On singularly perturbed small step size difference-differential nonlinear PDEs

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1 On singularly perturbed small step size difference-differential nonlinear PDEs Stéphane Malek Université de Lille 1 UFR de mathématiques Villeneuve d Ascq Cedex France Stephane.Malek@math.univ-lille1.fr Tel May, Abstract We study a family of singularly perturbed small step size difference-differential nonlinear equations in the complex domain. We construct formal solutions to these equations with respect to the perturbation parameter ɛ which are asymptotic expansions with 1 Gevrey order of actual holomorphic solutions on some sectors in ɛ near the origin in C. However, these formal solutions can be written as sums of formal series with a corresponding decomposition for the actual solutions which may possess a different Gevrey order called 1 + Gevrey in the literature. This phenomenon of two levels asymptotics has been already observed in the framework of difference equations, see [6]. The proof rests on a new version of the so-called Ramis-Sibuya theorem which involves both 1 Gevrey and 1 + Gevrey orders. Namely, using classical and truncated Borel-Laplace transforms (introduced by G. Immink in [18], we construct a set of neighboring sectorial holomorphic solutions and functions whose difference have exponentially and super-exponentially small bounds in the perturbation parameter. Key words: asymptotic expansion, Borel-Laplace transform, Cauchy problem, difference equation, integrodifferential equation, nonlinear partial differential equation, singular perturbation. 2 MSC: 35C1, 35C2. 1 Introduction We consider a family of singularly perturbed small step size difference-differential nonlinear problem of the form (1 ɛ t z S X i (t, z, ɛ + a z S X i (t, z, ɛ b k (z, ɛ = t s+1 ( k t k 1 z X i (t + k 2 ɛ, z, ɛ + P (z, ɛ, X i (t, z, ɛ k=(s,k,k 1,k 2 A 1 The author is partially supported by the french ANR-1-JCJC 15 project and the PHC Polonium 213 project No SG.

2 2 for given initial data (2 ( j zx i (t,, ɛ = ϕ j,i (t, ɛ, i ν 1, j S 1, where ɛ is a complex parameter, S is some positive integer, a C is some complex number with arg(a, ν is some integer larger than 2 and A 1 is a finite subset of N 4 which satisfies the constraints (115. The coefficients b k (z, ɛ, k A 1 of the linear part belong to O{z, ɛ} and P (z, ɛ, X O{z, ɛ}[x] where O{z, ɛ} denotes the space of holomorphic functions in (z, ɛ near the origin in C 2. The initial data ϕ j,i (t, ɛ are assumed to be holomorphic on products of two sectors (T { t > h} E i C 2, for some h > large enough, where T is a fixed open unbounded sector centered at and E = {E i } i ν 1 is a family of open bounded sectors centered at the origin whose union form a covering of V \ {}, where V is some neighborhood of. In the paper [24], we have considered singular singularly perturbed nonlinear problems (3 ɛt 2 t S z u i (t, z, ɛ = F (t, z, ɛ, t, z u i (t, z, ɛ + P (t, z, ɛ, u i (t, z, ɛ for given initial data (4 ( j zu i (t,, ɛ = φ j,i (t, ɛ, i ν 1, j S 1, where F is some differential operator with polynomial coefficients and P some polynomial. The initial data φ j,i (t, ɛ were assumed to be holomorphic on products (T { t < h } E i, for some h > small enough. Under suitable constraints on the shape of the equation (3 and on the initial data (4, we have shown the existence of a formal series û(ɛ = k h kɛ k /k! with coefficients h k belonging to the Banach space F of bounded holomorphic functions on (T { t < h } D(, δ (for some δ > equipped with the supremum norm, solution of (3, which is the 1 Gevrey asymptotic expansion of actual holomorphic solutions u i of (3, (4 on E i as F valued functions, for all i ν 1 (see Definition 4 1. We mention also the work [21] of A. Lastra and the author, where a q analog of the problem (3, (4 was investigated. The discretization was performed with respect to the variable t meaning that t was replaced by the q difference operator (f(qt f(t/(qt t for a complex q C with q > 1 (which formally tends to t as q tends to 1. In this work we address the same question as in our previous papers [21], [24], namely our main goal is the construction of actual holomorphic solutions X i (t, z, ɛ to the problem (1, (2 on domains (T { t > h} D(, δ E i for some small disc D(, δ and the analysis of their asymptotic expansions as ɛ tends to. More precisely, we can present our main statements as follows. Main results We choose a set of directions d i R, i ν 1, such that d i arg(a with the property that d i + arg(t arg(ɛ ( π/2, π/2 for all ɛ E i and t T { t > h}. We make the assumption that the initial Cauchy data ϕ j,i (t, ɛ (given in (2 can be written as Laplace transforms ϕ j,i (t, ɛ = V j,i (τ, ɛe tτ ɛ dτ L di on (T { t > h} E i along halflines L di = R + e 1d i for directions d i ( π/2, π/2 and as truncated Laplace transforms ϕ j,i (t, ɛ = Γi log(ω i t/ɛ V j,i (τ, ɛe tτ ɛ dτ

3 3 on (T { t > h} E i, for well chosen complex numbers Ω i C and Γ i L di, for directions d i [ π, π/2 (π/2, π], where V j,i (τ, ɛ are holomorphic functions satisfying the growth constraints (117 and (118. Then, in Theorem 1, we construct a family of holomorphic and bounded functions X i (t, z, ɛ, i ν 1 on the products (T { t > h} D(, δ E i for some radius δ > small enough with the property that ( j zx i (t,, ɛ = ϕ j,i (t, ɛ, i ν 1, j S 1, and whose differences X i+1 (t, z, ɛ X i (t, z, ɛ satisfy the exponential and super-exponential flatness estimates (121 and (122 on E i+1 E i. Moreover, for all integers i {,..., ν 1} with d i ( π/2, π/2, we prove that X i (t, z, ɛ is an actual solution of the problem (1, (2. In a second step (described in Theorem 2, we show the existence of a formal series ˆX(ɛ = k H k ɛ k k! whose coefficients H k belong to the Banach space E of bounded holomorphic functions on (T { t > h} D(, δ, which solves the equation (1 and is the 1 Gevrey asymptotic expansion of X i on E i as E valued functions. However, this formal series ˆX(ɛ and the corresponding functions X i own a fine structure which involves two levels of asymptotics. Namely, ˆX(ɛ and X i (t, z, ɛ can be written as sums ˆX(ɛ = a(ɛ + ˆX 1 (ɛ + ˆX 2 (ɛ, X i (t, z, ɛ = a(ɛ + X 1 i (ɛ + X 2 i (ɛ where a(ɛ is a convergent series near ɛ = with coefficients in E and ˆX 1 (ɛ (resp. ˆX2 (ɛ is a formal series with coefficients in E which is the 1 Gevrey asymptotic expansion (resp. the 1 + Gevrey asymptotic expansion of the E valued function Xi 1(ɛ (resp. X2 i (ɛ on E i in the sense of Definition 4, for all i ν 1. In particular, the coefficients Hk 2, k 2, of the formal series ˆX 2 (ɛ satisfy estimates of the following form: there exist two constants C, M > such that (5 H 2 k E CM k (k/ log k k for all k 2. We stress the fact that this kind of phenomenon with two levels of asymptotics has already been observed in the framework of linear difference systems of the form Y (s + 1 = A(sY (s for meromorphic matrices A(s at in [6] (see especially Example 1 therein. These authors denote the estimates of the form (5 what they called 1 + Gevrey type growth since a sequence which is s Gevrey for any s > 1 is 1 + Gevrey but a 1 Gevrey sequence is not necessary 1 + Gevrey (recall that s Gevrey order means that estimates of the form CM k (k/e k/s hold for some constants C, M >. More recently G. Immink has extended their study to the nonlinear situation and has also proposed a resummation procedure which constructs actual holomorphic solutions of difference systems from formal series solutions using modified Laplace transforms and acceleration kernels, see [18]. The Cauchy problem (1, (2 we consider in this work comes within the framework of the asymptotic analysis of singularly perturbed difference-differential equations with small advance or delay, which becomes a growing domain of research these last years and has numerous applications to engineering problems and biology, see for instance [13] and references therein.

4 4 In the context of differential equations most of the statements in the literature are dedicated to problems of the form ɛ t x(t, ɛ = f(t, ɛ, x(t, ɛ, x(t ± δ, ɛ for some vector valued function f, where ɛ > is a small parameter and δ > may or not depend on ɛ, and concern the study of asymptotic behaviour of their solutions x(t, ɛ as ɛ tends to. For general and abstract convergence results we quote [1] and references therein for a historical overview. For the construction of solutions x(t, ɛ having asymptotic expansions of the form n 1 (6 x(t, ɛ = x l (tɛ i + R n (t, ɛ l= with error bounds estimates for the remainder R n for some integers n 2, we refer to [16] and [31]. In the framework of partial differential equations, we mention [28] and [32] where formal asymptotic expansion like (6 have been obtained for solutions to reaction-diffusion equations with small delay. Nevertheless, these problems when working with a complex parameter ɛ and with solutions in analytic functions spaces are still quite unstudied although some important and interesting results have been obtained for small step size difference equations, see [12], [14], [15] and for singularly perturbed elliptic partial differential equations, see [27]. In the following, we explain our main results and the principal arguments needed in their proofs. In a first part, we construct a holomorphic function V (τ, z, ɛ near the origin with respect to (τ, z and on a punctured disc with respect to ɛ which solves an integro-differential problem whose coefficients are meromorphic functions with respect to (τ, ɛ with a pole at ɛ =, see (18, (19. The main novelty compared to our previous studies on singular perturbation problems, see [21], [23], [24], [25], is that the coefficients of (18 now have at most polynomial growth with respect to τ on the half plane C + = {τ C/Re(τ } but exponential growth on the half plane C = {τ C/Re(τ < }. For suitable initial data satisfying the conditions (117, (118, we show that V (τ, z, ɛ can be analytically continued to functions V i (τ, z, ɛ defined on products U di D(, δ E i where U di, i ν 1, are suitable open unbounded sectors with small aperture (see Definition 3 and below. If U di is contained in C +, then V i (τ, z, ɛ has at most exponential growth rate with respect to (τ, ɛ, namely there exist C, K > such that (7 sup V i (τ, z, ɛ C exp(k τ / z D(,δ for all τ U di, all ɛ E i. When U di belongs to C, V i (τ, z, ɛ owns an exponential growth rate with respect to ɛ but a super-exponential growth rate with respect to τ, more precisely there exist C, K 1, K 2 > with τ (8 sup V i (τ, z, ɛ C exp(k 1 z D(,δ + exp(k 2 τ for all τ U di, all ɛ E i. In a second part, we construct actual solutions X i (t, z, ɛ of our problem (1, (2 as Laplace transforms (9 X i (t, z, ɛ = V i (τ, z, ɛe tτ ɛ L γi dτ

5 5 along halflines L γi U di, when U di C +, which defines a bounded holomorphic function on the product (T { t > h} D(, δ E i, provided that h > is large enough. On the other hand, when U di is located in C, we construct bounded holomorphic functions X i (t, z, ɛ as truncated Laplace transform (as introduced by G. Immink in [18] (1 X i (t, z, ɛ = Γi log(ω i t/ɛ V i (τ, z, ɛe tτ ɛ dτ for some complex numbers Ω i C and Γ i U di, for all (t, z, ɛ (T { t > h} D(, δ E i. We stress the fact that these functions (1 do not solve the equation (1, but they share with (9 the following crucial properties for our scope: the functions (cocyle G i (ɛ = X i+1 (t, z, ɛ X i (t, z, ɛ, for i ν 1 (with the convention X ν = X are exponentially flat as ɛ tends to on E i+1 E i as E valued functions, where E is the Banach space of bounded holomorphic functions on (T { t > h} D(, δ equipped with the supremum norm. Moreover, when U di U di+1 and U di, U di+1 C, the function G i (ɛ is even super-exponentially flat, meaning that there exist K, L, M > such that (11 G i (ɛ E K exp( M log L for all ɛ E i+1 E i. In the proof, we use as in [24] deformations of the integration s paths in X i with the help of the estimates (7 and (8 (Theorem 1. In the last part, we establish a new version of the Ramis-Sibuya theorem (Theorem (RS in Section 5.1 with two levels of 1 Gevrey and of 1 + Gevrey type estimates. It is worthwhile noting that the classical Ramis-Sibuya criterion has shown to be very useful to investigate the phenomenon of the so-called multi-summability (which involves several Gevrey orders of solutions to systems of meromorphic linear differential equations, see [26], nonlinear equations, see [3], and nonlinear systems of difference equations, see [5]. By applying this criterion to our given cocycle G i, we deduce the main result of this paper (Theorem 2, namely the existence of a formal series solution of (1 ˆX(ɛ = k H k ɛ k k! E[[ɛ]] which is the 1 Gevrey asymptotic expansion of X i (t, z, ɛ on E i for all i ν 1 and such that the couple ( ˆX, X i shares a three terms decomposition ˆX(ɛ = a(ɛ + ˆX 1 (ɛ + ˆX 2 (ɛ, X i (t, z, ɛ = a(ɛ + X 1 i (ɛ + X 2 i (ɛ where a(ɛ is a convergent series at ɛ =, ˆX1 (ɛ is the 1 Gevrey asymptotic expansion of X i (ɛ on E i and ˆX 2 (ɛ is the 1 + Gevrey asymptotic expansion of X 2 i (ɛ on E i, for any i ν 1. The paper is organized as follows. In Section 2, we consider parameter depending nonlinear convolution differential Cauchy problems with singular and exponential growing coefficients. We construct solutions of these equations in parameter depending Banach spaces of holomorphic functions on sectors with exponential and super-exponential growths. In Section 3, we construct holomorphic solutions X i to our problem (1, (2 on some sectors E i as Laplace transform of solutions to nonlinear convolution problems studied in the section 2. In the section 4, we show flatness estimates of exponential type for the cocycle G i = X i+1 X i

6 6 where the X i are constructed in Section 3. We complete this cocycle on the full family E with the help of truncated Laplace transform of solutions to singular problems studied in Section 2. Moreover, we give flatness estimates that can be of super-exponential growth on some intersections E i+1 E i. In the last section, we first state a new version of the Ramis-Sibuya theorem in the framework of 1 Gevrey and 1 + Gevrey type estimates and we finally prove our main result. 2 A global Cauchy problem with singular complex parameter 2.1 Banach spaces of holomorphic functions on sectors with exponential and super-exponential growths We denote D(, r the open disc centered at with radius r > in C. Let S d be an open bounded or unbounded sector with bisecting direction d R and E be an open sector with finite radius r E, both centered at in C. By convention, these sectors do not contain the origin in C. For any open set D C, we denote O(D the vector space of holomorphic functions on D. In this section 2, we set Ω = (S d D(, r E. Let b > 1 a real number and let r b (β = β n= 1/(n+1b for all integers β. Definition 1 Let ɛ E and σ >, σ be real numbers. We denote SE β,ɛ,σ,σ,ω the vector space of all functions v O(S d D(, r such that v(τ β,ɛ,σ,σ,ω := ( sup v(τ (1 + τ 2 τ S d D(,r 2 exp σ r b(β τ exp(σ r b (β τ is finite. Let δ > be a real number. We define SG(ɛ, σ, σ, δ, Ω to be the vector space of all functions v(τ, z = β v β(τz β /β! that belong to O(S d D(, r{z} such that v(τ, z (ɛ,σ,σ,δ,ω := β δ β v β (τ β,ɛ,σ,σ,ω β! is finite. One can check that the normed space (SG(ɛ, σ, σ, δ, Ω,. (ɛ,σ,σ,δ,ω is a Banach space. Remark: These norms are appropriate modifications of the norms introduced by G. Immink in [18] and those of the author introduced in the work [24]. Notice that for σ =, the space SG(ɛ, σ,, δ, Ω coincides with the space G(ɛ, σ, δ, Ω defined in [24]. In the next proposition, we study the rate of growth of the functions belonging to the latter Banach spaces. Proposition 1 Let v(τ, z SG(ɛ, σ, σ, δ, Ω. Let < δ 1 < 1. There exists a constant C > (depending on v (ɛ,σ,σ,δ,ω and δ 1 such that ( (12 v(τ, z C(1 + τ 2 σζ(b 2 1 exp τ + exp(σ ζ(b τ for all τ S d D(, r, all z C such that z δ < δ 1, where ζ(b = n= 1/(n + 1b.

7 7 Proof Let v(τ, z = β v β(τz β /β! be in SG(ɛ, σ, σ, δ, Ω. constant c 1 > (depending on v (ɛ,σ,σ,δ,ω such that By definition, there exists a ( v β (τ c 1 (1 + τ 2 σ 2 1 exp r b(β τ + exp(σ r b (β τ β!( 1 δ β for all β, all τ S d D(, r. Let < δ 1 < 1. From the definition of ζ(b, we deduce that (13 v(τ, z c 1 (1 + τ 2 ( σ 2 1 exp β r b(β τ + exp(σ r b (β τ (δ 1 β c 1 (1 + τ exp ( σζ(b 1 τ + exp(σ ζ(b τ, 1 δ 1 for all z C such that z δ < δ 1 < 1, all τ S d D(, r. Finally, from (13, we deduce the estimates (12. In the next proposition, we study some linear operators of multiplication by polynomials and exponential functions acting on the space SG(ɛ, σ, σ, δ, Ω. Proposition 2 Let k be a non negative integer and let k 1, k 2 N. 1 Let σ >. Assume that the condition (14 k 1 bk + k 2b σ holds. Moreover, we assume that the function τ exp( k 2 τ is unbounded on the sector S d. Then, for all ɛ E, the operator v(τ, z exp( k 2 ττ k k 1 z v(τ, z is a bounded linear operator from (SG(ɛ, σ, σ, δ, Ω,. (ɛ,σ,σ,δ,ω into itself. Moreover, there exists a constant C 1 > (depending on k, k 1, k 2, σ, σ, b, which does not depend on ɛ E, such that (15 exp( k 2 ττ k k 1 z v(τ, z (ɛ,σ,σ,δ,ω C 1 k δ k 1 v(τ, z (ɛ,σ,σ,δ,ω for all v SG(ɛ, σ, σ, δ, Ω, all ɛ E. 2 Let σ. Assume that the condition (16 k 1 bk holds. Then, for all ɛ E, the operator v(τ, z τ k k 1 z v(τ, z is a bounded linear operator from (SG(ɛ, σ, σ, δ, Ω,. (ɛ,σ,σ,δ,ω into itself. Moreover, there exists a constant Č1 > (depending on k, k 1, σ, b, which does not depend on ɛ E, such that (17 τ k k 1 z v(τ, z (ɛ,σ,σ,δ,ω Č1 k δ k 1 v(τ, z (ɛ,σ,σ,δ,ω for all v SG(ɛ, σ, σ, δ, Ω, all ɛ E. Proof 1 Let v(τ, z SG(ɛ, σ, σ, δ, Ω. By definition, we have (18 exp( k 2 ττ k k 1 z v(τ, z (ɛ,σ,σ,δ,ω = exp( k 2 ττ k v β k1 (τ β,ɛ,σ,σ,ω β!. β k 1 δ β

8 8 Lemma 1 There exists a constant C 1.1 > (depending on k, k 1, k 2, σ, σ such that (19 exp( k 2 ττ k v β k1 (τ β,ɛ,σ,σ,ω C 1.1 k (1 + β bk + k 2 b σ v β k1 (τ β k1,ɛ,σ,σ,ω for all β k 1. Proof We can write exp( k 2 ττ k v β k1 (τ = exp( k 2 ττ k ( σ exp r b(β k 1 τ + exp(σ r b (β k 1 τ (1 + τ 2 v β k1 (τ (1 + τ 2 2 exp for all τ S d D(, r. From this latter equality, we deduce 2 1 ( σ r b(β k 1 τ exp(σ r b (β k 1 τ (2 exp( k 2 ττ k v β k1 (τ β,ɛ,σ,σ,ω A(ɛ, β v β k1 (τ β k1,ɛ,σ,σ,ω where ( (21 A(ɛ, β = sup exp(k 2 τ τ k σ exp τ S d D(,r r b(β k 1 τ + exp(σ r b (β k 1 τ ( exp σ r b(β τ exp(σ r b (β τ Lemma 2 The following inequality holds (22 sup τ k exp( σ τ S d D(,r r b(β k 1 τ σ r b(β τ ( k e 1 k k (1 + β bk σk 1 for all β k 1. Proof From the fact that (23 r b (β r b (β k 1 for all β k 1, we deduce that k 1 (β + 1 b (24 τ k exp( σ r b(β k 1 τ σ r b(β τ τ k exp( σ for all τ S d D(, r. From (24 and the classical equality (25 sup x m 1 exp( m 2 x = (m 1 /m 2 m 1 e m 1 x k 1 (1 + β b τ for any real numbers m 1, m 2 >, we get (22. Lemma 3 There exists a constant d > (depending on k 1,k 2 and σ such that (26 exp ( k 2 τ + exp(σ r b (β k 1 τ exp(σ r b (β τ exp( k 2b σ log(1 + β + d for all β k 1, all τ S d D(, r.

9 9 Proof With the help of the inequality (23 and from the Taylor formula we know that for all τ S d D(, r, all β there exists a constant c (σ r b (β k 1 τ, σ r b (β τ such that (27 exp(σ r b (β k 1 τ exp(σ r b (β τ = σ τ (r b (β k 1 r b (βe c σ τ (1 + β b exp(σ r b (β k 1 τ The next inequality will the useful. Let a, b, c > be real numbers. Then, (28 sup cx axe bx max(c, c x b (log( c ab 1. Indeed, for all x 1, we have cx axe bx ψ(x = cx ae bx. If log( c ab /b 1, the function ψ gets it s maximum value c b (log( c ab 1 on [1, + at x = log( c ab /b. If log( c ab /b < 1, ψ gets it s maximum value c ae b c on [1, + at x = 1. On the other hand, for all x 1, we have that cx axe bx c. Using (28, we deduce that sup τ S d D(,r k 1 k 2 τ σ τ (1 + β b exp(σ r b (β k 1 τ max(k 2, k 1 k 2 σ r b (β k 1 (log( k 2 (σ 2 k 1 r b (β k 1 (1 + βb 1 for all β k 1. From this latter inequality, we deduce the existence of a constant d > (depending on k 1,k 2, and σ such that sup τ S d D(,r for all β k 1. The inequality (26 follows. k 1 k 2 τ σ τ (1 + β b exp(σ r b (β k 1 τ k 2b σ log(1 + β + d Gathering the estimates (22 and (26 yields a constant C 1.1 > (depending on k, k 1, k 2, σ, σ such that (29 A(ɛ, β C 1.1 k (1 + β bk + k 2 b σ for all β k 1. From (2 and (29 the lemma 1 follows. Now, from (18 and (19, we deduce (3 exp( k 2 ττ k k 1 z v(τ, z (ɛ,σ,σ,δ,ω C 1.1 k (1 + β bk + k 2 b (β k 1! σ v β k1 (τ β k1,ɛ,σ,σ β!,ωδ k δ β k 1 1 (β k 1!. β k 1 Now, from the assumption (14, we get a constant C 1.2 > (depending on b, k, k 1, k 2, σ such that (31 (1 + β bk + bk 2 (β k 1! σ C 1.2 β! for all β k 1. Finally, from (3 together with (31 we get (15. 2 Let v(τ, z SG(ɛ, σ, σ, δ, Ω. By definition, we have (32 τ k k 1 z v(τ, z (ɛ,σ,σ,δ,ω = τ k v β k1 (τ β,ɛ,σ,σ,ω β!. β k 1 δ β

10 1 Lemma 4 There exists a constant Č1.1 > (depending on k, k 1, σ such that (33 τ k v β k1 (τ β,ɛ,σ,σ,ω Č1.1 k (1 + β bk v β k1 (τ β k1,ɛ,σ,σ,ω for all β k 1. Proof The explanation relies on the beginning of Lemma 1. If one puts k 2 = in the estimates (2 one gets (34 τ k v β k1 (τ β,ɛ,σ,σ,ω Ǎ(ɛ, β v β k 1 (τ β k1,ɛ,σ,σ,ω where (35 Ǎ(ɛ, β = sup exp τ S d D(,r ( τ k σ exp ( σ r b(β τ exp(σ r b (β τ r b(β k 1 τ + exp(σ r b (β k 1 τ sup τ k exp( σ τ S d D(,r r b(β k 1 τ exp( σ r b(β τ. Therefore, Lemma 4 is a consequence of Lemma 2 together with (34, (35. Now, from (32 and (33, we get (36 τ k k 1 z v(τ, z (ɛ,σ,σ,δ,ω Č 1.1 k (1 + β bk (β k 1! v β k1 (τ β k1,ɛ,σ,σ β!,ωδ k δ β k 1 1 (β k 1!. β k 1 Now, from the assumption (16, we get a constant Č1.2 > (depending on b, k, k 1 such that (37 (1 + β bk (β k 1! β! Č1.2 for all β k 1. Finally, the result follows from (36 and (37. Proposition 3 Let k be a non negative integer and let k 2 N. 1 Let σ > σ >, σ > σ > be real numbers. We assume that the function τ exp( k 2 τ is unbounded on the sector S d. Then, there exists a constant C 1 > (depending on k, k 2, σ, σ, σ, σ such that (38 exp( k 2 ττ k v(τ, z (ɛ,σ,σ,δ,ω C 1 k v(τ, z (ɛ, σ, σ,δ,ω for all v SG(ɛ, σ, σ, δ, Ω, all ɛ E. 2 Let σ > σ > be real numbers. Then, there exists a constant C 1.1 > (depending on k, σ, σ such that (39 τ k v(τ, z (ɛ,σ,,δ,ω C 1.1 k v(τ, z (ɛ, σ,,δ,ω for all v SG(ɛ, σ,, δ, Ω, all ɛ E.

11 11 Proof 1 Let v(τ, z SG(ɛ, σ, σ, δ, Ω. By definition, we have (4 exp( k 2 ττ k v(τ, z (ɛ,σ,σ,δ,ω = β exp( k 2 ττ k δ β v β (τ β,ɛ,σ,σ,ω β!. Lemma 5 There exists a constant C 1 > (depending on k, k 2, σ, σ, σ, σ such that (41 exp( k 2 ττ k v β (τ β,ɛ,σ,σ,ω C 1 k v β (τ β,ɛ, σ, σ,ω for all β. Proof We can write exp( k 2 ττ k v β (τ = exp( k 2 ττ k ( σ exp r b(β τ + exp( σ r b (β τ (1 + τ 2 v β (τ (1 + τ 2 2 exp for all τ S d D(, r. From this latter equality, we get 2 1 (42 exp( k 2 ττ k v β (τ β,ɛ,σ,σ,ω B(ɛ, β v β (τ β,ɛ, σ, σ,ω where ( (43 B(ɛ, β = sup exp(k 2 τ τ k σ exp τ S d D(,r ( σ r b(β τ exp( σ r b (β τ r b(β τ + exp( σ r b (β τ exp for all β. Moreover, using the estimates (25, we deduce that (44 sup τ k exp( (σ σr b(β τ k ( k e 1 τ S d D(,r σ σ k ( σ r b(β τ exp(σ r b (β τ for all β. On the other hand, there exists a constant d > (depending on k 2, σ, σ such that (45 exp ( k 2 τ + exp( σ r b (β τ exp(σ r b (β τ exp( d for all β, all τ S d D(, r. Indeed, from the Taylor formula, we know that for all τ S d D(, r, all β there exists a constant c ( σ r b (β τ, σ r b (β τ such that (46 exp( σ r b (β τ exp(σ r b (β τ From (28 we deduce that = ( σ σ r b (β τ e c (σ σ r b (β τ exp( σ r b (β τ (47 sup k 2 τ (σ σ r b (β τ exp( σ r b (β τ τ S d D(,r ( k 2 k 2 max(k 2, σ log( r b (β (σ σ σ (r b (β 2 1

12 12 for all β. From (46 and (47, one deduces (45. Finally, from (44 and (45, one obtains a constant C 1 > (depending on k, k 2, σ, σ, σ, σ such that B(ɛ, β C 1 k for all β. Hence, the lemma 5 follows. The estimates (38 result from (4 and (41. 2 Let v(τ, z SG(ɛ, σ,, δ, Ω. By definition, we have (48 τ k v(τ, z (ɛ,σ,,δ,ω = β τ k v β (τ β,ɛ,σ,,ω δ β β!. Lemma 6 There exists a constant C 1.1 > (depending on k, σ, σ such that (49 τ k v β (τ β,ɛ,σ,,ω C 1.1 k v β (τ β,ɛ, σ,,ω for all β. Proof The arguments relies on the beginning of Lemma 5. If one puts σ =, σ = and k 2 = in (42, one gets (5 τ k v β (τ β,ɛ,σ,,ω B(ɛ, β v β (τ β,ɛ, σ,,ω where (51 B(ɛ, β = sup τ k exp( σ τ S d D(,r r b(β τ exp( σ r b(β τ for all β. Gathering (5 and (44 yields Lemma 6. The estimates (39 follow from (48 and (49. In the next proposition, we study linear operators of multiplication by bounded holomorphic functions. Proposition 4 Let h(τ, z, ɛ be a holomorphic function on (S d D(, r D(, ρ E, for some ρ >, bounded by some constant M >. Let < δ < ρ. Then, the linear operator of multiplication by h(τ, z, ɛ is continuous from (SG(ɛ, σ, σ, δ, Ω,. (ɛ,σ,σ,δ,ω into itself, for all ɛ E. Moreover, there exists a constant C 2 (depending on M,δ,ρ, independent of ɛ, such that (52 h(τ, z, ɛv(τ, z (ɛ,σ,σ,δ,ω C 2 v(τ, z (ɛ,σ,σ,δ,ω for all v(τ, z SG(ɛ, σ, σ, δ, Ω, for all ɛ E. Proof Let h(τ, z, ɛ = β h β(τ, ɛz β /β! be holomorphic on (S d D(, r D(, ρ E such that there exists M > with sup h(τ, z, ɛ M. τ S d D(,r,z D(,ρ,ɛ E Let v(τ, z = β v β(τz β /β! SG(ɛ, σ, σ, δ, Ω. By construction, we have that (53 h(τ, z, ɛv(τ, z (ɛ,σ,σ,δ,ω β ( β 1 +β 2 =β β! h β1 (τ, ɛv β2 (τ β,ɛ,σ,σ,ω β 1!β 2! δβ β!.

13 13 From the Cauchy formula, we have sup h β (τ, ɛ M( 1 τ S d D(,r,ɛ E δ β β! for any δ < δ < ρ, for all β. By definition, we deduce that (54 h β1 (τ, ɛv β2 (τ β,ɛ,σ,σ,ω Mβ 1!( 1 δ β 1 v β2 (τ β,ɛ,σ,σ,ω Mβ 1!( 1 δ β 1 v β2 (τ β2,ɛ,σ,σ,ω for all β 1, β 2 such that β 1 + β 2 = β. From (53 and (54, we deduce that h(τ, z, ɛv(τ, z (ɛ,σ,σ,δ,ω M( β ( δ δ β v(τ, z (ɛ,σ,σ,δ,ω which yields (52. In the next proposition, we give norm estimates for the convolution product. Proposition 5 Let f,g be in SG(ɛ, σ, σ, δ, Ω. Then, the function (f g(τ, z = τ f(τ s, zg(s, zds belongs to SG(ɛ, σ, σ, δ, Ω. Moreover, there exists a (universal constant C 3 > such that (55 (f g(τ, z (ɛ,σ,σ,δ,ω C 3 f(τ, z (ɛ,σ,σ,δ,ω g(τ, z (ɛ,σ,σ,δ,ω for all f, g SG(ɛ, σ, σ, δ, Ω. Proof Let f(τ, z = β f β (τz β /β!, g(τ, z = β g β (τz β /β! be in SG(ɛ, σ, σ, δ, Ω. By construction of f g, we have that (56 τ f(τ s, zg(s, zds (ɛ,σ,σ,δ,ω β ( β 1 +β 2 =β Lemma 7 There exists a (universal constant C 3 > such that (57 τ β! τ β 1!β 2! f β1 (τ sg β2 (sds β,ɛ,σ,σ,ω δβ β!. f β1 (τ sg β2 (sds β,ɛ,σ,σ,ω C 3 f β1 (τ β1,ɛ,σ,σ,ω g β2 (τ β2,ɛ,σ,σ,ω for all β and all β 1, β 2 with β 1 + β 2 = β. Proof We write τ τ τ s 2 f β1 (τ sg β2 (sds = f β1 (τ s(1 + 2 ( exp σ r b(β 1 τ s exp(σ r b (β 1 τ s g β2 (s(1 + s 2 2 exp ( σ r b(β 2 s exp(σ r b (β 2 s exp ( σ (r b(β 1 τ s + r b (β 2 s + exp(σ r b (β 1 τ s + exp(σ r b (β 2 s (1 + s 2 2 (1 + τ s 2 2 ds

14 14 for all τ S d D(, r. We deduce that τ (58 1 τ exp f β1 (τ sg β2 (sds f β1 (τ β1,ɛ,σ,σ,ω g β2 (τ β2,ɛ,σ,σ,ω ( σ τ (r b(β 1 (1 h + r b (β 2 h + exp(σ r b (β 1 τ (1 h + exp(σ r b (β 2 τ h (1 + τ 2 2 (1 h 2 (1 + τ 2 2 h 2 In the next step we will show that there exists a constant C 3 > such that (59 I( τ,, β, β 1, β 2 = (1 + τ 2 1 τ exp ( 2 exp σ r b(β τ exp(σ r b (β τ ( σ τ (r b(β 1 (1 h + r b (β 2 h + exp(σ r b (β 1 τ (1 h + exp(σ r b (β 2 τ h (1 + τ 2 2 (1 h 2 (1 + τ 2 2 h 2 dh. dh C 3 for all τ S d D(, r, all ɛ E, for all β, all β 1, β 2 with β 1 + β 2 = β. Indeed, from the fact that r b is increasing, we first have that (6 r b (β 1 (1 h + r b (β 2 h r b (β for all h 1, all β 1, β 2 with β 1 + β 2 = β. Then, from (6, we get that (61 exp( σ r b(β τ exp( σ τ (r b(β 1 (1 h + r b (β 2 h 1 for all τ S d D(, r, all ɛ E, for all β, all β 1, β 2 with β 1 + β 2 = β and all h [, 1]. On the other hand, since r b is increasing, we get that (62 exp(σ r b (β 1 τ (1 h + exp(σ r b (β 2 τ h exp(σ r b (β τ exp(σ r b (β τ (1 h + exp(σ r b (β τ h exp(σ r b (β τ = ϕ(h for all h [, 1], for all τ S d D(, r, all ɛ E, for all β, all β 1, β 2 with β 1 + β 2 = β. By construction, one has ϕ( = ϕ(1 = 1. Moreover, by direct computation, one can check that ϕ (h if h 1/2 and ϕ (h if 1/2 h 1. So that ϕ(h 1 for all h [, 1]. From (61 and (62, we deduce that (63 I( τ,, β, β 1, β 2 J( τ, = 1 e(1 + τ 2 τ 2 (1 + τ 2 (1 h 2 (1 + τ 2 h 2 2 dh 2 for all τ S d D(, r, all ɛ E. On the other hand, we have that (64 J( τ, = 1 e(1 + τ 2 τ (1 + τ 2 (1 h 2 (1 + τ 2 h 2 dh. From Corollary 4.9 of [1], we know that the right hand side of (64 is a bounded function of τ on R +. We deduce that there exists a (universal constant C 3 > such that J( τ, J( τ, (65 sup = sup C 3 τ τ

15 15 for all ɛ E. We get from (63 and (65 that the inequality (59 holds. Finally, the inequality (57 follows from (58 and (59. From (56 and (57, we get that (55 holds with the constant C 3 from Lemma 7. Corollary 1 Let s, k be non negative integers and let k 1, k 2 N. 1 Let σ >. Assume that the condition (66 k 1 bk + k 2b σ holds. Moreover, we assume that the function τ exp( k 2 τ is unbounded on the sector S d. Then, for all ɛ E, the operator v(τ, z τ (τ h s exp( k 2 hh k k 1 z v(h, zdh is a bounded linear operator from (SG(ɛ, σ, σ, δ, Ω,. (ɛ,σ,σ,δ,ω into itself. Moreover, there exists a constant C 4 > (depending on s, k, k 1, k 2, σ, σ, b, which does not depend on ɛ E, such that (67 τ (τ h s exp( k 2 hh k k 1 z v(h, zdh (ɛ,σ,σ,δ,ω C 4 s+k+1 δ k 1 v(τ, z (ɛ,σ,σ,δ,ω for all v SG(ɛ, σ, σ, δ, Ω, all ɛ E. 2 Assume that the condition (68 k 1 bk holds. Let q(τ be a holomorphic function bounded by some constant M > on S d D(, r. Then, for all ɛ E, the operator v(τ, z τ (τ h s q(hh k k 1 z v(h, zdh is a bounded linear operator from (SG(ɛ, σ,, δ, Ω,. (ɛ,σ,,δ,ω into itself. Moreover, there exists a constant C 4.1 > (depending on s, k, k 1, σ, b, M, which does not depend on ɛ E, such that (69 τ (τ h s q(hh k k 1 z v(h, zdh (ɛ,σ,,δ,ω C 4.1 s+k+1 δ k 1 v(τ, z (ɛ,σ,,δ,ω for all v SG(ɛ, σ,, δ, Ω, all ɛ E. Proof 1 Using Proposition 5, there exists a universal constant C 3 > for which (7 τ (τ h s exp( k 2 hh k k 1 z v(h, zdh (ɛ,σ,σ,δ,ω C 3 τ s (ɛ,σ,σ,δ,ω exp( k 2 ττ k k 1 z v(τ, z (ɛ,σ,σ,δ,ω

16 16 holds. Moreover, using the estimates (25 we notice that (71 τ s (ɛ,σ,σ,δ,ω = τ s,ɛ,σ,σ,ω τ s,ɛ,σ,,ω = s (( se 1 (s + σ s 2e 1 + ( s+2. σ Finally, gathering (15, (7 and (71 yields (67. 2 Again, due to Proposition 5, we get a universal constant C 3 > such that (72 τ (τ h s q(hh k k 1 z v(h, zdh (ɛ,σ,,δ,ω C 3 τ s (ɛ,σ,,δ,ω q(ττ k k 1 z v(τ, z (ɛ,σ,,δ,ω Using (17 and (52 together with (71 and (72 give the result. 2.2 A global Cauchy problem We keep the same notations as in the previous section. In the following, we introduce some definitions. Let A 1 (resp. A 2 be a finite subset of N 4 (resp. N. For all k = (s, k, k 1, k 2 A 1, we denote I k a finite subset of N. For all n I k, we denote a n,k (τ, z, ɛ some bounded holomorphic function on (S d D(, r D(, ρ E, for some ρ >. For all k A 1, we consider a k (τ, z, ɛ = n I k a n,k (τ, z, ɛɛ n which are holomorphic functions on (S d D(, r D(, ρ E. For all l A 2, we denote α l (τ, z, ɛ some bounded holomorphic function on (S d D(, r D(, ρ E. Let S 1 be an integer. We consider the following equation (73 S z V (τ, z, ɛ = k A 1 a k (τ, z, ɛ τ (τ h s exp( k 2 hh k k 1 z V (h, z, ɛdh + l A 2 α l (τ, z, ɛv l (τ, z, ɛ where V 1 = V and V l 1, l 1 2, stands for the convolution product of V applied l 1 1 times with respect to τ. We state the main result of this section. Proposition 6 1 We make the following assumptions. There exist real numbers σ > σ > such that, for all k A 1, all n I k, we have (74 S k 1 + bk + k 2b σ, s + k + 1 n, S > k 1. For all l A 2, we have (75 l 2. For all j S 1, we consider a function τ V j (τ, ɛ that belongs to SE,ɛ, σ, σ,ω, for some σ > and all ɛ E. We assume that the function τ exp( k 2 τ is unbounded on the sector S d.

17 17 Then, there exist constants I >, R > and δ > (independent of ɛ such that if we assume that (76 S 1 h j= δ j V j+h (τ, ɛ,ɛ, σ, σ,ω j! I, for all h S 1, for all ɛ E, the equation (73 with initial data (77 ( j zv (τ,, ɛ = V j (τ, ɛ, j S 1, has a unique solution V (τ, z, ɛ in the space SG(ɛ, σ, σ, δ, Ω, for some σ > σ, for all ɛ E, which satisfies moreover the estimates for all ɛ E. 2 We assume that the next conditions hold. For all k A 1, all n I k, V (τ, z, ɛ (ɛ,σ,σ,δ,ω δ S R + I, (78 S k 1 + bk, s + k + 1 n, S > k 1 hold and for all l A 2, we have (79 l 2. For all j S 1, we consider a function τ V j (τ, ɛ that belongs to SE,ɛ, σ,,ω, for some σ > and all ɛ E. We make the assumption that the function τ exp( k 2 τ is bounded on the sector S d. Then, there exist constants I >, R > and δ > (independent of ɛ such that if we assume that (8 S 1 h j= V j+h (τ, ɛ,ɛ, σ,,ω δ j j! I, for all h S 1, for all ɛ E, the equation (73 with initial data (81 ( j zv (τ,, ɛ = V j (τ, ɛ, j S 1, has a unique solution V (τ, z, ɛ in the space SG(ɛ, σ,, δ, Ω, for some σ > σ, for all ɛ E, which satisfies moreover the estimates for all ɛ E. V (τ, z, ɛ (ɛ,σ,,δ,ω δ S R + I, Proof We consider S 1 w(τ, z, ɛ = V j (τ, ɛ zj j! j=

18 18 where V j (τ, ɛ are given in (77 or (81. For all ɛ E, we define a map A ɛ from O(S d D(, r{z} into itself by A ɛ (U(τ, z = k A 1 a k (τ, z, ɛ + τ k A 1 a k (τ, z, ɛ (τ h s exp( k 2 hh k k 1 S z U(h, zdh τ (τ h s exp( k 2 hh k k 1 z w(h, z, ɛdh + α l (τ, z, ɛ(( z S U(τ, z + w(τ, z, ɛ l 1 l A 2 In the following, we only plan to give details for the point 1 since exactly the same lines of arguments apply for the point 2 of the Proposition 6 with the help of the point 2 of Propositions 2,3 and Corollary 1 instead. In the next lemma, we show that A ɛ is a Lipschitz shrinking map from and into a small ball in a neighborhood of the origin of SG(ɛ, σ, σ, δ, Ω, for some σ > σ, σ > σ. Lemma 8 Under the conditions (74, (75, let a real number I be such that S 1 h j= δ j V j+h (τ, ɛ,ɛ, σ, σ,ω j! I, for all h S 1, for all ɛ E. Then, for a good choice of I >, a there exist real numbers < δ < ρ, σ > σ, σ > σ and R > (not depending on ɛ such that (82 A ɛ (U(τ, z (ɛ,σ,σ,δ,ω R for all U(τ, z B(, R, for all ɛ E, where B(, R is the closed ball centered at with radius R in SG(ɛ, σ, σ, δ, Ω, b we have (83 A ɛ (U 1 (τ, z A ɛ (U 2 (τ, z (ɛ,σ,σ,δ,ω 1 2 U 1(τ, z U 2 (τ, z (ɛ,σ,σ,δ,ω for all U 1, U 2 B(, R, for all ɛ E. Proof First of all, for all h S 1, j S 1 h, we have that V j+h (τ, ɛ j,ɛ, σ, σ,ω V j+h (τ, ɛ,ɛ, σ, σ,ω. We deduce that h z w(τ, z, ɛ SG(ɛ, σ, σ, δ, Ω and that (84 h z w(τ, z, ɛ (ɛ, σ, σ,δ,ω for all h S 1. We first show the estimates (82. S 1 h j= δ j V j+h (τ, ɛ,ɛ, σ, σ,ω j! I

19 19 Let σ > σ, σ > σ, R > and U(τ, z SG(ɛ, σ, σ, δ, Ω with U(τ, z (ɛ,σ,σ,δ,ω R. Under the assumptions (74, from Proposition 4 and Corollary 1, we get a constant C 5 > (independent of ɛ such that (85 a n,k (τ, z, ɛɛ n τ (τ h s exp( k 2 hh k k 1 S z U(h, zdh (ɛ,σ,σ,δ,ω C 5 s+k +1 n δ S k 1 U(τ, z (ɛ,σ,σ,δ,ω for all k A 1, all n I k. Again under the assumptions (74 with the help of Propositions 3, 1 and Proposition 5 and the estimates (71, (84, we get constants C 6, C 7 > (independent of ɛ such that (86 a n,k (τ, z, ɛɛ n τ (τ h s exp( k 2 hh k k 1 z w(h, z, ɛ (ɛ,σ,σ,δ,ω C 6 1 n τ s (ɛ,σ,σ,δ,ω exp( k 2 ττ k k 1 z w(τ, z, ɛ (ɛ,σ,σ,δ,ω C 7 s+k +1 n k 1 z w(τ, z, ɛ (ɛ, σ, σ,δ,ω C 7 I s+k +1 n for all k A 1, all n I k. On the other hand, since the convolution product is commutative, from the binomal formula, we can write ( S z U(τ, z + w(τ, z, ɛ l = ( S z U(τ, z l + (w(τ, z, ɛ l l! + l 1!l 2! ( S z U(τ, z l1 (w(τ, z, ɛ l2 l 1 +l 2 =l,l 1 1,l 2 1 for all l 2. From Proposition 2, 2 and Proposition 5 we get a constant C 8 > (independent of ɛ such that (87 ( z S U(τ, z + w(τ, z, ɛ l (ɛ,σ,σ,δ,ω C 8 l 1 (δ Sl R l + I l + l 1 +l 2 =l,l 1 1,l 2 1 l! l 1!l 2! δsl1 R l1 I l2 = C 8 l 1 (δ S R + I l for all l A 2. From Proposition 4 we get a constant C 9 > (independent of ɛ such that (88 α l (τ, z, ɛ( z S U(τ, z + w(τ, z, ɛ l (ɛ,σ,σ,δ,ω for all l A 2. From (87 and (88, we get that C 9 ( z S U(τ, z + w(τ, z, ɛ l (ɛ,σ,σ,δ,ω (89 α l (τ, z, ɛ( z S U(τ, z + w(τ, z, ɛ l (ɛ,σ,σ,δ,ω C 8 C 9 l 1 (δ S R + I l for all l A 2. Now, we choose δ, R, I > such that (9 s+k+1 n (C 5 δ S k1 R + C 7 I + C 8 C 9 l 1 (δ S R + I l R n I k l A 2 k A 1 for all ɛ E. From the inequalities (85, (86 and (89, we deduce that A ɛ (U(τ, z (ɛ,σ,σ,δ,ω R

20 2 for all ɛ E. We prove now the estimates (83. Let R > and let U 1, U 2 B(, R. Under the assumptions (74, from Proposition 4 and Corollary 1, 1 we get a constant C 1 > (independent of ɛ with (91 a n,k (τ, z, ɛɛ n τ (τ h s exp( k 2 hh k k 1 S z (U 1 (h, z U 2 (h, zdh (ɛ,σ,σ,δ,ω C 1 s+k +1 n δ S k 1 U 1 (τ, z U 2 (τ, z (ɛ,σ,σ,δ,ω for all k A 1, all n I k. As in the part a, we can write from the binomial formula (92 ( S z U 1 (τ, z + w(τ, z, ɛ l ( z S U 2 (τ, z + w(τ, z, ɛ l = ( S + z l 1 +l 2 =l,l 1 1,l 2 1 U 1 (τ, z l ( z S U 2 (τ, z l l! l 1!l 2! (( S z for all l 2. On the other hand, we have that (93 ( S z U 1 (τ, z 2 ( z S U 2 (τ, z 2 and, for all l 1 3, we can write (94 ( S z = ( S z U 1 (τ, z S z U 1 (τ, z l1 ( z S U 2 (τ, z l1 = ( z S ( ( S z U 1 (τ, z l1 ( z S U 2 (τ, z l1 (w(τ, z, ɛ l2 U 2 (τ, z ( S z U 1 (τ, z S z U 2 (τ, z U 1 (τ, z l1 1 + ( z S U 2 (τ, z l1 1 l k=1 ( S z U 2 (τ, z k ( S z U 1 (τ, z + z S U 2 (τ, z U 1 (τ, z l1 k 1. Using (93 and (94, from Propositions 2, 2 and 5 we get a constant C 11 > (independent of ɛ such that (95 ( S z U 1 (τ, z l1 ( z S U 2 (τ, z l1 (ɛ,σ,σ,δ,ω (C 11 l1 1 δ Sl1 R l1 1 U 1 (τ, z U 2 (τ, z (ɛ,σ,σ,δ,ω for all l 1 A 2. From (92, (95, we get a constant C 12 > (independent of ɛ such that (96 ( S z U 1 (τ, z + w(τ, z, ɛ l ( z S U 2 (τ, z + w(τ, z, ɛ l (ɛ,σ,σ,δ,ω C 12 l 1 (δ Sl R l 1 + l 1 +l 2 =l,l 1 1,l 2 1 l! l 1!l 2! δsl1 R l1 1 I l2 U 1 (τ, z U 2 (τ, z (ɛ,σ,σ,δ,ω = C 12 l 1 R 1 ((δ S R + I l I l U 1 (τ, z U 2 (τ, z (ɛ,σ,σ,δ,ω for all l A 2. From Proposition 4 we get a constant C 13 > (independent of ɛ such that (97 α l (τ, z, ɛ(( z S U 1 (τ, z + w(τ, z, ɛ l ( z S U 2 (τ, z + w(τ, z, ɛ l (ɛ,σ,σ,δ,ω C 13 ( z S U 1 (τ, z + w(τ, z, ɛ l ( z S U 2 (τ, z + w(τ, z, ɛ l (ɛ,σ,σ,δ,ω

21 21 for all l A 2. Gathering (96 and (97, we get that (98 α l (τ, z, ɛ(( S z U 1 (τ, z + w(τ, z, ɛ l ( z S U 2 (τ, z + w(τ, z, ɛ l (ɛ,σ,σ,δ,ω for all l A 2. Now, we choose δ, R, I > such that (99 k A 1 C 12 C 13 l 1 R 1 ((δ S R + I l I l U 1 (τ, z U 2 (τ, z (ɛ,σ,σ,δ,ω C 1 s+k+1 n δ S k1 + C 12 C 13 l 1 R 1 ((δ S R + I l I l 1 2 n I k l A 2 for all ɛ E. From the inequalities (91, (98, we deduce that A ɛ (U 1 (τ, z A ɛ (U 2 (τ, z (ɛ,σ,σ,δ,ω 1 2 U 1(τ, z U 2 (τ, z (ɛ,σ,σ,δ,ω, for all ɛ E. Finally, we choose δ, R, I > is such a way that the conditions (9 and (99 hold simultaneously. This yields Lemma 8. Now, let the assumptions (74, (75 hold. We choose the constants I, R, δ as in the lemma 8. Assume that S 1 h δ j V j+h (τ, ɛ,ɛ, σ, σ,ω j! I, j= for all h S 1, for all ɛ E. From Lemma 8 and the classical contractive mapping theorem on complete metric spaces, we deduce that the map A ɛ has a unique fixed point (called U(τ, z, ɛ in the closed ball B(, R SG(ɛ, σ, σ, δ, Ω, for all ɛ in E, which means that A ɛ (U(τ, z, ɛ = U(τ, z, ɛ with U (ɛ,σ,σ,δ,ω R. Finally, we get that the function V (τ, z, ɛ = S z U(τ, z, ɛ + w(τ, z, ɛ satisfies the Cauchy problem (73, (77, for all τ S d D(, r, all z D(, δ, all ɛ E. Moreover, from Proposition 2, 2 we deduce that V (τ, z, ɛ (ɛ,σ,σ,δ,ω δ S R + I, for all ɛ E. 3 Analytic solutions in a complex parameter of a singularly perturbed Cauchy problem Definition 2 Let V (τ, ɛ be a holomorphic function on some punctured polydisc Ω τ,ɛ = D(, τ (D(, ɛ \ {} where < τ < a and < ɛ < 1, with a C such that arg(a. We make the assumption that the function τ V (τ, ɛ belongs to SE,ɛ, σ, σ,ω τ,ɛ, for some σ, σ >, all ɛ D(, ɛ \{}. Let U d be an open unbounded sector centered at, with bisecting direction d and with small opening. Let E be an open sector centered at such that E D(, ɛ. We denote by Ω(d, E = (U d D(, τ E.

22 22 We also assume that there exists an open unbounded sector T centered at such that d + arg(t arg(ɛ ( π/2, π/2 for all t T. 1 Let d ( π/2, π/2 with d arg(a. We assume that a / U d. We assume that the function (τ, ɛ V (τ, ɛ can be extended to an analytic function (τ, ɛ V Ud,E(τ, ɛ on Ω(d, E and that the function τ V Ud,E(τ, ɛ belongs to SE,ɛ, σ,,ω(d,e, for all ɛ E. 2 Let d [ π, π/2 (π/2, π] with d arg(a. We assume that a / U d. We assume that the function (τ, ɛ V (τ, ɛ can be extended to an analytic function (τ, ɛ V Ud,E(τ, ɛ on Ω(d, E and that the function τ V Ud,E(τ, ɛ belongs to SE,ɛ, σ, σ,ω(d,e, for all ɛ E. If 1 holds, we say that the set {V, V Ud,E, a, σ, σ, T } is ( σ, admissible and if 2 holds, we say that the set {V, V Ud,E, a, σ, σ, T } is ( σ, σ admissible. Let A 1 (resp. A 2 be a finite subset of N 4 (resp. N. For all k = (s, k, k 1, k 2 A 1, we denote by b k (z, ɛ some bounded holomorphic function on D(, ρ E, for some ρ >. For all l A 2, we denote by c l (z, ɛ some bounded holomorphic function on D(, ρ E. Let a C be a complex number with arg(a, S 1 be an integer and d ( π/2, π/2 with d arg(a. We consider the following singularly perturbed Cauchy problem (1 ɛ t z S X Ud,E(t, z, ɛ + a z S X Ud,E(t, z, ɛ b k (z, ɛ = t s+1 ( k t k 1 z X Ud,E(t + k 2 ɛ, z, ɛ + c l (z, ɛxu l d,e(t, z, ɛ k=(s,k,k 1,k 2 A 1 l A 2 for given initial data (11 ( j zx Ud,E(t,, ɛ = ϕ j,ud,e(t, ɛ, j S 1, where the functions ϕ j,ud,e(t, ɛ are constructed in the following manner. For all j S 1, let {V j, V j,ud,e, a, σ, σ, T } be a ( σ, admissible set, then we consider the function ϕ j,ud,e(t, ɛ = V j,ud,e(τ, ɛ exp( tτ L γ ɛ dτ where L γ = R + e iγ U d {} is a halfline where γ depends on t and ɛ in such a way that there exists δ 1 > with cos(γ + arg(t arg(ɛ δ 1 for all t T with t > σ/δ 1 and ɛ E. By construction, the function ϕ j,ud,e(t, ɛ is holomorphic and bounded on (T { t > σ/δ 1 } E. Proposition 7 We make the following assumptions. There exist real numbers σ > σ > such that, for all k A 1, we have (12 S k 1 + bk + k 2b σ, S > k 1, and for all l A 2, we have (13 l 2.

23 23 Then, there exist constants I > and δ > (independent of ɛ such that if we assume that (14 S 1 h j= δ j S 1 h V j+h (τ, ɛ,ɛ, σ,,ωτ,ɛ j! I, δ j V j+h,ud,e(τ, ɛ,ɛ, σ,,ω(d,e j! I, j= for all h S 1, for all ɛ E, the Cauchy problem (1, (11 has a solution (t, z, ɛ X Ud,E(t, z, ɛ which is holomorphic and bounded on a domain (t, z, ɛ (T { t > σζ(b/δ 1 } D(, δ/2 E for some σ > σ. The function X Ud,E(t, z, ɛ can be written as a Laplace transform (15 X Ud,E(t, z, ɛ = V Ud,E(τ, z, ɛ exp( tτ L γ ɛ dτ where V Ud,E(τ, z, ɛ is a holomorphic function on the domain (U d D(, τ D(, δ/2 E and satisfies the following estimates: there exists a constant C Ω(d,E > (independent of ɛ such that (16 V Ud,E(τ, z, ɛ C Ω(d,E (1 + τ exp( σζ(b τ for all (τ, z, ɛ (U d D(, τ D(, δ/2 E. Moreover, the function V Ud,E(τ, z, ɛ is the analytic continuation of a function V (τ, z, ɛ which is holomorphic on a punctured polydisc D(, τ D(, δ/2 (D(, ɛ \ {} and fulfills the next estimates: there exists a constant C Ωτ,ɛ > (independent of ɛ such that (17 V (τ, z, ɛ C Ωτ,ɛ (1 + τ exp( σζ(b τ for all τ D(, τ, all z D(, δ/2 and all ɛ D(, ɛ \ {}. Proof We consider the following Cauchy problem (18 z S V (τ, z, ɛ = ( 1k b k (z, ɛ (a τɛ 1+s+k s! k=(s,k,k 1,k 2 A 1 for the given initial data τ (τ h s exp( k 2 hh k k 1 z V (h, z, ɛdh (19 ( j zv (τ,, ɛ = V j (τ, ɛ, j S 1. + c l (z, ɛ a τ V l (τ, z, ɛ l A 2 From the assumptions (12, (13 together with (14 we deduce that the conditions (78, (79 and (8 from Proposition 6, 2 are fulfilled for the problem (18, (19. We deduce that the problem (18, (19 has a unique solution V (τ, z, ɛ that belongs to the space SG(ɛ, σ,, δ, Ω τ,ɛ, for some σ > σ. In particular, V (τ, z, ɛ is holomorphic on the punctured polydisc D(, τ D(, δ/2 (D(, ɛ \ {} and using Proposition 1, it satisfies also (17. In the second step of the proof, we show that the function V (τ, z, ɛ can be analytically continued to a function V Ud,E(τ, z, ɛ on (U d D(, τ D(, δ/2 E which satisfies (16.

24 24 Indeed, by construction, the function V (τ, z, ɛ solves also the problem (11 z S V (τ, z, ɛ = ( 1k b k (z, ɛ (a τɛ 1+s+k s! k=(s,k,k 1,k 2 A 1 with the given initial conditions τ (τ h s exp( k 2 hh k k 1 z V (h, z, ɛdh (111 ( j zv (τ,, ɛ = V j,ud,e(τ, ɛ, j S 1, + c l (z, ɛ a τ V l (τ, z, ɛ l A 2 for all τ D(, τ, z D(, δ/2 and ɛ E. Gathering the assumptions (12, (13 and (14 we deduce that the conditions (78, (79 and (8 from Proposition 6, 2 are fulfilled for the problem (11, (111. We get that the problem (11, (111 has a unique solution V Ud,E(τ, z, ɛ that belongs to the space SG(ɛ, σ,, δ, Ω(d, E for some σ > σ. In particular, V Ud,E(τ, z, ɛ defines a holomorphic function on (U d D(, τ D(, δ/2 E, coincides with V (τ, z, ɛ on D(, τ D(, δ/2 E and fills (16 due to Proposition 1. In the last part of the proof, it remains to show that the Laplace transform X Ud,E(t, z, ɛ = V Ud,E(τ, z, ɛ exp( tτ L γ ɛ dτ satisfies the problem (1, (11 on the domain (T { t > σζ(b/δ 1 } D(, δ/2 E. This is a consequence of the classical properties of the Laplace transform that we recall in the next lemma, see [2], [9] for references. Lemma 9 Let m be an integer. Let w 1 (τ, w 2 (τ be holomorphic functions on the unbounded sector U d such that there exist C, K > with for all τ U d. We denote w j (τ C exp(k τ, j = 1, 2 w 1 w 2 (τ = τ w 1 (τ sw 2 (sds their convolution product on U d. We denote by D an unbounded sector centered at for which there exists δ 1 > with d + arg(t ( π/2, π/2, cos(d + arg(t δ 1, for all t D. Then the following identities hold for the Laplace transforms τ m exp( tτdτ = m! L d t m+1, t ( w 1 (τ exp( tτdτ = ( τw 1 (τ exp( tτdτ, L d L d w 1 w 2 (τ exp( tτdτ = ( w 1 (τ exp( tτdτ( w 2 (τ exp( tτdτ L d L d L d where L d = R + e id U d {}, for all t D { t > K/δ 1 }.

25 25 4 Construction of a Banach valued cocyle We keep the notations of Section 3. We recall the definition of a good covering. Definition 3 For all i ν 1, we consider open sectors E i centered at, with radius ɛ such that E i E i+1, for all i ν 1 (with the convention that E ν = E, which are three by three disjoint and such that ν 1 i= E i = U \ {}, where U is some neighborhood of in C. Such a set of sectors {E i } i ν 1 is called a good covering in C. Let {E i } i ν 1 be a good covering in C. For all j S 1, for all i ν 1, we consider directions d i R with d i arg(a, i ν 1, for some a C and a family of sets {V j, V j,udi,e i, a, σ, σ, T } which are ( σ, admissible when d i ( π/2, π/2 and ( σ, σ admissible when d i [ π, π/2 (π/2, π]. We make the assumption that there exists a least one integer i ν 1 such that d i, d i +1 [ π, π/2 (π/2, π] with U di U. di +1 For d i ( π/2, π/2, we consider the Laplace transform (112 ϕ j,udi,e i (t, ɛ = V j,udi,e i (τ, ɛ exp( tτ L γi ɛ dτ where L γi = R + e 1γ i U di {} is a halfline where γ i may depend on t and ɛ in such a way that there exists δ 1 > with cos(γ i + arg(t arg(ɛ δ 1 for all t T with t > σ/δ 1 and ɛ E i. We choose a real number σ > σ and a real number θ i such that arg(t arg(ɛ + θ i π for all t T, all ɛ E i. The function log(z denotes also the principal branch of the logarithm of z which is holomorphic on {z C /arg(z π}. For d i [ π, π/2 (π/2, π], we choose γ i with e 1γ i U di that may depend on t and ɛ in such a way that there exists δ 1 > with (113 cos(γ i + arg(t arg(ɛ δ 1 for all t T, all ɛ E i. Due to the formula arg(log( t ɛ e 1θ i = arctan( arg(t arg(ɛ + θ i log t ɛ for all t T with t > µ T provided that µ T > is large enough, all ɛ E i, we notice that e 1γ i log( t ɛ e 1θ i U di for all ɛ E i, all t T for t > µ T. We consider the truncated Laplace transform (introduced in [18] (114 ϕ j,udi,e i,γ i,σ,θ i (t, ɛ = e 1γi log( t ɛ e 1θ i/(σ ζ(b V j,udi,e i (τ, ɛ exp( tτ ɛ dτ where the integration is made along the segment [, e 1γ i log( t ɛ e 1θ i /(σ ζ(b], for all t T, all ɛ E i.

26 26 The following theorem is crucial in order to establish our main result (Theorem 2 that will be stated in the next section. In this theorem we construct by means of Laplace transforms and truncated Laplace transforms a family of bounded holomorphic functions X Udi,E i (t, z, ɛ on products (T { t > h} D(, δ/2 E i, for h > large enough and δ > small enough, for all i ν 1, whose differences i (ɛ = X Udi+1,E i+1 (t, z, ɛ X Udi,E i (t, z, ɛ satisfy exponential and super-exponential flatness estimates on the intersections E i+1 E i. The sequence of functions i (ɛ, i ν 1, viewed as functions from E i into the Banach space E of bounded holomorphic functions on (T { t > h} D(, δ/2 equipped with the supremum norm will be called a (Banach valued cocycle according to the terminology of [26]. Theorem 1 We make the following assumptions. There exists σ > σ such that, for all k A 1, we have (115 S k 1 + bk + k 2b σ, S > k 1, and for all l A 2, we have (116 l 2. Then, there exist constants I > and δ > (independent of ɛ such that if we assume that for all d i ( π/2, π/2, (117 S 1 h j= δ j S 1 h V j+h (τ, ɛ,ɛ, σ,,ωτ,ɛ j! I, j= δ j V j+h,udi,e i (τ, ɛ,ɛ, σ,,ω(di,e i j! I, for all h S 1, for all ɛ E i and for all d i [ π, π/2 (π/2, π], (118 S 1 h j= δ j S 1 h V j+h (τ, ɛ,ɛ, σ, σ,ω τ,ɛ j! I, j= δ j V j+h,udi,e i (τ, ɛ,ɛ, σ, σ,ω(d i,e i j! I, for all h S 1, for all ɛ E i, there exists a family of holomorphic and bounded functions X Udi,E i (t, z, ɛ, i ν 1, on the product (T { t > µ T } D(, δ/2 E i with the following properties: 1 If d i ( π/2, π/2, then X Udi,E i (t, z, ɛ is the solution of the equation (1 with initial data (119 ( j zx Udi,E i (t,, ɛ = ϕ j,udi,e i (t, ɛ, j S 1. 2 If d i [ π, π/2 (π/2, π], then X Udi,E i (t, z, ɛ satisfies (12 ( j zx Udi,E i (t,, ɛ = ϕ j,udi,e i,γ i,σ,θ i (t, ɛ, j S 1. 3 For all i ν 1, there exist constants h µ T, K i, M i > such that (121 sup X Udi+1,E i+1 (t, z, ɛ X Udi,E i (t, z, ɛ K i e M i t T { t >h},z D(,δ/2

27 27 for all ɛ E i E i+1 where by convention E ν = E, d ν = d. 4 If d i+1, d i [ π, π/2 (π/2, π] and if moreover U di U di+1, then there exist constants h µ T, K i, M i > and L i > 1 with (122 sup X Udi+1,E i+1 (t, z, ɛ X Udi,E i (t, z, ɛ K i exp( M i t T { t >h},z D(,δ/2 log L i for all ɛ E i E i+1. Proof If d i ( π/2, π/2, we consider the solution X Udi,E i (t, z, ɛ of the equation (1 with initial data ( j zx Udi,E i (t,, ɛ = ϕ j,udi,e i (t, ɛ, j S 1. constructed as in Proposition 7. Recall that the function X Udi,E i (t, z, ɛ can be written as a Laplace transform (123 X Udi,E i (t, z, ɛ = V Udi,E i (τ, z, ɛ exp( tτ L γi ɛ dτ where V Udi,E i (τ, z, ɛ is a holomorphic function on the domain (U di D(, τ D(, δ/2 E i and satisfies the following estimates: there exists a constant C Ω(di,E i > (independent of ɛ such that (124 V Udi,E i (τ, z, ɛ C Ω(di,E i (1 + τ exp( σζ(b τ for all (τ, z, ɛ (U di D(, τ D(, δ/2 E i, for some σ > σ. Moreover, the function V Udi,E i (τ, z, ɛ is the analytic continuation of a function V (τ, z, ɛ which is holomorphic on a punctured polydisc D(, τ D(, δ/2 (D(, ɛ \ {} and fulfills the next estimates: there exists a constant C Ωτ,ɛ > (independent of ɛ such that (125 V (τ, z, ɛ C Ωτ,ɛ (1 + τ exp( σζ(b τ for all τ D(, τ, all z D(, δ/2 and all ɛ D(, ɛ \ {}. If d i [ π, π/2 (π/2, π], we consider the following Cauchy problem (126 S z V Udi,E i (τ, z, ɛ = ( 1k b k (z, ɛ (a τɛ 1+s+k s! k=(s,k,k 1,k 2 A 1 with the given initial conditions τ (τ h s exp( k 2 hh k k 1 z V Udi,E i (h, z, ɛdh (127 ( j zv Udi,E i (τ,, ɛ = V j,udi,e i (τ, ɛ, j S 1, + c l (z, ɛ a τ V U l di,e i (τ, z, ɛ l A 2 for all τ D(, τ U di, z D(, δ/2 and ɛ E i. Gathering the assumptions (115, (116 and (118 we deduce that the conditions (74, (75 and (76 from Proposition 6, 1 are fulfilled for the problem (126, (127. We get that the problem (126, (127 has a unique solution