On singular solutions to PDEs with turning point involving a quadratic nonlinearity

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1 On singular solutions to PDEs with turning point involving a quadratic nonlinearity Stéphane Malek University of Lille, Laboratoire Paul Painlevé, Villeneuve d Ascq cedex, France, Stephane.Malek@math.univ-lille.fr November, Abstract We study a singularly perturbed PDE with quadratic nonlinearity depending on a complex perturbation parameter ɛ. The problem involves an irregular singularity in time, as in a previous work of the author and A. Lastra [2], but possess also, as a new feature, a turning point at the origin in C. We construct a family of sectorial meromorphic solutions obtained as a small perturbation in ɛ of a slow curve of the equation in some time scale. We show that the non singular part of these solutions share a common formal power series that generally diverge in ɛ as Gevrey asymptotic expansion of some order depending on data both arising from the turning point and from the irregular singular point of the main problem. Key words: asymptotic expansion, Borel-Laplace transform, Fourier transform, Cauchy problem, formal power series, nonlinear integro-differential equation, nonlinear partial differential equation, singular perturbation. 2 MSC: 35C, 35C2. Introduction In this work, we consider a family of nonlinear singularly perturbed equations of the form Q z P t, ɛut, z, ɛ P 2 t, ɛu 2 t, z, ɛ = ft, z, ɛ P 3 t, ɛ, t, z ut, z, ɛ where Q, P, P 2, P 3 are polynomials with complex coefficients and f is an analytic function in the vicinity of the origin w.r.t t and ɛ in C and holomorphic w.r.t z on an horizontal strip in C of the form H β = {z C/ Imz < β} for some β >. Here we consider the case when P, ɛ vanishes identically near. The point t = is know to be called a turning point in that situation, see [27] and [9] for a more detailed description of this terminology in the linear and nonlinear settings. Let us recall the definition of the valuation val t f of an analytic function near t = as the smallest integer k with the factorization ft = t k ft for an analytic function f near t = with f. The most interesting case examined in this work is when the valuation val t P of P t, ɛ w.r.t t is larger than the valuation val t P 2 or val t ft, z, ɛ since the problem cannot be reduced to the case P, by dividing the equation by a suitable power of t and ɛ, see Remark 4.

2 2 In our previous study [2], we already have considered a similar problem which corresponds to the situation when P, for our equation. Namely, we focused on the following problem 2 Q z t yt, z, ɛ = Q z yt, z, ɛq 2 z yt, z, ɛ Ht, ɛ, t, z yt, z, ɛ ft, z, ɛ for given vanishing initial data y, z, ɛ, where Q, Q, Q 2, H are polynomials with complex coefficients and ft, z, ɛ is a forcing term constructed as above. Under appropriate assumptions on the shape of 2, we established the existence of a family of actual bounded holomorphic solutions y p t, z, ɛ, p ς, for some integer ς 2, defined on domains T H β E p, for some fixed bounded sector T with vertex at and E = {E p } pς a set of bounded sectors whose union covers a full neighborhood of in C. These solutions are obtained by means of Laplace and inverse Fourier transforms. On each sector E p, they share w.r.t ɛ a common asymptotic expansion ŷt, z, ɛ = n y nt, zɛ n which defines a formal series with bounded holomorphic coefficients on T H β. Moreover, this asymptotic expansion is shown to be of Gevrey order at most /k that appears in the highest order term of the operator H which is of irregular type in the sense of [23] outlined as ɛ δd k t δd k δ D t R D z, for some integer δ D 2 and a polynomial R D with complex coefficients. Conjointly, since the aperture of the sectors E p can be chosen slightly larger than π/k, the functions ɛ y p t, z, ɛ can be viewed as k sums of the formal series ŷ as defined in [2]. In this work, our goal is to achieve a similar statement namely the existence of sectorial holomorphic solutions and asymptotic expansions as ɛ tends to. However, the main contrast with the problem 2 is that, due to the presence of the turning point, our solutions are no longer bounded in the vicinity of the origin, being meromorphic in both time variable t and parameter ɛ. Namely, we build a set of actual meromorphic solutions u dp t, z, ɛ to the problem of the form u dp t, z, ɛ = ɛ β U ɛ α t ɛ α t γ v dp t, z, ɛ where α >, β are some rational numbers, γ is an integer and U T is a non identically vanishing root of a second order algebraic equation with polynomial coefficients related to the polynomials P, P 2, see 37 and where v dp t, z, ɛ is a bounded holomorphic function on products T H β E p similar to the ones mentioned above, which can be expressed as a Laplace transform of some order and Fourier inverse transform v dp t, z, ɛ = 2π /2 L dp ω dp u u, m, ɛ exp ɛ χα t e izm du u dm along some halfline L dp = R e idp, for some positive rational number χ >, where ω dp u, m, ɛ represents a function with at most exponential growth of order on a sector containing L dp w.r.t u, with exponential decay w.r.t m on R and with analytic dependence on ɛ near see Theorem. Furthermore, we show that these functions v dp t, z, ɛ own w.r.t ɛ a common asymptotic expansion ˆvt, z, ɛ = n v nt, zɛ n which represents a formal series with bounded holomorphic coefficients on T H β. We specify also the nature of this asymptotic expansion which turns out to be of Gevrey order at most χα. Besides, since the aperture of the sectors E p may π be selected slightly larger than χα, the functions vdp can be identified as χ α sums of the formal series ˆv Theorem 2. By construction, the integer shows up in the highest order term of the operator P 3 which is of irregular type of the form ɛ Dt δ Dk δ D t R D z, with k = val t P, for some integers D, δ D 2 and a polynomial R D with complex coefficients. The rational number χ is built with the help of the integers D, δ D, k, and

3 3 the rational numbers α, β, see 52. According to the fact that α, β are mainly related to constraints assumed on the polynomials P and P 2 see 34, 35, we observe that the Gevrey order χα of the asymptotic expansion involves informations both coming from the highest irregular term and from the two polynomials P, P 2 that shape the turning point at t =, whereas in our previous contribution [2], the Gevrey order was exclusively stemming from the irregular singularity at t =. The kind of equations with quadratic nonlinearity we investigate in this work are strongly related to singularly perturbed ODEs which are nonsingular at the origin of the form ɛ σ dy/dt = F t, y, a, ɛ for some analytic functions F, small complex parameter ɛ and a complex additional parameter a, described in the seminal joint paper by M. Canalis-Durand, J.P. Ramis, R. Schäfke, Y. Sibuya, see [4], where they study asymptotic properties of actual overstable solutions near a slow curve φ t meaning that F t, φ t, a, in the case when the Jacobian y F t, φ t, a, is not invertible at t =. The main notable difference is that we assume the origin to be at the same time a turning point and an irregular singularity. More precisely, with the rescaling map t, ɛ T = ɛt, ɛ the transformed equation 32 possess a rational slow curve U T and T = remains a turning point and an irregular singularity for this new equation. The construction of the distinguished solution performed in Section 4 and the parametric Borel/Laplace summable character of these solutions shown in Section 7 are also intimately linked to recent developments of exact WKB analysis of formal and analytic solutions to second order linear ODEs of Schrödinger type. Namely, let 3 ɛ 2 ψ t, ɛ = Qtψt, ɛ be a singularly perturbed ODE where ɛ is a small complex parameter and Qt is some polynomial with complex coefficients. WKB solutions of 3 are known as special solutions that are described t as an exponential ˆψt, ɛ = exp t Ŝs, ɛds where the expression Ŝt, ɛ satisfies a so-called Riccati equation ɛ 2 Ŝ t, ɛ ɛ 2 Ŝ 2 t, ɛ = Qt. This last equation possess formal power series solutions Ŝt, ɛ = S t/ɛ n S ntɛ n where S t satisfies the quadratic equation S 2 t = Qt. Once S t = ± Qt is fixed, we get two formal solutions Ŝ±t, ɛ = S t/ɛ ˆT ± t, ɛ, where ˆT ± t, ɛ C[[ɛ]] for any t U = {t C/Qt }. Notice that ˆT ± t, ɛ solves the first order Riccati equation ɛ ˆT ±t, ɛ 2S t ˆT ± t, ɛ ɛ ˆT 2 ±t, ɛ S t = with turning points at the roots of Qt. Our main PDE resembles this last one provided that S t is a polynomial and with the significant distinction that our equation only involves differential operators with irregular singularity at t =. An essential feature of the theory is that the formal series ˆT ± t, ɛ are summable in suitable directions d R w.r.t ɛ that are related to the function t t S sds for any fixed t U. Different proofs of this fact can be found in [25], [], [8], [5]. Our second main statement Theorem 2 can be considered as a similar contribution for some higher order PDEs of this latter result. Furthermore, in our study we are also able to described the behaviour of our specific solutions near t, ɛ =,. For more recent and advanced works related to WKB analysis and local/global studies of solutions to linear ODEs near turning points, we refer to contributions related to the D Schrödinger equation with simple poles [8], with merging pairs of simple poles and turning points [5], with merging triplet of poles and turning points [6], [7] and for analytic continuation properties of the Borel transform resurgence of WKB expansions in the problem of confluence of two

4 4 simple turning points we quote [6]. Concerning the structure of singular formal solutions to singularly perturbed linear systems of ODEs with turning points we point out [24] solving an old question of W. Wasow. We mention also preeminent studies on WKB analysis for higher order differential equations which reveal new Stokes phenomena giving rise to so-called virtual turning points, [2], []. In the framework of linear PDEs, normal forms for completely integrable systems near a degenerate point where two turning points coalesce have been obtained in [], which is a first step toward the so-called Dubrovin conjecture which concerns the question of universal behaviour of generic solutions near gradient catastrophe of singularly Hamiltonian perturbations of first order hyperbolic equations, see [7]. We mention also that sectorial analytic transformations to normal forms have been obtained for systems of singularly perturbed ODEs near a turning point with multiplicity using the recent approach of composite asymptotic expansions developped in [9], see [4]. The paper is organized as follows. In Section 2, we recall the definition introduced in the work [2] of some weighted Banach spaces of continuous functions with exponential growth on unbounded sectors in C and with exponential decay on R. We analyze the continuity of specific multiplication and linear/nonlinear convolution operators acting on these spaces. In Section 3, we remind the reader basic statements concerning m k Borel-Laplace transforms, a version of the classical Borel-Laplace maps already used in previous works [9], [2], [2] and Fourier transforms acting on exponentially flat functions. In Section 4, we display our main problems and explain the leading strategy in order to solve them. It consists in four operations. In a first step, we restrict our inquiry for the sets of solutions to time rescaled function spaces, see 3. Then, we consider candidates for solutions to the resulting auxiliary problem 32 that are small perturbations of a so-called slow curve which solves a second order algebraic equation and which may be singular at the origin in C. In a third step, we search again for time rescaled functions solutions for the associated problem 5 solved by the small perturbation of the slow curve, see 5. In the last step, we write down the convolution problem 58 solved by a suitable m Borel transform of a formal solution to the attached problem 53. In Section 5, we solve the main convolution problem 58 within the Banach spaces described in Section 2 using some fixed point theorem argument. In Section 6, we provide a set of actual meromorphic solutions to our initial equation 29 by executing backwards the operations described in Section 4. In particular, we show that our singular functions actually solve the problem 2 which is a factorized part of equation 29 with a more restrictive forcing term. Furthermore, the difference of any two neighboring solutions tends to as ɛ tends to faster than a function with exponential decay of order χ α. In Section 7, we show the existence of a common asymptotic expansion of Gevrey order χα for the non singular part of these solutions of 29, 2 based on the flatness estimates obtained in Section 6 using a theorem by Ramis and Sibuya. 2 Banach spaces with exponential growth and exponential decay We denote D, ρ the open disc centered at with radius ρ > in C and by D, ρ its closure. Let S d be an open unbounded sector in 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.

5 5 We first give definitions of Banach spaces which already appear in our previous work [2]. Definition Let β > and µ > be real numbers. functions h : R C such that We denote E β,µ the vector space of hm β,µ = sup m µ expβ m hm m R is finite. The space E β,µ endowed with the norm. β,µ becomes a Banach space. As a direct consequence of Proposition 5 from [2], we notice that Proposition The Banach space E β,µ,. β,µ is a Banach algebra for the convolution product f gm = fm m gm dm Namely, there exists a constant C > depending on µ such that for all f, g E β,µ. f gm β,µ C fm β,µ gm β,µ Definition 2 Let ν, ρ > and β >, µ > be real numbers. Let and χ, α be integers. Let ɛ E. We denote Fν,β,µ,χ,α,,ɛ d the vector space of continuous functions τ, m hτ, m on D, ρ S d R, which are holomorphic w.r.t τ on D, ρ S d and such that hτ, m ν,β,µ,χ,α,,ɛ = sup τ D,ρ S d,m R m µ expβ m τ ɛ χα 2 τ ɛ χα τ exp ν ɛ χα hτ, m is finite. One can check that the normed space F d ν,β,µ,χ,α,,ɛ,. ν,β,µ,χ,α,,ɛ is a Banach space. Throughout the whole section, we keep the notations of Definitions and 2. In the next lemma, we check that some parameter depending functions with polynomial growth w.r.t the variable τ and exponential decay w.r.t the variable m, that will appear later on in our study Section 5, belong to the Banach spaces described above. Lemma Let γ, γ 2 be integers. Let RX be a polynomial that belongs to C[X] such that Rim for all m R. We take a function Bm located in E β,µ and we consider a continuous function a γ,τ, m on D, ρ S d R, holomorphic w.r.t τ on D, ρ S d such that a γ,τ, m τ γ Rim for all τ D, ρ S d, all m R. Then, the function ɛ χγ 2 τ γ 2 Bmaγ,τ, m belongs to Fν,β,µ,χ,α,,ɛ d. Moreover, there exists a constant C > depending on and γ 2 such that 4 ɛ χγ 2 τ γ 2 Bmaγ,τ, m ν,β,µ,χ,α,,ɛ C Bm β,µ inf m R Rim ɛ γ 2α for all ɛ E.

6 6 Proof By definition of the norm and bearing in mind the constraint on the polynomial RX, we can write ɛ χγ 2 τ γ 2 Bmaγ,τ, m ν,β,µ,χ,α,,ɛ = sup m µ expβ m Bm τ D,ρ S d,m R τ 2 ɛ χα τ τ exp ν ɛ ɛ χα ɛ χγ 2 τ ɛ χα γ 2 ɛ γ 2χα τ χα ɛ ɛ χα γ χα Rim Bm β,µ inf m R Rim ɛ γ 2α x 2 sup x γ e νx 2 x x x ɛ χα γ Bm β,µ inf m R Rim ɛ γ 2α sup x which yields the lemma since an exponential grows faster than any polynomial. x 2 x γ 2 e νx x The next proposition provides norms estimates for some linear convolution operators acting on the Banach spaces introduced above. These bounds are more accurate than the one supplied in Proposition 2 from [2]. These new estimates will be essential in Section 5 in order to solve the problem 58. The improvements are due to the use of thorough upper bounds estimates of a generalized Mittag-Leffler function described in the proofs of Propositions and 5 from [2]. Proposition 2 Let γ j, j 3, be real numbers with γ. Let RX, R D X be polynomials with complex coefficients such that deg R deg R D and with R D im for all m R. We consider a continuous function a γ,τ, m on D, ρ S d R, holomorphic w.r.t τ on D, ρ S d such that a γ,τ, m τ γ RD im for all τ D, ρ S d, all m R. We make the next assumptions 5 γ 3 >, γ 2 γ 3 2, γ 2 >. If γ 3, then there exists a constant C 2 > depending on ν,, γ 2, γ 3 and RX, R D X such that τ 6 ɛ γ a γ,τ, m Rimτ τ s γ 2 s γ 3 fs /, mds ν,β,µ,χ,α,,ɛ C 2 ɛ χαγ 2γ 3 2 γ fτ, m ν,β,µ,χ,α,,ɛ for all fτ, m F d ν,β,µ,χ,α,,ɛ. 2 If γ 3 > and γ γ 3, then there exists a constant C 2 > depending on ν,, γ, γ 2, γ 3 and RX, R D X such that τ 7 ɛ γ a γ,τ, m Rimτ τ s γ 2 s γ 3 fs /, mds ν,β,µ,χ,α,,ɛ for all fτ, m F d ν,β,µ,χ,α,,ɛ. C 2 ɛ χαγ 2γ 3 2 γ χαγ fτ, m ν,β,µ,χ,α,,ɛ

7 7 Proof By definition of the norm, we can write τ 8 A = ɛ γ a γ,τ, m Rimτ τ s γ 2 s γ 3 fs /, mds ν,β,µ,χ,α,,ɛ = sup τ D,ρ S d,m R where τ τ m µ expβ m τ ɛ χα 2 τ ɛ χα s 2 { m µ ɛ expβ m χα2 s / ɛ χα Aτ, s, m, ɛ = m µ expβ m τ exp ν ɛ χα ɛ γ Rim τ γ R D im s exp ν ɛ χα fs/, m}aτ, s, m, ɛds expν s ɛ χα s / s 2 ɛ χα2 ɛ χα τ s γ 2 s γ 3. Again by the definition of the norm of f and by the constraints on the polynomials R, R D, we deduce that Rim 9 A C 2. ɛ sup m R R D im fτ, m ν,β,µ,χ,α,,ɛ where C 2. ɛ = sup τ D,ρ S d τ ɛ χα 2 τ ɛ χα τ exp ν ɛ χα ɛ γ τ γ h τ expν τ ɛ χα h/ h 2 ɛ χα τ h γ 2 h γ 3 dh. ɛ χα2 We perform the change of variable h = ɛ χα h inside the integral which is a part of C 2. ɛ that yields C 2. ɛ = sup τ D,ρ S d τ ɛ χα 2 τ ɛ χα τ τ ɛ χα As a result, we obtain the bounds τ exp ν ɛ χα ɛ γ τ γ e νh h 2 h / τ ɛ χα h γ 2 h γ 3 dh ɛ χαγ 2γ 3. C 2. ɛ ɛ χαγ 2γ 3 γ χα x 2 x sup x x / e νx ɛ χα x γ Gx where Gx = x e νh h 2 h γ 3 x h γ 2 dh. We now proceed as in Proposition of [2]. We split the function Gx into two pieces and study them separately. Namely, we decompose Gx = G x G 2 x where G x = x/2 e νh x h 2 h γ 3 x h γ 2 dh e νh, G 2 x = x/2 h 2 h γ 3 x h γ 2 dh.

8 8 We first provide estimates for G x. a Assume that < γ 2 <. We see that x h γ 2 x/2 γ 2 for all h x/2, for x >. Hence, from the first constraint of 5, we get G x x x/2 2 γ 2 e νx/2 h γ 3 dh = x 2 γ 2 e νx/2 x/2 γ 3 for all x. Subsequently, we obtain x 2 x sup x x / e νx ɛ χα x γ G x sup x γ 3 x 2 x / e νx xg x which is finite due to the second assumption of 5. b Assume that γ 2 >. We notice that x h γ 2 x γ 2 for all h x/2, for x >. Therefore, again from the first constraint of 5 we get x/2 G x x γ 2 e νx/2 h γ 3 dh = x γ 2 e νx/2 x/2 γ 3 for all x. Consequently, we obtain x 2 x 2 sup x x / e νx ɛ χα x γ G x sup x which is finite due to the second assumption of 5. In a second step, we study G 2 x. We see that h 2 x/2 2 for all x/2 h x. Hence, 3 G 2 x x 2 2 where x x/2 G 2. x = e νh h γ 3 x h γ 2 dh x e νh h γ 3 x h γ 2 dh γ 3 x 2 x / e νx xg x x 2 2 G 2.x for all x. Taking account of the estimates 8 in [2] which are deduced from the asymptotic behaviour for large x of the generalized Mittag-Leffler function E α,β x = n xn /Γβ nα, for α, β >, we get a constant K 2. > that depends on ν,, γ 2, γ 3 such that 4 G 2. x K 2. x γ 3 e νx for all x, provided the first and last constraints of 5 hold. We consider the first case when γ 3. Bearing in mind 3 and 4, we deduce that x 2 x 5 sup x x / e νx ɛ χα x γ G x 2 2x sup x x/2 2 K 2.x γ 3 which is finite. On the other hand, when x <, we make the change of variable h = xu inside G 2. x and taking 3 into account, we get x 2 x 6 sup x< x / e νx ɛ χα x γ G x 2 2x sup x x< x/2 2 e νx x / x γ 3γ 2 e νxu u γ 3 u γ 2 du

9 9 which is finite provided that the constraints 5 are fulfilled. 2 We examine the second case when γ 3 > and γ γ 3. We use this time the fact that ɛ χα x ɛ χα x for all x and the bounds 4 in order to get x 2 x 7 sup x x / e νx ɛ χα x γ G 2x ɛ χαγ x 2 x γ3 K 2. sup x x/2 2 x γ On the other hand, the bounds on the domain x < have already been treated above owing to 6. Finally, gathering 9,,, 2, 5, 6 and 7 yields the statement of Proposition 2. The forthcoming proposition present norms estimates for some bilinear convolution operators acting on the aforementioned Banach spaces. Proposition 3 There exists a constant C 3 > depending on µ and such that 8 τ τ for all fτ, gτ F d ν,β,µ,χ,α,,ɛ. fτ /, m m g /, m τ ds dm ν,β,µ,χ,α,,ɛ C 3 ɛ χα fτ, m ν,β,µ,χ,α,,ɛ gτ, m ν,β,µ,χ,α,,ɛ Proof We follow the same guidelines as in the proof of Proposition 3 from [2]. By definition of the norm, we can write 9 B = τ τ fτ /, m m g /, m τ ds dm ν,β,µ,χ,α,,ɛ = sup τ D,ρ S d,m R τ τ τ 2 ɛ χα2 τ / ɛ χα m µ expβ m τ ɛ χα 2 s 2 { m µ ɛ expβ m χα2 / ɛ χα where Bτ, s, m, m = exp β m m exp β m m m µ m µ τ ɛ χα τ exp ν ɛ χα { m m µ expβ m m τ exp ν ɛ χα fτ /, m m } exp ν ɛ χα gs /, m } Bτ, s, m, m d dm / τ / ɛ 2χα τ 2 ɛ χα2 s 2 ɛ χα2 expν τ ɛ χα expν ɛ χα τ.

10 By definition of the norms of f and g and according to the triangular inequality m m m m for all m, m R, we deduce that 2 B C 3 ɛ fτ, m ν,β,µ,χ,α,,ɛ gτ, m ν,β,µ,χ,α,,ɛ where C 3 ɛ = τ sup τ D,ρ S d,m R m µ τ ɛ χα 2 τ ɛ χα τ exp ν ɛ χα τ h / τ h / m m µ m µ ɛ 2χα τ h 2 h 2 ɛ χα2 ɛ χα2 We provide upper bounds that can be split in two parts, 2 C 3 ɛ C 3. C 3.2 ɛ where expν τ h ɛ χα expν h ɛ χα τ h h dh dm. 22 C 3. = sup m µ m R m m µ m µ dm is finite under the condition that µ > according to Lemma 4 of [22] and C 3.2 ɛ = sup τ D,ρ S d τ 2 ɛ χα τ τ ɛ χα τ h / τ h / ɛ 2χα τ h 2 ɛ χα2 h 2 ɛ χα2 τ h h dh We carry out the change of variable h = ɛ χα h inside the integral piece of C 3.2 ɛ which yields the bounds 23 C 3.2 ɛ = sup τ D,ρ S d τ ɛ χα τ 2 ɛ χα τ τ ɛ χα τ h / h / ɛ χα dh τ h ɛ 2 h 2 τ hh ɛ χα ɛ χα ɛ χα sup χα x Bx where Bx = x x2 x 2/ x h / x h / x h 2 h 2 x hh dh. A change of variable h = xu in this last expression followed by a partial fraction decomposition allow us to write 24 Bx = x 2 = x2 x 2 4 x 2 u 2 x 2 u 2 du u u 3 2u x 2 u 2 du x2 u u x 2 4 2u x 2 u 2 du u u

11 which acquaints us that Bx is finite provided that and bounded on R w.r.t x. At last, collecting 9, 2, 2, 22, 23 and 24 leads to the statement of Proposition 3. 3 Borel-Laplace and Fourier transforms In this section, we review some basic statements concerning a k Borel summability method of formal power series which is a slightly modified version of the more classical procedure see [2], Section 3.2. This novel version has already been used in works such as [9] and [2] when studying Cauchy problems under the presence of a small perturbation parameter. We remind also the reader the definition of Fourier inverse transform acting on functions with exponential decay. Definition 3 Let k be an integer. Let m k n n be the sequence n m k n = Γ = k t n k e t dt, n. Let E, E be a complex Banach space. We say a formal power series ˆXT = a n T n T E[[T ]] n= is m k summable with respect to T in the direction d [, 2π if the following assertions hold:. There exists ρ > such that the m k Borel transform of ˆX, Bmk ˆX, is absolutely convergent for τ < ρ, where B mk ˆXτ = n= a n Γ n τ n τe[[τ]]. k 2. The series B mk ˆX can be analytically continued in a sector S = {τ C : d argτ < δ} for some δ >. In addition to this, the extension is of exponential growth at most k in S, meaning that there exist C, K > such that B mk ˆXτ Ce K τ k, τ S. E Under these assumptions, the vector valued Laplace transform of B mk ˆX along direction d is defined by L d m k B mk ˆX T = k B mk ˆXue u/t du k L γ u, where L γ is the path parametrized by u [, ue iγ, for some appropriate direction γ depending on T, such that L γ S and coskγ argt > for some >. The function L d m k B mk ˆX is well defined and turns out to be a holomorphic and bounded function in any sector of the form S d,θ,r /k = {T C : T < R /k, d argt < θ/2}, for some π k < θ < π k 2δ and < R < /K. This function is known as the m k sum of the formal power series ˆXT in the direction d.

12 2 The following are some elementary properties concerning the m k sums of formal power series which will be crucial in our procedure. The function L d m k B mk ˆXT admits ˆXT as its Gevrey asymptotic expansion of order /k with respect to T in S d,θ,r /k. More precisely, for every π k < θ < θ, there exist C, M > such that Ld m k B mk ˆXT n a p T p CM n Γ n k T n, E p= for every n 2 and T S d,θ,r/k. Watson s lemma see Proposition p.75 in [3] allows us to affirm that L d m k B mk ˆXT is unique provided that the opening θ is larger than π k. 2 Whenever E is a Banach algebra, the set of holomorphic functions having Gevrey asymptotic expansion of order /k on a sector with values in E turns out to be a differential algebra see Theorem 8, 9 and 2 in [3]. This, and the uniqueness provided by Watson s lemma allow us to obtain some properties on m k summable formal power series in direction d. By we denote the product in the Banach algebra and also the Cauchy product of formal power series with coefficients in E. Let ˆX, ˆX2 T E[[T ]] be m k summable formal power series in direction d. Let q q 2 be integers. Then ˆX ˆX 2, ˆX ˆX 2 and T q q 2 ˆX T, which are elements of T E[[T ]], are m k summable in direction d. Moreover, one has L d m k B mk ˆX T L d m k B mk ˆX 2 T = L d m k B mk ˆX ˆX 2 T, L d m k B mk ˆX T L d m k B mk ˆX 2 T = L d m k B mk ˆX ˆX 2 T, T q q 2 T Ld m k B mk ˆX T = L d m k B mk T q q 2 T ˆX T, for every T S d,θ,r /k. The next proposition is written without proof for it can be found in [2], Proposition 6. Proposition 4 Let ˆft = n f nt n and ĝt = n g nt n that belong to E[[t]], where E, E is a Banach algebra. Let k, m be integers. The following formal identities hold. B mk t k t ˆftτ = kτ k B mk ˆftτ, and B mk t m ˆftτ = τ k Γ m k τ k τ k s m k B mk ˆfts /k ds s τ k B mk ˆft ĝtτ = τ k B mk ˆftτ k s /k B mk ĝts /k τ k ss ds. In the last part of the section, we recall without proofs some properties of the inverse Fourier transform acting on continuous functions with exponential decay on R, see [2], Proposition 7 for more details. Proposition 5 Let f : R R be a continuous function with a constant C > such that fm C exp β m for all m R, for some β >. The inverse Fourier transform of f is defined by the integral representation F fx = 2π /2 fm expixmdm

13 3 for all x R. It turns out that the function F f extends to an analytic function on the horizontal strip 25 H β = {z C/ Imz < β}. Let φm = imfm. Then, we have the commuting relation 26 z F fz = F φz for all z H β. 2 Let f, g E β,µ and let ψm = 2π /2 f gm, the convolution product of f and g, for all m R. From Proposition, we know that ψ E β,µ. Moreover, the next formula 27 F fzf gz = F ψz holds for all z H β. 4 Layout of the main nonlinear PDE and related auxiliary problems Let q, M, Q, D 2 be integers. For all l q, let k l, m l be non negative integers and a l be complex numbers with a such that k l < k l for l {,..., q }. For all l M, we consider non negative integers h l, µ l and complex numbers c l with c such that h l < h l for l {,..., M }. For all l Q, we denote n l and b l non negative integers such that b l < b l for l {,..., Q }. For l D, we set nonnegative integers l, d l and δ l such that δ l < δ l for l {,..., D }. Let QX, R l X C[X], l D, be polynomials which can be factorized as QX = X v QX, Rl X = X v Rl X for some common integer v where QX and R l X are polynomial that satisfy 28 deg Q = deg R D deg R l, Qim, RD im for all m R, all l D. We consider the following nonlinear singularly perturbed PDE 29 Q z q a l ɛ m l t k l ut, z, ɛ c l ɛ µ l t h l u 2 t, z, ɛ l= l= = Q b j zɛ n j t b j j= D ɛ l t d l δ l t R l z ut, z, ɛ The coefficients b j z are constructed as follows. For all j Q, we consider functions m B j m that belong to the Banach space E β,µ for some µ > and β >. We define B j m = im v Bj m where v is the integer introduced above, for j Q. We set 3 b j z = F m B j mz, j Q, where F denotes the Fourier inverse transform defined in Proposition 5. From 26, it turns out by construction that one can write b j z = v z b j z where b j z is the inverse Fourier transform of B j m.

14 4 Remark. The reason why we make these factorizations hypotheses on the polynomials QX, R l X and the functions B j m will be explained later on in the remark 5 of next section and is related to the construction of the Banach spaces in Section 2 and their Fourier inverse transforms. Within this work, we will search for time rescaled solutions of 29 of the form 3 ut, z, ɛ = ɛ β Uɛ α t, z, ɛ where α, β Q are two rational numbers and α >. Then, the expression UT, z, ɛ needs to formally solve the next nonlinear PDE 32 Q z q a l ɛ m lβ αk l T k l UT, z, ɛ c l ɛ µ l2β αh l U 2 T, z, ɛ l= = l= Q b j zɛ n j αb j T b j j= 4. Construction of a distinguished solution D ɛ lαδ l d l β T d l R l z δ l T UT, z, ɛ We make the additional assumption that α, β set above can be chosen in such a way that the next inequalities 33 l αδ l d l β >, n j αb j > for all l D, j Q and 34 m l β αk l =, m j β αk j > for all l s and all s j q, for some integer s q, together with 35 µ l 2β αh l =, µ j 2β αh j > for all l and all j M, for some integer M, hold. Remark 2. In the case q =, k, k, the roots of the polynomial in t P t, ɛ = a ɛ m t k a ɛ m t k all have modulus equal to a /a k k ɛ m m k k except the trivial root. The constraints 34 imply in particular that m m > αk k. As a result, all the nonvanishing roots of P t, ɛ tend to as ɛ tends to and is therefore the only root with order k of P t, ɛ in the vicinity of as ɛ stays near the origin. Let us assume that the expression UT, z, ɛ is allowed to be written as a perturbation series w.r.t ɛ 36 UT, z, ɛ = U T n U n T, zɛ n. where the constant term U T is taken independent of z and not identically equal to. The coefficient U T is called the slow curve of the equation 32 in the terminology of [4].

15 5 In the following, we make the assumption that U T solves the next second order algebraic equation s 37 a l T k l U T c l U T 2 = l= l= As U T is not identically vanishing, it must be equal to s l= a lt k l/ l= c l. Bearing in mind that a, c, we get its asymptotic behaviour 38 U T a c T k h as T tends to. Remark 3. Under the hypotheses 28 and 3, we observe by factoring out the operator z v from 32, that UT, z, ɛ must solve the related PDE q 39 Q z a l ɛ m lβ αk l T k l UT, z, ɛ c l ɛ µ l2β αh l U 2 T, z, ɛ l= = l= Q bj zɛ n j αb j T b j F T, z, ɛ j= D ɛ lαδ l d l β T d l Rl z δ l T UT, z, ɛ where the forcing term F T, z, ɛ is a polynomial in z of degree less than v. According to the assumptions 33, 34, 35 and using the fact that Q, by taking ɛ = into equation 39 we see that the constraint 37 is equivalent to the fact that F T, z,. The precise shape of the term F T, z, ɛ will be given later in Section 6, see 34. In a first step, we express UT, z, ɛ as a small perturbation of U T that can be express in the form U T = a T k h a T k h JT c c where JT = j J jt j is a convergent series near T =, namely 4 UT, z, ɛ = a c T k h a c T k h JT T γ V T, z, ɛ for some integer γ Z and some expression V T, x, ɛ. By plugging this last expansion inside 32 and using the Leibniz rule, we get s q 4 Q z a l T k l a l ɛ m lβ αk l T k l a T k h a T k h JT T γ V T, z, ɛ c c l= c l a c l= l= l=s c l ɛ µ l2β αh l a c T k h a c T k h JT T γ V T, z, ɛ 2 = Q b j zɛ n j αb j T b j j= D ɛ lαδ l d l β T d l R l z a Π δ l c d= k h dt k h δ l δ l! q!q 2! q T T k h q 2 q q 2 =δ l T JT δ l! q!q 2! Πq d= γ dt γ q q 2 T V T, z, ɛ q q 2 =δ l

16 6 where we put Π d= γ d = by convention. At this level, we observe the important fact that the coefficient in front of Q z V T, z, ɛ contains the term a T kγ 2 a c T k h γ c T h = a T k γ that we want to set appart. As a result, we get the next equation satisfied by V T, z, ɛ, s q 42 Q z V T, z, ɛ a T kγ a l T k l a l ɛ m lβ αk l T k l T γ 2 a T k h γ c l c c l l= l= l= l=s Q z V 2 T, z, ɛt 2γ c l = l= c l ɛ µ l2β αh l 2 a c T k h γ JT c l ɛ µ l2β αh l l= Q b j zɛ n j αb j T b j j= c l ɛ µ l2β αh l D ɛ lαδ l d l β δ l! q!q 2! Πq d= γ dt d lγ q R l z q 2 T V T, z, ɛ q q 2 =δ l We now introduce some additional constraints on the integers γ, k l, h j, b h, for l q, j M, h Q and d l, δ l, for l D. Namely, we impose that the next inequalities hold 43 k h p γ for p M, together with 44 γ b j k for all j Q and finally 45 k d l δ l for all l D. Remark 4. For the case k > h, from 43, we need that γ k h >. As consequence of 44, we get that b j > k, for j Q. Let for instance q =, M =, Q = and D = 2. We set α = 2, β =, γ = 6, = and we choose the powers of t and ɛ in the coefficients of 29 as follows, 46 m = 3, k = 2, m = 6, k = 3, µ =, h =, µ = 3, h = 2, n = 9, b = 9, = 2, d = 5, δ =, 2 = 2, d 2 = 6, δ 2 = 2. For these data, we can check that the constraints 33, 34, 35, 43, 44, 45 above are fulfilled. Moreover, all the forthcoming requirements 54, 67, 68, 69 and 35 stated in Theorem are also verified. In this special case, the main equation 29 writes 47 Q z a ɛ 3 t 2 a ɛ 6 t 3 ut, z, ɛ c t c ɛ 3 t 2 u 2 t, z, ɛ = b zɛ 9 t 9 ɛ 2 t 5 t R z ut, z, ɛ ɛ 2 t 6 2 t R 2 z ut, z, ɛ.

17 7 We can divide this last equation by t, but not by ɛ, and the resulting equation still possess a turning point and an irregular singularity at t =. 2 For the case h > k, we may take γ < and hence one can choose some b j < k for some j Q. Let for instance q =, M =, Q = and D = 2. We choose α = 2, β =, γ = 2, = and we select the powers of t and ɛ in the coefficients of 29 as follows, 48 m = 5, k = 2, m =, k = 4, µ = 4, h = 6, µ = 9, h = 8, n = 3, b =, =, d = 5, δ =, 2 = 2, d 2 = 6, δ 2 = 2. For these data, we can figure out that the constraints 33, 34, 35, 43, 44, 45 above are satisfied. Moreover, all the forthcoming requirements 54, 67, 68, 69 and 35 stated in Theorem are also verified. In this particular case, the main equation 29 writes 49 Q z a ɛ 5 t 2 a ɛ t 4 ut, z, ɛ c ɛ 4 t 6 c ɛ 9 t 8 u 2 t, z, ɛ = b zɛ 3 t ɛ t 5 t R z ut, z, ɛ ɛ 2 t 6 2 t R 2 z ut, z, ɛ. We can divide this latter equation by ɛ 3 and by t. The corresponding equation still suffers the presence of a turning point and an irregular singularity at t =. In a second step, we divide the left and right handside of 42 by the monomial T k γ. We obtain the next equation 5 Q z V T, z, ɛ a s a l T k l 2 a T h c l c c l l= q l=s l= l= Q z V 2 T, z, ɛt kγ c l = l= a l ɛ m lβ αk l T k l T k c l ɛ µ l2β αh l 2 a c T h JT c l ɛ µ l2β αh l l= Q b j zɛ n j αb j T b j k γ j= c l ɛ µ l2β αh l D ɛ lαδ l d l β δ l! q!q 2! Πq d= γ dt d l k q R l z q 2 T V T, z, ɛ q q 2 =δ l Notice that the additional constraints 33, 34, 35 and 43, 44, 45 ensure that the coefficients of the PDE 5 are analytic with respect to T and ɛ on a neighborhood of the origin in C 2. Moreover, the coefficient of Q z V T, z, ɛ is invertible at T = since a. We will see later that this fact is essential in order to solve this equation within some function space of analytic functions. We look for solutions which are rescaled in time of the form 5 V T, z, ɛ = Vɛ χ T, z, ɛ

18 8 where 52 χ = D αδ D d D β d D k δ D. As a result, the expression VT, z, ɛ is supposed to solve the next equation 53 Q z VT, z, ɛ a s a l ɛ χk l k T k l k 2 a c l ɛ χh l h h c l= c l ɛ χh l h h l= q l=s a l ɛ m lβ αk l χk l k T k l k c l ɛ µ l2β αh l χh l h h 2 a c Jɛ χ T l= Q z V 2 T, z, ɛɛ χ kγ T kγ c l ɛ χh l = l= c l ɛ µ l2β αh l χh l h h l= Q b j zɛ n j αb j χb j k γ T b j k γ j= D ɛ lαδ l d l β c l ɛ µ l2β αh l χh l δ l! q!q 2! Πq d= γ dɛ χd l k q T d l k q R l z ɛ χq 2 q 2 T VT, z, ɛ q q 2 =δ l ɛ Dαδ D d D β δ D! q!q 2! Πq d= γ dɛ χd D k q T d D k q q q 2 =δ D,q R D z ɛ χq 2 q 2 T VT, z, ɛ Td D k R D z δ D T VT, z, ɛ We make further assumptions on the coefficients d l and δ l for l D which are stronger than the constraint 45. Assume the existence of integers and d l, such that 54 d D k = δ D, d l k = δ l d l, for all l D. Then, for all l D, and all integers q, q 2 with q q 2 = δ l we deduce the existence of an nonnegative integer d l,q,q 2 which is larger than except d D,,δD = such that 55 d l k q = q 2 d l,q,q 2. Indeed, if one puts d D, =, from 54, we can write d l,q,q 2 = d l k q q 2 = δ l d l, q q 2 = q q 2 d l, q q 2 = q d l,. According to 54 and 55, with the help of the formula 8.7 from [26] p. 363, we can expand

19 9 the following pieces appearing in 53 satisfied by VT, z, ɛ, 56 T d D k δ D T VT, z, ɛ = T T δ D pδ D A δd,pt δd p T T p VT, z, ɛ, T d l k δ l T VT, z, ɛ = T d l,δ l, T T VT, z, ɛ, T d l k q q 2 T VT, z, ɛ = Td l,q,q 2 T q 2 q 2 T VT, z, ɛ = T d l,q,q 2 T T q 2 A q2,pt q2 p T T p VT, z, ɛ pq 2 for all l D, all integers q and q 2 2 such that q q 2 = δ l, for some real constants A δd,p, p δ D and A q2,p, p q 2. In a third step, let us assume that the expression VT, z, ɛ has a formal power series expansion 57 VT, z, ɛ = n V n z, ɛt n where each coefficient V n z, ɛ is defined as an inverse Fourier transform V n z, ɛ = F m ω n m, ɛz for some function m ω n m, ɛ belonging to the Banach space E β,µ and depending holomorphically on ɛ on some punctured disc D, ɛ \{} centered at with radius ɛ >. We consider the formal power series ω n m, ɛ Γ n τ n ω τ, m, ɛ = n obtained by formally applying a m Borel transform w.r.t T and Fourier transform w.r.t z to the power series 57. The constraints 54 are introduced in such a way that ω τ, m, ɛ satisfies some integral equation by making use of the properties of the m Borel transform of formal series and Fourier inverse transforms described in Propositions 4 and 5 with the help of the prepared expansions 56. Namely, after division by the power im v, which is by construction a common factor of the functions Qim, R l im and B j m for l D, j Q, we get the new problem 58 L τ,m,ɛ ω τ, m, ɛ = R τ,m,ɛ ω τ, m, ɛ

20 2 with vanishing initial data ω, m, ɛ, where 59 L τ,m,ɛ ω τ, m, ɛ = Qim a ω τ, m, ɛ q l=s s a l ɛ χk l k a l ɛ m lβ αk l χk l k 2 a c l= l= c l ɛ χh l h c l ɛ µ l2β αh l ɛ χh l h j τ Γ k l k τ τ Γ k l k τ τ Γ h l h τ Γ h l h τ 2 a c l J j ɛ χh l h j τ c Γ h l h j l= {s {s j τ τ s k l k ω s /, m, ɛ ds s τ s k l k ω s /, m, ɛ ds s τ s h l h ω s /, m, ɛ ds s τ c l J j ɛ µ l2β αh l ɛ χh l h j τ Γ h l h j s l= s τ s h l h ω s /, m, ɛ ds s τ τ s h l h j ω s /, m, ɛ ds s τ s h l h j τ ω s /, m, ɛ ds s Qim c l ɛ χ k γh l τ Γ k τ s k γh l γh l l= 2π /2 ω s /, m m, ɛω /, m, ɛ s ds dm } ds s c l ɛ µ l2β αh l χh l k γ τ τ Γ k τ s k γh l γh l 2π /2 ω s /, m m, ɛω /, m, ɛ s ds dm } ds s

21 2 and 6 R τ,m,ɛ ω τ, m, ɛ = Q = B j mɛ n j αb j χb j k γ pq 2 pq 2 j= { A q2,p τ b j k γ Γ b j k γ D ɛ lαδ l d l β δ l! q!q 2! Πq d= γ dɛ χd l k q q 2 Rl im q q 2 =δ l τ Γ d l,q,q 2 τ τ Γ d l,q,q 2 q 2 p q q 2 =δ D,q τ τ { A q2,p Γ d D,q,q 2 τ Γ d D,q,q 2 q 2 p τ s τ d l,q,q 2 q 2 s q 2 ω s /, m, ɛ ds s τ s d l,q,q 2 q 2 p ɛ Dαδ D d D β δ D! q!q 2! Πq d= γ dɛ χd D k q q 2 RD im τ s d D,q,q 2 q 2 s q 2 ω s /, m, ɛ ds s τ R D im{τ δ D ω τ, m, ɛ p s p ω s /, m, ɛ ds s } τ s d D,q,q q 2 2 p p s p ω s /, m, ɛ ds s } pδ D τ A δd,p τ Γ δ D p τ s δ D p p s p ω s /, m, ɛ ds s }. By convention, the two sums pq 2 [...] appearing in 6 are vanishing provided that q 2 {, }. Remark 5. The hypotheses 28 and 3 ensure that the equation 53 does not contain terms that involve isolated polynomials in T which are not inverse Fourier transformable. 5 Analytic solutions of a convolution problem with complex parameters Our main goal in this section is the construction of a unique solution of the problem 58 within the Banach spaces introduced in Section 2. We make the following further assumptions. The conditions below are very similar to the ones proposed in Section 4 of [2]. Namely, we demand that there exists an unbounded sector S Q, RD = {z C/ z r Q, RD, argz d Q, RD η Q, RD } with direction d Q, RD R, aperture η Q, RD > for some radius r Q, RD > such that 6 Qim R D im S Q, RD

22 22 for all m R. The polynomial Pm τ = Qima R D im δ Dτ δ D can be factorized in the form 62 Pm τ = R D im δ D Π δ D l= τ q l m where 63 q l m = a Qim R δ D exp arg a Qim D im δ D R D im δ D δ D 2πl δ D for all l δ D, all m R. We select an unbounded sector S d centered at, a small closed disc D, ρ and we require the sector S Q, to fulfill the next conditions. RD There exists a constant M > such that 64 τ q l m M τ for all l δ D, all m R, all τ S d D, ρ. Indeed, from 6 and the explicit expression 63 of q l m, we first observe that q l m > 2ρ for every m R, all l δ D for an appropriate choice of r Q, and of ρ >. We also see that for all m R, all l δ RD D, the roots q l m remain in a union U of unbounded sectors centered at that do not cover a full neighborhood of the origin in C provided that η Q, is small enough. Therefore, one can RD choose an adequate sector S d such that S d U = with the property that for all l δ D the quotients q l m/τ lay outside some small disc centered at in C for all τ S d, all m R. This yields 64 for some small constant M >. 2 There exists a constant M 2 > such that 65 τ q l m M 2 q l m for some l {,..., δ D }, all m R, all τ S d D, ρ. Indeed, for the sector S d and the disc D, ρ chosen as above in, we notice that for any fixed l δ D, the quotient τ/q l m stays outside a small disc centered at in C for all τ S d D, ρ, all m R. Hence 65 must hold for some small constant M 2 >. By construction of the roots 63 in the factorization 62 and using the lower bound estimates 64, 65, we get a constant > such that 66 P m τ M δ D M 2 R D im δ D a Qim R δ D τ δ D D im δ D M δ D δ D a M 2 min x δ D δ D δ D r Q, RD δ D R D im x δ D τ δ D x δ D = r Q, RD δ D R D im τ δ D for all τ S d D, ρ, all m R. In the next proposition, we provide sufficient conditions under which the main convolution equation 58 possess solutions ωτ, d m, ɛ in the Banach space Fν,β,µ,χ,α,,ɛ d described in Section 2.

23 23 Proposition 6 Under the additional assumptions 67 δ D 2, k γ h l >, b j k γ, for all l M, j Q, χ α k γ h l δ D χ k γ h l 68 δ D δ l, l αδ l d l β χ α d l,q,q 2 for all q, q 2 such that q q 2 = δ l, for l D and 69 D αδ D d D β χ α d D,q,q 2 q 2 δ D χd l k δ l q 2 δ D χd D k δ D for all q, q 2 such that q q 2 = δ D, there exist a radius r Q, >, ɛ RD > and a constant ϖ > such that the equation 58 has a unique solution ωτ, d m, ɛ in the Banach space Fν,β,µ,χ,α,,ɛ d which suffers the bounds ω d τ, m, ɛ ν,β,µ,χ,α,,ɛ ϖ for all ɛ D, ɛ \ {}, where the direction d R can be chosen for any sector S d that fulfills the constraints 64 and 65 above. Proof We undertake the proof with a lemma that studies some shrinking map on the Banach spaces mentioned above and reduces the main convolution problem 58 to the existence of a unique fixed point for this map. Lemma 2 Taking for granted that the assumptions 67, 68 and 69 hold, one can select the constant r Q, > large enough and a constant ϖ > small enough such that for all RD ɛ D, ɛ \ {}, the map H ɛ defined as 7 H ɛ = H ɛ H 2 ɛ H 3 ɛ

24 24 where 7 H ɛ wτ, m := Q B j mɛ n j αb j χb j k γ j= s Qim a l ɛ χk l k q l=s τ P m τγ k l k τ a l ɛ m lβ αk l χk l k P m τγ k l k 2 a c l ɛ χh l h c l= l= c l ɛ µ l2β αh l ɛ χh l h 2 a c l J j ɛ χh l h j c j l= τ P m τγ h l h j τ P m τγ h l h τ τ b j k γ P m τγ b j k γ τ τ τ P m τγ h l h τ τ P m τγ h l h j τ s k l k ws /, m ds s τ s k l k ws /, m ds s τ s h l h ws /, m ds s τ c l J j ɛ µ l2β αh l ɛ χh l h j j τ τ s h l h ws /, m ds s τ s h l h j ws /, m ds s τ s h l h j ws /, m ds s and 72 Hɛ 2 wτ, m := Qim c l ɛ χ k γh l τ P l= m τγ k τ s k γh l γh l s {s 2π /2 ws s /, m m w /, m s ds dm } ds s c l ɛ µ l2β αh l χh l k γ τ τ P l= m τγ k τ s k γh l γh l s {s 2π /2 ws s /, m m w /, m s ds dm } ds s D ɛ lαδ l d l β pq 2 { A q2,p τ P m τγ d l,q,q 2 τ τ δ l! q!q 2! Πq d= γ dɛ χd l k q q 2 Rl im q q 2 =δ l τ τ s P m τγ d l,q,q 2 q 2 p τ d l,q,q 2 q 2 s q 2 ws /, m ds s τ s d l,q,q 2 q 2 p p s p ws /, m ds s }

25 25 along with 73 H 3 ɛ wτ, m := ɛ Dαδ D d D β pq 2 { A q2,p R D im{ q q 2 =δ D,q τ P m τγ d D,q,q 2 τ δ D! q!q 2! Πq d= γ dɛ χd D k q q 2 RD im τ P m τγ d D,q,q 2 q 2 p pδ D A δd,p satisfies the next properties. i The following inclusion holds τ s d D,q,q 2 q 2 s q 2 ws /, m ds s τ τ P m τγ δ D p τ s d D,q,q q 2 2 p p s p ws /, m ds s } τ 74 H ɛ B, ϖ B, ϖ τ s δ D p p s p ws /, m ds s } where B, ϖ is the closed ball of radius ϖ > centered at in Fν,β,µ,χ,α,,ɛ d, for all ɛ D, ɛ \ {}. ii We have 75 H ɛ w H ɛ w 2 ν,β,µ,χ,α,,ɛ 2 w w 2 ν,β,µ,χ,α,,ɛ for all w, w 2 B, ϖ, for all ɛ D, ɛ \ {}. Proof We first deal with the property 74. Let ɛ D, ɛ \ {} and consider wτ, m Fν,β,µ,χ,α,,ɛ d. We take ϖ > such that wτ, m ν,β,µ,χ,α,,ɛ ϖ. We start providing norms estimates for each piece of the map H ɛ. From Lemma, we deduce the existence of a constant C > depending on, γ, k and b j for j Q such that 76 B j mɛ χb j k γ τ b j k γ P m τ ν,β,µ,χ,α,,ɛ C r Q, RD δ D B j m β,µ inf m R R D im ɛ b j k γα According to Proposition 2, we obtain a constant C 2 > depending on ν,, k l, for l q, h l, for l M, QX, RD X and a constant C 2 j depending on ν,, h l for l M,

26 26 QX, R D X and j such that 77 ɛ χk l k Qimτ P m τ τ τ s k l k ws /, m ds s ν,β,µ,χ,α,,ɛ C 2 r Q, RD δ D ɛ χh l h Qimτ P m τ C 2 τ r Q, RD δ D ɛ αk l k wτ, m ν,β,µ,χ,α,,ɛ, τ s h l h ws /, m ds s ν,β,µ,χ,α,,ɛ ɛ αh l h wτ, m ν,β,µ,χ,α,,ɛ, ɛ Qim τ χh l h j P m τ τ τ s h l h j ws /, m ds s ν,β,µ,χ,α,,ɛ C 2 j r Q, RD δ D ɛ αh l h j wτ, m ν,β,µ,χ,α,,ɛ. Moreover, it appears from the proof of Proposition 2 that the next bounds hold for C 2 j : there exist a constant Ĉ2 > depending on ν,, h l for l M, Q, RD and a constant A 2 > depending on ν,, h l for l M such that 78 C 2 j Ĉ2A j 2 Γh l h j for all j. In the following, we will make use of the notations from the proof of Proposition 2. From the classical estimates 79 sup x m e m2x = m m e m x m 2 for any real numbers m, m 2 >, we deduce that for all j such that h l h j > sup x x 2 x / e νx xg x sup x 2 x h l h j e ν 2 x 2 / x h l h j h l h j ν/2 h l h j 2 ν/2 exp h l h j h l h j 2 exp h l h j 2 2 / Furthermore, according to the Stirling formula Γx 2πx x 2 e x as x and bearing in mind the functional relation Γx = xγx for all x >, we get two constants Č2 > and A 2 > independent of j such that 8 sup x x 2 x / e νx xg x Č2A j 2 Γh l h j Č2A j 2 Γ h l h j h l h j Γ h l h j 2 h l h j Γ h l h j

27 27 On the other hand, by direct inspection, we observe that there exists a constant Č2. > independent of j and ɛ such that x 2 x 8 sup x< x / e νx ɛ χα x γ G 2x Č2. Furthermore, there exists a constant K 2. j depending on ν,, h l for l M and j, such that x 2 x 82 sup x x / e νx ɛ χα x γ G x 2 2x sup x x K 2.j 2 2 Now, after a thorough examination of the proof of Proposition out of [2], one can check that there exists a constant Ǩ2. > independent of j such that 83 K 2. j Ǩ2.Γ h l h j for all j. Finally, gathering 8, 8, 82 and 83 yields the estimates 78. Besides, we choose the radius r Q, RD > large enough and ϖ in such a manner that 84 Q j= C r Q, RD δ D Γ b j k γ q B j m β,µ inf m R R D im ɛ n j αb j b j k γα s C 2 a l ɛ αk l k ϖ r Q, δ D Γ k l k RD C 2 a l ɛ m lβ αk l αk l k ϖ l=s r Q, δ D Γ k l k RD 2 a C 2 c l ɛ αh l h ϖ c r Q, δ D Γ h l h RD l= c l 2 a c C 2 r Q, δ D Γ h l h RD c l Ĉ 2 A j 2 J j j r Q, δ D RD l= l= ɛ µ l2β αh l ɛ αh l h ϖ ɛ αh l h j ϖ c l Ĉ 2 A j 2 J j ɛ µ l2β αh l ɛ αh l h j ϖ ϖ j r Q, δ D 3. RD Notice that the infinite sums over the integers j are convergent in the left handside of the above inequality 84, provided that ɛ > is small enough, according to the fact that there exist two constants J, J 2 > such that J j J J 2 j for all j since JT = j J jt j is a convergent series near T =. From the definition of H ɛ given by 7, we deduce the next inequality 85 H ɛ wτ, m ν,β,µ,χ,α,,ɛ ϖ 3.

28 28 Hereafter, we focus on norms estimates for each part of the map Hɛ 2. We set hτ, m = τ τ wτ /, m m w /, m τ ds dm. Regarding Proposition 3, we get a constant C 3 > depending on µ, such that 86 hτ, m ν,β,µ,χ,α,,ɛ C 3 ɛ χα wτ, m 2 ν,β,µ,χ,α,,ɛ. On the other hand, using Proposition 2, 2, we grab a constant C 2 > depending on ν,, γ, δ D, k, h l for l M and QX, R D X such that 87 ɛ Qim τ χ k γh l P m τ τ τ s k γh l s hs /, mds ν,β,µ,χ,α,,ɛ C 2 ɛ χα k γh l r Q, χ k γh l χαδ D hτ, m ν,β,µ,χ,α,,ɛ. δ D RD Therefore, gathering 86 and 87 returns 88 ɛ Qim τ χ k γh l P m τ τ τ s k γh l {s s ws /, m m w /, m s ds dm } ds s ν,β,µ,χ,α,,ɛ C 2 C 3 r Q, RD δ D ɛ χα k γh l δ D χ k γh l wτ, m 2 ν,β,µ,χ,α,,ɛ. Bearing in mind Proposition 2,, we get a constant C 2 > depending on ν,, d l, δ l and R l X, R D X for l D, such that 89 ɛ R χd l k δ l l im τ P m τ τ τ d l,δl, s ws /, m ds s ν,β,µ,χ,α,,ɛ C 2 r Q, RD δ D ɛ χαd l,δ l, χd l k δ l wτ, m ν,β,µ,χ,α,,ɛ = C 2 r Q, RD δ D ɛ αd l k δ l wτ, m ν,β,µ,χ,α,,ɛ. Likewise, we can apply Proposition 2, 2 in order to exhibit a constant C 2 ν,, d l, δ l, k, δ D and R l X, R D X for l D with > depending on 9 ɛ R χd l k δ l l im τ P m τ τ τ d l,q,q 2 s s q 2 ws /, m ds s ν,β,µ,χ,α,,ɛ C 2 r Q, RD δ D ɛ χα d l,q,q 2 q 2 δ D χd l k δ l wτ, m ν,β,µ,χ,α,,ɛ

29 29 for all q and q 2 with q q 2 = δ l. Besides, 9 ɛ R χd l k δ l l im τ P m τ τ τ d l,q,q 2 q s 2 p s p ws /, m ds s ν,β,µ,χ,α,,ɛ C 2 r Q, RD δ D provided that q 2 2 and p q 2, with q q 2 = δ l. Now, we choose r Q, RD > and ϖ in such a way that ɛ χα d l,q,q 2 q 2 δ D χd l k δ l wτ, m ν,β,µ,χ,α,,ɛ 92 C 2 c l C 3 ɛ χα kγhl δd χ kγhl ϖ 2 l= r Q, δ D Γ k γh l RD 2π /2 C 2 c l C 3 l= r Q, δ D Γ k γh l RD 2π /2 D ɛ χα k γh l δ D χ k γh l ɛ µ l2β αh l ϖ 2 ɛ lαδ l d l β q q 2 =δ l,q 2 Π δ l d= γ d C 2 ɛ αdl k δl ϖ r Q, δ D Γ d l,δ l, RD δ l! q!q 2! Πq d= γ d C 2 q 2 r Q, δ D Γ d l,q,q 2 RD ɛ χα d l,q,q 2 q 2 δ D χd l k δ l ϖ C 2 A q2,p p pq 2 r Q, δ D Γ d l,q,q 2 RD q 2 p ɛ χα d l,q,q 2 q 2 δ D χd l k δ l ϖ ] ϖ 3. With the help of the definition of H 2 ɛ given by 72, we deduce that 93 H 2 ɛ wτ, m ν,β,µ,χ,α,,ɛ ϖ/3. Ultimately, we attract our attention to norms estimates for H 3 ɛ. Taking notice of Proposition 2,, we get a constant C 2 > depending on ν,, k, δ D, d D, such that 94 ɛ R χd D k δ D D im τ P m τ τ τ d D,δD, s ws /, m ds s ν,β,µ,χ,α,,ɛ C 2 r Q, RD δ D ɛ χαd D,δ D, χd D k δ D wτ, m ν,β,µ,χ,α,,ɛ = C 2 ɛ αd D k δ D wτ, m r Q, ν,β,µ,χ,α,,ɛ. δ D RD

30 3 Moreover, we can apply Proposition 2, 2 in order to exhibit a constant C 2 > depending on ν,, k, d D, δ D with 95 ɛ R χd D k δ D D im τ P m τ τ τ s d D,q,q 2 s q 2 ws /, m ds s ν,β,µ,χ,α,,ɛ C 2 r Q, RD δ D for all q and q 2 with q q 2 = δ D. Besides, ɛ χα d D,q,q 2 q 2 δ D χd D k δ D wτ, m ν,β,µ,χ,α,,ɛ 96 ɛ R χd D k δ D D im τ P m τ τ τ s d D,q,q 2 q 2 p s p ws /, m ds s ν,β,µ,χ,α,,ɛ C 2 r Q, RD δ D ɛ χα d D,q,q 2 q 2 δ D χd D k δ D wτ, m ν,β,µ,χ,α,,ɛ provided that q, q 2 2 and p q 2 with q q 2 = δ D. Finally, we can select a constant C 2 > depending on ν,, δ D such that 97 R D im τ P m τ τ τ s δd p s p ws /, m ds s ν,β,µ,χ,α,,ɛ C 2 ɛ χα wτ, m r Q, ν,β,µ,χ,α,,ɛ δ D RD for all p δ D. We make the choice for the size of radius r Q, and ϖ in such a manner RD that 98 ɛ Dαδ D d D β Π δ D d= γ d C 2 r Q, δ D Γ d D,δ D, RD δ D! q!q 2! Πq d= γ d q q 2 =δ D,q,q 2 ɛ χα d D,q,q 2 q 2 δ D χd D k δ D ϖ ɛ αd D k δ D ϖ C 2 q 2 r Q, RD δ D Γ d D,q,q 2 A q2,p C 2 p pq 2 r Q, δ D Γ d D,q,q 2 RD q 2 p ] ɛ χα d D,q,q 2 q 2 δ D χd D k δ D ϖ pδ D A δd,p From the construction of the map H 3 ɛ, it is now clear that 99 H 3 ɛ wτ, m ν,β,µ,χ,α,,ɛ ϖ/3. Eventually, gathering 85, 93 and 99 yields the first claim 74. C 2 p r Q, RD δ D Γδ D p ɛ χα ϖ ϖ 3