhood of the origin for τ τ in C which can be prolonged to some neighborhood of the origin together with an infinite sector of bisecting direction d times an infinite sector with bisecting direction d ; under certain concrete monomial exponential growth at infinity. More precisely ω dτ τ m is a continuous function in D ρ S d S d R D \{} and holomorphic with respect to τ τ in D ρ S d S d and on D \ {} with respect to the perturbation parameter. In addition to this there exist constants ϖ µ β ν ν > such that τ ω d τ τ τ m ϖ + m µ + τ + τ exp βm + ν τ τ + ν for every τ τ m D ρ S d S d R D \{}. Laplace and Fourier transforms mae sense in order to get 3. At this point we are able to construct a family of solutions {u p p t t z } p ς of where u p p t t z is a holomorphic function defined in p ς T T H β E p p with T and T being finite sectors in C with vertex at the origin and where {E p p } p ς conforms a good covering at see Definition 3. p ς The distinction of < and < provide Gevrey asymptotics or multisummability results in Theorem resp. Theorem 3. It is worth mentioning that these results lean on the application of a cohomological criteria nown as Ramis-Sibuya Theorem; resp. a multilevel version of such result. The fact that a different behavior can be observed with respect to the variables in time is due to the domain of definition of ω d with respect to such variables: a neighborhood of the origin for τ τ C which can only be prolonged up to a neighborhood of the origin together with an infinite sector with respect to the first variable; whereas it can not be defined on any neighborhood of the origin with respect to the second time variable but it does on some infinite sector. This causes the impossibility of application of a deformation path at the

6 6 time of estimating the difference of two consecutive solutions in order to apply the multilevel version Ramis-Sibuya Theorem. With respect to the study of the main problem in [] the main difficulty at this point comes due to the fact that Case in Theorem of [] is no longer available. We also find it necessary to justify the fact that ω d can not be defined with respect to τ τ in sets of the form 4 S S D ρ for some infinite sectors S and S and for some ρ >. In order to solve the main equation one needs to divide by P m τ τ see 4 for its definition. However as stated in Section 3.. the roots of such polynomial lie on sets of the form 4 for any ρ >. Therefore a small divisor phenomena is observed which does not allow a summability procedure. This occurrence has already been noticed in another context in previous wors: in the framewor of q differencedifferential equations [6]; in the context of multilevel Gevrey solutions of PDEs in the complex domain in [5] etc. The layout of the paper is as follows. After recalling the definition and the action of Fourier transform in the first part of Section we describe the main problem under study in Section. and reduce it to the research of a solution of an auxiliary convolution equation. Such solution is obtained following a fixed point argument in appropriate Banach spaces see Section 3. whose main properties are provided in Section 3.. Section 3.. is devoted to motivate the domain of definition of the solution in contrast to that studied in []. The first main result of our wor is Theorem where the existence of a family of analytic solutions of the main problem is obtained. In Section 5. we recall the Borel summability procedure and two cohomological criteria: Ramis-Sibuya Theorem and a multilevel version of Ramis-Sibuya Theorem. We conclude the present wor with the existence of a formal solution to the problem and two asymptotic results which connect the formal and the analytic solutions: Theorem states a result on Gevrey asymtotics in a subfamily of equations; Theorem 3 states a multisummability result in another different subfamily of equations under study. Layout of the main and auxiliary problems This section is devoted to describe the main problem under study. We first recall some facts on the action of Fourier transform on certain Banach spaces of functions.. Fourier transform on exponentially decreasing function spaces In order to transform the main problem under study into an auxiliary one easier to handle we first describe the action of Fourier transform in certain Banach spaces of rapidly decreasing functions. Definition Let β µ R. E βµ stands for the vector space of continuous functions h : R C such that hm βµ = sup + m µ expβmhm is finite. E βµ turns out to be a Banach space when endowed with the norm. βµ.

7 7 The following result is stated without proof which can be found in [3] Proposition 7. Proposition Let f E βµ with β > µ >. The inverse Fourier transform of f F fx = π / + can be extended to an analytic function on the strip fm expixmdm x R H β := {z C/Imz < β}. Let φm = imfm E βµ. Then it holds that z F fz = F φz for z H β. Let g E βµ and put ψm = π / f gm the convolution product of f and g for all m R. ψ belongs to E βµ. Moreover we have F fzf gz = F ψz for z H β.. Layout of the main problem Let and D D be integer numbers. We also consider non negative integer numbers d d j j δ D δ Dj for j { 3}. For all l D and l D let d l d l δ l d l l l be non negative integers. We assume the previous elements satisfy the next identities: 5 < δ D δ D3 and δ l < δ l + δ l < δ l + for all l D and l D 6 + d d + δ D + δ D = 3 d 3 + δ D3 = d = δ D d j = δ Dj + j = 3 Moreover for every l D and l D we assume 7 d l > δ l + dl > δ l + l l > δ D + δ D. Let QX R D D R D3 C[X] and for l D and l D we tae R l l X C[X]. We consider polynomials P P with coefficients belonging to OD such that 8 degq degr l l for l D and l D. Moreover we choose these polynomials satisfying 9 degq degp j j = Qim m R degq = degr D3 = degr D D. More precisely we assume there exist sectorial annulus E D3 Q and E D D D 3 such that R D3 im Qim E D3 Q R D D im Qim E D D D 3

8 8 for every m R. In other words there exist real numbers < r j < R j and α j < β j for j = such that E D3 Q := {x C : r < x < R argx α β } E D D Q := {x C : r < x < R argx α β }. Throughout the whole wor we denote the pairs of variables in bold letters: t := t t T := T T τ := τ τ etc. We consider the following nonlinear initial value problem Q z t + t d δ D t t d δ D t R D D z + 3 t d δ 3 D3 t R D3 z ut z = P z ut z P z ut z + l j D j j= with null initial data ut z u t z. l l t d l t d l δ l t δ l t R l l z ut z + ft z The forcing term ft z is constructed as follows. For n n let m F n n m be a family of functions belonging to the Banach space E βµ for some β > µ > maxdegp + degp + and which depend holomorphically on D. We assume there exist constants K T > such that 3 F n n m βµ K T n +n for all n n and D. We deduce that FT z = n n F m F n n m zt n T n represents a bounded and holomorphic function on D T / H β D for any < β < β. We define 4 ft z = Ft t z. Observe the function f is holomorphic and bounded on D ρ H β D where ρ < T /. We search for solutions of the main problem which are time scaled and expressed as a Fourier transform with respect to z variable in the form ut z = Ut t z = π / The symbol UT m satisfies the next equation. Ut t m expizmdm.

9 9 5 Qim T + + d d T d T d = π / + l j D j j= + 3 d 3 T d 3 δ D + δ D δ D T δ D3 δd3 T D δ T R D D im R D3 im UT m P im m UT m m P im UT m dm l l d l d l +δ l + δ l T d l T d l δ l T δ l T R l l imut m + Fz FT z m Our goal is to provide solutions of 5 in the form of a Laplace transform. Namely we search for solutions of the form 6 UT m = ω d u u m e u T u T du du L γ u u L γ where L γj = R + e iγ j for some appropriate direction γ j R for j = which depend on T j. The function ω d τ m is constructed in the incoming sections as the fixed point of a map defined in certain Banach spaces studied in the forthcoming sections. For j = let S dj be infinite sectors with vertex at the origin and bisecting direction d j such that L γj S dj. We fix a positive real number ρ >. In the present section we depart from a function ω dτ m continuous on D ρ S d S d R D \ {} holomorphic with respect to τ in D ρ S d S d D and such that 7 ω d τ m ϖ d + m µ e βm τ τ + τ + τ expν τ + ν τ for all τ D ρ S d S d every m R and D \ {}. In order to construct the solution we present a refined form of the problem. For that purpose we need some preliminary results. We mae use of the following relations which can be found in [4] p. 4: 8 T δ D + δ D T = T + T δ D + 9 T δ Dj + δ Dj T = T + T δ Dj + p δ D p j δ Dj A δd p T δ D p T + T p = T + T δ D + AδD T T à δdj p j T δ Dj p j T + T p j = T + T δ Dj + à δdj T T for some real numbers A δd p p =... δ D and à δdj p j p j =... δ Dj for j = 3. We write A D resp. à Dj for j = 3 in the place of A δd resp. à δdj for the sae of simplicity.

10 We divide by and multiply by T + at both sides of 5. Under the assumptions displayed in 6 one may apply 8 and 9 in order to rewrite equation 5. This step is important to exhibit the equations as an expression where some operators algebraically well-behaved with respect to Laplace transform appear. The resulting equation is as follows: QimT + = T + T + T δ D T + T δ D RD D im + T + [ T + T δ D ÃD T T R D D im T + T δ D3 RD3 im UT m T δ D AD T T R D D im ] A D T T ÃD T T R D D im ÃD 3 T T R D3 im UT m + T + π / + l j D j j= P im m UT m m P im UT m dm l l d l d l +δ l + δ l T d l T d l δ l T δ l T R l l imut m + T + Fz FT z m. The following result allows to establish a one-to-one correspondence between solutions of equation and an auxiliary equation in the Borel plane 3. The last equation will be presented afterwards in this same section. Lemma Let UT m be the function constructed in 6. Then the following statements hold: T j+ j Tj UT m = j u u u j j ωd T du u m e T du j =. L γ L γ u u T m UT m = T m UT m = L γ L γ L γ L γ u Γ m u Γ m u u u s m ω d s / u m ds e u T u T du u du u m N. u s m ω d u s / s m ds e u T u T du u du u m N. s UT m m UT m dm = L γ L γ u u u u ω d u s u s m m ω d s s m u s s u s ds ds e u u T du T du s u u

11 Proof The first statement is a direct application of the derivation under the integral symbol. The second and third statements are equivalent so we only give details for the second one. In order to give proof for the second statement we first apply Fubini theorem at the inner and outer integrals. That expression can be rewritten in the next form: A := L γ L γ L / s γ u Γ m u s m u e T du u ω d s/ u m e u du T ds u s where L γ := {re i γ : r } and L s / γ = {re iγ : r s / }. We proceed by applying two consecutive deformation paths at the inner integral in the previous expression: first we apply h = u and then h s = h. We arrive at A = L γ Γ m e h m h T dh ω d s/ L γ L γ The deformation path ũ = s / followed by h = h T A = UT m T m Γ m u m e yields L γ argt s T m h e h dh. u T du u ds s A deformation path and the definition of Gamma function allow us to conclude that A = UT m. The proof of the third formula follows the same lines of arguments involving Fubini theorem and it is omitted for the sae of brevity. Remar: Lemma provides the equivalence of existence of solutions of different equation and 3 related by Laplace transformation. We define the operators A δd ω τ m = à δdj ω τ m = p δ D p j δ Dj A δd p τ Γδ D p à δdj p j τ Γ δ Dj p j τ s δ D p s p ω s / τ m ds s τ s δ Dj p j s p j ω τ s / m ds s for j = 3. Observe that they turn out to be the m resp. m Borel transform of the operator A D T T resp. à Dj T T for j = 3 see Section 5. for more details on this.

12 by In view of the assumptiondescribed in 7 we define the natural numbers d l and d l d l = δ l + + d l dl = δ l + + d l for all l D and l D. Taing into account Lemma we see that UT m satisfies iff ω d τ m is a solution of the next equation. 3 Qim + R D D im τ δ D τ δ D + R D3 im τ δ D3 ωτ m = τ Ã δ D D τ τ R D D imωτ m τ δ D A D τ R D D imωτ m A D τ ÃD τ τ + π τ Γ + R D D imωτ m ÃD 3 τ τ R D3 imωτ m τ s τ ωτ s s x m m P im ωs x τ + l j D j j= l l d l d l +δ l + δ l R l l im τ d l s τ s s m P im m τ Γ dl Γ d l δ l sδ l δ l s δ l + Γ + ωs τ s dx ds dm ds τ s s s x x dl s m ds ds s s ψ τ s m ds s where ψ is the formal m -Borel transform with respect to T and the formal m -Borel transform with respect to T of F T m i.e. ψ τ m = n n F n n m τ n Γ n τ n Γ n. Observe that ψ is an entire function with respect to τ. Moreover regarding the construction of ψ and 3 one has ψ τ m νβµ n n F n n m βµ + τ sup + τ τ Dρ S d S d τ τ exp ν τ ν τ τ n τ n Γ n Γ n for all D \{} any unbounded sectors S d and S d centered at and bisecting directions d R and d R respectively for some ν = ν ν +. Remar: According to classical estimates and Stirling formula we observe that ψ τ m. See Definition. F d νβµ

13 3 We write 4 P m τ = Qim + R D D im τ δ D τ δ D + R D3 im τ δ D3. 3 Construction of the solution for a convolution equation The main aim in this section is to provide with a solution of 3 which belongs to certain Banach space of functions satisfying bounds in the form 7. Such function is obtained as a fixed point of an operator acting on Banach spaces introduced and studied in the incoming section. 3. Banach spaces of exponencial growth We consider the open disc D ρ for some ρ >. Let S dj be open unbounded sectors with bisecting directions d j R for j = and let E be an open sector with finite radius r E all with vertex at in C. The following norm is inspired from that considered by the authors in []. It is an adecquate modification of that described in [3] adapted to the framewor of two complex time variables. Definition Let ν ν β µ > and ρ > be positive real numbers. Let be integer numbers and let E. We put ν = ν ν = d = d d and denote Fνβµ d the vector space of continuous functions τ m hτ m on the set D ρ S d S d R which are holomorphic with respect to τ on D ρ S d S d and such that 5 hτ m νβµ = sup τ Dρ S d S d + m µ + τ τ + τ is finite. The normed space F d νβµ. νβµ is a Banach space. τ expβm ν τ ν τ hτ m We fix E µ β > in the whole subsection. We also choose ν = ν ν d = d d R and = N. We first state some technical results. The first one follows directly from the definition of the norm of the Banach space. Lemma Let τ m aτ m be a bounded continuous function on D ρ S d S d R holomorphic with respect to τ on D ρ S d S d. Then aτ mhτ m νβµ for all hτ m F d νβµ. sup aτ m τ Dρ S d S d hτ m νβµ Lemma 3 Let σ = σ σ and assume that a σ is a holomorphic function of D ρ S d S d continuous up to D ρ S d S d such that a σ τ + τ σ + τ σ for every τ D ρ S d S d. We tae σ j σ j for j =. Assume that one of the following hold:

14 4 σ 3 and σ 3 + σ 4 σ σ σ 3 = ξ and σ 3 + σ σ where ξ >. Then there exists C > depending on ν σ j σ l j = l =... 4 such that a στ τ σ τ σ for every f F d νβµ. τ s σ 3 s σ 4 fτ s mds νβµ C +σ 3 +σ 4 σ + σ fτ m νβµ Proof There exists C. > only depending on σ σ such that a σ τ τ σ τ σ C. + τ σ σ for every τ D ρ S d S d. We apply the definition of the norm of Fνβµ d to arrive at a στ τ σ τ σ τ s σ 3 s σ 4 + C. fτ m νβµ sup τ S d τ fτ s mds τ τ exp νβµ ν τ τ h σ 3 h σ 4 + τ σ σ h + h exp ν h dh. The proof concludes with the steps providing a bound for C in the proof of Proposition in [3]. An analogous result holds by interchanging the role of the time variables. Lemma 4 Under the same hypotheses as in Lemma 3 assume that σ 3 and σ 3 + σ 4 σ σ σ 3 = ξ and σ 3 + σ σ where ξ >. Then there exists C > depending on ν σ j σ l for j = and l =... 4 such that a στ τ σ τ σ for every f F d νβµ. τ s σ 3 s σ 4 fs τ mds νβµ C +σ 3 +σ 4 σ + σ fτ m νβµ

15 5 Grouping the integral operators in Lemma 3 and Lemma 4 the following result is attained. Lemma 5 Let σ. Assume that a σ is given as in Lemma 3. Let σ j σ j for j = and σ 3 σ 3 σ 4 σ 4 be real numbers such that σ 3j and σ 3j + σ 4j σ j σ j σ 3j = ξ j j and σ 3j + j σ j σ j for j = and where ξ j >. Then there exists C > depending on ν σ j σ j σ 3j σ 4j for j = such that a στ τ σ τ σ for every f F d νβµ. τ s σ 3 s σ 4 τ s σ 3 s σ 4 fs s mds ds νβµ C +σ 3 +σ 4 σ + σ + +σ 3 +σ 4 σ + σ fτ m νβµ The proof of Proposition in [3] can be adapted with minor modifications to the Banach spaces under study. Lemma 6 Let γ >. Assume that / γ. Then there exists C > depending on ν γ such that τ for every fτ m F d νβµ. τ s γ fτ s mds C γ fτ m νβµ νβµ s The symmetric statement of Lemma 6 obtained by interchanging the role of τ and τ is derived straightforward from Lemma 6. We finally state the following auxiliary lemma. Lemma 7 Let σ and a σ be as in Lemma 3. Assume that P P R C[X] such that degr degp degr degp Rim for every m R. Assume that µ > max{degp + degp + }. We tae σ j σ j for j =. Then there exists a constant C 3 > depending on Q Q R µ ν such that a σ τ Rim τ σ τ σ τ s s P im m fτ s s x m m P im gs x m dx ds dm ds τ s s s x x for every fτ m gτ m F d νβµ. C 3 fτ m νβµ gτ m νβµ

16 6 Proof We follow analogous estimates as in the proof of Proposition 3 in [3] to arrive at a σ τ τ Rim τ σ τ σ τ τ s s P im m fτ s s x m m P im gs x m dx ds dm ds τ s s s x x τ τ τ h h dh τ τ h h s τ σ + sup τ Dρ S d + τ σ h x sup τ S d + h x τ σ + + τ σ h exp ν x τ τ + x exp On the one hand the expression τ σ / + τ σ τ + A := sup τ S d τ exp ν τ + τ h τ ν τ + h τ h dx dh h x x fτ m νβµ gτ m νβµ. τ s is bounded. Moreover τ h h x + h x h exp ν x + x dx dh h x x can be estimated following the same steps as in the study of upper bounds for C 3. in formula 35 of [3]. We get the existence of C. > such that A C 3.. It only rests to prove that A is upper bounded where A := τ σ + sup τ Dρ S d + τ σ = sup Ã. τ Dρ S d τ τ τ τ h + τ h h + h dh τ h h We distinguish two cases. First we assume that τ C for some C >. Then it holds that τ σ + τ σ is upper bounded and by putting x = τ / one can estimate à from above by + x dh sup x C x / + x h + h.

17 7 for some C >. We apply Corollary 4.9 in [7] to conclude that + x j à sup x C x / x + for some j >. The previous expression is upper bounded by a positive constant. Second in the case that τ < C we have + τ σ. We put x = τ / to get that sup τ Dρ S d τ C A partial fraction decomposition yields x x h h + x h à sup x + x x x x / + h dh h x h x h h + x h j x x h dh h x h. x for some j > valid for. This concludes the existence of a positive upper bound for A and the proof follows from this point. 3.. Domain of existence for the solution The purpose of this section is twofold. On the one hand we motivate the fact that any actual holomorphic solution ωτ m of 3 is not well defined on sets of the form S d S d D ρ for d d R and any choice of ρ >.This is due to a small divisor phenomenon observed which does not allow to proceed with a summability procedure. On the second hand we aim to display geometric conditions on the natural domains in which the solution is defined. In order to motivate that the natural domains of definition of a solution cannot be of the form S d S d D ρ for d d R and ρ > let ρ >. We rewrite the equation P m τ = see 4 for the definition of P m in the form 6 τ δ = Qim R D D im δ D D δ τ δ D D3 + R D3 im δ τ δ D3 δ D We put T = τ δ D and write 6 in the form ΨT = T where 7 ΨT := Qim R D D im δ D δ D τ δ D + R D3 im δ D3 T. δ D3 δ D δ D. Lemma 8 Let d d R. Under the assumption that δ D3 δ D δ D such that the following statements hold: N \ {} there exists τ S d. Ψ is a map from E := D ρ δ D into itself.. Ψ : E E is a shrining map.

18 8 Proof Let τ S d with large enough modulus in such a way that δ D3 δ D δ D 8 R D D im δ D D D3 T δ + R D3 im δ τ δ D R D3 im for every m R and all T Ŝd D ˆρ. Here Ŝd stand for the infinite sector defined by Ŝ d := {T C : τ δ D S d } and ˆρ = ρ δ. The assumption on the geometry of the problem and 8 yield ΨT Qim τ δ D R D D im δ D D3 T + R D3 im δ δ D δ D3 δ D δ D τ δ D Qim τ δ D R D3 im ρ δ D for large enough τ. As a result we get the fist statement in the result. We have Ψ z δ D3 δ R D3 δ D D3 δ D ρ δ D δ D3 im δ D δ D Qim τ δ D R D3 im δ D3 δ D ρ δ D3 δ D + Qim δ D R D3 im δ D3 τ δ D for every z D ρ δ D m R and large enough τ. We get that ΨT ΨT sup Ψ zt T z [T T ] T T for every T T D ρ δ D. The application of the mean value theorem entails the second statement of the result. As a consequence of Lemma 8 we deduce that Ψ has a unique fixed point in E hence there exists a unique solution of ΨT = T for T E say T. The solutions of 6 are the solutions of τ δ D = T. As a matter of fact the δ D roots of T belong the disc D ρ. Remar: Observe that in the case that δ D = δ D3 the equation ΨT = T can be solved directly in terms of τ. In this case the δ D roots of P m τ = lay on D ρ and we can not define ωτ m in any set of the form S d S d D ρ. In the next paragraphs we display geometric conditions on the problem which allow us to attain lower estimates on P m τ defined in 4. On the way the choice of directions d and d is made accordingly with the geometry of the problem. We write P m τ Qim = + τ δd RD D im δ D D δ τ δ D + R D 3 im δ D3 τ δ D3 δ D. Qim Qim We distinguish different cases.

19 9. In case that τ D ρ for some small enough ρ >... If τ D ρ for small enough ρ >. Regarding there exist r D D Q r D 3 Q > such that τ δ D RD D im δ D D δ τ δ D Qim ρ δ D + R D 3 im δ D3 τ δ D3 δ D Qim rd D Q δ D D δ ρ δ D + rd D3 3 Q δ ρ δ D3 δ D 4 for every m R every τ D ρ and τ D ρ. We conclude that 9 P m τ Qim C for some positive constant C common for all m R τ D ρ and τ D ρ... Assume that τ S d with τ ρ for some fixed ρ >. We write R D D im δ D D δ τ δ D + R D 3 im δ D3 τ δ D3 δ D Qim Qim where Am τ := R D D im R D3 im From the assumptions made in we get that Am τ rd δ D D δ D D 3 D3 δ = R D 3 im Qim δ D3 τ δ D3 δ D + Am τ δ D D δ τ δ D D3 δ τ δ D3 δ D ρ δ D ρ δ D3 δ D for some r D D D 3 >. Taing small enough ρ > we can write + Am τ = ρ mτ e iθmτ with ρ mτ close to and θ mτ close to uniformly for every m R and all τ S d with τ > ρ and τ D ρ. Therefore we have Let τ be the roots satisfying P m τ Qim = + δ D3 R D3 im Qim τ δd3 ρ mτ e iθmτ. τ δ D3 = δ D3 ρ mτ Qim R D3 im e iθmτ.

20 for =... δ D3. We select the sector S d in such a way that if τ S d then it can be expressed as τ = ρe iθ τ for some fixed some θ close to θ and any ρ >. We get P m τ Qim = ρ δ D3 e iθ δ D3 = ρ δ D3 e iθ δ D3 +. Now there exists C > such that eiθ δ D3 + for every ρ. ρ δ D3 C ρ δ D3 By construction we also have ρ δ D3 = τ δ D3 /τ δ D3. We deduce the existence of C > such that ρ δ D3 C τ δ D3. As a result we see that 3 P m τ Qim C C τ δ D3 for every τ S d with τ ρ and τ D ρ for some small enough ρ >.. Assume that τ S d with τ ρ for some fixed ρ > and τ S d. We select S d in such a way that for τ S d one can write τ = ξ e iθ τ δ d3 δ D δ D RD3 im δ D R D D im here we have chosen any particular / δ D root for some ξ > and θ close to when τ S d. Since τ ρ we have that ξ > ν > for some fixed ν >. Remar: This factorization is a particular case of a so-called blow up in the desingularization procedure. We refer to the excellent textboo of Y. Ilyasheno and S. Yaoveno [] Chapter Section 8 for an introduction to the geometric aspects. We write P m τ Qim = + τ δd3 ξ δ D R D3 im δ D D δ e iθ δ δd3 D +. Qim ξ δ D Again taing into account one can select a sector S d which additionally satisfies + R D 3 im τ δ D3 ξ δ D Qim δ D D δ e iθ δ δd3 D + ξ δ D C > for some C > valid for every τ S d and ξ > ν >. As a result we get 3 P m τ Qim Cξ δ D τ δ D3 = Cτ δ D τ δ D R D D im R D3 im Cτ δ D τ δ D for some C > and all m R. As a summary we have achieved the following result.

21 Proposition There exist d d R and ρ > such that for every m R and all τ D ρ S d τ S d one has 3 P m τ Qim C + τ δ D fτ for some C > and where fτ is defined by { + τ fτ = δ D3 if τ ρ + τ δ D if τ > ρ. Remar: Without loss of generality we may assume that ρ ρ where ρ > is the radius of the disc of holomorphy with respect to the first time variable appearing in Section.. 3. Fixed point of a convolution operator in Banach spaces The main purpose of this section is to obtain the existence of a fixed point on certain operator defined in a Banach space. It will allow us to construct the analytic solution of the main problem under study. For every D \ {} we consider the operator H defined by 33 H ωτ m = τ δ D à D τ P m τ τ R D D imωτ m τ P m τ P m τ A D τ ÃD τ τ R D D imωτ m + τ P m τ π Γ + δ D τ s A D τ R D D imωτ m à D3 τ P m τ τ R D3 imωτ m ωτ s s x m m P im ωs x τ + P m τ l j D j j= l l d l d l +δ l + δ l R l l im τ d l s τ s + P m τ Γ + s m d l δ l sδ l δ l s δ l P im m dx ds dm ds τ s s s x x τ Γ dl Γ ωs τ s dl s mds ds s s ψ τ s m ds. s Proposition 3 Assume that the hypotheses 5- hold. There exist ϖ ξ R > such that if ψ τ m νβµ ξ max{r R } R where R R are the geometric conditions determined in for all D \ {}. Then the operator H defined in 33 admits a unique fixed point ω dτ m F νβµ d such that ω dτ m νβµ ϖ for all D \ {}.

22 Proof Tae d d R and ρ > determined in Proposition. First we apply Lemma and Lemma 3 to get that 34 τ δ D R D D im P m τ τ s δ D p s p for every p δ D. In view of Lemma and Lemma 4 we have ωτ s mds C C sup R D D im Qim νβµ ωτ m νβµ 35 τ δd R D D im τ P m τ τ s δ D p s p ωs τ mds νβµ C C sup τ δd τ S d + τ sup δd for every p δ D. Moreover from Lemma and Lemma 5 we have Qim ωτ m νβµ 36 τ R D D im P m τ τ s δ D p s p τ s δ D p s p ωs s mds ds νβµ C C sup R D D im Qim ωτ m νβµ for every p δ D and p δ D. We apply Lemma and Lemma 3 to get 37 P m τ τ s δ D 3 p 3 s p 3 ωτ s mds for every p 3 δ D3. Regarding Lemma and Lemma 7 we deduce that C C sup νβµ Qim ωτ m νβµ τ 38 P m τ τ τ τ s s P im m ωτ s s x m m P im ωs x m dx ds dm ds τ s s s x x C 3 max Qim ωτ m νβµ

23 3 39 We apply Lemma and Lemma 5 to get R l l im τ τ P m τ τ d l s τ s ωs s mds ds νβµ C C sup for every l j D j for j =. Finally the application of Lemma and Lemma 6 yield 4 P m τ τ s d l δ l l s s δ ψ τ s m ds C C sup νβµ s R l l im Qim Tae small enough ϖ ξ > and assume that sup R D D im Qim R and sup R D3 im Qim R in such a way that ωτ m νβµ Qim ψ τ m νβµ 4 δ D C R C + C C C R C p δ D C R C p 3 δ D3 l j D j j= A δd p Γ δ D p ϖ + δ D C R C p δ D p δ D A δd3 p 3 Γ δ D3 p 3 ϖ + C 3 sup l l δ D δ D + sup p δ D A δd p Γδ D p ϖ A δd p Γδ D p Γ δ D p ϖ A δd p Qim π R l l im Qim + C Γ Γ + ϖ δ l δ l dl Γ dl Γ C + sup ϖ Qim ξ ϖ. Taing into account 34-4 and 4 we get that H D ϖ D ϖ. Here D ϖ stands for the closed disc of radius ϖ centered at the origin in the Banach space F d νβµ. Now let ω ω F d νβµ with ω jτ m νβµ ϖ. We now prove that 4 H ω H ω νβµ ω ω νβµ. At this point the classical contractive mapping theorem acting on the complete metric space D ϖ F d νβµ guarantees the existence of a fixed point for H. Let us chec 4. Analogous estimates as in the first part of the proof yield

24 4 43 τ δ D R D D im P m τ τ s δ D p s p C C sup ω τ s m ω τ s mds R D D im Qim for every p δ D. Also Lemma and Lemma 4 yield νβµ ω τ m ω τ m νβµ 44 τ δd R D D im τ P m τ C C τ s δ D p s p sup τ δd τ S d + τ sup δd for every p δ D. Lemma and Lemma 5 guarantee that 45 τ R D D im P m τ ω s τ m ω s τ mds Qim ω τ m ω τ m νβµ τ s δ D p s p τ s δ D p s p ω s s m ω s s mds ds νβµ C C sup R D D im Qim wτ m νβµ for every p δ D and p δ D. We apply Lemma and Lemma 3 to get τ 46 τ s δ D3 p 3 s p 3 ω τ s P m τ m ω τ s mds νβµ C C sup Qim ω τ m ω τ m νβµ for every p 3 δ D3. In order to study the convolution operator we need to give some details on the procedure. Put W := ω τ s / s x / m m ω τ s / s x / m m and W := ω s / x / m ω s / x / m. Then taing into account that 47 P im m ω τ s / s x / m m P im ω s / x / m P im m ω τ s / s x / m m P im ω s / x / m = P im m W P im ω s / x / m + P im m ω τ s / s x / m m P im W νβµ

25 5 and due to Lemma and Lemma 7 we proceed with analogous estimates as in 38 to get that 48 P m τ τ τ s s P im m ω τ s s x m m ω τ s s x m m P im ω s x m ω s x m dx ds dm ds τ s s s x x C 3 ω τ m νβµ + ω τ m νβµ ω τ m ω τ m νβµ max Qim 49 Finally we apply Lemma and Lemma 5 to get R l l im τ τ P m τ τ d l s τ s d l δ l l s s δ ω s s m ω s s mds ds νβµ C C sup R l l im Qim ω τ m ω τ m νβµ for every l j D j for j =. We choose small enough ϖ > and assume that sup R D D im Qim R and sup R D3 im Qim R to satisfy δ D C R C p δ D + C C C R C A δd p Γ δ D p + δ D C R C C R C p 3 δ D3 l j D j j= p δ D p δ D A δd3 p 3 Γ δ D3 p 3 + C 3 sup l l δ D δ D + sup p δ D A δd p A δd p Γδ D p A δd p Γδ D p Γ δ D p Qim π R l l im Qim Γ + ϖ δ l δ l Γ dl Γ dl. Then 4 holds and the proof is complete. The following is a direct consequence of the previous result.

26 6 Corollary The function ω d τ m obtained in Proposition 3 is a continuous function in D ρ S d S d R D \ {} and holomorphic with respect to τ in the set D ρ S d S d and on D \ {} with respect to the perturbation parameter. Moreover it turns out to be a solution of 3 which satisfies there exists ϖ > such that 5 ω d τ m ϖ + τ τ m µ + τ + τ exp βm + ν τ + ν τ for every τ m D ρ S d S d R D \ {}. 4 Family of analytic solutions of the main problem In this section we consider the main problem under study namely under the conditions 5-7 on the parameters involved and also on the geometry of the problem 8-. In order to construct the analytic solution of the problem we recall the definition of a good covering in C. Definition 3 Let ς ς be integer numbers. Let {E p p } p ς be a finite family of open p ς sectors with vertex at and radius. In addition to this we assume the opening of every sector is chosen to be slightly larger than π/ in the case that < and slightly larger than π/ in case <. We assume that the intersection of three different sectors in the good covering is empty and = U \ {} for some neighborhood of U C. Such set of sectors is called a p ς E p p p ς good covering in C. Definition 4 Let ς ς and {E p p } p ς be a good covering in C. Let T j be open p ς bounded sectors centered at with radius r Tj for j { } and consider two families of sectors as follows: let S dp θ r T = {T C /T < r T d p argt < θ /} S d p θ r T = {T C /T < r T d p argt < θ /} with opening θ j > π/ j and where d p d p R for all p ς and p ς is the couple of directions d d R mentioned in Proposition whenever E p p is the domain of definition of the perturbation parameter. In addition to that the sectors S dp θ r T and S d p θ r T are such that for all p ς p ς t T T and E p p one has t S dp θ r T and t S d p θ r T. We say that the family {S dp θ r T p ς S d p θ r T p ς T T } is associated to the good covering {E p p } p ς. p ς Let ς ς and {E p p } p ς p ς be a good covering in C. We assume the family {S dp θ r T p ς S d p θ r T p ς T T } is associated to the previous good covering.

27 7 The existence of a solution ω d τ m of the auxiliary problem 3 turns out to provide an actual solution of the main problem via Laplace and Fourier transform in view of the constraints satisfied by ω dτ m see 5. More precisely for every p ς and p ς the function 5 u p p t z = π / + L γp ω dp d p L γp u u u m e t u t e izm du du dm u u is holomorphic on the domain T D h T D h H β E p p for any < β < β and some h >. The first main result of the present wor is devoted to the construction of a family of actual holomorphic solutions to the equation for null initial data. Each of the elements in the family of solutions is associated to an element of a good covering with respect to the complex parameter. The strategy leans on the control of the difference of two solutions defined in domains with nonempty intersection with respect to the perturbation parameter. The construction of each analytic solution in terms of two Laplace transforms in different time variables requires to distinguish different cases depending on the coincidence of the integration paths or not. Theorem Let the hypotheses of Proposition 3 hold. Then for every element E p p in the good covering in C there exists a solution u p p t z of the main problem under study defined and holomorphic on T D h T D h H β E p p for any < β < β and some h >. Moreover for every two different multiindices p p p p {... ς } {... ς } one of the following situations hold: Case : E p p E p p =. Case : E p p E p p. The path L γ p coincides with L γp but L γp does not coincide with L γp. Then it holds that 5 sup u p p t z u p p t z K pe Mp t T Dh T Dh z H β for every E p p E p p. In that case we say that p p p p belongs to the subset U of {... ς } {... ς }. Case 3: E p p E p p. Neither the path L γ p coincides with L γp nor L γp coincides with L γp. Then it holds that 53 } sup u p p t z u p t T Dh T Dh p t z K p max {e Mp e Mp z H β for every E p p E p p. Proof The existence of the solution u p p t z for every p ς and p ς is guaranteed from the construction described previously.

28 8 We now give proof for the second statement of the result namely the existence of an exponential decay to with respect to the perturbation parameter of the difference of two consecutive solutions in the good covering uniformly with respect to t z. The proof is close to that of Theorem in [] but for the sae of clarity we give a complete description. Case : Assume that the path L γp coincides with L γp and L γp does not coincide with L γp. Then using that u ω dp d p u u m exp u t /u is holomorphic on D ρ for all m R D \ {} and every u L γp one can deform one of the integration paths and write I = in the form ω dp d p L γp ω dp d p L ρ /γp u u m e u du t u u u u m e t du u ω dp d p L ρ /γ p ω dp d p L γp u u u m e t du + ω dp d p C ρ /γ p γp u u m e u du t u u u u u m e t du. u where L ρ /γ p = [ρ / + e iγp L ρ /γ p = [ρ / + e iγ p and C ρ /γ p γ p is an arc of circle connecting ρ /e iγ p and ρ /e iγp with the adequate orientation. The positive real number ρ is determined in Proposition. We get the existence of constants C p p M p p > such that I C p p ϖ d p d p + m µ e βm u + u expν u M e p p for t T D h and E p p E p p and u L γp. We have 54 u p p t z u p p t z π / C p p u L γp + u expν u + m µ e βm e mimz dm u exp t du u e M p p. The last integral is estimated via the reparametrization u = re γp and the change of variable r = s by + s e δ s ds for some δ > whenever t T D h. The estimates given in the enunciate of Case follows from here.

29 9 Figure : Path deformation in Case Case 3: Assume that neither L γp coincides with L γp nor L γp coincides with L γp. Owing to the fact that u ω dp d p u u m exp u t /u is holomorphic on D ρ for all m R D \ {} and every u L γp we deform the integration paths with respect to the first time variable and write where J = π / L γp L γp J = π / L γp L γp u p p t z u p p t z = J J + J 3 ω dp d u p u u m e t u t e izm dm du du. u u ω d p d p u u u m e t u t e izm dm du du. u u J 3 = ρ e iθ π / ω dp u d p u u m e t du L γp u ω dp d p L γp u u u m e t du e izm dm e u t du u where ρ e iθ is such that θ is an argument between γ p and γ p. The path L γp resp. L γp consists of the concatenation of the arc of circle connecting ρ e iθ with ρ e iγp resp. with ρ e iγ p and the half line [ ρ e iγp resp. [ ρ e iγ p. We first give estimates for J. We have ω dp d p L γp u u u m e L γp t du u ϖ d p d p + m µ e βm u + u expν u u + u expν u e u t du u u ϖ dp d p C p + m µ e βm u + u expν u

30 3 Figure : Path deformation in Case 3 for some C p > and t T D h. Using the parametrization u = re γp and the change of variable r = s. Using analogous estimations as in the Case we arrive at J C p e M p for some C p M p > for all E p p E p p where t T D h and t T D h z H β. Analogous calculations yield to J C p e M p for some C p M p > for all E p p E p p where t T D h and t T D h z H β. In order to give upper bounds for J 3 we consider ω dp d p L γp u u u m e t du u ω dp d p L γp Since u belongs to the disc D ρ we now that the function u ω dp d u p u u m e t u u u u m e t du u. is holomorphic on the disc D ρ. In this framewor one is able to deform the integration path in order to write the difference as the next sum

31 3 ω dp d p L ρ /γp u u u m e t du u ω dp d p L ρ /γ p u u u m e t du + ω dp d p C ρ /γ p γp u u u u m e t du. u We get the previous expression is upper estimated by ϖ dp d p C p p + m µ e βm u + u expν u exp M p p for E p p E p p t T D h u [ ρ /e iθ ]. We finally get J 3 π / C p p ϖ d p d p + m µ e βm e mimz dm ρ /e iθ u + u expν u e u t du u exp M p p. We conclude that J 3 K p3 e M p3 uniformly for t t T D h T D h for some h > and z H β for any fixed β < β where K p3 M p3 are positive constants. Remar: Observe that in case that the path L γp coincides with L γp but L γp does not coincide with L γp then it is not possible to obtain estimates on the difference of two solutions in the form exp M/ as it happens in Case. The reason is that we can not deform the path L γp L γp since the function w dp d p τ m and w dp d p τ m are not holomorphic on a disc centered at respect to τ. 5 Asymptotics of the problem in the perturbation parameter 5. Summable formal series and Ramis-Sibuya Theorem For the sae of completeness we recall the definition of Borel summability of formal series with coefficients in a Banach space and Ramis-Sibuya Theorem. A reference for the details on the first part is [] whilst the second part of this section can be found in [] p. and [] Lemma XI--6. Definition 5 Let be an integer. A formal series ˆX = j= a j j! j F[[]]

32 3 with coefficients in a Banach space F. F is said to be summable with respect to in the direction d R if i there exists ρ R + such that the following formal series called formal Borel transform of ˆX of order B ˆXτ a j τ j = j!γ + j F[[τ]] is absolutely convergent for τ < ρ j= ii there exists δ > such that the series B ˆXτ can be analytically continued with respect to τ in a sector S dδ = {τ C : d argτ < δ}. Moreover there exist C > and K > such that B ˆXτ F Ce Kτ for all τ S dδ. If this is so the vector valued Laplace transform of order of B ˆXτ in the direction d is defined by L d B ˆX = u/ B ˆXue u du L γ along a half-line L γ = R + e iγ S dδ {} where γ depends on and is chosen in such a way that cosγ arg δ > for some fixed δ for all in a sector S dθr / = { C : < R / d arg < θ/} where π < θ < π + δ and < R < δ /K. The function L d B ˆX is called the sum of the formal series ˆXt in the direction d. It is bounded and holomorphic on the sector S dθr / and has the formal series ˆX as Gevrey asymptotic expansion of order / with respect to on S dθr /. This means that for all π < θ < θ there exist C M > such that for all n all S dθ R /. L d B ˆX n p= a p p! p F CM n Γ + n n Multisummability of a formal power series is a recursive process that allows to compute the sum of a formal power series in different Gevrey orders. One of the approaches to multisummability is that stated by W. Balser which can be found in [] Theorem p.57. Roughly speaing given a formal power series ˆf which can be decomposed into a sum ˆfz = ˆf z ˆf m z such that each of the terms ˆf j z is j -summable with sum given by f j then ˆf turns out to be multisummable and its multisum is given by f z f m z. More precisely one has the following definition. Definition 6 Let F F be a complex Banach space and let < <. Let E be a bounded open sector with vertex at and opening π + δ for some δ > and let F be a bounded open sector with vertex at the origin in C with opening π + δ for some δ > and such that E F holds. A formal power series ˆf F[[]] is said to be summable on E if there exist ˆf F[[]] which is summable on F with -sum given by f : F F and ˆf F[[]] which