On parametric multiummable formal olution to ome nonlinear initial value Cauchy problem A. Latra, S. Malek Univerity of Alcalá, Departamento de Fíica y Matemática, Ap. de Correo 2, E-2887 Alcalá de Henare (Madrid, Spain, Univerity of Lille, Laboratoire Paul Painlevé, 59655 Villeneuve d Acq cedex, France, alberto.latra@uah.e Stephane.Malek@math.univ-lille.fr June, 2 25 Abtract We tudy a nonlinear initial value Cauchy problem depending upon a complex perturbation parameter ɛ whoe coefficient depend holomorphically on (ɛ, t near the origin in C 2 and are bounded holomorphic on ome horizontal trip in C w.r.t the pace variable. In our previou contribution [4], we aumed the forcing term of the Cauchy problem to be analytic near. Preently, we conider a family of forcing term that are holomorphic on a common ector in time t and on ector w.r.t the parameter ɛ whoe union form a covering of ome neighborhood of in C, which are aked to hare a common formal power erie aymptotic expanion of ome Gevrey order a ɛ tend to. We contruct a family of actual holomorphic olution to our Cauchy problem defined on the ector in time and on the ector in ɛ mentioned above. Thee olution are achieved by mean of a verion of the o-called accelero-ummation method in the time variable and by Fourier invere tranform in pace. It appear that thee function hare a common formal aymptotic expanion in the perturbation parameter. Furthermore, thi formal erie expanion can be written a a um of two formal erie with a correponding decompoition for the actual olution which poe two different aymptotic Gevrey order, one teming from the hape of the equation and the other originating from the forcing term. The pecial cae of multiummability in ɛ i alo analyzed thoroughly. The proof lean on a verion of the o-called Rami-Sibuya theorem which entail two ditinct Gevrey order. Finally, we give an application to the tudy of parametric multi-level Gevrey olution for ome nonlinear initial value Cauchy problem with holomorphic coefficient and forcing term in (ɛ, t near and bounded holomorphic on a trip in the complex pace variable. Key word: aymptotic expanion, Borel-Laplace tranform, Fourier tranform, Cauchy problem, formal power erie, nonlinear integro-differential equation, nonlinear partial differential equation, ingular perturbation. 2 MSC: 35C, 35C2. The author i partially upported by the project MTM22-3439 of Miniterio de Ciencia e Innovacion, Spain The author i partially upported by the french ANR--JCJC 5 project and the PHC Polonium 23 project No. 2827SG.
2 Introduction We conider a family of parameter depending nonlinear initial value Cauchy problem of the form ( Q( z ( t u dp (t, z, ɛ = c,2 (ɛ(q ( z u dp (t, z, ɛ(q 2 ( z u dp (t, z, ɛ D + ɛ (δ D ( + δ D + t (δ D ( + δ D t R D ( z u dp (t, z, ɛ + ɛ l t d l δ l t R l( z u dp (t, z, ɛ l= + c (t, z, ɛr ( z u dp (t, z, ɛ + c F (ɛf dp (t, z, ɛ for given vanihing initial data u dp (, z, ɛ, where D 2 and δ D,, l, d l, δ l, l D are nonnegative integer and Q(X,Q (X,Q 2 (X,R l (X, l D are polynomial belonging to C[X]. The coefficient c (t, z, ɛ i a bounded holomorphic function on a product D(, r H β D(, ɛ, where D(, r (rep. D(, ɛ denote a dic centered at with mall radiu r > (rep. ɛ > and H β = {z C/ Im(z < β} i ome trip of width β >. The coefficient c,2 (ɛ and c F (ɛ define bounded holomorphic function on D(, ɛ vanihing at ɛ =. The forcing term f dp (t, z, ɛ, p ς, form a family of bounded holomorphic function on product T H β E p, where T i a mall ector centered at contained in D(, r and {E p } p ς i a et of bounded ector with aperture lightly larger than π/ covering ome neighborhood of in C. We make aumption in order that all the function ɛ f dp (t, z, ɛ, een a function from E p into the Banach pace F of bounded holomorphic function on T H β endowed with the upremum norm, hare a common aymptotic expanion ˆf(t, z, ɛ = m f m(t, zɛ m /m! F[[ɛ]] of Gevrey order / on E p, for ome integer <, ee Lemma. Our main purpoe i the contruction of actual holomorphic olution u dp (t, z, ɛ to the problem ( on the domain T H β E p and to analye their aymptotic expanion a ɛ tend to. Thi work i a continuation of the tudy initiated in [4] where the author have tudied initial value problem with quadratic nonlinearity of the form (2 Q( z ( t u(t, z, ɛ = (Q ( z u(t, z, ɛ(q 2 ( z u(t, z, ɛ D + ɛ (δ D (k+ δ D + t (δd (k+ δ D t R D ( z u(t, z, ɛ + ɛ l t d l δ l t R l( z u(t, z, ɛ l= + c (t, z, ɛr ( z u(t, z, ɛ + f(t, z, ɛ for given vanihing initial data u(, z, ɛ, where D, l, d l, δ l are poitive integer and Q(X, Q (X, Q 2 (X, R l (X, l D, are polynomial with complex coefficient. Under the aumption that the coefficient c (t, z, ɛ and the forcing term f(t, z, ɛ are bounded holomorphic function on D(, r H β D(, ɛ, one can build, uing ome Borel-Laplace procedure and Fourier invere tranform, a family of holomorphic bounded function u p (t, z, ɛ, p ς, olution of (2, defined on the product T H β E p, where E p ha an aperture lightly larger than π/k. Moreover, the function ɛ u p (t, z, ɛ hare a common formal power erie û(t, z, ɛ = m h m(t, zɛ m /m! a aymptotic expanion of Gevrey order /k on E p. In other word, u p (t, z, ɛ i the k um of û(t, z, ɛ on E p, ee Definition 9. In thi paper, we oberve that the aymptotic expanion of the olution u dp (t, z, ɛ of ( w.r.t ɛ on E p, defined a û(t, z, ɛ = m h m(t, zɛ m /m! F[[ɛ]], inherit a finer tructure which involve the two Gevrey order / and /. Namely, the order / originate from the equation ( itelf and it highet order term ɛ (δ D ( + δ D + t (δ D ( + δ D t R D ( z a it
3 wa the cae in the work [4] mentioned above and the cale / arie, a a new feature, from the aymptotic expanion ˆf of the forcing term f dp (t, z, ɛ. We can alo decribe condition for which u dp (t, z, ɛ i the (, um of û(t, z, ɛ on E p for ome p ς, ee Definition. More pecifically, we can preent our two main tatement and it application a follow. Main reult Let > be integer. We chooe a family {E p } p ς of bounded ector with aperture lightly larger than π/ which define a good covering in C (ee Definition 7 and a et of adequate direction d p R, p ς for which the contraint (52 and (53 hold. We alo take an open bounded ector T centered at uch that for every p ς, the product ɛt belong to a ector with direction d p and aperture lightly larger than π/, for all ɛ E p, all t T. We make the aumption that the coefficient c (t, z, ɛ can be written a a convergent erie of the pecial form c (t, z, ɛ = c (ɛ n c,n (z, ɛ(ɛt n on a domain D(, r H β D(, ɛ, where H β i a trip of width β, uch that T D(, r, p ς E p D(, ɛ and < β < β are given poitive real number. The coefficient c,n (z, ɛ, n, are uppoed to be invere Fourier tranform of function m C,n (m, ɛ that belong to the Banach pace E (β,µ (ee Definition 2 for ome µ > max(deg(q +, deg(q 2 + and depend holomorphically on ɛ in D(, ɛ and c (ɛ i a holomorphic function on D(, ɛ vanihing at. Since we have in view our principal application (Theorem 3, we chooe the forcing term f dp (t, z, ɛ a a m k2 Fourier-Laplace tranform f dp (t, z, ɛ = k 2 (2π /2 ψ dp (u, m, ɛe ( u ɛt k2 e izm du L γp u dm, where the inner integration i made along ome halfline L γp S dp and S dp i an unbounded ector with biecting direction d p, with mall aperture and where ψ dp (u, m, ɛ i a holomorphic function w.r.t u on S dp, defined a an integral tranform called acceleration operator with indice m k2 and m k, ψ dp (u, m, ɛ = ψ dp (h, m, ɛg(u, h dh L h γ p where G(u, h i a kernel function with exponential decay of order κ = (, ee (4. The integration path L γ p i a halfline in an unbounded ector U dp with biecting direction d p and ψ dp (h, m, ɛ i a function with exponential growth of order w.r.t h on U dp D(, ρ and exponential decay w.r.t m on R, atifying the bound (56. The function f dp (t, z, ɛ repreent a bounded holomorphic function on T H β E p. Actually, it turn out that f dp (t, z, ɛ can be imply written a a m k Fourier-Laplace tranform of ψ dp (h, m, ɛ, f dp (t, z, ɛ = (2π /2 L γp ψ dp (u, m, ɛe ( u ɛt k e izm du u dm, ee Lemma 3. Our firt reult tated in Theorem read a follow. We make the aumption that the integer δ D,, l, d l, δ l, l D atify the inequalitie (47, (48 and (6. The polynomial Q(X, Q (X, Q 2 (X and R l (X, l D are ubmitted to the contraint (49 on their degree. We require the exitence of contant r Q,Rl > uch that Q(im R l (im r Q,R l
4 for all m R, all l D (ee (5 and moreover that the quotient Q(im/R D (im belong to ome uitable unbounded ector S Q,RD for all m R (ee (5. Then, if the up norm of the coefficient c,2 (ɛ/ɛ, c (ɛ/ɛ and c F (ɛ/ɛ on D(, ɛ are choen mall enough and provided that the radii r Q,Rl, l D, are taken large enough, we can contruct a family of holomorphic bounded function u dp (t, z, ɛ, p ς, defined on the product T H β E p, which olve the problem ( with initial data u dp (, z, ɛ. Similarly to the forcing term, u dp (t, z, ɛ can be written a a m k2 Fourier-Laplace tranform u dp (t, z, ɛ = (2π /2 L γp ω dp (u, m, ɛe ( u ɛt k2 e izm du u dm where ω dp (u, m, ɛ denote a function with at mot exponential growth of order in u on S dp and exponential decay in m R, atifying (66. The function ω dp (u, m, ɛ i hown to be the analytic continuation of a function Acc dp, (ω dp (u, m, ɛ defined only on a bounded ector Sd b p with aperture lightly larger than π/κ w.r.t u, for all m R, with the help of an acceleration operator with indice m k2 and m k, Acc dp, (ω dp (u, m, ɛ = L γ p ω dp (h, m, ɛg(u, h dh h. We how that, in general, ω dp (h, m, ɛ uffer an exponential growth of order larger than (and actually le than κ w.r.t h on U dp D(, ρ, and obey the etimate (68. At thi point u dp (t, z, ɛ cannot be merely expreed a a m k Fourier-Laplace tranform of ω dp and i obtained by a verion of the o-called accelero-ummation procedure, a decribed in [], Chapter 5. Our econd main reult, decribed in Theorem 2, aert that the function u dp, een a map from E p into F, for p ς, turn out to hare on E p a common formal power erie û(ɛ = m h mɛ m /m! F[[ɛ]] a aymptotic expanion of Gevrey order /. The formal erie û(ɛ formally olve the equation ( where the analytic forcing term f dp (t, z, ɛ i replaced by it aymptotic expanion ˆf(t, z, ɛ F[[ɛ]] of Gevrey order / (ee Lemma. Furthermore, the function u dp and the formal erie û own a fine tructure which actually involve two different Gevrey order of aymptotic. Namely, u dp and û can be written a um û(ɛ = a(ɛ + û (ɛ + û 2 (ɛ, u dp (t, z, ɛ = a(ɛ + u dp (ɛ + udp 2 (ɛ where a(ɛ i a convergent erie near ɛ = with coefficient in F and û (ɛ (rep. û 2 (ɛ belong to F[[ɛ]] and i the aymptotic expanion of Gevrey order / (rep. / of the F valued function u dp (ɛ (rep. udp 2 (ɛ on E p. Beide, under a more retrictive aumption on the covering {E p } p ς and the unbounded ector {U dp } p ς (ee Aumption 5 in Theorem 2, one get that u dp (t, z, ɛ i even the (, um of û(ɛ on ome ector E p, with p ς, meaning that u dp (ɛ can be analytically continued on a larger ector S π/k, containing E p, with aperture lightly larger than π/ where it become the um of û (ɛ and by contruction u dp 2 (ɛ i already the um of û 2 (ɛ on E p, ee Definition. A an important application (Theorem 3, we deal with the pecial cae when the forcing term f dp (t, z, ɛ themelve olve a linear partial differential equation with a imilar hape a (2, ee (22, whoe coefficient are holomorphic function on D(, r H β D(, ɛ. When thi hold, it turn out that u dp (t, z, ɛ and it aymptotic expanion û(t, z, ɛ olve a nonlinear
5 ingularly perturbed PDE with analytic coefficient and forcing term on D(, r H β D(, ɛ, ee (224. We tre the fact that our application (Theorem 3 relie on the factorization of ome nonlinear differential operator which i an approach that belong to an active domain of reearch in ymbolic computation thee lat year, ee for intance [6], [7], [2], [28], [29], [33]. We mention that a imilar reult ha been recently obtained by H. Tahara and H. Yamazawa, ee [3], for the multiummability of formal erie û(t, x = n u n(xt n O(C N [[t]] with entire coefficient on C N, N, olution of general non-homogeneou time depending linear PDE of the form t m u + a j,α (t j t α x u = f(t, x j+ α L for given initial data ( j t u(, x = ϕ j(x, j m (where m L, provided that the coefficient a j,α (t together with t f(t, x are analytic near and that ϕ j (x with the forcing term x f(t, x belong to a uitable cla of entire function of finite exponential order on C N. The different level of multiummability are related to the lope of a Newton polygon attached to the main equation and analytic acceleration procedure a decribed above are heavily ued in their proof. It i worthwhile noticing that the multiummable tructure of formal olution to linear and nonlinear meromorphic ODE ha been dicovered two decade ago, ee for intance [2], [5], [8], [8], [2], [27], but in the framework of PDE very few reult are known. In the linear cae in two complex variable with contant coefficient, we mention the important contribution of W. Baler, [4] and S. Michalik, [22], [23]. Their trategy conit in the contruction of a multiummable formal olution written a a um of formal erie, each of them aociated to a root of the ymbol attached to the PDE uing the o-called Puieux expanion for the root of polynomial with holomorphic coefficient. In the linear and nonlinear context of PDE that come from a perturbation of ordinary differential equation, we refer to the work of S. Ouchi, [25], [26], which are baed on a Newton polygon approach and accelero-ummation technic a in [3]. Our reult concern more peculiarly multiummability and multiple cale analyi in the complex parameter ɛ. Alo from thi point of view, only few advance have been performed. Among them, we mut mention two recent work by K. Suzuki and Y. Takei, [3] and Y. Takei, [32], for WKB olution of the Schrödinger equation ɛ 2 ψ (z = (z ɛ 2 z 2 ψ(z which poee a fixed turning point and z ɛ = ɛ 2 a movable turning point tending to infinity a ɛ tend to. In the equel, we decribe our main intermediate reult and the ketch of the argument needed in their proof. In a firt part, we depart from an auxiliary parameter depending initial value differential and convolution equation which i regularly perturbed in it parameter ɛ, ee (7. Thi equation i formally contructed by making the change of variable T = ɛt in the equation ( and by taking the Fourier tranform w.r.t the variable z (a done in our previou contribution [4]. We contruct a formal power erie Û(T, m, ɛ = n U n(m, ɛt n olution of (7 whoe coefficient m U n (m, ɛ depend holomorphically on ɛ near and belong to a Banach pace E (β,µ of continuou function with exponential decay on R introduced by O. Cotin and S. Tanveer in []. A a firt tep, we follow the trategy recently developed by H. Tahara and H. Yamazawa in [3], namely we multiply each hand ide of (7 by the power T + which tranform it into an
6 equation (75 which involve only differential operator in T of irregular type at T = of the form T β T with β + due to the aumption (72 on the hape of the equation (7. Then, we apply a formal Borel tranform of order, that we call m k Borel tranform in Definition 4, to the formal erie Û with repect to T, denoted by τ n ω k (τ, m, ɛ = U n (m, ɛ Γ(n/. n Then, we how that ω k (τ, m, ɛ formally olve a convolution equation in both variable τ and m, ee (83. Under ome ize contraint on the up norm of the coefficient c,2 (ɛ/ɛ, c (ɛ/ɛ and c F (ɛ/ɛ near, we how that ω k (τ, m, ɛ i actually convergent for τ on ome fixed neighborhood of and can be extended to a holomorphic function ω d (τ, m, ɛ on unbounded ector U d centered at with biecting direction d and tiny aperture, provided that the m k Borel tranform of the formal forcing term F (T, m, ɛ, denoted by ψ k (τ, m, ɛ i convergent near τ = and can be extended on U d w.r.t τ a a holomorphic function ψ d (τ, m, ɛ with exponential growth of order le than. Beide, the function ω d (τ, m, ɛ atifie etimate of the form: there exit contant ν > and ϖ d > with ω d (τ, m, ɛ ϖ d ( + m µ e β m τ + τ 2 eν τ κ for all τ U d, all m R, all ɛ D(, ɛ, ee Propoition. The proof lean on a fixed point argument in a Banach pace of holomorphic function F(ν,β,µ,k d,κ tudied in Section 2.. Since the exponential growth order κ of ωk d i larger than, we cannot take a m k Laplace tranform of it in direction d. We need to ue a verion of what i called an accelero-ummation procedure a decribed in [], Chapter 5, which i explained in Section 4.3. In a econd tep, we go back to our eminal convolution equation (7 and we multiply each handide by the power T k2+ which tranform it into the equation (2. Then, we apply a m k2 Borel tranform to the formal erie Û w.r.t T, denoted by ˆω (τ, m, ɛ. We how that ˆω k2 (τ, m, ɛ formally olve a convolution equation in both variable τ and m, ee (23, where the formal m k2 Borel tranform of the forcing term i et a ˆψ k2 (τ, m, ɛ. Now, we oberve that a verion of the analytic acceleration tranform with indice and contructed in Propoition 3 applied to ψk d (τ, m, ɛ, tanding for ψk d 2 (τ, m, ɛ, i the κ um of ˆψ k2 (τ, m, ɛ w.r.t τ on ome bounded ector Sd,κ b with aperture lightly larger than π/κ, viewed a a function with value in E (β,µ. Furthermore, ψk d 2 (τ, m, ɛ can be extended a an analytic function on an unbounded ector S d,κ with aperture lightly larger than π/κ where it poee an exponential growth of order le than, ee Lemma 4. In the equel, we focu on the olution ωk d 2 (τ, m, ɛ of the convolution problem (29 which i imilar to (23 but with the formal forcing term ˆψ k2 (τ, m, ɛ replaced by ψk d 2 (τ, m, ɛ. Under ome ize retriction on the up norm of the coefficient c,2 (ɛ/ɛ, c (ɛ/ɛ and c F (ɛ/ɛ near, we how that ωk d 2 (τ, m, ɛ define a bounded holomorphic function for τ on the bounded ector Sd,κ b and can be extended to a holomorphic function on unbounded ector S d with direction d and tiny aperture, provided that S d tay away from the root of ome polynomial P m (τ contructed with the help of Q(X and R D (X in (, ee (3. Moreover, the function ωk d 2 (τ, m, ɛ atifie etimate of the form: there exit contant ν > and υ d > with ωk d 2 (τ, m, ɛ υ d ( + m µ e β m τ τ k2 + τ 2 eν for all τ S d, all m R, all ɛ D(, ɛ, ee Propoition 4. Again, the proof ret on a fixed point argument in a Banach pace of holomorphic function F(ν d,β,µ, outlined in Section 2.2.
7 In Propoition 5, we how that ω d (τ, m, ɛ actually coincide with the analytic acceleration tranform with indice m k2 and m k applied to ω d (τ, m, ɛ, denoted by Acc d, (ω d (τ, m, ɛ, a long a τ lie in the bounded ector S b d,κ. A a reult, ome m Laplace tranform of the analytic continuation of Acc d, (ω d (τ, m, ɛ, et a U d (T, m, ɛ, can be conidered for all T belonging to a ector S d,θk2,h with biecting direction d, aperture θ k2 lightly larger than π/ and radiu h >. Following the terminology of [], Section 6., U d (T, m, ɛ can be called the (m k2, m k um of the formal erie Û(T, m, ɛ in direction d. Additionally, U d (T, m, ɛ olve our primary convolution equation (7, where the formal forcing term ˆF (T, m, ɛ i interchanged with F d (T, m, ɛ which denote the (m k2, m k um of ˆF in direction d. In Theorem, we contruct a family of actual bounded holomorphic olution u dp (t, z, ɛ, p ς, of our original problem ( on domain of the form T H β E p decribed in the main reult above. Namely, the function u dp (t, z, ɛ (rep. f dp (t, z, ɛ are et a Fourier invere tranform of U dp, u dp (t, z, ɛ = F (m U dp (ɛt, m, ɛ(z, f dp (t, z, ɛ = F (m F dp (ɛt, m, ɛ(z where the definition of F i pointed out in Propoition 9. One prove the crucial property that the difference of any two neighboring function u d p+ (t, z, ɛ u dp (t, z, ɛ tend to zero a ɛ on E p+ E p fater than a function with exponential decay of order k, uniformly w.r.t t T, z H β, with k = when the interection U dp+ U dp i not empty and with k =, when thi interection i empty. The ame etimate hold for the difference f d p+ (t, z, ɛ f dp (t, z, ɛ. The whole ection 6 i devoted to the tudy of the aymptotic behaviour of u dp (t, z, ɛ a ɛ tend to zero. Uing the decay etimate on the difference of the function u dp and f dp, we how the exitence of a common aymptotic expanion û(ɛ = m h mɛ m /m! F[[ɛ]] (rep. ˆf(ɛ = m f mɛ m /m! F[[ɛ]] of Gevrey order / for all function u dp (t, z, ɛ (rep. f dp (t, z, ɛ a ɛ tend to on E p. We obtain alo a double cale aymptotic for u dp a explained in the main reult above. The key tool in proving the reult i a verion of the Rami-Sibuya theorem which entail two Gevrey aymptotic order, decribed in Section 6.. It i worthwhile noting that a imilar verion wa recently brought into play by Y. Takei and K. Suzuki in [3], [32], in order to tudy parametric multiummability for the complex Schrödinger equation. In the lat ection, we tudy the particular ituation when the formal forcing term F (T, m, ɛ olve a linear differential and convolution initial value problem, ee (24. We multiply each handide of thi equation by the power T k+ which tranform it into the equation (28. Then, we how that the m k Borel tranform ψ k (τ, m, ɛ formally olve a convolution equation in both variable τ and m, ee (29. Under a ize control of the up norm of the coefficient c (ɛ/ɛ and c F (ɛ/ɛ near, we how that ψ k (τ, m, ɛ i actually convergent near w.r.t τ and can be holomorphically extended a a function ψ dp (τ, m, ɛ on any unbounded ector U dp with direction d p and mall aperture, provided that U dp tay away from the root of ome polynomial P m (τ contructed with the help of Q(X and R D (X in (24. Additionally, the function ψ dp (τ, m, ɛ atifie etimate of the form: there exit a contant υ > with (τ, m, ɛ υ( + m µ e β m τ + τ 2 eν τ k ψ dp for all τ U dp, all m R, all ɛ D(, ɛ, ee Propoition 8. The proof i once more baed upon a fixed point argument in a Banach pace of holomorphic function F(ν,β,µ,k d, defined in Section 2.. Thee latter propertie on ψ dp (τ, m, ɛ legitimize all the aumption made above on the forcing term F (T, m, ɛ. Now, we can take the m k Laplace tranform L dp m k (ψ dp (T of
8 ψ dp (τ, m, ɛ w.r.t τ in direction d p, which yield an analytic olution of the initial linear equation (24 on ome bounded ector S dp,θ k,h with aperture θ k lightly larger than π/. It come to light in Lemma 3, that L dp m k (ψ dp (T coincide with the analytic (m k2, m k um F dp (T, m, ɛ of ˆF in direction d p on the maller ector S dp,θ k2,h with aperture lightly larger than π/. We deduce conequently that the analytic forcing term f dp (t, z, ɛ olve the linear PDE (22 with analytic coefficient on D(, r H β D(, ɛ, for all t T, z H β, ɛ E p. In our lat main reult (Theorem 3, we ee that thi latter iue implie that u dp (t, z, ɛ itelf olve a nonlinear PDE (224 with analytic coefficient and forcing term on D(, r H β D(, ɛ, for all t T, z H β, ɛ E p. The paper i organized a follow. In Section 2, we define ome weighted Banach pace of continuou function on (D(, ρ U R with exponential growth of different order on unbounded ector U w.r.t the firt variable and exponential decay on R w.r.t the econd one. We tudy the continuity propertie of everal kind of linear and nonlinear operator acting on thee pace that will be ueful in Section 4.2, 4.4 and 7.2. In Section 3, we recall the definition and the main analytic and algebraic propertie of the m k ummability. In Section 4., we introduce an auxiliary differential and convolution problem (7 for which we contruct a formal olution. In Section 4.2, we how that the m k Borel tranform of thi formal olution atifie a convolution problem (83 that we can uniquely olve within the Banach pace decribed in Section 2. In Section 4.3, we decribe the propertie of a variant of the formal and analytic acceleration operator aociated to the m k Borel and m k Laplace tranform. In Section 4.4, we ee that the m k2 Borel tranform of the formal olution of (7 atifie a convolution problem (23. We how that it formal forcing term i κ ummable and that it κ um i an acceleration of the m k Borel tranform of the above formal forcing term. Then, we contruct an actual olution to the correponding problem with the analytic continuation of thi κ um a nonhomogeneou term, within the Banach pace defined in Section 2. We recognize that thi actual olution i the analytic continuation of the acceleration of the m k Borel tranform of the formal olution of (7. Finally, we take it m k2 Laplace tranform in order to get an actual olution of (46. In Section 5, with the help of Section 4, we build a family of actual holomorphic olution to our initial Cauchy problem (. We how that the difference of any two neighboring olution i exponentially flat for ome integer order in ɛ (Theorem. In Section 6, we how that the actual olution contructed in Section 5 hare a common formal erie a Gevrey aymptotic expanion a ɛ tend to on ector (Theorem 2. The reult i built on a verion of the Rami-Sibuya theorem with two Gevrey order tated in Section 6.. In Section 7, we inpect the pecial cae when the forcing term itelf olve a linear PDE. Then, we notice that the olution of ( contructed in Section 5 actually olve a nonlinear PDE with holomorphic coefficient and forcing term near the origin (Theorem 3. 2 Banach pace of function with exponential growth and decay The Banach pace introduced in the next ubection 2. (rep. ubection 2.2 will be crucial in the contruction of analytic olution of a convolution problem invetigated in the forthcoming ubection 4.2 (rep. ubection 4.4.
9 2. Banach pace of function with exponential growth κ and decay of exponential order We denote D(, r the open dic centered at with radiu r > in C and D(, r it cloure. Let U d be an open unbounded ector in direction d R centered at in C. By convention, the ector we conider do not contain the origin in C. Definition Let ν, β, µ > and ρ > be poitive real number. Let k, κ be integer and d R. We denote F(ν,β,µ,k,κ d the vector pace of continuou function (τ, m h(τ, m on ( D(, ρ U d R, which are holomorphic with repect to τ on D(, ρ U d and uch that h(τ, m (ν,β,µ,k,κ = up ( + m µ + τ 2k exp(β m ν τ κ h(τ, m τ D(,ρ U d,m R τ i finite. One can check that the normed pace (F d (ν,β,µ,k,κ,. (ν,β,µ,k,κ i a Banach pace. Remark: Thee norm are appropriate modification of thoe introduced in the work [4], Section 2. Throughout the whole ubection, we aume µ, β, ν, ρ >, k, κ and d R are fixed. In the next lemma, we check the continuity property under multiplication operation with bounded function. Lemma Let (τ, m a(τ, m be a bounded continuou function on ( D(, ρ U d R by a contant C >. We aume that a(τ, m i holomorphic with repect to τ on D(, ρ U d. Then, we have (3 a(τ, mh(τ, m (ν,β,µ,k,κ C h(τ, m (ν,β,µ,k,κ for all h(τ, m F d (ν,β,µ,k,κ. In the next propoition, we tudy the continuity property of ome convolution operator acting on the latter Banach pace. Propoition Let χ 2 > be a real number. Let ν 2 be an integer. We aume that + χ 2 + ν 2. If κ k( ν 2 χ 2 + +, then there exit a contant C 2 > (depending on ν, ν 2, χ 2 uch that (4 τ k for all f(τ, m F d (ν,β,µ,k,κ. (τ k χ 2 ν 2 f( /k, md (ν,β,µ,k,κ C 2 f(τ, m (ν,β,µ,k,κ Proof Let f(τ, m F(ν,β,µ,k,κ d. By definition, we have (5 τ k (τ k χ 2 ν 2 f( /k, md (ν,β,µ,k,κ = up ( + m µ + τ 2k exp(β m ν τ κ τ D(,ρ U d,m R τ τ k {( + m µ e β m exp( ν κ/k + 2 /k f(/k, m} B(τ,, md
where Therefore, (6 where B(τ,, m = τ k ( + m µ e β m exp(ν κ/k + 2 /k (τ k χ 2 ν 2. (τ k χ 2 ν 2 f( /k, md (ν,β,µ,k,κ C 2 f(τ, m (ν,β,µ,k,κ + (7 C 2 = up τ 2k exp( ν τ κ τ D(,ρ U d τ τ k exp(νh κ/k + h 2 h k ( τ k h χ 2 h ν 2 dh = up B(x x where B(x = + x x2 exp(νh κ/k x /k exp( νxκ/k + h 2 h k +ν 2 (x h χ 2 dh. We write B(x = B (x + B 2 (x, where B (x = + x/2 x2 exp(νh κ/k x /k exp( νxκ/k + h 2 h k +ν 2 (x h χ 2 dh, B 2 (x = + x2 x /k exp( νxκ/k x x/2 exp(νh κ/k + h 2 h k +ν 2 (x h χ 2 dh. Now, we tudy the function B (x. We firt aume that < χ 2 <. In that cae, we have that (x h χ 2 (x/2 χ 2 for all h x/2 with x >. Since ν 2, we deduce that (8 B (x + x/2 x2 x /k (x 2 χ 2 e νhκ/k e νxκ/k + h 2 h k +ν 2 dh ( + x 2 2 /k ( k + ν 2 + (x 2 +χ 2+ν 2 exp( ν( 2 κ/k xκ/k for all x >. Since κ k and + χ 2 + ν 2, we deduce that there exit a contant K > with (9 up B (x K. x We aume now that χ 2. In thi ituation, we know that (x h χ 2 x χ 2 for all h x/2, with x. Hence, ince ν 2, ( B (x ( + x 2 2 /k ( k + ν 2 + xχ 2 (x/2 ν2+ exp( ν( 2 κ/k xκ/k for all x. Again, we deduce that there exit a contant K. > with ( up B (x K.. x In the next tep, we focu on the function B 2 (x. Firt, we oberve that + h 2 + (x/2 2 for all x/2 h x. Therefore, there exit a contant K 2 > uch that (2 B 2 (x + x2 + ( x 2 2 x /k exp( νxκ/k x x/2 exp(νh κ/k h k +ν 2 (x h χ 2 dh K 2 x /k exp( νxκ/k x exp(νh κ/k h k +ν 2 (x h χ 2 dh
for all x >. It remain to tudy the function B 2. (x = x exp(νh κ/k h k +ν 2 (x h χ 2 dh for x. By the uniform expanion e νhκ/k = n (νhκ/k n /n! on every compact interval [, x], x, we can write (3 B 2. (x = n ν n n! x h nκ k + k +ν 2 (x h χ 2 dh. Uing the Beta integral formula (ee [3], Appendix B3 and ince χ 2 >, k + ν 2 >, we can write (4 B 2. (x = n for all x. Bearing in mind that ν n Γ(χ 2 + Γ( nκ n! Γ( nκ k + k + ν 2 + χ 2 + 2 k + k + ν 2 + (5 Γ(x/Γ(x + a /x a x nκ k + k +ν 2+χ 2 + a x +, for any a > (ee for intance, [3], Appendix B3, from (4, we get a contant K 2. > uch that (6 B 2. (x K 2. x k +ν 2+χ 2 + n (n + χ 2+ n! (νxκ/k n for all x. Uing again (5, we know that /(n + χ2+ Γ(n + /Γ(n + χ 2 + 2 a n +. Hence, from (6, there exit a contant K 2.2 > uch that (7 B 2. (x K 2.2 x k +ν 2+χ 2 + n Γ(n + χ 2 + 2 (νxκ/k n for all x. Remembering the aymptotic propertie of the generalized Mittag-Leffler function (known a Wiman function in the literature E α,β (z = n zn /Γ(β + αn, for any α, β > (ee [3], Appendix B4 or [], expanion (22 p. 2, we get from (7 a contant K 2.3 > uch that (8 B 2. (x K 2.3 x k +ν 2+χ 2 + x κ k (χ 2+ e νxκ/k for all x. Under the aumption that ν 2 + χ 2 + κ k (χ 2 + and gathering (2, (8, we get a contant K 2.4 > uch that (9 up B 2 (x K 2.4. x Finally, taking into account the etimate (6, (7, (9, (, (9, the inequality (4 follow. Propoition 2 Let k, κ be integer uch that κ k. Let Q (X, Q 2 (X, R(X C[X] uch that (2 deg(r deg(q, deg(r deg(q 2, R(im
2 for all m R. Aume that µ > max(deg(q +, deg(q 2 +. Let m b(m be a continuou function on R uch that b(m R(im for all m R. Then, there exit a contant C 3 > (depending on Q, Q 2, R, µ, k, κ, ν uch that (2 b(m τ k (τ k k ( for all f(τ, m, g(τ, m F d (ν,β,µ,k,κ. Q (i(m m f(( x /k, m m Q 2 (im g(x /, m ( xx dxdm d (ν,β,µ,k,κ C 3 f(τ, m (ν,β,µ,k,κ g(τ, m (ν,β,µ,k,κ Proof Let f(τ, m, g(τ, m F d (ν,β,µ,k,κ. For any τ D(, ρ U d, the egment [, τ k ] i uch that for any [, τ k ], any x [, ], the expreion f(( x /k, m m and g(x /k, m are well defined, provided that m, m R. By definition, we can write b(m where τ k τ k (τ k k ( (τ k /k ( Q (i(m m f(( x /k, m m Q 2 (im g(x /, m ( xx dxdm d (ν,β,µ,k,κ = up ( + m µ + τ 2k exp(β m ν τ κ τ D(,ρ U d,m R τ {( + m m µ e β m m + x 2 x /k exp( ν x κ/k f(( x /k, m m } {( + m µ e β m + x 2 x /k exp( ν x κ/k g(x /k, m } C(, x, m, m = exp( β m exp( β m m ( + m m µ ( + m µ b(mq (i(m m Q 2 (im Now, we know that there exit Q, Q 2, R > with C(, x, m, m dxdm d x /k x /k ( + x 2 ( + x 2 exp(ν x κ/k exp(ν x κ/ ( xx. (22 Q (i(m m Q ( + m m deg(q, Q 2 (im Q 2 ( + m deg(q 2, R(im R( + m deg(r
3 for all m, m R. Therefore, (23 b(m where τ k (τ k k ( Q (i(m m f(( x /k, m m Q 2 (im g(x /, m ( xx dxdm d (ν,β,µ,k,κ C 3 f(τ, m (ν,β,µ,k,κ g(τ, m (ν,β,µ,k,κ (24 C 3 = up ( + m µ + τ 2k exp(β m ν τ κ τ D(,ρ U d,m R τ R( + m deg(r τ k ( τ k h /k ( h exp( β m exp( β m m ( + m m µ ( + m µ Q Q 2 ( + m m deg(q ( + m deg(q 2 (h x /k x /k ( + (h x 2 ( + x 2 exp(ν(h x κ/k exp(νx κ/ (h xx dxdm dh. Now, ince κ k, we have that (25 h κ/k (h x κ/k + x κ/k for all h, all x [, h]. Indeed, let x = hu where u [, ]. Then, the inequality (25 i equivalent to how that (26 ( u κ/k + u κ/k for all u [, ]. Let ϕ(u = ( u κ/k + u κ/k on [, ]. We have ϕ (u = κ k (u κ k ( u κ k. Since, κ k, we know that the function ψ(z = z κ k i increaing on [, ], and therefore we get that ϕ (u < if u < /2, ϕ (u =, if u = /2 and ϕ (u > if /2 < u. Since ϕ( = ϕ( =, we get that ϕ(u for all u [, ]. Therefore, (26 hold and (25 i proved. Uing the triangular inequality m m + m m, for all m, m R, we get that C 3 C 3. C 3.2 where (27 C 3. = Q Q 2 R up ( + m µ deg(r m R ( + m m µ deg(q ( + m µ deg(q 2 dm which i finite whenever µ > max(deg(q +, deg(q 2 + under the aumption (2 uing the ame etimate a in Lemma 4 of [2] (ee alo Lemma 2.2 from [], and where + (28 C 3.2 = up τ 2k exp( ν τ κ τ D(,ρ U d τ τ k ( τ k h /k exp(νh κ/k h (h x /k x / ( + (h x 2 ( + x 2 (h xx dxdh.
4 From (28 we get that C 3.2 C 3.3, where + x 2 (29 C 3.3 = up x x /k exp( νxκ/k ( x h (x h /k exp(νh κ/k By the change of variable x = h u, for u [, ], we can write ( + (h x 2 ( + x 2 dx dh. (h x k x k (3 h ( + (h x 2 ( + x 2 dx (h x k x k = h 2 k du = J ( + h 2 ( u 2 ( + h 2 u 2 ( u k u k (h. k Uing a partial fraction decompoition, we can plit J k (h = J,k (h + J 2,k (h, where (3 J,k (h = h 2 k (h 2 + 4 3 2u du ( + h 2 ( u 2 ( u k u k J 2,k (h = h 2 k (h 2 + 4 2u + du. ( + h 2 u 2 ( u k u k From now on, we aume that k 2. By contruction of J,k (h and J 2,k (h, we ee that there exit a contant j k > uch that (32 J k (h j k h 2 k (h 2 + 4 for all h >. From (29 and (32, we deduce that C 3.3 up x C3.3 (x, where x (33 C3.3 (x = ( + x 2 exp( νx κ/k j k exp(νh κ/k h 2 k (h 2 + 4 dh. From L Hopital rule, we know that lim C 3.3 (x = x + lim x + j k x 2 k (+x 2 2 x 2 +4 ν κ k x κ k ( + x 2 2x which i finite if κ k and when k 2. Therefore, we get a contant C 3.3 > uch that (34 up x C 3.3 (x C 3.3. Taking into account the etimate for (24, (27, (28, (29, (33 and (34, we obtain the reult (2. It remain to conider the cae k =. In that cae, we know from Corollary 4.9 of [9] that there exit a contant j > uch that (35 J (h j h 2 +
5 for all h. From (29 and (35, we deduce that C 3.3 up x C3.3. (x, where (36 C3.3. (x = ( + x 2 exp( νx κ From L Hopital rule, we know that lim C 3.3. (x = x + lim x + x j exp(νh κ h 2 dh. + ( + x 2 j νκx κ ( + x 2 2x which i finite whenever κ. Therefore, we get a contant C 3.3. > uch that (37 up x C 3.3. (x C 3.3.. Taking into account the etimate for (24, (27, (28, (29, (36 and (37, we obtain the reult (2 for k =. Definition 2 Let β, µ R. We denote E (β,µ the vector pace of continuou function h : R C uch that h(m (β,µ = up( + m µ exp(β m h(m m R i finite. The pace E (β,µ equipped with the norm. (β,µ i a Banach pace. Propoition 3 Let k, κ be integer uch that κ k. Let Q(X, R(X C[X] be polynomial uch that (38 deg(r deg(q, R(im for all m R. Aume that µ > deg(q +. Let m b(m be a continuou function uch that b(m R(im for all m R. Then, there exit a contant C 4 > (depending on Q, R, µ, k, κ, ν uch that (39 b(m τ k (τ k k for all f(m E (β,µ, all g(τ, m F d (ν,β,µ,k,κ. f(m m Q(im g( /k, m dm d (ν,β,µ,k,κ C 4 f(m (β,µ g(τ, m (ν,β,µ,k,κ Proof The proof follow the ame line of argument a thoe of Propoition and 2. Let f(m E (β,µ, g(τ, m F(ν,β,µ,k,κ d. We can write (4 N 2 := b(m τ k (τ k k f(m m Q(im g( /k, m dm d (ν,β,µ,k,κ = up ( + m µ + τ 2k exp(β m ν τ κ τ D(,ρ U d,m R τ b(m τ k {( + m m µ exp(β m m f(m m } {( + m µ exp(β m exp( ν κ/k + 2 /k g(/k, m } D(τ,, m, m dm d
6 where D(τ,, m, m = Q(im e β m e β m m ( + m m µ ( + m µ exp(ν κ/k + 2 /k (τ k /. Again, we know that there exit contant Q, R > uch that Q(im Q( + m deg(q, R(im R( + m deg(r for all m, m R. By mean of the triangular inequality m m + m m, we get that (4 N 2 C 4. C 4.2 f(m (β,µ g(τ, m (ν,β,µ,k,κ where and C 4. = + τ up τ 2k k exp( ν τ κ exp(νh κ/k τ D(,ρ U d τ + h 2 h k ( τ k h /k dh C 4.2 = Q R up ( + m µ deg(r m R ( + m m µ ( + m µ deg(q dm. Under the hypothei κ k and from the etimate (7, ( and (9 in the pecial cae χ 2 = /k and ν 2 =, we know that C 4. i finite. From the etimate for (27, we know that C 4.2 i finite under the aumption (38 provided that µ > deg(q +. Finally, gathering thee latter bound etimate together with (4 yield the reult (39. In the next propoition, we recall from [4], Propoition 5, that (E (β,µ,. (β,µ i a Banach algebra for ome noncommutative product introduced below. Propoition 4 Let Q (X, Q 2 (X, R(X C[X] be polynomial uch that (42 deg(r deg(q, deg(r deg(q 2, R(im, for all m R. Aume that µ > max(deg(q +, deg(q 2 +. Then, there exit a contant C 5 > (depending on Q, Q 2, R, µ uch that (43 R(im Q (i(m m f(m m Q 2 (im g(m dm (β,µ C 5 f(m (β,µ g(m (β,µ for all f(m, g(m E (β,µ. Therefore, (E (β,µ,. (β,µ become a Banach algebra for the product defined by f g(m = R(im Q (i(m m f(m m Q 2 (im g(m dm. A a particular cae, when f, g E (β,µ with β > and µ >, the claical convolution product belong to E (β,µ. f g(m = f(m m g(m dm
7 2.2 Banach pace of function with exponential growth k and decay of exponential order In thi ubection, we mainly recall ome functional propertie of the Banach pace already introduced in the work [4], Section 2. The Banach pace we conider here coincide with thoe introduced in [4] except the fact that they are not depending on a complex parameter ɛ and that the function living in thee pace are not holomorphic on a dic centered at but only on a bounded ector centered at. For thi reaon, all the propoition are given without proof except Propoition 5 which i an improved verion of Propoition and 2 of [4]. We denote Sd b an open bounded ector centered at in direction d R and S d b it cloure. Let S d be an open unbounded ector in direction d. By convention, we recall that the ector we conider throughout the paper do not contain the origin in C. Definition 3 Let ν, β, µ > be poitive real number. Let k be an integer and let d R. We denote F d (ν,β,µ,k the vector pace of continuou function (τ, m h(τ, m on ( S b d S d R, which are holomorphic with repect to τ on S b d S d and uch that h(τ, m (ν,β,µ,k = up ( + m µ + τ 2k exp(β m ν τ k h(τ, m τ S d b S τ d,m R i finite. One can check that the normed pace (F d (ν,β,µ,k,. (ν,β,µ,k i a Banach pace. Throughout the whole ubection, we aume that µ, β, ν > and k, d R are fixed. In the next lemma, we check the continuity property by multiplication operation with bounded function. Lemma 2 Let (τ, m a(τ, m be a bounded continuou function on ( S d b S d R, which i holomorphic with repect to τ on Sd b S d. Then, we have ( (44 a(τ, mh(τ, m (ν,β,µ,k for all h(τ, m F d (ν,β,µ,k. up a(τ, m τ S d b S d,m R h(τ, m (ν,β,µ,k In the next propoition, we tudy the continuity property of ome convolution operator acting on the latter Banach pace. Propoition 5 Let γ and χ 2 > be real number. Let ν 2 be an integer. We conider a holomorphic function a γ,k(τ on S b d S d, continuou on S b d S d, uch that for all τ S b d S d. a γ,k(τ ( + τ k γ If + χ 2 + ν 2 and γ ν 2, then there exit a contant C 6 > (depending on ν, ν 2, χ 2, γ uch that (45 a γ,k(τ τ k for all f(τ, m F d (ν,β,µ,k. (τ k χ 2 ν 2 f( /k, md (ν,β,µ,k C 6 f(τ, m (ν,β,µ,k
8 Proof The proof follow imilar argument to thoe in Propoition. Indeed, let f(τ, m F(ν,β,µ,k d. By definition, we have (46 a γ,k(τ τ k (τ k χ 2 ν 2 f( /k, md (ν,β,µ,k = up ( + m µ + τ 2k exp(β m ν τ k τ S d b S τ d,m R a γ,k(τ τ k {( + m µ e β m exp( ν + 2 /k f(/k, m} F(τ,, md where Therefore, F(τ,, m = exp(ν e β m ( + m µ + 2 /k (τ k χ 2 ν 2. (47 a γ,k(τ where C 6 = τ k + τ 2k up exp( ν τ k τ S d b S τ d (τ k χ 2 ν 2 f( /k, md (ν,β,µ,k C 6 f(τ, m (ν,β,µ,k ( + τ k γ τ k exp(νh + h 2 h k ( τ k h χ 2 h ν 2 dh = up F (x x where F (x = + x2 x /k exp( νx x exp(νh ( + x γ + h 2 h k +ν 2 (x h χ 2 dh. We write F (x = F (x + F 2 (x, where F (x = + x2 x /k exp( νx x/2 exp(νh ( + x γ + h 2 h k +ν 2 (x h χ 2 dh, F 2 (x = + x2 x /k exp( νx ( + x γ x x/2 exp(νh + h 2 h k +ν 2 (x h χ 2 dh. Now, we tudy the function F (x. We firt aume that < χ 2 <. In that cae, we have that (x h χ 2 (x/2 χ 2 for all h x/2 with x >. We deduce that (48 F (x + x2 x /k (x 2 χ 2 e νx x/2 e νh ( + x γ + h 2 h k +ν 2 dh ( + x 2 2 /k ( k + ν 2 + (x 2 +χ 2+ν 2 ( + x γ exp( νx 2 for all x >. Bearing in mind that + χ 2 + ν 2 and ince + x for all x, we deduce that there exit a contant K > with (49 up F (x K. x
9 We aume now that χ 2. In thi ituation, we know that (x h χ 2 x χ 2 for all h x/2, with x. Hence, (5 F (x ( + x 2 2 /k ( k + ν 2 + xχ 2 (x/2 ν 2+ ( + x γ exp( νx 2 for all x. Again, we deduce that there exit a contant K. > with (5 up F (x K.. x In the next tep, we focu on the function F 2 (x. Firt, we oberve that + h 2 + (x/2 2 for all x/2 h x. Therefore, there exit a contant K 2 > uch that (52 F 2 (x + x2 + ( x 2 2 x /k exp( νx x ( + x γ exp(νhh k +ν 2 (x h χ 2 dh x/2 x K 2 x /k ( + x γ exp( νx exp(νhh k +ν 2 (x h χ 2 dh for all x >. Now, from the etimate (8, we know that there exit a contant K 2.3 > uch that (53 F 2. (x = x exp(νhh k +ν 2 (x h χ 2 dh K 2.3 x k +ν 2 e νx for all x. From (52 we get the exitence of a contant F 2 > with (54 up F 2 (x F 2. x [,] On the other hand, we alo have that + x x for all x. Since γ ν 2 and due to (52 with (53, we get a contant ˇF 2 > with (55 up F 2 (x ˇF 2. x Gathering the etimate (47, (49, (5, (54 and (55, we finally obtain (45. The next two propoition are already tated a Propoition 3 and 4 in [4]. Propoition 6 Let k be an integer. Let Q (X, Q 2 (X, R(X C[X] uch that (56 deg(r deg(q, deg(r deg(q 2, R(im for all m R. Aume that µ > max(deg(q +, deg(q 2 +. Let m b(m be a continuou function on R uch that b(m R(im for all m R. Then, there exit a contant C 7 > (depending on Q, Q 2, R, µ, k, ν uch that (57 b(m τ k (τ k k ( for all f(τ, m, g(τ, m F d (ν,β,µ,k. Q (i(m m f(( x /k, m m Q 2 (im g(x /, m ( xx dxdm d (ν,β,µ,k C 7 f(τ, m (ν,β,µ,k g(τ, m (ν,β,µ,k
2 Propoition 7 Let k be an integer. Let Q(X, R(X C[X] be polynomial uch that (58 deg(r deg(q, R(im for all m R. Aume that µ > deg(q +. Let m b(m be a continuou function uch that b(m R(im for all m R. Then, there exit a contant C 8 > (depending on Q, R, µ, k, ν uch that (59 b(m τ k (τ k k for all f(m E (β,µ, all g(τ, m F d (ν,β,µ,k. f(m m Q(im g( /k, m dm d (ν,β,µ,k C 8 f(m (β,µ g(τ, m (ν,β,µ,k 3 Laplace tranform, aymptotic expanion and Fourier tranform We recall a definition of k Borel ummability of formal erie with coefficient in a Banach pace which i a lightly modified verion of the one given in [], Section 3.2, that wa introduced in [4]. All the propertie tated in thi ection are already contained in our previou work [4]. Definition 4 Let k be an integer. Let m k (n be the equence defined by for all n. A formal erie m k (n = Γ( n k = t n k e t dt ˆX(T = a n T n T E[[T ]] n= with coefficient in a Banach pace (E,. E i aid to be m k ummable with repect to T in the direction d [, 2π if i there exit ρ R + uch that the following formal erie, called a formal m k Borel tranform of ˆX B mk ( ˆX(τ a n = Γ( n τe[[τ]], k τn i abolutely convergent for τ < ρ. n= ii there exit δ > uch that the erie B mk ( ˆX(τ can be analytically continued with repect to τ in a ector S d,δ = {τ C : d arg(τ < δ}. Moreover, there exit C > and K > uch that B mk ( ˆX(τ E Ce K τ k for all τ S d,δ.
2 If thi i o, the vector valued m k Laplace tranform of B mk ( ˆX(τ in the direction d i defined by L d m k (B mk ( ˆX(T = k B mk ( ˆX(ue (u/t du k L γ u, along a half-line L γ = R + e iγ S d,δ {}, where γ depend on T and i choen in uch a way that co(k(γ arg(t δ >, for ome fixed δ. The function L d m k (B mk ( ˆX(T i well defined, holomorphic and bounded in any ector S d,θ,r /k = {T C : T < R /k, d arg(t < θ/2}, where π k < θ < π k + 2δ and < R < δ /K. Thi function i called the m k um of the formal erie ˆX(T in the direction d. In the next propoition, we give ome identitie for the m k Borel tranform that will be ueful in the equel. Propoition 8 Let ˆf(t = n f nt n, ĝ(t = n g nt n be formal erie whoe coefficient f n, g n belong to ome Banach pace (E,. E. We aume that (E,. E i a Banach algebra for ome product. Let k, m be integer. The following formal identitie hold. (6 B mk (t k+ t ˆf(t(τ = kτ k B mk ( ˆf(t(τ (6 B mk (t m ˆf(t(τ = τ k and Γ( m k τ k (τ k m k B mk ( ˆf(t( /k d τ k (62 B mk ( ˆf(t ĝ(t(τ = τ k B mk ( ˆf(t((τ k /k B mk (ĝ(t( / (τ k d. In the following propoition, we recall ome propertie of the invere Fourier tranform Propoition 9 Let f E (β,µ with β >, µ >. The invere Fourier tranform of f i defined by F + (f(x = (2π /2 f(m exp(ixmdm for all x R. The function F (f extend to an analytic function on the trip (63 H β = {z C/ Im(z < β}. Let φ(m = imf(m E (β,µ. Then, we have (64 z F (f(z = F (φ(z for all z H β. Let g E (β,µ and let ψ(m = (2π /2 f g(m, the convolution product of f and g, for all m R. From Propoition 4, we know that ψ E (β,µ. Moreover, we have (65 F (f(zf (g(z = F (ψ(z for all z H β.
22 4 Formal and analytic olution of convolution initial value problem with complex parameter 4. Formal olution of the main convolution initial value problem Let,, D 2 be integer uch that >. Let δ l be integer uch that (66 = δ, δ l < δ l+, for all l D. For all l D, let d l, l be nonnegative integer uch that (67 d l > δ l, l d l + δ l. Let Q(X, Q (X, Q 2 (X, R l (X C[X], l D, be polynomial uch that (68 deg(q deg(r D deg(r l, deg(r D deg(q, deg(r D deg(q 2, Q(im, R l (im, R D (im for all m R, all l D. We conider equence of function m C,n (m, ɛ, for all n and m F n (m, ɛ, for all n, that belong to the Banach pace E (β,µ for ome β > and µ > max(deg(q +, deg(q 2 + and which depend holomorphically on ɛ D(, ɛ for ome ɛ >. We aume that there exit contant K, T > uch that (69 C,n (m, ɛ (β,µ K ( T n for all n, for all ɛ D(, ɛ. We define C (T, m, ɛ = n C,n(m, ɛt n which i a convergent erie on D(, T /2 with value in E (β,µ and F (T, m, ɛ = n F n(m, ɛt n which i a formal erie with coefficient in E (β,µ. Let c,2 (ɛ, c (ɛ, c, (ɛ and c F (ɛ be bounded holomorphic function on D(, ɛ which vanih at the origin ɛ =. We conider the following initial value problem (7 Q(im( T U(T, m, ɛ T (δ D ( + δ D T R D (imu(t, m, ɛ = ɛ c,2(ɛ (2π /2 Q (i(m m U(T, m m, ɛq 2 (im U(T, m, ɛdm D + + ɛ c (ɛ (2π /2 + ɛ c,(ɛ (2π /2 l= for given initial data U(, m, ɛ. R l (imɛ l d l +δ l T d l δ l T U(T, m, ɛ Propoition There exit a unique formal erie C (T, m m, ɛr (im U(T, m, ɛdm C, (m m, ɛr (im U(T, m, ɛdm + ɛ c F (ɛf (T, m, ɛ Û(T, m, ɛ = n U n (m, ɛt n olution of (7 with initial data U(, m, ɛ, where the coefficient m U n (m, ɛ belong to E (β,µ for β > and µ > max(deg(q +, deg(q 2 + given above and depend holomorphically on ɛ in D(, ɛ.
23 Proof From Propoition 4 and the condition tated above, we get that the coefficient U n (m, ɛ of Û(T, m, ɛ are well defined, belong to E (β,µ for all ɛ D(, ɛ, all n and atify the following recurion relation (7 (n + U n+ (m, ɛ + ɛ Q(im = R D(im Q(im Πδ D j= (n + δ D (δ D ( + ju n+δd (δ D ( +(m, ɛ c,2 (ɛ + (2π /2 Q (i(m m U n (m m, ɛq 2 (im U n2 (m, ɛdm n +n 2 =n,n,n 2 + ɛ Q(im D + l= R l (im ( ɛ l d l +δ l Π δ l j= Q(im (n + δ l d l j U n+δl d l (m, ɛ n +n 2 =n,n,n 2 + ɛ c, (ɛ (2π /2 Q(im c (ɛ (2π /2 C,n (m m, ɛr (im U n2 (m, ɛdm C, (m m, ɛr (im U n (m, ɛdm + ɛ c F (ɛ Q(im F n(m, ɛ for all n max(max l D d l, (δ D ( +. 4.2 Analytic olution for an auxiliary convolution problem reulting from a m k Borel tranform applied to the main convolution initial value problem We make the additional aumption that (72 d l > (δ l ( + for all l D. Uing the formula (8.7 from [3], p. 363, we can expand the operator T δ l( + δ l T in the form (73 T δ l( + δ l T = (T k+ T δ l + A δl,pt (δ l p (T k+ T p p δ l where A δl,p, p =,..., δ l are real number, for all l D. We define integer d l, > in order to atify (74 d l + + = δ l ( + + d l, for all l D. We alo rewrite (δ D ( + = (δ D ( + + (δ D (. Multiplying the equation (7 by T + and uing (73, we can rewrite the equation (7 in
24 the form (75 Q(im(T + T U(T, m, ɛ + R D (im + ɛ T + c,2(ɛ (2π /2 + ɛ T + c,(ɛ (2π /2 + = R D (imt (δ D ( (T k+ T δ D U(T, m, ɛ A δd,pt (δ D ( T (δ D p (T k+ T p U(T, m, ɛ p δ D D + R l (im l= p δ l + ɛ T + c (ɛ (2π /2 Q (i(m m U(T, m m, ɛq 2 (im U(T, m, ɛdm ( ɛ l d l +δ l T d l, (T + T δ l U(T, m, ɛ A δl,p ɛ l d l +δ l T (δ l p+d l, (T k+ T p U(T, m, ɛ C (T, m m, ɛr (im U(T, m, ɛdm C, (m m, ɛr (im U(T, m, ɛdm + ɛ c F (ɛt + F (T, m, ɛ. We denote ω k (τ, m, ɛ the formal m k Borel tranform of Û(T, m, ɛ with repect to T, ϕ (τ, m, ɛ the formal m k Borel tranform of C (T, m, ɛ with repect to T and ψ k (τ, m, ɛ the formal m k Borel tranform of F (T, m, ɛ with repect to T. More preciely, τ n ω k (τ, m, ɛ = U n (m, ɛ Γ( n n, ϕ (τ, m, ɛ = C,n (m, ɛ Γ( n n τ n τ n ψ k (τ, m, ɛ = F n (m, ɛ Γ( n n. Uing (69 we get that for any κ, the function ϕ k (τ, m, ɛ belong to F(ν,β,µ,k d,κ for all ɛ D(, ɛ, any unbounded ector U d centered at with biecting direction d R, for ome ν >. Indeed, we have that (76 ϕ k (τ, m, ɛ (ν,β,µ,k,κ By uing the claical etimate + C,n (m, ɛ (β,µ ( up τ 2k n τ D(,ρ U d τ (77 up x m exp( m 2 x = ( m m e m x m 2 exp( ν τ κ τ n Γ( n. for any real number m, m 2 > and Stirling formula Γ(n/ (2π /2 (n/ n 2 e n/ a n tend to +, we get two contant A, A 2 > depending on ν,, κ uch that + (78 up τ 2k exp( ν τ κ τ n τ D(,ρ U d τ Γ( n = up x ( ( n νκ n κ e n κ + ( n νκ + 2 νκ n ( + x 2k/κ x n κ κ + 2 κ e νx Γ( n e ( n κ + 2 κ /Γ(n/ A (A 2 n
25 for all n. Therefore, if the inequality A 2 < T hold, we get the etimate (79 ϕ k (τ, m, ɛ (ν,β,µ,k,κ A n C,n (m, ɛ (β,µ (A 2 n A A 2 K T A. 2 T On the other hand, we make the aumption that ψ k (τ, m, ɛ F(ν,β,µ,k d,, for all ɛ D(, ɛ, for ome unbounded ector U d with biecting direction d R, where ν i choen above. We will make the convention to denote ψk d the analytic continuation of the convergent power erie ψ k on the domain U d D(, ρ. In particular, we get that ψk d (τ, m, ɛ F(ν,β,µ,k d,κ for any κ. We alo aume that there exit a contant ζ ψk > uch that (8 ψ d (τ, m, ɛ (ν,β,µ,k, ζ ψk for all ɛ D(, ɛ. In particular, we notice that (8 ψ d (τ, m, ɛ (ν,β,µ,k,κ ζ ψk for any κ. We require that there exit a contant r Q,Rl > uch that (82 Q(im R l (im r Q,R l for all m R, all l D. Uing the computation rule for the formal m k Borel tranform in Propoition 8, we deduce
26 the following equation atified by ω k (τ, m, ɛ, (83 Q(im( τ ω k (τ, m, ɛ + = R D (im D + p δ l ( c (ɛ (2π /2 + ɛ τ Γ( + τ Γ( (δ D ( τ + R D (im ( c,2 (ɛ (2π /2 τ p δ D (τ (δ D ( k δ D δ D ω k ( /, m, ɛ d A δd,p τ Γ( (δ D ( + (δ D p (τ (δ D ( + (δ D p k p p ω k ( /, m, ɛ d + ɛ τ Γ( + τ (τ / Q (i(m m ω k (( x /, m m, ɛ Q 2 (im ω k (x / d, m, ɛ ( xx dxdm R l (im ɛ l d l +δ l τ k τ (τ l= Γ( d l, A δl,pɛ l d l +δ l τ τ Γ( d l, + δ l p τ (τ τ d l, δ (k l δ l ω k ( /, m, ɛ d d l, +δ k l p (k p p ω k ( /, m, ɛ d + ɛ τ Γ( + (τ / ϕ k (( x /, m m, ɛr (im ω k (x / d, m, ɛ ( xx dxdm (τ / c,(ɛ (2π /2 ( + ɛ c F (ɛ Γ( + C, (m m, ɛr (im ω k ( /, m, ɛdm d τ τ (τ / ψ d ( /, m, ɛ d. In the next propoition, we give ufficient condition under which the equation (83 ha a olution ωk d (τ, m, ɛ in the Banach pace F(ν,β,µ,k d,κ where β, µ are defined above and for well choen κ >. Propoition Under the aumption that (84 κ =, d l + ( δ l d l + ( δ l ( + for all l D, there exit radii r Q,Rl >, l D, a contant ϖ > and contant ζ,2, ζ,, ζ, ζ, ζ,, ζ F, ζ 2 > (depending on Q, Q 2,, µ, ν, ɛ, R l, l, δ l, d l for l D