Multisummability of formal solutions to the Cauchy problem for some linear partial differential equations

J. Differential Equations 255 (2013) 3592–3637

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Journal of Differential Equations www.elsevier.com/locate/jde

Multisummability of formal solutions to the Cauchy problem for some linear partial differential equations Hidetoshi Tahara a,∗,1 , Hiroshi Yamazawa b a b

Department of Information and Communication Sciences, Sophia University, Kioicho, Chiyoda-ku, Tokyo 102-8554, Japan College of Engineer and Design, Shibaura Institute of Technology, Minuma-ku, Saitama-shi, Saitama 337-8570, Japan

a r t i c l e

i n f o

Article history: Received 1 February 2013 Revised 25 July 2013 Available online 13 August 2013 MSC: primary 35C10 secondary 35A10, 35A20

a b s t r a c t The paper considers the Cauchy problem for linear partial differential equations of non-Kowalevskian type in the complex domain. It is shown that if the Cauchy data are entire functions of a suitable order, the problem has a formal solution which is multisummable. The precise bound of the admissible order of entire functions is described in terms of the Newton polygon of the equation. © 2013 Elsevier Inc. All rights reserved.

Keywords: Multisummability Formal solution Non-Kowalevskian equation Linear partial differential equation

1. Introduction The summability or the multisummability of formal power series has been studied quite well in the theory of ordinary differential equations: in particular, by Balser, Braaksma, Ramis and Sibuya [3] and Braaksma [6,7] it is shown that every formal power series solution of ordinary differential equation is multisummable. Recently, this theory has come to be applied to some partial differential equations. Lutz, Miyake and Schäfke [18] have given a necessary and sufficient condition for a formal solution of the complex heat equation to be Borel summable: this result is extended to the case of variable coefficients by Balser and Loday [5], and to general partial differential equations with constant coefficients by

*

Corresponding author. E-mail addresses: [email protected] (H. Tahara), [email protected] (H. Yamazawa). 1 The first author is partially supported by the Grant-in-Aid for Scientific Research No. 22540206 of Japan Society for the Promotion of Science. 0022-0396/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jde.2013.07.061

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Miyake [26], Balser [4] and Michalik [24]. Ouchi [28,29] showed that formal power series solutions of general partial differential equations are multisummable under the condition that the equation is a perturbation type of ordinary differential equations. Hibino [12–14] showed the Borel summability of formal solutions for some singular first-order partial differential equations, Malek [19–21] discussed summability of formal solutions for various linear and nonlinear partial differential equations, and Yoshino [36] showed the summability for some Hamiltonian systems. Chen, Luo and Zhang [9] and Luo, Zhang and Chen [17] have shown the summability of formal power series solution for some singular partial differential equations discussed by Chen, Luo and Tahara [8]. In these papers, a formal solution or a formal power series solution is considered in the space O R [[t ]] which is the ring of formal power series in t with coefficients in O R , where O R denotes the set of all holomorphic functions on D R = {x ∈ C N ; |x| < R }. In this paper, we will consider the Cauchy problem for linear partial differential equation

 ⎧ j m ⎪ a j ,α (t )∂t ∂xα u = f (t , x), ⎨ ∂t u + ⎪ ⎩

j +|α | L

j  ∂t u t =0

(1.1)

= ϕ j (x),

j = 0, 1 , . . . , m − 1

(where 1  m  L) with holomorphic coefficients a j ,α (t ) ( j + |α |  L) in a neighborhood of t = 0 under the following assumption:

ordt (a j ,α )  max{0, j − m + 1} for any ( j , α ) (where ordt (a j ,α ) denotes the order of the zero of the function a j ,α (t ) at t = 0). It is easy to see that for any holomorphic functions f (t , x) and ϕ j (x) ( j = 0, 1, . . . , m − 1) in a neighborhood of (t , x) = (0, 0) and x = 0, respectively, the Cauchy problem (1.1) has a unique formal solution

uˆ (t , x) =

∞ 

un (x)t n ∈ O R [[t ]]

(1.2)

n =0

for some R > 0. In the case m  L this formal solution is convergent, but in the case m < L this is not convergent in general. The purpose of this paper is to show the following result: if f (t , x) and ϕ j (x) ( j = 0, 1, . . . , m − 1) belong to a suitable class of entire functions, this formal solution is multisummable in a suitable direction. The motivation comes from the following examples. Example 1.1. Let us consider the complex heat equation:

∂t u = ∂x2 u , We have a unique formal solution uˆ (t , x) =



n0

u |t =0 = ϕ (x).

(1.3)

ϕ (2n) (x)t n /n!. We know:

(1) (See [16].) The formal solution uˆ (t , x) is convergent in a neighborhood of (0, 0) ∈ C2 , if and only if ϕ (x) is an entire function of order at most 2. (2) (See [18].) The formal solution uˆ (t , x) is Borel summable in a direction d if and only if ϕ (x) can be analytically continued to infinity in directions d/2 and π + d/2, and is of exponential order at most 2 when x is going to infinity in these directions. Example 1.2. Let us consider

∂t u = ∂x2 u + t (t ∂t )3 u ,

u |t =0 = ϕ (x).

(1.4)

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This equation has a unique formal solution uˆ (t , x) of type (1.2). We have: (1) In the case ϕ (x) = exp(x2 ), the formal solution uˆ (t , x) is divergent. (2) If ϕ (x) is an entire function of finite order, the formal solution uˆ (t , x) is Borel summable in any direction d ∈ (−π , 0) ∪ (0, π ). See also Yamazawa [35]. Example 1.3. Let us consider

∂t u = ∂x3 u + t (t ∂t )3 u ,

u |t =0 = ϕ (x).

(1.5)

This equation has a unique formal solution uˆ (t , x) of type (1.2). We have: (1) In the case ϕ (x) = exp(x2 ), the formal solution uˆ (t , x) is divergent. (2) If ϕ (x) is an entire function of order less than 3, the formal solution uˆ (t , x) is Borel summable in any direction d ∈ (−π , 0) ∪ (0, π ). In the case (1.3) the admissible exponential order is at most 2, in the case (1.4) any exponential order is admissible, and in the case (1.5) the admissible exponential order is less than 3. By looking at these examples, the authors have come to be interested in the following problem: what determine the bound of the admissible order (2, ∞ or 3)? In this paper, we will give an answer to this problem. The paper is organized as follows. In Section 2, we will give a brief survey of Laplace transform, Borel transform and multisummability of formal series. In Section 3, we will state the main theorem which gives a sufficient condition for the formal solution of (1.1) to be multisummable. In this theorem, the t-Newton polygon plays an important role to describe the class of admissible exponents of entire functions. The rest Sections 4–8 are used to prove this main theorem. We will use the notations: N = {0, 1, 2, . . .} and N∗ = {1, 2, . . .}. 2. Preliminaries In this section, we will recall some basic results on Laplace transform, Borel transform and multisummability of formal series. For the details of these topics and the proofs of some results, readers can refer to Malgrange [22], Martinet and Ramis [23], Braaksma [6,7], Balser [1,2] and Ouchi [28,29]. As to Laplace and Borel transforms, we will use the definitions in [28]. If K = (φ1 , φ2 ) is an open interval, we write | K | = φ2 − φ1 : for a > 0 we write K + [a] = (φ1 − a, φ2 + a) and K − [a] = (φ1 + a, φ2 − a) (if 0 < a < | K |/2). We denote by R(C \ {0}) the universal covering space of C \ {0}. For two subsets A and B in the complex plain, we write A  B if the closure of A is contained in the interior of B. For an open subset U of CxN we denote by O (U ) the set of all holomorphic functions f (x) on U , and by O (U )[[t ]] the ring of formal power series in t with holomorphic coefficients on U . 2.1. Laplace transform Let I = (θ1 , θ2 ) be a nonempty open interval, and let U be an open subset of C N : we set S I = {ξ ∈ R(C \ {0}); θ1 < arg ξ < θ2 } and S I (r ) = {ξ ∈ S I ; 0 < |ξ | < r } for r > 0. For k > 0 we denote by E k ( S I × U ) the set of all holomorphic functions f (ξ, x) on S I × U satisfying the estimate

 

 f (ξ, x)  A |ξ |a−k exp b|ξ |k on S I × U

(2.1)

for some A > 0, a > 0 and b > 0. For f (ξ, x) ∈ E k ( S I × U ), we define the k-Laplace transform Lk [ f ](t , x) of f (ξ, x) with respect to ξ as follows:

H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

∞  eiθ



3595



exp −(ξ/t )k f (ξ, x) dξ k

Lk [ f ](t , x) = 0

where θ and t are taken so that θ ∈ I and |θ − arg t | < π /2k hold. Then, Lk [ f ](t , x) is well-defined as a holomorphic function on W I ,k (1/b) × U , where b > 0 is the one in (2.1) and

W I ,k (1/b) =











t ∈ R C \ {0} ; t k − (1/2b)e ikθ  < 1/2b .

θ ∈I

By the definition we can see: for any sufficiently small > 0 there is an r > 0 such that S I +[π /2k− ] (r ) ⊂ W I ,k (1/b). Moreover, if f (ξ, x) satisfies (2.1) we have the estimate |Lk [ f ](t , x)|  C |t |a on S I +[π /2k− ] (r ) × U for some C > 0. 2.2. Borel transform Conversely, if F (t , x) is a holomorphic function on W I ,k (c ) × U for some c > 0 and if the estimate

   F (t , x)  C |t |a on W I ,k (c ) × U

(2.2)

holds for some C > 0 and a > 0, we define k-Borel transform Bk [ F ](ξ, x) of F (t , x) with respect to t as follows:

Bk [ F ](ξ, x) =



1 2π i





exp (ξ/t )k F (t , x) dt −k

C (ξ )

where C (ξ ) is a contour in W I ,k (c ) that starts from 0e i (arg ξ +π /2k+δ2 ) and ends to 0e i (arg ξ −π /2k−δ1 ) with 0 < δ2 < min{θ2 − arg ξ, π /k} and 0 < δ1 < min{arg ξ − θ1 , π /k}. Then, Bk [ F ](ξ, x) is well-defined as a holomorphic function on S I × U and we have the following estimate: for any > 0 (sufficiently small) and b > 1/c there is an A > 0 such that |Bk [ F ](ξ, x)|  A |ξ |a−k exp(b|ξ |k ) on S I −[ ] × U . The following result is well known: Theorem 2.1 (Inversion formula). (1) For any f (ξ, x) ∈ E k ( S I × U ) we have (Bk ◦ Lk [ f ])(ξ, x) = f (ξ, x) on S I × U . (2) For any holomorphic function F (t , x) on W I ,k (c ) × U satisfying the estimate (2.2), we have (Lk ◦ Bk [ F ])(t , x) = F (t , x) on W I ,k (c ) × U . Let a > 0: for a formal power series

uˆ (t , x) =



un (x)t a+n ∈ t a O (U )[[t ]]

n0

we define the formal k-Borel transform Bˆk [uˆ ](ξ, x) of uˆ (t , x) as follows:

Bˆk [uˆ ](ξ, x) =

 n0

un (x)

ξ a+n−k ∈ ξ a−k O(U )[[ξ ]]. Γ ((a + n)/k)

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2.3. Acceleration operator Let 0 < k1 < k2 . We define the operator Ak2 ,k1 by

Ak2 ,k1 [ f ](ξ, x) = Bk2 ◦ Lk1 [ f ](ξ, x),

f (ξ, x) ∈ E k1 ( S I × U ).

(2.3)

This operator is called the acceleration operator (or accelerator): it was introduced by Ecalle [10]. We define κ > 0 by the relation 1/κ = 1/k1 − 1/k2 . If we substitute for Lk1 and Bk2 their integral expressions and then change the order of integration we have ∞  eiθ

G k2 ,k1 (ξ, τ ) f (τ , x) dτ k1

Ak2 ,k1 [ f ](ξ, x) =

(2.4)

0

where θ ∈ I and

G k2 ,k1 (ξ, τ ) =

1 2π i







exp (ξ/t )k2 − (τ /t )k1 dt −k2

C (ξ )

which is the k2 -Borel transform of exp(−(τ /t )k1 ) with respect to t. Thus, instead of (2.3) we can adopt (2.4) as a definition of the acceleration operator Ak2 ,k1 . About the kernel function G k2 ,k1 (ξ, τ ) we can see that G k2 ,k1 (ξ, τ ) is a holomorphic function on R(Cξ \ {0}) × R(Cτ \ {0}) satisfying the following: for any > 0 there are C = C > 0 and b = b > 0 such that the estimate

 G k

2 ,k1



κ

C (ξ, τ )  k exp −b |τ |/|ξ | 2 |ξ |

holds on {(ξ, τ ) ∈ R(Cξ \ {0}) × R(Cτ \ {0}); |arg(ξ/τ )|  π /2κ − }. The following result is due to Ecalle. Proposition 2.2. Let f (ξ, x) be a holomorphic function on S I × U satisfying the estimate

 

 f (ξ, x)  A |ξ |a−k1 exp c |ξ |κ on S I × U for some A > 0, a > 0 and c > 0. Then, for any sufficiently small > 0 there is an r > 0 such that Ak2 ,k1 [ f ](ξ, x) is well-defined as a holomorphic function on S I +[π /2κ − ] (r ) × U , and we have the estimate

  Ak ,k [ f ](ξ, x)  C |ξ |a−k2 2 1

on S I +[π /2κ − ] (r ) × U .

2.4. Multisummability Let q ∈ N∗ and

0 < k1 < k2 < · · · < kq < kq+1 = ∞, 1/κi = 1/ki − 1/ki +1 We have

(1  i  q).

κi > 0 (i = 1, 2, . . . , q) and κq = kq . Let μ ∈ N∗ , I be an open interval, and let

(2.5)

H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

uˆ (t , x) =



3597

un (x)t n ∈ t μ O (U )[[t ]]:

nμ

then the formal k1 -Borel transform w 1 (ξ, x) of uˆ (t , x) is given by

w 1 (ξ, x) =

 nμ

un (x) n−k1 ∈ ξ μ−k1 C[[ξ ]]. ξ Γ (n/k1 )

(2.6)

Definition 2.3. We say that the series uˆ (t , x) is (kq , . . . , k1 )-summable in the I -direction, if there are intervals I i (i = 1, . . . , q) with I q = I satisfying

I i +1  I i + [π /2κi ] (i = 1, 2, . . . , q − 1)

(2.7)

and if the following two requirements are satisfied. 1) The formal power series ξ k1 −μ w 1 (ξ, x) is convergent on D δ × U for some δ > 0 and so w 1 (ξ, x) defines a holomorphic function on R( D δ \ {0}) × U . 2) For i = 1, 2, . . . , q, respectively, the function w i (ξ, x) has an analytic continuation w ∗i (ξ, x) on S I i × U satisfying the estimate

 ∗ 

 w (ξ, x)  A i |ξ |μ−ki exp c i |ξ |κi on S I × U i i

(2.8)

for some A i > 0 and c i > 0, and if i = q we define





w i +1 (ξ, x) = Aki+1 ,ki w ∗i (ξ, x) on S I i+1 (r i +1 ) × U for some r i +1 > 0 (see the explanation given below). The situation is as follows:

w 1 (ξ, x) = Bˆk1 [uˆ ](ξ, x)







w 2 (ξ, x) = Ak2 ,k1 w ∗1 (ξ, x)

.. .





on R D δ \ {0} × U ,



on S I 2 (r2 ) × U ,

w q (ξ, x) = Akq ,kq−1 w q∗−1 (ξ, x)

on S I q (rq ) × U .

If w i (ξ, x) is a holomorphic function on S I i × U and if it satisfies (2.8), by Proposition 2.2 we see: for any > 0 there is an r > 0 such that w i +1 (ξ, x) = Aki+1 ,ki [ w ∗i ](ξ, x) is well-defined on S I i +[π /2κi − ] (r ) × U and satisfies

   w i +1 (ξ, x)  C i +1 |ξ |μ−ki+1

on S I i +[π /2κi − ] (r ) × U

for some C i +1 > 0. Since I i +1  I i + [π /2κi ] is supposed in (2.7) we have I i +1 ⊂ I i + [π /2κi − ] for some > 0: this shows that the function w i +1 (ξ, x) is well-defined on S I i+1 (r i +1 ) × U for some r i +1 > 0. Thus, we have seen that the above requirement 2) makes sense. Since κq = kq and I q = I hold, by (2.8) with i = q we have

 ∗ 

 w (ξ, x)  A q |ξ |μ−kq exp cq |ξ |kq on S I × U ; q

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hence we can define the kq -Laplace transform





u ∗ (t , x) = Lkq w q∗ (t , x) on W I ,kq (1/cq ) of w q∗ (ξ, x). This function u ∗ (t , x) is called the (kq , . . . , k1 )-sum of uˆ (t , x). It is a holomorphic function on W I ,kq (1/cq ). Roughly, it is given by

u ∗ (t , x) = Lkq ◦ Akq ,kq−1 ◦ · · · ◦ Ak2 ,k1 [ w 1 ](t , x) where w 1 (ξ, x) is the one in (2.6). The following result is a consequence of Theorem 22 and Theorem 55 in [2]: Proposition 2.4. Let uˆ (t , x) be the formal series in (2.5) and suppose that it is (kq , . . . , k1 )-summable in the I -direction. Let u ∗ (t , x) be the (kq , . . . , k1 )-sum of uˆ (t , x). Then, for any sufficiently small > 0 there are r > 0, C > 0 and h > 0 such that the asymptotic relation

  N −1     ∗  un (x)t n   Ch N Γ ( N /k1 )|t | N u (t , x) −   n =μ holds on S I +[π /2kq − ] (r ) × U for any N  μ.



We may say that the formal series uˆ (t , x) = n0 un (x)t n ∈ O (U )[[t ]] is (kq , . . . , k1 )-summable in I -direction, if the formal series nμ un (x)t n is (kq , . . . , k1 )-summable in I -direction for some μ ∈ N∗ . In the case q  2, (kq , . . . , k1 )-summability is often called (kq , . . . , k1 )-multisummability. In the case q = 1, (k1 )-summability is the same as the k1 -summability of Ramis [30]. We say that uˆ (t , x) is Borel summable in the direction d if it is 1-summable in I -direction for some interval I with d ∈ I . 3. Main theorem In this section, we will give a statement of the main theorem of this paper. First, we explain our problem, then we define the t-Newton polygon of the equation and the characteristic polynomials on the boundary of the Newton polygon, and finally we state our main result. 3.1. Problem Let (t , x) = (t , x1 , . . . , x N ) be the complex variable in Ct × CxN . We write D r = {t ∈ C; |t | < r }; for W = D r , C N or D r × C N we denote by O ( W ) the set of all holomorphic functions on W . For a function g (t ) ∈ O ( D r ) we denote by ordt ( g ) the order of the zero of the function g (t ) at t = 0. We consider the Cauchy problem for linear partial differential equation

 ⎧ j m ⎪ a j ,α (t )∂t ∂xα u = f (t , x), ⎨ ∂t u + (E)

⎪ ⎩

( j ,α )∈Λ

j  ∂t u t =0

= ϕ j (x),

j = 0, 1 , . . . , m − 1 ,

under the following assumptions: m  1 is an integer, Λ is a finite subset of N × N N , a j ,α (t ) ∈ O ( D r ) and a j ,α (t ) ≡ 0 (( j , α ) ∈ Λ), f (t , x) ∈ O ( D r × C N ), and ϕ j (x) ∈ O (C N ) ( j = 0, 1, . . . , m − 1). It is easy to see: if the condition (A1 )

ordt (a j ,α )  max{0, j − m + 1} for any ( j , α ) ∈ Λ

H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

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Fig. 1. t-Newton polygon of (E).

holds, Eq. (E) has a unique formal solution

u (t , x) =







un (x)t n ∈ O C N [[t ]].

(3.1)

n0

If Λ ⊂ {( j , α ) ∈ N × N N ; j + |α |  m}, Eq. (E) is Kowalevskian and the formal solution u (t , x) is convergent in a neighborhood of (0, 0) ∈ Ct × CxN . If otherwise, that is, if

j + |α | > m

for some ( j , α ) ∈ Λ,

(3.2)

Eq. (E) is non-Kowalevskian and the formal solution is not convergent in general, as is seen in Examples 1.2 and 1.3. Thus, our problem is: Problem 3.1. Discuss the summability (or the multisummability) of this formal solution in the case (3.2). 3.2. Newton polygon with respect to t Let us define the t-Newton polygon of Eq. (E) (or the Newton polygon of Eq. (E) with respect to t). For (a, b) ∈ R2 , we write C (a, b) = {(x, y ); x  a, y  b}. The t-Newton polygon N t (E) of Eq. (E) is defined by the convex hull of the union of sets C (m, −m) and C ( j , ordt (a j ,α ) − j ) (( j , α ) ∈ Λ); that is,

 N t (E) = the convex hull of C (m, −m) ∪





C j , ordt (a j ,α ) − j



.

( j ,α )∈Λ

The figure of N t (E) can be drawn as in Fig. 1. As is seen in Fig. 1, the vertices of N t (E) consist of p ∗ + 1 points

(l0 , e 0 ), (l1 , e 1 ), (l2 , e 2 ), . . . , (l p ∗ −1 , e p ∗ −1 ), (l p ∗ , e p ∗ ), and the boundary of N t (E) consists of a horizontal half line Γ0 , p ∗ -segments Γ1 , Γ2 , . . . , Γ p ∗ , and a vertical half line Γ p ∗ +1 . We denote the slope of Γi by ki (i = 0, 1, 2, . . . , p ∗ + 1); then we have

k0 = 0 < k1 < k2 < · · · < k p ∗ < k p ∗ +1 = ∞.

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Since (A1 ) is supposed, we have ordt (a j ,α )  j − m + 1 and so ( j , ordt (a j ,α ) − j ) ∈ N t (E) ∩ {(x, y ); y  −m + 1}: this means that (l0 , e 0 ) = (m, −m) holds. We denote by ( N t (E))◦ the interior of the set N t (E). From now on, we suppose the following condition: (A2 )

( j , α ) ∈ Λ and |α | > 0



⇒





◦

j , ordt (a j ,α ) − j ∈ N t (E)

which is equivalent to





j , ordt (a j ,α ) − j ∈

p ∗ +1



Γi

⇒

|α | = 0 .

i =1

Remark 3.2. We remark that our definition of the t-Newton polygon is different from that of the Newton polygon in Miyake [25] and from that of the characteristic polygon in Ouchi [28]: in our j j definition, the operator t q ∂t ∂xα corresponds to C ( j , q − j ); while in [25] and [28] the operator t q ∂t ∂xα corresponds to C ( j + |α |, q − j ). 3.3. Characteristic polynomial on Γi In the case p ∗  1, let us define the characteristic polynomial on Γi (i = 1, 2, . . . , p ∗ ). For i = 1, 2, . . . , p ∗ , we set



Ii = ( j , 0) ∈ Λ; j , ordt (a j ,0 ) − j ∈ Γi ,

i = 1, 2, . . . , p ∗ .

For ( j , 0) ∈ I1 ∪ I2 ∪ · · · ∪ I p ∗ we set n j ,0 = ordt (a j ,0 ); then we have

a j ,0 (t ) = t n j,0 a0j ,0 (t )

with a0j ,0 (0) = 0

for some holomorphic function a0j ,0 (t ) ∈ O ( D r ). We set

p1 (λ) =

 ( j ,0)∈I1



P 1 (λ) =

a0j ,0 (0)λ j −m + 1 = al01 ,0 (0)λl1 −m + · · · + 1, a0j ,0 (0)λ j + λm = p1 (λ) × λm

( j ,0)∈I1

(the last term 1 of p1 (λ) corresponds to ∂tm ), and for 2  i  p ∗ we set

pi (λ) =

 ( j ,0)∈Ii

P i (λ) =



a0j ,0 (0)λ j −li−1 = al0,0 (0)λli −li−1 + · · · + al0 ,0 (0), i i −1 a0j ,0 (0)λ j = pi (λ) × λli−1 .

( j ,0)∈Ii

We call pi (λ) the characteristic polynomial on Γi and we denote by

λi ,1 , . . . , λi ,li −li−1 the roots of pi (λ) = 0 that are called the characteristic roots on Γi . Since al0,0 (0) = 0 and al0 ,0 (0) = 0 i i −1 hold, we have

H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

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λi ,d = 0 for all 1  i  p ∗ and 1  d  li − li −1 . We set li −li −1 

Ξi =



arg λi ,d + 2π j ki

d =1

 ; j = 0, ±1, ±2, . . . ,

1  i  p∗ ,

1  i  p∗ .

Zi = R \ Ξi ,

(3.3) (3.4)

These sets will play a very important role to determine the direction in which the formal solution (3.1) is summable. We denote by p : R(C \ {0}) → C the natural projection. It is easy to see: Lemma 3.3. Let I be a nonempty open interval. For i = 1, 2, . . . , p ∗ , the following two conditions (1) and (2) are equivalent. (1) I  Zi . (2) 0 < | I | < 2π /ki and λi ,d ∈ C \ p ( S ki I ) for d = 1, . . . , li − li −1 . 3.4. Statement of the main theorem Let γ > 0. A function f (x) on C N is said to be an entire function of order function on C N satisfying

γ if it is a holomorphic

 

 f (x)  A exp a|x|γ on C N for some A > 0 and a > 0. We denote by Exp{γ } (C N ) the set of all entire functions of order γ . Similarly, for δ > 0 and γ > 0 we denote by Exp{γ } ( D δ × C N ) the set of all holomorphic functions u (t , x) on D δ × C N satisfying the estimate

 

u (t , x)  B exp b|x|γ on D δ × C N for some B > 0 and b > 0. In the problem (E), we suppose:



(A3 )





f (t , x) ∈ Exp{γ } D r × C N , {γ }

ϕ j (x) ∈ Exp



C

N



In order to state our condition on the exponent nents γ . We set

,

and

j = 0, 1 , . . . , m − 1 .

γ , let us define the set C of admissible expo-



 / Nt ( E ) . Λ∗ = ( j , α ) ∈ Λ; j + |α |, ordt (a j ,α ) − j ∈ If ( j , α ) ∈ Λ∗ , by the definition of N t ( E ) we have |α | > 0 and by the assumption (A1 ) we have ordt (a j ,α )− j  −m + 1 > e 0 (= −m). Therefore, if we set Λ∗i = {( j , α ) ∈ Λ∗ ; e i −1 < ordt (a j ,α )− j  e i } (i = 1, 2, . . . , p ∗ + 1 with e p ∗ +1 = ∞), we have the expression of Λ∗ as a disjoint union

Λ∗ = Λ∗1 ∪ · · · ∪ Λ∗p ∗ ∪ Λ∗p ∗ +1 . For ( j , α ) ∈ Λ∗i we set

L j ,α = j + |α | − li −1 −

ordt (a j ,α ) − j − e i −1 ki

.

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Fig. 2. Case: 1  i  p ∗ .

Fig. 3. Case: i = p ∗ + 1.

Lemma 3.4. We have

0 < L j ,α < |α |

for any ( j , α ) ∈ Λ∗ .

(3.5)

Proof. Let ( j , α ) ∈ Λ∗i and set





P j ,α = j , ordt (a j ,α ) − j ,





Q j ,α = j + |α |, ordt (a j ,α ) − j :

we have P j ,α ∈ ( N t ( E ))◦ and Q j ,α ∈ / N t ( E ). Let us first consider the case: 1  i  p ∗ . In this case, the line containing the segment Γi is given by the equation y = e i −1 + ki (x − li −1 ). Since P j ,α ∈ ( N t ( E ))◦ holds, we see that the point P j ,α is / Nt ( E ) located in the domain {(x, y ); y > e i −1 + ki (x − li −1 )}: similarly, since the condition Q j ,α ∈ holds, we see that the point Q j ,α is located in the domain {(x, y ); y < e i −1 + ki (x − li −1 )}. Therefore, we have





e i −1 + ki ( j − li −1 ) < ordt (a j ,α ) − j < e i −1 + ki j + |α | − li −1 : this is equivalent to the condition (3.5). In the case i = p ∗ + 1, we have ordt (a j ,α ) − j > e p ∗ and

j < l p ∗ < j + | α |: this follows from the conditions P j ,α ∈ ( N t ( E ))◦ and Q j ,α ∈ / N t ( E ). Since k p ∗ +1 = ∞ holds, we have the condition (3.5). 2 Remark 3.5. Let ( j , α ) ∈ Λ∗i . In the case 1  i  p ∗ , the relation of P j ,α , Q j ,α , |α | and L j ,α is as in Fig. 2; in the case i = p ∗ + 1, we have L j ,α = j + |α | − l p ∗ and the relation of P j ,α , Q j ,α , |α | and L j ,α is as in Fig. 3.

H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

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Now, let us define Definition 3.6 (Definition of C ). If Λ∗ = ∅ we set C = (0, ∞). If Λ∗ = ∅, first we set

C j ,α =

⎧ ⎨ (0, L|α | ), if ( j , α ) ∈ Λ∗1 ∪ · · · ∪ Λ∗p ∗ , j ,α ⎩ (0, |α | ], if ( j , α ) ∈ Λ∗ ∗ p +1 L j ,α

(which is an open (or half-open and half-closed) interval in R), and then we define C by



C=

C j ,α .

( j ,α )∈Λ∗

(We note: by (3.5) we have (0, 1 + ) ⊂ C for some

> 0.)

If K = (φ1 , φ2 ) is an open interval, for a > 0 we write K + [a] = (φ1 − a, φ2 + a). Let U be an open subset of R which can  be expressed as a disjoint union of nonempty open intervals K j ( j = 1, 2, . . .); then we have U = j 1 K j . In this case, the notation I  U ⊕ [a] means that I  K j + [a] holds for some j  1. For an open set A we write I  A ∩ (U ⊕ [a]) if I  A and I  U ⊕ [a] hold. Similarly, we write









I  A ∩ U 1 ⊕ [a1 ] ∩ U 2 ⊕ [a2 ] if I  A, I  U 1 ⊕ [a1 ] and I  U 2 ⊕ [a2 ] hold. The following is the main result of this paper.

Theorem 3.7 (Main theorem). Suppose the conditions (A1 ), (3.2) and (A2 ). Suppose also that the data f (t , x) and ϕ j (x) ( j = 0, 1, . . . , m − 1) satisfy (A3 ) for some γ ∈ C . Let uˆ (t , x) be the unique formal solution of (E) in (3.1), and let the t-Newton polygon of (E) be as in Fig. 1. Then we have: (1) If p ∗ = 0, the formal solution uˆ (t , x) is convergent on D δ × C N for some δ > 0. (2) If p ∗  1, the formal solution uˆ (t , x) is (k p ∗ , . . . , k1 )-summable in the I -direction for any nonempty open interval I satisfying ∗

I

p 

Zi ⊕ [π /2κi , p ∗ ],

(3.6)

i =1

where Zi (1  i  p ∗ ) are the ones in (3.4), (for 1  i  p ∗ − 1), and κ p ∗ , p ∗ = ∞.

κi, p∗ > 0 is defined by the relation 1/κi, p∗ = 1/ki − 1/k p∗

The meaning of the condition γ ∈ C will be explained in Section 4.3, and the meaning of the condition (3.6) will be explained in Section 4.4. We note that if p ∗ = 1 the condition (3.6) is nothing but I  Z1 . Example 3.8. Let us consider

∂t u = f (t , x) + ∂x2 u + t (t ∂t )3 u ,

u (0, x) = ϕ (x),

(3.7)

∂t u = f (t , x) + ∂x3 u + t (t ∂t )3 u ,

u (0, x) = ϕ (x),

(3.8)

∂t u =

f (t , x) + t 2 ∂x4 u

3

+ t (t ∂t ) u ,

u (0, x) = ϕ (x),

(3.9)

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where (t , x) ∈ C2 , f (t , x) ∈ Exp{γ } ( D r × C) and solution uˆ (t , x) ∈ Exp{γ } (C)[[t ]] and

ϕ (x) ∈ Exp{γ } (C). In any case, we have a unique formal

Z = R \ {π j ; j = 0, ±1, ±2, . . .}. Moreover, we have C = (0, ∞) in the case (3.7), C = (0, 3) in the case (3.8), and C = (0, 4] in the case (3.9). Thus, the formal solution uˆ (t , x) is Borel summable in I -direction for any nonempty open interval I  Z , if γ > 0 in the case (3.7), if 0 < γ < 3 in the case (3.8), or if 0 < γ  4 in the case (3.9). The rest part of this paper is organized as follows. In the next Section 4 we will present some preparatory discussions which are needed in the proof of Theorem 3.7. In Section 5 we will give Gevrey type estimates of the coefficients of the formal solution: this proves that the formal k1 -Borel transform Bˆk1 [uˆ ](ξ, x) of the formal solution uˆ (t , x) is convergent in a neighborhood of (ξ, x) = (0, 0). In Sections 6 and 7, we will discuss some convolution partial differential equation in the Borel plane and show the possibility of analytic continuation of local solution to a sector as a holomorphic function with exponential growth. By using the result of this analytic continuation, we will give a proof of Theorem 3.7 in the last Section 8. 4. Some preparatory discussions In this section, we will present some basic tools and preparatory discussions which are needed in the proof of Theorem 3.7. 4.1. Estimates of entire functions For γ > 0 we set σ = 1 − 1/γ ; then we have σ < 1 and γ = 1/(1 − σ ). As to the estimates of derivatives of an entire function of order γ , the following result is well known: Proposition 4.1. Let γ > 0. If f (x) belongs to the class Exp{γ } (C N ), we can find constants C > 0, h > 0 and a > 0 such that

 α 

∂ f (x)  Ch|α | |α |!σ exp a|x|γ on C N , ∀α ∈ N N x

(4.1)

holds for σ = 1 − 1/γ . Proof. For the reader’s convenience, we will give here the proof of the case N = 1. By the assumption we have the estimate | f (x)|  A exp(a|x|γ ) on C for some A > 0 and a > 0, and so by Cauchy’s integral formula we have

  (n)   n!  f (x) =   2π i 

n!

ρ

 |z−x|=ρ

f ( z)

( z − x)n+1



  dz

max A exp a| z|γ 

n | z−x|=ρ

n!

ρ

n



A exp a



for any ρ > 0:  n A exp a2γ ρ γ + |x|γ n!

ρ

γ

ρ + |x|

(4.2)

in the last inequality we have used the fact (a + b)γ  2γ (aγ + bγ ). Since the minimum of the function (1/t n ) exp(ct γ ) (with c = a2γ ) on t > 0 is equal to (ec γ /n)n/γ which is attained at t = (n/c γ )1/γ , by taking ρ = (n/c γ )1/γ in (4.2) we have the estimate

H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

 n/γ  (n) 

 f (x)  n! A exp a2γ |x|γ ec γ n

Thus, by Stirling’s formula we have (4.1).

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on C, ∀n ∈ N∗ .

2

4.2. Formal norms We remark that the estimate (4.1) is very close to that of a function of the Gevrey class of order σ : this indicates that various arguments in the theory of partial differential equations in Gevrey classes (in the real domain) can be applied to the present case. Let us give an analogue of the divergent formal norm which was often used in the theory of hyperbolic equations in the real domain (see Komatsu [15] and Tahara [32]). For a function f (x) ∈ C ∞ (C N ) we write

    f (x) = ρ

q0

   α  q ∂ f (x) ρ x q! |α |=q

which is a formal power series in ρ with coefficients in C 0 (C N ). We write if |aq |  bq holds for all q  0. If x is fixed, we have



q0 aq

ρq 



q0 bq

ρq

       f (x) g (x)   f (x) ×  g (x) ; ρ ρ ρ N      ∂x f (x) ; ∂ρ  f (x)ρ  i ρ i =1

    ∂x f (x)  ∂ρ  f (x) i ρ ρ For a real number

for i = 1, . . . , N .

σ we set θσ (ρ ) =



(q!)σ

ρq

q0

q!

=



Γ (q + 1)σ

q0

ρq q!

which is a formal power series in ρ . In the case σ = 1, this is nothing but the function 1/(1 − ρ ), and in the case σ < 1 this is an entire function in ρ of order 1/(1 − σ ). By Proposition 4.1 we have Proposition 4.2. Let γ > 0. If f (x) belongs to the class Exp{γ } (C N ), we can find constants C > 0, h > 0 and a > 0 such that

 

 f (x)  C θσ (hρ ) exp a|x|γ on C N ρ holds for σ = 1 − 1/γ . For a  0 we set (a)

θσ (ρ ) =



Γ (q + a + 1)σ

q0

Lemma 4.3. Suppose σ  0; then we have the following properties. (a)

(1) If a is an integer, we have θσ (ρ ) = (d/dρ )a θσ (ρ ). (a ) (a ) (2) If 1  a1 < a2 holds, we have θσ 1 (ρ )  θσ 2 (ρ ).

ρq q!

.

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(a1 )

(3) If 0  a1 < 1 and a1 < a2 hold, we have θσ (4) If H > 0, a > 0 and b  0 hold, we have

(a )

(ρ )  2θσ 2 (ρ ).

 H

aσ

(b)

θσ

(ρ )  e σ

aσ

eH a+b+1

θσ(a+b) (ρ ).

(4.3)

Proof. (1), (2) and (3) are clear. Let us show (4). Let H > 0, a > 0 and b  0. We recall that a sharp form of Stirling’s formula for the Γ -function guarantees



Γ (x)

1



1< √ < exp 12x 2π xx−1/2 e −x

< e for x  1

(see [34]). Therefore, we have

H a Γ (q + b + 1)

Γ (q + a + b + 1)

√

Ha

√

2π (q + a + b + 1)q+a+b+1/2 e −q−a−b−1

 =e  e this leads us to (4.3).

2π (q + b + 1)q+b+1/2 e −q−b−1 e

a 

eH q+a+b+1

a

eH a+b+1

q+b+1

q+b+1/2

q+a+b+1

:

2

4.3. Discussion on the condition γ ∈ C In Theorem 3.7 we have supposed that the data f (t , x) and ϕ j (x) ( j = 0, 1, . . . , m − 1) satisfy (A3 ) for some γ ∈ C : without loss of generality we may assume that γ > 1 holds. In this section, we will give some consequences of the condition

γ ∈ C and γ > 1.

(4.4)

The following lemma explains the meaning of the condition (4.4). Lemma 4.4. Let γ satisfy the condition (4.4) and set σ = 1 − 1/γ . Then we have 0 < σ < 1 and







◦

j + σ |α |, ordt (a j ,α ) − j ∈ N t ( E )





∪ Γ p ∗ +1 \ (l p ∗ , e p ∗ )

for any ( j , α ) ∈ Λ with |α | > 0. Proof. We set





R j ,α = j + σ |α |, ordt (a j ,α ) − j . If ( j , α ) ∈ Λ∗i for some 1  i  p ∗ , we have 1 < γ < |α |/ L j ,α which is equivalent to the condition: ordt (a j ,α ) − j > ki ( j + σ |α | − li −1 ) + e i −1 . This means that the point R j ,α is located in the domain {(x, y ); y > ki (x − li −1 ) + e i −1 }. Therefore, we have R j ,α ∈ ( N t ( E ))◦ . If ( j , α ) ∈ Λ∗p ∗ +1 , we have 1 < γ  |α |/ L j ,α which is equivalent to the condition: j + σ |α |  l p ∗ . Since ordt (a j ,α ) − j > e p ∗ holds, we have R j ,α ∈ ( N t ( E ))◦ ∪ (Γ p ∗ +1 \ {(l p ∗ , e p ∗ )}). 2

H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

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Let ki (0  i  p ∗ + 1) and (li , e i ) (0  i  p ∗ ) be as in Fig. 1. We set

h0 = m

and h i = ki li −1 − e i −1





i = 1, 2, . . . , p ∗ .

(4.5)

By Fig. 1 and Lemma 4.4 we have Lemma 4.5. Let γ satisfy the condition (4.4) and set σ = 1 − 1/γ . Then we have: (1) 0 < m = h0 < h1 < · · · < h p ∗ and h1 = k1 m + m. (2) The line containing the segment Γi is given by the equation y = ki x − h i (i = 1, 2, . . . , p ∗ ). In particular, the line containing Γ1 is given by the equation y = k1 x − (k1 m + m). (3) h i > (ki + 1)m holds for i = 2, . . . , p ∗ . (4) Let i = 1, 2, . . . , p ∗ : we have



h i + ordt (a j ,0 ) = ki j + j ,



if ( j , 0) ∈ Ii ,



h i + ordt (a j ,α ) > ki j + σ |α | + j , if ( j , α ) ∈ Λ \ Ii . (5) Moreover, if 1  i  p ∗ − 1 we have

k i +1 =

h i + ordt (a j ,α ) − j − ki li

min

j + σ |α | − l i

( j ,α )∈Λ, j +σ |α |>li

(4.6)

.

Proof. (1), (2) and (3) are clear from Fig. 1. Let us show (4). If ( j , 0) ∈ Ii we have ( j , ordt (a j ,0 ) − j ) ∈ Γi : this means that ordt (a j ,0 ) − j = ki j − h i and so we have h i + ordt (a j ,0 ) = ki j + j. If ( j , α ) ∈ Λ \ Ii we see that the point ( j + σ |α |, ordt (a j ,α ) − j ) is located in {(x, y ); y > ki x − hi }: this means that ordt (a j ,α ) − j > ki ( j + σ |α |) − h i and so we have h i + ordt (a j ,α ) > ki ( j + σ |α |) + j. This proves (4). Let us show (5). Suppose that 1  i  p ∗ − 1: take any ( j , α ) ∈ Λ with j + σ |α | > li . By the definition of N t ( E ) and Lemma 4.4 we see that the point ( j + σ |α |, ordt (a j ,α ) − j ) is located in {(x, y ); y  ki +1 (x − li ) + e i }: this means that ordt (a j ,α ) − j  ki +1 ( j + σ |α | − li ) + e i and so we have

k i +1 

ordt (a j ,α ) − j − e i j + σ |α | − l i

.

Since e i = ki li − h i , by substituting this to the above formula we have

k i +1 

h i + ordt (a j ,α ) − j − ki li j + σ |α | − l i

.

Moreover, by Fig. 1 and (A2 ) we know that the equality is attained by a suitable ( j , 0) ∈ Ii +1 with j > li . This proves (4.6). 2 4.4. Discussion on the condition (3.6) Let 0 = k0 < k1 < k2 < · · · < k p ∗ < k p ∗ +1 = ∞ be as in Fig. 1. For 1  i < j  p ∗ we define by the relation 1/κi , j = 1/ki − 1/k j : we set κ p ∗ , p ∗ = ∞. It is easy to see:

1

κi , p ∗

=

1

κ i , i +1

+

1

κ i +1 , i +2

+ ··· +

1

κ p ∗ −1 , p ∗



κi , j > 0



i = 1, 2, . . . , p ∗ − 1 .

In the definition of the (k p ∗ , . . . , k1 )-summability we have required that there are nonempty open intervals I i (i = 1, . . . , p ∗ ) satisfying the condition (2.7): nevertheless, in Theorem 3.7 we

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have supposed only the condition (3.6) and do not suppose the existence of such open intervals I i (i = 1, . . . , p ∗ ). This gap is filled as follows: Proposition 4.6. Let I be a nonempty open interval. The following two conditions (1) and (2) are equivalent: (1) We have I  Z p ∗ , and we can find nonempty open intervals I i  Zi (i = 1, 2, . . . , p ∗ − 1) such that

I i +1  I i + [π /2κi ,i +1 ],

i = 1, 2, . . . , p ∗ − 1

(4.7)

with I p ∗ = I . Moreover, we may suppose that I i  I (1  i  p ∗ − 1) hold. (2) I satisfies ∗

I

p 

Zi ⊕ [π /2κi , p ∗ ].

(4.8)

i =1

Proof. Suppose the condition (1): by I i  Zi (i = 1, 2, . . . , p ∗ − 1) and (4.7) we have

I = I p ∗  I p ∗ −1 + [π /2κ p ∗ −1, p ∗ ]

 I p ∗ −2 + [π /2κ p ∗ −2, p ∗ −1 ] + [π /2κ p ∗ −1, p ∗ ]  ···  I i + [π /2κi ,i +1 ] + · · · + [π /2κ p ∗ −2, p ∗ −1 ] + [π /2κ p ∗ −1, p ∗ ]  Zi + [π /2κi , p ∗ ],

i = 1, 2, . . . , p ∗ − 1.

Since I  Z p ∗ is supposed, we have (4.8). This proves that (1) implies (2). Let us show the converse. To do so, we note: Lemma 4.7.

∞

(1) Let Ξ be a discrete subset of R, set Z = R \ Ξ , and let Z = j =1 K j be the expression of Z as a disjoint union of nonempty open intervals K j ( j = 1, 2, . . .). Let a > 0, let I be a nonempty open interval, and suppose the condition

I  Z ⊕ [a].

(4.9)

Let x0 be the middle point of I . If x0 ∈ Z we have x0 ∈ K j 0 for some j 0 , and if x0 ∈ / Z we have x0 ∈ Ξ and so we have K j 0 = (x0 , x0 + b) for some j 0 and 0 < b  ∞; in any case, for this K j 0 we have

I  K j 0 + [a].

(4.10)

(2) Moreover, we can find a nonempty open interval K 0 ⊂ K j 0 (⊂ Z ) and a positive constant > 0 such that

(x0 , x0 + ) ⊂ K 0  I  K 0 + [a]. Proof. Suppose (4.9). Let K j 0 be as above, and set c = | I |/2: I is expressed as I = (x0 − c , x0 + c ). We can show: 1) if x0 ∈ K j 0 and a  c, we have (4.10); 2) if x0 ∈ K j 0 and 0 < a < c, we have I ⊂ K j + [a] for any j = j 0 ;

H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

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3) if K j 0 = (x0 , x0 + b) and a > c, we have (4.10); 4) if K j 0 = (x0 , x0 + b) and 0 < a  c we have a contradiction to (4.9). This proves the part (1). Since the case 4) does not occur, to show (2) we have only to discuss the cases 1), 2) and 3). In the case 1), we take 0 < δ < c sufficiently small and set K 0 = (x0 − δ, x0 + δ): then we have K 0 ⊂ K j 0 and

K 0 = (x0 − δ, x0 + δ)  (x0 − c , x0 + c ) = I ⊂ (x0 − a, x0 + a)

 (x0 − a − δ, x0 + a + δ) = K 0 + [a]. In the case 2), we have I  K j 0 + [a]; therefore, if we set K j 0 = (ξ, η) we have ξ − a < x0 − c < x0 + c < η + a. In this case, we take 0 < δ < a sufficiently small so that ξ − a < x0 − c − δ and x0 + c + δ < η + a hold, and we set K 0 = (x0 − (c − a) − δ, x0 + (c − a) + δ): then we have K 0 ⊂ (ξ, η) = K j 0 and



K 0 = x0 − (c − a) − δ, x0 + (c − a) + δ



 x0 − (c − a) − a, x0 + (c − a) + a = (x0 − c , x0 + c ) = I  (x0 − c − δ, x0 + c + δ) = K 0 + [a].

In the case 3), we have K j 0 = (x0 , x0 + b) and a > c. In this case, we take δ > 0 sufficiently small so that δ < min{b, c , a − c } holds, and we set K 0 = (x0 , x0 + δ): then we have K 0 ⊂ K j 0 and

K 0 = (x0 , x0 + δ)  (x0 − c , x0 + c ) = I

 (x0 − c − δ, x0 + c + δ)  (x0 − a, x0 + a) ⊂ (x0 − a, x0 + a + δ) = K 0 + [a].

2

Lemma 4.8. (1) Let K 1 and K 2 be nonempty open intervals, and let a > 0. If K 1 ∩ K 2 = ∅ holds, we have ( K 1 + [a]) ∩ ( K 2 + [a]) = ( K 1 ∩ K 2 ) + [a]. (2) Let K i (i = 1, . . . , q) be nonempty open intervals, and let ai > 0 (i = 1, . . . , q). Suppose that K 1 ∩ · · · ∩ K q = ∅ and

I

q 



K i + [ai + · · · + aq ] .

(4.11)

i =1

Then, we can find nonempty open intervals I i (i = 1, . . . , q) such that I i  K i (i = 1, . . . , q) and I i +1  I i +[ai ] (i = 1, . . . , q) hold, where I q+1 = I . Proof. (1) is clear. Let us show (2). By (4.11) we have











I  K q ∩ K q−1 + [aq−1 ] ∩ · · · ∩ K 1 + [a1 + · · · + aq−1 ]

+ [aq ]:

therefore, we can find a nonempty open interval I q such that I  I q + [aq ] and I q  K q ∩ ( K q−1 +

[aq−1 ]) ∩ · · · ∩ ( K 1 + [a1 + · · · + aq−1 ]). This implies I q  K q and

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q −1

Iq 





K i + [ai + · · · + aq−1 ] .

i =1

Thus, by induction on q we have the result (2).

2

Now, let us complete the proof of Proposition 4.6. Let I be a nonempty open interval, and suppose the condition (4.8). We set ai = π /2κi ,i +1 (i = 1, . . . , p ∗ − 1); then we have π /2κi , p ∗ = ai + · · · + a p ∗ −1 and so (4.8) implies I  Z p ∗ and I  Zi + [ai + · · · + a p ∗ −1 ] (i = 1, . . . , p ∗ − 1). By (2) of Lemma 4.7 we have nonempty open intervals K i0  Zi (i = 1, . . . , p ∗ − 1) such that (x0 , x0 + ) ⊂ K i0  I  K i0 + [ai + · · · + a p ∗ −1 ] (i = 1, . . . , p ∗ − 1) for some > 0, where x0 is the middle point of I . Therefore, we have K 10 ∩ · · · ∩ K p0∗ −1 = ∅ and p ∗ −1

I





K i0 + [ai + · · · + a p ∗ −1 ] .

i =1

Thus, by (2) of Lemma 4.8 we have nonempty open intervals I i  K i0 (i = 1, . . . , p ∗ − 1) such that I i +1  I i + [ai ] holds for i = 1, . . . , p ∗ − 1 under I p ∗ = I . This proves (1) of Proposition 4.6. Thus, we have shown that (2) implies (1). 2 5. Estimate of the formal solution Let us recall that we are considering the equation

∂tm u +





a j ,α (t )∂t ∂xα u = f (t , x) ∈ Exp{γ } D r × C N j



(5.1)

( j ,α )∈Λ

with data ∂t u (0, x) = ϕ j (x) ∈ Exp{γ } (C N ) ( j = 0, 1, . . . , m − 1) under the assumptions (A1 ), (3.2), (A2 ) and γ ∈ C with γ > 1. Let j

uˆ (t , x) =







un (x)t n ∈ Exp{γ } C N [[t ]]

(5.2)

n0

be the unique formal solution of (5.1). Theorem 5.1. Under the above situation, we can find constants C > 0, H > 0 and b > 0 such that the following estimates hold:

 

un (x)  C H n n!1/k1 exp b|x|γ on C N , n = 0, 1, 2, . . . ,

(5.3)

where k1 is the lowest slope of the boundary of the t-Newton polygon in Fig. 1. This type of theorem is called a Maillet type theorem in Gérard and Tahara [11]. Similar results are obtained in Miyake [25] for linear equations, and in Ouchi [27], Shirai [31] and Tahara [33] for nonlinear equations. If p ∗ = 0, we have k1 = ∞ and so we have Corollary 5.2. If p ∗ = 0, the formal solution uˆ (t , x) is convergent on D δ × C N for some δ > 0.

H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

3611

5.1. Proof of Theorem 5.1 Let us multiply (5.1) by t m : we have



[t ∂t ]m u +

t m− j a j ,α (t )[t ∂t ] j ∂xα u = t m f (t , x)

(5.4)

( j ,α )∈Λ

where [λ]0 = 1 and [λ] j = λ(λ − 1) · · · (λ − j + 1) for j  1. Since uˆ (t , x) is a formal solution of (5.1), it is also a formal solution of this equation (5.4). We set q j ,α = m − j + ordt (a j ,α ) (( j , α ) ∈ Λ): by the assumption (A1 ) we have q j ,α  1 (( j , α ) ∈ Λ). Let

t m f (t , x) =



f n (x)t n ,

t m− j a j ,α (t ) =

nm



c j ,α , p t p



( j, α) ∈ Λ



p q j ,α

be Taylor expansions of t m f (t , x) and t m− j a j ,α (t ) (( j , α ) ∈ Λ), respectively. Since t m f (t , x) ∈ Exp{γ } ( D r × C N ) is supposed, by Proposition 4.2 we have

 

 f n (x)  A θσ (hρ ) exp b|x|γ on C N , n = 0, 1, 2, . . . , ρ n

(5.5)

r

for some A > 0, h > 0 and b > 0. Since t m− j a j ,α (t ) (( j , α ) ∈ Λ) are all holomorphic functions on D r we have the estimate |c j ,α , p |  A j ,α /r p (for p  q j ,α ) for some A j ,α > 0 (( j , α ) ∈ Λ). By an easy calculation we see that the coefficients un (x) (n = 0, 1, 2, . . .) of the formal solution (5.2) are determined by the following recurrent formulas: u j (x) = ϕ j (x)/ j ! ( j = 0, 1, . . . , m − 1) and for n  m

un (x) =



1

f n (x) −

[n]m





 c j ,α , p [n − p ] j ∂xα un− p (x) .

(5.6)

( j ,α )∈Λ q j,α  p n

We note: Lemma 5.3. We set σ = 1 − 1/γ : we have 0 < σ < 1 and

1 k1

   j + σ |α | − m . = max 0, max ( j ,α )∈Λ

q j ,α

(5.7)

Proof. In the case p ∗ = 0, by the shape of N t ( E ) and Lemma 4.4 we have j + σ |α |  m for any ( j , α ) ∈ Λ: since k1 = ∞ in this case, we have the result (5.7). In the case p ∗  1, by (4) of Lemma 4.5 with i = 1 we have



This proves (5.7).

(k1m + m) + ordt (a j ,0 ) = k1 j + j , if ( j , 0) ∈ I1 ,

(k1m + m) + ordt (a j ,α ) > k1 j + σ |α | + j , if ( j , α ) ∈ Λ \ I1 . 2

By using this lemma, let us show

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Lemma 5.4. Let d = (m + 1/k1 )/σ : then we can find C > 0, H > 0, h > 0 and b > 0 such that the estimate n  

un (x)  C H θσ(dn) (hρ ) exp b|x|γ on C N ρ n!m

(5.8)

holds for any n = 0, 1, 2, . . . . Proof. Since u 0 (x), . . . , um−1 (x) ∈ Exp{γ } (C N ) is assumed, by Proposition 4.2 we see that if we take C > 0, H > 0, h > 0 and b > 0 sufficiently large we have the estimate (5.8) for n = 0, 1, . . . , m − 1. We will show the general case by induction on n. We take c > 0 so that [n]m  cnm holds for any n  m. For simplicity we set



Φ(x) = exp b|x|γ . Let n  m and suppose that (5.8) (with n replaced by i) is already proved for i = 0, 1, . . . , n − 1: then by (5.5), (5.6) and the induction hypothesis we have

  un (x)  1 ρ m cn



A rn

θσ (hρ )Φ(x)



+



A j ,α

( j ,α )∈Λ q j,α  p n

=

1 cnm

rp



C H n− p h|α | (d(n− p )+|α |) [n − p ] j θσ (hρ )Φ(x) (n − p )!m

[ I 1 + I 2 ] on C N ,

where I 1 = ( A /r n )θσ (hρ )Φ(x) and I 2 denotes the rest parts in the above [· · ·]. Since nm − m < nm  n(m + 1/k1 ) = ndσ , by (4) of Lemma 4.3 we have

nnm−m θσ (hρ )  nndσ θσ (hρ ) ( 0)



 eσ

en

ndσ

dn + 1

θσ(dn) (hρ )  e σ

 ndσ e

d

θσ(dn) (hρ ):

therefore, if we use n!  nn and if we take C and H so that C  2e σ A /c and H  (1/r )(e /d)σ d hold, we have

A nnm−m A 1 I  θσ (hρ )Φ(x)  n m e σ 1 cnm cr n n!m cr n! 1



1 2

×

C H n (dn) θσ (hρ )Φ(x) n!m

 ndσ e

d

(dn)

θσ

(hρ )Φ(x)

on C N .

(5.9)

On the other hand, by using the conditions n!/(n − p )!  n p and [n − p ] j  n j we have

 C Hn I  Φ(x) × 2 cnm n!m 1



( j ,α )∈Λ q j,α  p n

A j ,α h|α | c



1 Hr

p

(d(n− p )+|α |)

n j + pm−m θσ

(hρ ).

Since (5.7) implies the inequality p /k1  j + σ |α | − m for p  q j ,α , we have j + pm − m  (dp − |α |)σ and so by (4) of Lemma 4.3 we have

H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

(d(n− p )+|α |)

n j + pm−m θσ

3613

(d(n− p )+|α |)

(hρ )  n(dp −|α |)σ θσ (hρ )  (dp −|α |)σ  (dp −|α |)σ en e (dn) (dn) θσ (hρ )  e σ θσ (hρ ).  eσ dn + 1 d

Therefore, if δ( H ) defined by δ( H ) = (1/ Hr )(e /d)dσ is sufficiently small (that is, if H is sufficiently large) we have



 C Hn I  Φ(x) × 2 m m cn n! 1

e σ A j ,α h|α | c

( j ,α )∈Λ q j,α  p n



 C H n (dn) θσ (hρ )Φ(x) × m n!

e σ A j ,α h|α | c

( j ,α )∈Λ



C H n (dn) 1 θσ (hρ )Φ(x) × n!m 2

 σ |α |  d

1

e

Hr

 σ |α | d e

 dσ  p e

d

θσ(dn) (hρ )

δ( H )q j,α 1 − δ( H )

on C N .

(5.10)

2

Thus, by (5.9) and (5.10) we have (5.8). This proves Lemma 5.4.

Now, let us complete the proof of Theorem 5.1. Since d is a fixed number, by Stirling’s formula we have Γ (dn + 1)  A K n n!d (n = 0, 1, 2, . . .) for some A > 0 and K > 0. Therefore, by setting ρ = 0 in (5.8) and by using the condition dσ = m + 1/k1 we have |un (x)|  ( A σ C )( H K σ )n n!1/k1 Φ(x) on C N for any n = 0, 1, 2, . . . . This proves Theorem 5.1. 2 5.2. Estimates from below Let s  1. As in [11], we say that a formal series ˆf (t , x) =

formal Gevrey class E {s} of order s if



s−1



n0

f n (x)t n ∈ O R [[t ]] belongs to the

is convergent in a neighborhood of (0, 0) ∈ n0 f n (x)t /n! ˆ Theorem 5.1 says that the formal solution u (t , x) in (5.2) belongs to the class E {1+1/k1 } . Ct × CxN . Then,  α α N and b(x) = α bα x we write For a(x) = α aα x   a(x)  b(x) if |aα |  bα holds for all α ∈ N : similarly, for f (t , x) = n0 f n (x)t n and g (t , x) = n0 gn (x)t n we write f (t , x)  g (t , x) if f n (x)  gn (x) holds for all n = 0, 1, 2, . . . . n

Proposition 5.5. In addition to the situation in Theorem 5.1 we suppose: p ∗  1, −a j ,α (t )   0 (( j , α ) ∈ Λ), f (t , x)  0 and ϕ j (x)  0 ( j = 0, 1, . . . , m − 1). Let l1 be as in Fig. 1 and let uˆ (t , x) = n0 un (x)t n be

the formal solution in (5.2). If u K (x) ≡ 0 for some K  l1 and if uˆ (t , x) belongs to the class E {s} for some s  1, then we have s  1 + 1/k1 . This shows that the formal solution uˆ (t , x) is divergent in a neighborhood of (0, 0) ∈ C1 × CxN . Proof. By the assumption we have uˆ (t , x)  0. Since (l1 , ordt (al1 ,0 )− l1 ) ∈ Γ1 we have ordt (al1 ,0 )− l1 = k1 (l1 − m) − m and so

t m−l1 al1 ,0 (t ) = t k1 (l1 −m) al01 ,0 (t ) with al01 ,0 (0) = 0. We set c 1 = −al0 ,0 (0) > 0 and q = ql1 ,0 (= m − l1 + ordt (al1 ,0 ) = k1 (l1 − m)). By the assumption 1 −al1 ,0 (t )  0 we have c 1 > 0. Then, by (5.6) we have

un (x) 

1

[n]m

c 1 [n − q]l1 un−q (x)

Therefore, using this formula n-times we have

for any n  m.

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H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

u K +nq (x) 

c 1 n [ K + (n − 1)q]l1 · · · [ K + q]l1 [ K ]l1

[ K + nq]m · · · [ K + 2q]m [ K + q]m

 C 1n n!l1 −m u K (x),

u K (x)

n = 1, 2, . . .

(5.11)

for some C 1 > 0. On the other hand, the assumption uˆ (t , x) ∈ E {s} implies

u K +nq (x) 

Ch K +nq ( K + nq)!s−1 1 − (x1 + · · · + x N )/ R 1

n = 1, 2, . . . ,

,

(5.12)

for some C > 0, h > 0 and R 1 > 0. Thus, by (5.11), (5.12) and the condition u K (x) ≡ 0 we obtain q(s − 1)  l1 − m, that is, s  1 + (l1 − m)/q = 1 + 1/k1 . This proves Proposition 5.5. 2 Example 5.6. (1) Let us consider ∂t u = ∂x2 u + t (t ∂t )3 u with u (0, x) = ϕ (x). In this case we have p ∗ = 1, k1 = 1 and l1 = 3. If ϕ (x)  0 and ϕ (2) (x) ≡ 0 hold, we can see that u 3 (x)  ϕ (2) (x)/3 and so we have the following result: the formal solution uˆ (t , x) belongs to E {s} if and only if s  2. This means that the formal solution uˆ (t , x) is divergent. (2) Let us consider ∂t u = ∂x3 u + t (t ∂t )3 u with u (0, x) = ϕ (x). In this case we have p ∗ = 1, k1 = 1 and l1 = 3. If ϕ (x)  0 and ϕ (3) (x) ≡ 0, we can see that u 3 (x)  ϕ (3) (x)/3 and so we have the following result: the formal solution uˆ (t , x) belongs to E {s} if and only if s  2. This means that the formal solution uˆ (t , x) is divergent. 6. Convolution partial differential equations Let I = (θ1 , θ2 ) be a nonempty open interval and r > 0. For k > 0 and two functions f (ξ, x) and g (ξ, x) on S I × C N (resp. on S I (r ) × C N ) we define the k-convolution ( f ∗k g )(ξ, x) of f (ξ, x) and g (ξ, x) with respect to ξ by

ξ



f (τ , x) g ξ k − τ k

( f ∗k g )(ξ, x) =

1/k k , x dτ

0

for (ξ, x) ∈ S I × C N (resp. for (ξ, x) ∈ S I (r ) × C N ). For basic properties of k-convolutions, see Balser [1,2], Braaksma [6] and Ouchi [28]. In particular, we will often use the formula:



Bk [ F G ](ξ, x) = Bk [ F ] ∗k Bk [G ] (ξ, x). Let k > 0 and

γ > 1: we set σ = 1 − 1/γ . For ( j , α ) ∈ N × NN we write  M j ,α [ W ] =

ξ kσ |α |−k Γ (σ |α |)

∗k ((kξ k ) j W ), if |α | > 0, if |α | = 0.

(kξ k ) j W ,

In this section, we will consider the following model of convolution partial differential equations:





P kξ k w = f (ξ, x) +







a j ,α (ξ ) ∗k M j ,α ∂xα w

( j ,α )∈Λ

where

P (λ) = λm + c 1 λm−1 + · · · + cm−1 λ + cm



(6.1)

H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

3615

is a polynomial of degree m with complex coefficients. We suppose: c1) c2) c3) c4) c5)

k > 0 is a real number, and 0 < | I | < 2π /k; γ > 1 and σ = 1 − 1/γ (we have 0 < σ < 1); m  1 is an integer, and Λ is a finite subset of N × N N ; f (ξ, x) ∈ O ( S I × CxN ) and a j ,α (ξ ) ∈ O ( S I ) (( j , α ) ∈ Λ); moreover, there are real numbers μ > 0 and q j ,α > 0 (( j , α ) ∈ Λ) such that the estimates

   f (ξ, x)    a j ,α (ξ ) 

F

Γ (μ/k)



|ξ |μ−k exp c |ξ |k + b|x|γ on S I × C N ,

A j ,α

Γ (q j ,α /k)





|ξ |q j,α −k exp c |ξ |k on S I ( j , α ) ∈ Λ

hold for some F  0, c > 0, b > 0 and A j ,α  0 (( j , α ) ∈ Λ). 6.1. Analytic continuation Let λ1 , . . . , λm be the roots of the equation P (λ) = 0, and let us consider Eq. (6.1) under the conditions c1)–c5) and

λi = 0 or λi ∈ C \ p ( S kI ) for i = 1, 2, . . . , m.

(6.2)

We define 0 < k1  ∞ by





k1 = min ∞,

min

( j ,α )∈Λ, j +σ |α |>m

q j ,α + k( j + σ |α | − m)



j + σ |α | − m

.

(6.3)

If j + σ |α |  m holds for all ( j , α ) ∈ Λ, we have k1 = ∞: in this case we have 1/k1 = 0. By the definition (6.3) we have 0 < k < k1  ∞: hence we can define κ0,1 > 0 by the relation 1/κ0,1 = 1/k − 1/k1 . If k1 = ∞ we have κ0,1 = k. The following result is the key to the proof of Theorem 3.7. Theorem 6.1. Suppose the conditions c1)–c5) and (6.2). Let ε > 0. If w (ξ, x) is a holomorphic solution of (6.1) on S I (ε ) × C N and if it satisfies

 

 w (ξ, x)  C |ξ |μ−k exp b|x|γ on S I (ε ) × C N

(6.4)

for some C > 0 and b > 0, then w (ξ, x) has an analytic continuation w ∗ (ξ, x) on S I × C N as a holomorphic solution of (6.1) that satisfies the following estimate: for any I 1  I there are M > 0, c 1 > 0 and b1 > 0 such that μ−k  ∗ 

 w (ξ, x)  M |ξ | exp c 1 |ξ |κ0,1 + b1 |x|γ on S I 1 × C N . (|ξ |k + 1)m

The rest part of this section is used to prove Theorem 6.1. We note that if

λi ∈ C \ p ( S kI ) for i = 1, 2, . . . , m holds, we can find a δ > 0 such that | P (kξ k )|  δ(|ξ |k + 1)m for any ξ ∈ S I . In the case where λi = 0 occurs for some i, we cannot get such an estimate on S I . To avoid the difficulties near ξ = 0, we

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H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

will use an argument similar to the argument in Ouchi [28]: we will take a suitable ξ0 ∈ S I so that ξ0k + S kI ⊂ S kI , and we will consider Eq. (6.1) on (ξ0k + S kI )1/k , where (ξ0k + S kI )1/k = {ξ ∈ S I ; ξ =

(ξ0k + ζ )1/k for some ζ ∈ S kI }. It is easy to see:

Lemma 6.2. Let I = (θ1 , θ2 ) with 0 < | I | < 2π /k, and set ξ0 = re i ϕ for r > 0 and ϕ = (θ1 + θ2 )/2. Then, we have: ξ0 ∈ S I , ξ0k + S kI ⊂ S kI and dist(0, (ξ0k + S kI )1/k ) > 0, where dist(0, A ) denotes the distance from 0 to the set A in the complex plane. Moreover, we can take a positive constant δ > 0 such that

 k



 P k ξ + ξ k   δ |ξ0 |k + |ξ |k + 1 m for all ξ ∈ S I . 0

(6.5)

Thus, if we consider Eq. (6.1) only on (ξ0k + S kI )1/k , we can use the estimate (6.5). The proof of Theorem 6.1 will be done in the following three steps. (1) First, we will rewrite our equation (6.1) into an equation on (ξ0k + S kI )1/k by the change of variable

ξ → (ξ0k + ξ k )1/k .

(2) Then, we will discuss the reduced equation on (ξ0k + S kI )1/k , and show the existence and uniqueness results of a solution. (3) Lastly, we will prove Theorem 6.1 by using the results in step (2). 6.2. Reduction of Eq. (6.1) First, let us rewrite our equation (6.1) into an equation on (ξ0k + S kI )1/k . Let w (ξ, x) ∈ O ( S I (ε )×C N ) be a holomorphic solution of (6.1) on S I (ε ) × C N that satisfies (6.4). We take ξ0 ∈ S I (ε ) as in Lemma 6.2 with 0 < r = |ξ0 | < ε . Set

1/k

 Sξ0 = ξ ∈ S I ; ξ0k + ξ k ∈ S I (ε ) . It is clear that ξ ∈ Sξ0 if and only if ξ k ∈ S kI and ξ0k + ξ k ∈ S kI (εk ). In addition, we have S I ((εk −

r k )1/k ) ⊂ Sξ0 ⊂ S I (ε ). For a function f (ξ, x) we write f ξ0 (ξ, x) = f ((ξ0k + ξ k )1/k , x). If f (ξ, x) is a holomorphic function on S I × C N (resp. on S I (ε ) × C N ), then f ξ0 (ξ, x) is a holomorphic function on S I × C N (resp. on Sξ0 × C N ). Moreover, we have Lemma 6.3. Let f (ξ, x) and g (ξ, x) be holomorphic functions on S I (ε ) × C N . Then, ( f ∗k g )ξ0 (ξ, x) is a holomorphic function on Sξ0 × C N and we have

( f ∗k g )ξ0 (ξ, x) = ( f ∗k g ξ0 )(ξ, x) + ( f ξ ∗k g )(ξ0 , x) on Sξ0 × C N . Proof. Under

τ = (ξ k + yk )1/k we have (ξ0k +ξ k )1/k





f (τ , x) g ξ0k + ξ k − τ k

( f ∗k g )ξ0 (ξ, x) =

1/k k , x dτ

0

ξ



f (τ , x) g ξ0k + ξ k − τ k

= 0

1/k k , x dτ

H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

(ξ0k +ξ k )1/k





f (τ , x) g ξ0k + ξ k − τ k

+

3617

1/k k , x dτ

ξ

ξ0 = ( f ∗k g ξ0 )(ξ, x) +

f

k

1/k k

1/k k ξ + yk , x g ξ0 − y k , x dy

0

= ( f ∗k g ξ0 )(ξ, x) + ( f ξ ∗k g )(ξ0 , x).

2

For ( j , α ) ∈ N × N N we write

⎧ ⎨ ξ kσ |α|−k ∗ (k j (ξ k + ξ k ) j W ), if |α | > 0, k 0 M j ,α ,ξ0 [ W ] = Γ (σ |α |) ⎩ k j (ξ k + ξ k ) j W , if |α | = 0. 0 For ( j , α ) ∈ N × N N with |α | > 0 we set

 K j ,α ,ξ0 [ W ](η, x) =

 (ξ k + ηk )σ |α |−1 k j

∗k kξ W (ξ0 , x), Γ (σ |α |)

where the k-convolution is taken in the variable ξ . By Lemma 6.3 we have Lemma 6.4. Let w (ξ, x) be a holomorphic function on S I (ε ) × C N : then the function (M j ,α [ w ])ξ0 (ξ, x) is a holomorphic function on Sξ0 × C N and we have the following equality on Sξ0 × C N :



M j ,α [ w ] ξ (ξ, x) =



0

M j ,α ,ξ0 [ w ξ0 ](ξ, x) + K j ,α ,ξ0 [ w ](ξ, x), if |α | > 0, M j ,α ,ξ0 [ w ξ0 ](ξ, x),

if |α | = 0.

Let us return to the situation in Theorem 6.1. Since w (ξ, x) is a holomorphic solution of (6.1) on S I (ε ) × C N , we see that w ξ0 (ξ, x) is a holomorphic function on Sξ0 × C N and it satisfies the equation



P k ξ0k + ξ k





w ξ0 = f 0 (ξ, x) +





a j ,α (ξ ) ∗k M j ,α ,ξ0 ∂xα w ξ0



(6.6)

( j ,α )∈Λ

on Sξ0 × C N , where

f 0 (ξ, x) = f ξ0 (ξ, x) +



+



 

(a j ,α )ξ ∗k M j ,α ∂xα w (ξ0 )

(i ,α )∈Λ





a j ,α (ξ ) ∗k K j ,α ,ξ0 ∂xα w .

( j ,α )∈Λ,|α |>0

The following result is very important. Proposition 6.5. The above f 0 (ξ, x) is a holomorphic function on S I × C N and we have the estimate k k μ/k−1  



 f 0 (ξ, x)  F 0 (|ξ0 | + |ξ | ) exp c 0 |ξ0 |k + |ξ |k + b0 |x|γ Γ (μ/k)

on S I × C N for some F 0 > 0, c 0 > 0 and b0 > 0.

(6.7)

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To prove this, for a > 0 and c  0 we introduce the functions

φa (ξ ; c ) =



|ξ |a−k exp c |ξ |k , Γ (a/k)

ψa,ξ0 (ξ ; c ) =



(|ξ0 |k + |ξ |k )a/k−1 exp c |ξ0 |k + |ξ |k . Γ (a/k)

In the case ξ0 = 0 we have ψa,ξ0 (ξ ; c ) = φa (ξ ; c ). Then, the assumption c5) is expressed as | f (ξ, x)|  F φμ (ξ ; c ) exp(b|x|γ ) on S I × C N , and |a j ,α (ξ )|  A j ,α φq j,α (ξ ; c ) on S I (( j , α ) ∈ Λ). The conclusion (6.7) of Proposition 6.5 is nothing but the estimate

 

 f 0 (ξ, x)  F 0 ψμ,ξ (ξ ; c 0 ) exp b0 |x|γ on S I × C N . 0 Since w (ξ, x) satisfies the estimate (6.4), by Proposition 4.1 we can take C 0 > 0 and b0 > b so that μ−k  α 

∂ w (ξ, x)  C 0 |ξ | exp b0 |x|γ on S I (ε ) × C N , ( j , α ) ∈ Λ. x Γ (μ/k)

Thus, Proposition 6.5 is easily verified by the following lemma: Lemma 6.6. (1) Let f (ξ ) and g (ξ ) be two holomorphic functions on S I . If | f (ξ )|  A φa (ξ ; c ) and | g (ξ )|  B ψb,ξ0 (ξ ; c ) hold on S I for some A > 0, B > 0, a > 0, b > 0 and c  0, then ( f ∗k g )(ξ ) is a holomorphic function on S I and we have |( f ∗k g )(ξ )|  A B ψa+b,ξ0 (ξ ; c ) on S I . (2) Let G (ξ ) be a holomorphic function on S I satisfying |G (ξ )|  A φa (ξ ; c ) on S I for some A > 0, a > 0 and c  0, and let W (ξ ) be a holomorphic function on S I (ε ) satisfying | W (ξ )|  M |ξ |b−k /Γ (b/k) on S I (ε ) for some M > 0 and b > 0. Then, (G ξ ∗k (M j ,α [ W ]))(ξ0 ) is a holomorphic function on S I (as a function in ξ ) and we have

 

 G ξ ∗k M j ,α [ W ] (ξ0 )  k j AM Γ (b/k + j ) ψa+b+k( j +σ |α |),ξ (ξ ; c ) on S I . 0 Γ (b/k) (3) Under the same condition as in (2), we see: G (ξ ) ∗k (K j ,α ,ξ0 [ W ]) is a holomorphic function on S I and we have



 G (ξ ) ∗k K j ,α ,ξ [ W ]   k j AM Γ (b/k + j ) ψa+b+k( j +σ |α |),ξ (ξ ; c ) on S I . 0 0 Γ (b/k) (4) For a > 0 and b > 0 we have



b/k Γ (a/k)Γ (b/k)

ψa+b,ξ0 (ξ ; c )  √

2π Γ ((a + b)/k)

ψa,ξ0 (ξ ; c + 1) on C.

Proof. Set E (ξ ) = exp(c (|ξ0 |k + |ξ |k )): then (1) is verified by

  ( f ∗k g )(ξ ) 

AB

Γ (a/k)Γ (b/k)

|ξ | E (ξ ) 0



ya−k |ξ0 |k + |ξ |k − yk

b/k−1

dyk

H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637



(|ξ0 |k +|ξ |k )1/k

AB

Γ (a/k)Γ (b/k)

E (ξ )



3619

ya−k |ξ0 |k + |ξ |k − yk

b/k−1

dyk

0

=

A B (|ξ0 |k + |ξ |k )(a+b)/k−1

Γ (a/k)Γ (b/k)

1 E (ξ )

xa/k−1 (1 − x)b/k−1 dx

0

=

k (a+b)/k−1

A B (|ξ0 | + |ξ | ) k

Γ ((a + b)/k)

E (ξ ) = A B ψa+b,ξ0 (ξ ; c ).

(2) and (3) are verified in the same way. Since the maximum the function xα e −x on x > 0 is equal √of √ to α α e −α , by Stirling’s formula we have xα e −x  α α e −α  ( α / 2π )Γ (α ): therefore, (4) is verified by



(|ξ0 |k + |ξ |k )a/k−1 exp (c + 1) |ξ0 |k + |ξ |k Γ (a/k)

b/k

Γ (a/k) exp − |ξ0 |k + |ξ |k × × |ξ0 |k + |ξ |k Γ ((a + b)/k)  b/k Γ (a/k) 2  ψa,ξ0 (ξ ; c + 1) × × √ Γ (b/k). Γ ((a + b)/k) 2π

ψa+b,ξ0 (ξ ; c ) =

6.3. Results on the reduced equation (6.6) In this section we will consider the reduced equation (6.6) with w ξ0 (ξ, x) replaced by the unknown function W (ξ, x):



P k ξ0k + ξ k



W = f 0 (ξ, x) +







a j ,α (ξ ) ∗k M j ,α ,ξ0 ∂xα W



.

(6.8)

( j ,α )∈Λ

We already know: f 0 (ξ, x) is a holomorphic function on S I × C N , a j ,α (ξ ) (( j , α ) ∈ Λ) are holomorphic functions on S I , and

 k



 P k ξ + ξ k   δ |ξ0 |k + |ξ |k + 1 m on S I , 0  

 f 0 (ξ, x)  F 0 ψμ,ξ (ξ ; c 0 ) exp b0 |x|γ on S I × C N , 0  

a j ,α (ξ )  A j ,α φq (ξ ; c 0 ) on S I ( j , α ) ∈ Λ j ,α for some δ > 0, F 0 > 0, c 0 > 0, b0 > 0 and A i ,α > 0 ((i , α ) ∈ Λ). Without loss of generality we may suppose that c 0  m holds. Theorem 6.7. (1) (Existence of a solution.) Eq. (6.8) has a holomorphic solution W (ξ, x) on S I × C N which satisfies k k μ/k−1  

κ /k

 W (ξ, x)  M (|ξ0 | + |ξ | ) × exp c 1 |ξ0 |k + |ξ |k 0,1 + b1 |x|γ on S I × C N k k m (|ξ0 | + |ξ | + 1)

for some M > 0, c 1 > 0 and b1 > 0.

(6.9)

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H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

(2) (Local uniqueness of the solution.) Let 0 < r < ∞. If W i (ξ, x) ∈ O ( S I (r ) × C N ) (i = 1, 2) are two holomorphic solutions of Eq. (6.8) on S I (r ) × C N and if they satisfy the estimates

 



 W i (ξ, x)  M |ξ0 |k + |ξ |k μ/k−1 exp b1 |x|γ on S I (r ) × C N , i = 1, 2 for some M > 0 and b1 > 0, then we have W 1 (ξ, x) = W 2 (ξ, x) on S I (r ) × C N . We will give a proof of this theorem in the next Section 7, and we will admit this for a while. 6.4. Proof of Theorem 6.1 Lastly, let us give a proof of Theorem 6.1 by using Theorem 6.7. Let w (ξ, x) be a holomorphic solution of Eq. (6.1) on S I (ε ) × C N that satisfies (6.4): then we know that w ξ0 (ξ, x) is a holomorphic solution of Eq. (6.6) on Sξ0 × C N and by (6.4) we have

 



 w ξ (ξ, x)  C |ξ0 |k + |ξ |k μ/k−1 exp b|x|γ on Sξ × C N . 0 0 Let W (ξ, x) be the holomorphic solution of (6.8) on S I × C N that satisfies the estimate (6.9). Then, if we consider this only on S I ((εk − r k )1/k ) × C N (where r = |ξ0 |) we have the estimate

 







 W (ξ, x)  M 1 |ξ0 |k + |ξ |k μ/k−1 exp b1 |x|γ on S I εk − r k 1/k × C N for some M 1 > 0. Since S I ((εk − r k )1/k ) ⊂ Sξ0 , by applying the uniqueness of the solution in Theorem 6.7 we have w ξ (ξ, x) = W (ξ, x) on S I ((εk − r k )1/k ) × C N and by the unique continuation property of holomorphic functions we have w ξ0 (ξ, x) = W (ξ, x) on Sξ0 × C N . This shows that w (ξ, x) has an analytic continuation w ∗ (ξ, x) on ( S I (ε ) ∪ (ξ0k + S kI )1/k ) × C N and we have the estimates

 ∗ 

 w (ξ, x)  C |ξ |μ−k exp b|x|γ on S I (ε ) × C N ,

(6.10)

 ∗  M (|ξ0 |k + |ξ k − ξ0k |)μ/k−1  w (ξ, x)  (|ξ0 |k + |ξ k − ξ0k | + 1)m



1/k

κ /k × exp c 1 |ξ0 |k + |ξ k − ξ0k | 0,1 + b1 |x|γ on ξ0k + S kI × CN .

(6.11)

Set d0 = dist(0, (ξ0k + S kI )1/k ); by Lemma 6.2 we have d0 > 0 and so if ξ ∈ (ξ0k + S kI )1/k we have

|ξ |  d0 and |ξ |k  |ξ0 |k + |ξ k − ξ0k |  2|ξ0 |k + |ξ |k  (2r /d0 + 1)|ξ |k . Therefore, by (6.10) and (6.11) we

have

μ−k  ∗ 



1/k

 w (ξ, x)  M 2 |ξ | × CN exp c 2 |ξ |κ0,1 + b1 |x|γ on S I (ε ) ∪ ξ0k + S kI k m (|ξ | + 1)

for some M 2 > 0. This proves Theorem 6.1.

2

7. Proof of Theorem 6.7 In this section we will give a proof of Theorem 6.7. Instead of W (ξ, x) we will use the notation w (ξ, x); then our equation (6.8) is written as

H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637



P k ξ0k + ξ k



w = f 0 (ξ, x) +







3621



a j ,α (ξ ) ∗k M j ,α ,ξ0 ∂xα w .

(7.1)

( j ,α )∈Λ

Here, f 0 (ξ, x) is a holomorphic function on S I × C N , a j ,α (ξ ) (( j , α ) ∈ Λ) are holomorphic functions on S I , and

 k



 P k ξ + ξ k   δ |ξ0 |k + |ξ |k + 1 m on S I , 0  

 f 0 (ξ, x)  F ψμ,ξ (ξ ; c ) exp b|x|γ on S I × C N , 0  

a j ,α (ξ )  A j ,α φq (ξ ; c ) on S I ( j , α ) ∈ Λ j ,α

(7.2) (7.3) (7.4)

for some δ > 0, F > 0, μ > 0, c > 0, b > 0, A j ,α > 0 (( j , α ) ∈ Λ) and q j ,α > 0 (( j , α ) ∈ Λ): for simplicity, we have used F , b and c instead of F 0 , b0 and c 0 . Without loss of generality we may suppose that c  m holds. By (6.3), the constant k1 is well-defined so that 0 < k < k1  ∞ and we have

1 k1

  = max 0, max ( j ,α )∈Λ



j + σ |α | − m q j ,α + k[ j + σ |α | − m]+

(7.5)

,

where we used the notation: [x]+ = max{0, x} for x ∈ R. Recall that κ0,1 > 0 is defined by the relation 1/κ0,1 = 1/k − 1/k1 , and that κ0,1 = k if j + σ |α |  m holds for all ( j , α ) ∈ Λ. 7.1. Formal solution We set

 

 K = q j ,α + k j + σ |α | − m + ; ( j , α ) ∈ Λ : since this is a finite set, we can write K = {q1 , . . . , q }, where q1 , . . . , q are distinct positive real numbers. We define a set N of real numbers by

N =μ+

 

Nq j ;

(7.6)

j =1

that is, a real number n belongs to N if and only if n is expressed in the form n = μ + p 1 q1 + · · · + p  q for some p j ∈ N ( j = 1, . . . , ). Since N is a discrete set of positive real numbers, we can write it in the form N = {n0 , n1 , n2 , . . .} with n0 = μ, 0 < n0 < n1 < n2 < · · · , and nk → ∞ (as k → ∞). By the definition we see:

n∈N

⇒





n + q j ,α + k j + σ |α | − m + ∈ N ,

( j , α ) ∈ Λ.

First, let us look for a formal solution of (7.1) in the form

w (t , x) =



w n (ξ, x).

n∈N

By substituting this formal series into the equation we have formally

 n∈N



P k ξ0k + ξ k



w n = f 0 (ξ, x) +





( j ,α )∈Λ n∈N







a j ,α (ξ ) ∗k M j ,α ,ξ0 ∂xα w n .

(7.7)

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H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

Here, we give the following weights (we denote by w( f ) the weight of f ):



w P k ξ0k + ξ k









w n = n,

w M j ,α ,ξ0 ∂xα w n



w( f 0 ) = μ,

w(a j ,α ) = q j ,α

 

= n + k j + σ |α | − m + ( j , α ) ∈ Λ .



( j, α) ∈ Λ ,

Under these weights, we see that the weight of every term in the both sides of (7.7) belongs to N and so if we collect the terms of the same weight, our equation is decomposed into the following recurrent formulas:



P k ξ0k + ξ k



w μ = f 0 (ξ, x),

(7.8)

and for n ∈ N with n > μ



P k ξ0k + ξ k





wn =





a j ,α (ξ ) ∗k M j ,α ,ξ0 ∂xα w n



.

(7.9)

( j ,α )∈Λ, n ∈N   n +q j ,α +k j +σ |α |−m + =n

Since n ∈ N and n + q j ,α + k[ j + σ |α | − m]+ = n imply n = n − q j ,α − k[ j + σ |α | − m]+ ∈ N , we can express (7.9) in the form



P k ξ0k + ξ k





wn =



( j ,α )∈Λ  n−q j ,α −k j +σ |α |−m + ∈N



a j ,α (ξ ) ∗k M j ,α ,ξ0 ∂xα w n−q j,α −k j +σ |α |−m +



.

Thus, by the assumption (7.2) we have Proposition 7.1. By (7.8) and (7.9) we have a formal solution of Eq. (7.1) in the form

w (t , x) =







w n (ξ, x) ∈ O S I × C N (n ∈ N ).

w n (ξ, x),

(7.10)

n∈N

7.2. Some lemmas Let us present some lemmas which are needed in the proof of the convergence of the formal solution. First we note: Lemma 7.2. Suppose the condition c  m. Then, for any μ > 0 and L Λ > 0 there is a constant β > 0 which satisfies the following: if W (ξ ) is a holomorphic function on S I and if the estimate

   W (ξ ) 

A

(|ξ0 |k + |ξ |k + 1)m

ψ N ,ξ0 (ξ ; c ) on S I

holds for some A > 0 and N  μ, we have [ j +σ |α |−m]+   M j ,α ,ξ [ W ](ξ )  β N A ψ N +k[ j +σ |α |−m]+ ,ξ0 (ξ ; c ) on S I 0 σ |α |

N

for any j + σ |α |  L Λ .

H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

3623

Proof. The case |α | = 0 is easily verified. Let us show the case |α | > 0. Since c  m is supposed, we have (x + 1)m  e cx for any x  0, and so we have

  M j ,α ,ξ [ W ](ξ )  0

|ξ | 0

×

ykσ |α |−k exp(c yk )

Γ (σ |α |)

( yk

+ 1)m

×

kjA

Γ ( N /k)

(|ξ0 |k + |ξ |k − yk ) N /k+ j −1 exp(c (|ξ0 |k + |ξ |k − yk )) k dy . ((|ξ0 |k + |ξ |k − yk ) + 1)m

(7.11)

Since (x + 1)((t − x) + 1)  t + 1 holds for any 0  x  t, by setting x = yk and t = |ξ0 |k + |ξ |k we have

1

( yk

+ 1)((|ξ0

|k

+ |ξ |k

−

y k ) + 1)



1

|k

|ξ0 + |ξ |k + 1

for 0  y  (|ξ0 |k + |ξ |k )1/k . Therefore, by applying this to (7.11) and by the same calculation as in the proof of (1) of Lemma 6.6 we have j k k  

M j ,α ,ξ [ W ](ξ )  |ξ0 |k + |ξ |k N /k+[ j +σ |α |−m]+ −1 k A exp(c (|ξ0 | + |ξ | ))Γ ( N /k + j ) 0 Γ ( N /k)Γ ( N /k + j + σ |α |)

= k j A ψ N +k[ j +σ |α |−m]+ ,ξ0 (ξ ; c )

Γ ( N /k + j )Γ ( N /k + [ j + σ |α | − m]+ ) . Γ ( N /k)Γ ( N /k + j + σ |α |)

Thus, to complete the proof of Lemma 7.2 it is enough to notice the following fact:

Γ ( N /k + j )Γ ( N /k + [ j + σ |α | − m]+ ) C j ,α N [ j +σ |α |−m]+ ,  Γ ( N /k)Γ ( N /k + j + σ |α |) N σ |α | holds for some C j ,α > 0 (which also depends on may omit the details. 2

∀ N  μ,

μ). Since this is verified by Stirling’s formula, we

The following lemma is also very important. Lemma 7.3. (1) Let q > 0 and α > 0: then there are C > 0 and a > 0 such that for any t > 0 we have

 n0

tn

Γ ((α + n)/q)



 C exp at q .

(7.12)

(2) Let q i > 0 (i = 1, . . . , ), αi > 0 (i = 1, . . . , ) and δi > 0 (i = 1, . . . , ): then there are C > 0 and ai > 0 (i = 1, . . . , ) such that for any t > 0 we have

 n1 ,...,n 0

t δ1 n1 +···+δ n

Γ ((α1 + n1 )/q1 + · · · + (α + n )/q )

C

 





exp ai t δi qi .

(7.13)

i =1

Proof. By Stirling’s formula we have (1/Γ ((α + n)/q))  Ahn /n!1/q (n = 0, 1, 2, . . .) for some A > 0 and h > 0. Since t n /n!  exp(t ) holds for any n = 0, 1, 2, . . . we have

3624

H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

 n0

tn

Γ ((α + n)/q)



 1 n  ((2ht )q )n 1/q  Ahn t n = A 2 n! n!1/q

n0

A

 n 1

n0

2

n0





exp (2ht )q

1/q



= 2 A exp at q

for a = (2h)q /q. This proves (7.12). Since

1

Γ ((α1 + n1 )/q1 + · · · + (α + n )/q ) holds for some K > 0 (depending only on

K

  i =1

1

Γ ((αi + ni )/qi )

α1 /q1 , . . . , α /q ), by (7.12) we have the result (7.13). 2

7.3. Convergence of the formal solution We will show the convergence of the formal solution (7.10), and prove the existence of a solution in Theorem 6.7. By the assumption (7.3) we have

 

 f 0 (ξ, x)  F 0 ψμ,ξ (ξ ; c )θσ (hρ ) exp b1 |x|γ on S I × C N 0 ρ

(7.14)

for some F 0 > 0, h > 0 and b1 > 0. We take L > 0 sufficiently large so that

L

σ σ |α | and L > max : μ ( j ,α )∈Λ q j ,α

then we have ( L /σ )μ  1 and q j ,α L − σ |α | > 0 (for any ( j , α ) ∈ Λ). We have: Theorem 7.4. Set d = ( L + 1/k1 )/σ : then we can find C > 0 and H > 0 such that the estimate

   w n (ξ, x)  ρ

C Hn

Γ (n

+ 1) L



ψn,ξ0 (ξ ; c ) (dn) θσ (hρ ) exp b1 |x|γ k k m (|ξ0 | + |ξ | + 1)

(7.15)

holds on S I × C N for any n ∈ N . (dμ)

Proof. We write Φ1 (x) = exp(b1 |x|γ ). Since dμ  ( L /σ )μ  1 holds, we have θσ (hρ )  θσ therefore, by (7.2), (7.8) and (7.14) we have

(hρ ):

   w μ (ξ, x)  ( F 0 /δ)ψμ,ξ0 (ξ ; c ) θσ(dμ) (hρ )Φ1 (x) on S I × C N . ρ (|ξ0 |k + |ξ |k + 1)m Thus, if we take C and H sufficiently large so that C H μ /Γ (μ + 1) L  F 0 /δ holds, we have the condition (7.15) for n = μ. Let us show the general case by induction on n ∈ N . Let n ∈ N with n > μ, and suppose that (7.15) (with n replaced by p) is already proved for all |α | p ∈ N with p < n. Since |∂xα w p (ξ, x)|ρ  ∂ρ | w p (ξ, x)|ρ holds, by the induction hypothesis and Lemma 7.2 we have

H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

3625

    M j ,α ,ξ ∂ α w n−q −k[ j +σ |α |−m] (ξ, x) x 0 + j ,α ρ 

β(n − q j ,α − k[ j + σ |α | − m]+ )[ j +σ |α |−m]+ (n − q j ,α − k[ j + σ |α | − m]+ )σ |α | C H n−q j,α −k[ j +σ |α |−m]+ h|α |

×

Γ (n − q j ,α − k[ j + σ |α | − m]+ + 1) L (d(n−q j,α −k[ j +σ |α |−m]+ )+|α |)

× θσ

ψn−q j,α ,ξ0 (ξ ; c )

(hρ )Φ1 (x)

(7.16)

for any ( j , α ) ∈ Λ satisfying n − q j ,α − k[ j + σ |α | − m]+ ∈ N . For simplicity, we set p j ,α = q j ,α + k[ j + σ |α | − m]+ ; then by the condition n − p j,α ∈ N we have n − p j,α  μ. Since k1 is the one in (7.5) we have p j ,α /k1  [ j + σ |α | − m]+ for all ( j , α ) ∈ Λ. By using (7.2), (7.9), (7.16) and (1) of Lemma 6.6 we have

   w n (ξ, x)  ρ





1

| P (k(ξ0k

+ ξ k ))|

( j ,α )∈Λ n− p j ,α ∈N

1

|k

δ(|ξ0 + |ξ |k + 1)m 

×

C Hn

Γ (n

+ 1) L

(n − p j ,α

)σ |α |

C H n− p j,α h|α |

Γ (n − p j ,α + 1) L

(d(n− p j,α )+|α |)

θσ

(hρ )

ψn,ξ0 (ξ ; c ) Φ1 (x) (|ξ0 |k + |ξ |k + 1)m



×

ψn,ξ0 (ξ ; c )Φ1 (x)

A j ,α β(n − p j ,α )[ j +σ |α |−m]+

( j ,α )∈Λ n− p j ,α ∈N

=

    a j ,α (ξ ) ∗k M j ,α ,ξ ∂ α w n− p  x 0 j ,α ρ

A j ,α β h|α |

δH

( j ,α )∈Λ n− p j ,α ∈N

(d(n− p j,α )+|α |)

× θσ

p j ,α

×

(n − p j ,α )[ j +σ |α |−m]+ Γ (n + 1) L σ | α | (n − p j ,α ) Γ (n − p j ,α + 1) L

(hρ ).

(7.17)

Here, we note the following facts:

Γ (n + 1)  e (1 + p j ,α /μ)1/2n p j,α , Γ (n − p j ,α + 1)  

0 < p j ,α L + j + σ |α | − m + − σ |α |  dp j ,α − |α | σ .

(7.18) (7.19)

(7.18) is verified by Stirling’s formula and the condition n − p j ,α  μ. Since L is taken so that L > σ |α |/q j,α and since p j,α  q j,α holds, the first inequality of (7.19) is clear. Since p j,α /k1  [ j + σ |α | − m]+ is known, by the condition d = ( L + 1/k1 )/σ we have the second inequality of (7.19). Therefore, if we set β j ,α = e L (1 + p j ,α /μ)σ |α |+ L /2 we have

(n − p j ,α )[ j +σ |α |−m]+ Γ (n + 1) L (d(n− p j,α )+|α |) θσ (hρ ) σ | α | (n − p j ,α ) Γ (n − p j ,α + 1) L (d(n− p j,α )+|α |)

 β j ,α n p j,α L +[ j +σ |α |−m]+ −σ |α | θσ

(d(n− p j,α )+|α |)

 β j ,α n(dp j,α −|α |)σ θσ

(hρ )

(hρ )

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H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

  e σ β j ,α

(dp j,α −|α |)σ

en dn + 1

θσ(dn) (hρ )  e σ β j ,α

 (dp j,α −|α |)σ e

d

θσ(dn) (hρ ):

in the above we have used (4) of Lemma 4.3. Thus, by applying this to (7.17) and by taking H > 0 sufficiently large so that



A j ,α h|α | β

( j ,α )∈Λ

δH

p j ,α

× e σ β j ,α

we have the result (7.15). This proves Theorem 7.4.

 (dp j,α −|α |)σ e

d

 1,

2

Now, let us show that the formal solution

w (ξ, x) =



w n (ξ, x)

(7.20)

n∈N

is convergent on S I × C N and it defines a holomorphic solution of (7.1) on S I × C N . Since defined by 1/κ0,1 = 1/k − 1/k1 , we have dσ = L + 1/k − 1/κ0,1 and so

Γ (dn + 1)σ Ah1 n  , Γ (n + 1) L Γ (n/k) Γ (n/κ0,1 )

κ0,1 > 0 is

n∈N ,

for some A > 0 and h1 > 0. By combining this with (7.15), we have constants C 1 > 0 and H 1 > 0 such that

    w n (ξ, x)  n∈N

n∈N





(|ξ0 |k + |ξ |k )n/k−1 exp c |ξ0 |k + |ξ |k + b1 |x|γ k k m Γ (n/κ0,1 ) (|ξ0 | + |ξ | + 1) C 1 H 1n

(7.21)

holds on S I × C N for any n ∈ N . Let qi > 0 (i = 1, . . . , ) be the ones in (7.6). We choose αi > 0 (i = 1, . . . , ) so that μ = α1 q1 + · · · + α q . Since any n ∈ N is expressed in the form n = μ + p 1 q1 + p 2 q2 + · · · + p  q for some p 1 , . . . , p  ∈ N, we have n/κ0,1 = (α1 + p 1 )q1 /κ0,1 + · · · + (α + p  )q /κ0,1 and by (2) of Lemma 7.3 we have

 C 1 H 1 n t n−k  Γ (n/κ0,1 )

n∈N

 p 1 ,..., p  0

C 1 H 1 μ+ p 1 q1 +···+ p  q × t μ+ p 1 q1 +···+ p  q −k

Γ ((α1 + p 1 )q1 /κ0,1 + · · · + (α + p  )q /κ0,1 )



 C 1 H 1 μ t μ−k K exp a( H 1 t )κ0,1 for any t > 0

for some K > 0 and a > 0. Thus, by setting M 0 = C 1 H 1 μ K , c 2 = a( H 1 )κ0,1 and t = (|ξ0 |k + |ξ |k )1/k , and by applying this to (7.21) we have k k μ/k−1  

κ0,1 /k

 w n (ξ, x)  M 0 (|ξ0 | + |ξ | ) exp c 2 |ξ0 |k + |ξ |k (|ξ0 |k + |ξ |k + 1)m n∈N



× exp c |ξ0 |k + |ξ |k + b1 |x|γ on S I × C N .

This proves that the formal solution (7.20) is convergent on S I × C N . Since the following result.

κ0,1  k holds, we have

H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

3627

Theorem 7.5. Eq. (7.1) has a holomorphic solution w (ξ, x) on S I × C N which satisfies k k μ/k−1    w (ξ, x)  M (|ξ0 | + |ξ | ) (|ξ0 |k + |ξ |k + 1)m

κ /k

× exp c 1 |ξ0 |k + |ξ |k 0,1 + b1 |x|γ on S I × C N

for some M > 0, c 1 > 0 and b1 > 0. 7.4. Local uniqueness of the solution Lastly, let us show the local uniqueness of the solution in Theorem 6.7. To do so, it is enough to prove the following result. Theorem 7.6. Let 0 < r < ∞, and let w (t , x) be a holomorphic function on S I (r ) × C N that satisfies



P k ξ0k + ξ k



w=







a j ,α (ξ ) ∗k M j ,α ,ξ0 ∂xα w



on S I (r ) × C N ,

(7.22)

( j ,α )∈Λ

 



 w (ξ, x)  M |ξ0 |k + |ξ |k μ/k−1 exp b1 |x|γ on S I (r ) × C N

(7.23)

for some M > 0 and b1 > 0. Then we have w (ξ, x) = 0 on S I (r ) × C N . In the case 0 < r < ∞, we will use

ψa,ξ0 (ξ ; 0) =

(|ξ0 |k + |ξ |k )a/k−1 , Γ (a/k)

a > 0.

Then, Lemma 7.2 can be rewritten into the form Lemma 7.7. For any μ > 0 and L Λ > 0 there is a constant β > 0 which satisfies the following: if w (ξ ) is a holomorphic function on S I (r ) and if the estimate | w (ξ )|  A ψ N ,ξ0 (ξ ; 0) holds on S I (r ) for some A > 0 and N  μ, we have

  M j ,α ,ξ [ w ](ξ )  0

β N σ |α |

A ψ N ,ξ0 (ξ ; 0)

on S I (r )

for any j + σ |α |  L Λ . Proof. Since |ξ0 |k + |ξ |k  |ξ0 |k + r k holds for any ξ ∈ S I (r ), by (1) of Lemma 6.6 we have

 

M j ,α ,ξ [ w ](ξ )  k j |ξ0 |k + r k j A ψ N +kσ |α |,ξ (ξ ; 0) 0 0

j  k j |ξ0 |k + r k A ×

(|ξ0 |k + r k )σ |α | × (|ξ0 |k + |ξ |k ) N /k−1 Γ ( N /k) × . Γ ( N /k) Γ ( N /k + σ |α |)

Therefore, by Stirling’s formula we have the result (7.24).

2

(7.24)

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Proof of Theorem 7.6. We take q so that 0 < q  q j ,α holds for any ( j , α ) ∈ Λ: since 0 < r < ∞ is assumed, we may suppose that |a j ,α (ξ )|  A 0j ,α φq (ξ ; 0) on S I (r ) for any ( j , α ) ∈ Λ. By (7.23) and Proposition 4.2 we have

 

 w (ξ, x)  M 0 ψμ,ξ (ξ ; 0)θσ (hρ ) exp b2 |x|γ on S I (r ) × C N 0 ρ for some M 0 > 0, h > 0 and b2 > 0. We take L > 0 so that L > σ |α | holds for any ( j , α ) ∈ Λ, and set d = L /σ . Then, by the similar argument to the proof of Theorem 7.4 we have Lemma 7.8. There are C > 0 and H > 0 such that

   w (ξ, x)  ρ

C Hn

Γ (n + 1) L



ψμ+qn,ξ0 (ξ ; 0)θσ(dn) (hρ ) exp b2 |x|γ

(7.25)

on S I (r ) × C N for any n = 0, 1, 2, . . . . Since d = L /σ , by Stirling’s formula we have

σ Γ (dn + 1)σ  A 1 H 1 n n!d = A 1 σ H 1 σ n n! L , for some A 1 > 0 and H 1 > 0. Therefore, by setting

n = 0, 1, 2, . . . ,

ρ = 0 in (7.25) we have

 

 w (ξ, x)  C H n A 1 σ H 1 σ n ψμ+qn,ξ (ξ ; 0) exp b2 |x|γ 0 = C H n A1σ H 1σ n



(|ξ0 |k + |ξ |k )(μ+qn)/k−1 exp b2 |x|γ Γ ((μ + qn)/k)

on S I (r ) × C N . By letting n → ∞ we have | w (ξ, x)| = 0 for any (ξ, x) ∈ S I (r ) × C N . This proves Theorem 7.6. 2 8. Proof of Theorem 3.7 In this last section, we will give a proof of Theorem 3.7 (the main theorem). We recall that our equation in (E) is



∂tm u +

j

a j ,α (t )∂t ∂xα u = f (t , x),

(8.1)

( j ,α )∈Λ

and our assumptions are (A1 ), (3.2), (A2 ), (A3 ),

uˆ (t , x) =



γ ∈ C and γ > 1. Let



un (x)t n ∈ Exp{γ } C N [[t ]]

(8.2)

n0

be the unique formal solution of (E), and let the t-Newton polygon of (E) be as in Fig. 1. We set σ = 1 − 1/γ : we have 0 < σ < 1. Since the result in the case p ∗ = 0 is already known in Corollary 5.2, we may suppose: p ∗  1. Take any nonempty open interval I satisfying the condition (3.6): we take another open interval J  I very close to I so that J also satisfies the condition (3.6) (with I replaced by J ). In the case p ∗  2, by Proposition 4.6 we can choose nonempty open intervals I 1 , I 2 , . . . , I p ∗ −1 which satisfy I i  J (1  i  p ∗ − 1),

H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

I i  Zi

I i +1  I i + [π /2κi ,i +1 ]

and



3629

1  i  p∗ − 1

(where I p ∗ = J ). Since J satisfies (3.6) we have I p ∗ = J  Z p ∗ . 8.1. Change of the expression of (8.1)

μ > 0 be a positive integer with 1  μ 

We will give here another expression of Eq. (8.1). Let k1 m + m, and set

Uˆ (t , x) =

∞ 

un (x)t n ,



ϕ (t , x) =

n =μ

un (x)t n ,

0n<μ



F (t , x) = f (t , x) − ∂tm ϕ (t , x) −

j

a j ,α (t )∂t ∂xα ϕ (t , x):

(8.3)

( j ,α )∈Λ

we know that Uˆ (t , x) ∈ t μ × Exp{γ } (C N )[[t ]], F (t , x) ∈ Exp{γ } ( D r × C N ), and that Uˆ (t , x) is a formal solution of the equation

∂tm u +



j

a j ,α (t )∂t ∂xα u = F (t , x).

(8.4)

( j ,α )∈Λ

For q > 0 and r > 0 we denote by Xq ( D r ) the set of all functions f (t ) which can be expressed in the form

f (t ) = t q a0 (t ) + t q1 a1 (t ) + · · · + t qs as (t ) for some q < q1 < · · · < q s and ai (t ) ∈ O ( D r ) (i = 0, 1, . . . , s). Let P i (λ) (1  i  p ∗ ) be as in Section 3.3, let h i > 0 (1  i  p ∗ ) be the ones in (4.5), and set

 

(, α ); 1    j − 1 . Λ0 = (, 0); 1    m − 1 ∪ Λ ∪ ( j ,α )∈Λ

Proposition 8.1. (1) Let 1  i  p ∗ . By multiplying (8.4) by t hi we have the equation (Eq)i





P i t ki +1 ∂t u = t hi F (t , x) +





j

Ai , j ,α (t ) t ki σ |α | t ki +1 ∂t ∂xα u

( j ,α )∈Λ0

for some Ai , j ,α (t ) ∈ Xqi, j,α ( D r ) with q i , j ,α > 0 (( j , α ) ∈ Λ0 ) satisfying

 ki +1 = min ∞,

 min

qi , j ,α + ki ( j + σ |α | − li )

( j ,α )∈Λ0 , j +σ |α |>li

j + σ |α | − l i

 .

(8.5)

(2) Moreover, we have the following equalities:

t hi+1 −hi × (Eq)i = (Eq)i +1 ,

1  i  p ∗ − 1.

(8.6)

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Proof. For simplicity, we use the notation O (t q ) to denote a function belonging to Xq ( D r ). Note that the formula



t m(k+1) ∂tm = t k+1 ∂t

m



+



A m, t k(m−) t k+1 ∂t



(8.7)

1m−1

holds for some constants A m, (m  1 and 1    m − 1). By using this formula, let us give a proof of Proposition 8.1. By multiplying (8.4) by t h1 , and then by using h1 = (k1 + 1)m and the formula (8.7) we have



t k1 +1 ∂t

m



u+



A m, t k1 (m−) t k1 +1 ∂t

1m−1



+



t k1 +1 ∂t

j

( j ,α )∈Λ\I1





u+



A j , t k1 ( j −) t k1 +1 ∂t



 u



j

t h1 −(k1 +1) j −k1 σ |α | a j ,α (t ) t k1 σ |α | t k1 +1 ∂t ∂xα u



+

u

1 j −1

( j ,0)∈I1

+



t h1 −(k1 +1) j a j ,0 (t )







A j , t k1 ( j −) t k1 σ |α | t k1 +1 ∂t



 α

∂x u = t h1 F (t , x)

1 j −1

where I1 is the index set defined in Section 3.3. We set p 1, j ,α = h1 − (k1 + 1) j − k1 σ |α | + ordt (a j ,α ) (( j , α ) ∈ Λ). By (4) of Lemma 4.5 we know that if ( j , 0) ∈ I1 we have p 1, j ,0 = 0 and if ( j , α ) ∈ Λ \ I1 we have p 1, j ,α > 0: therefore, by the definition of P 1 (λ) we have







P 1 t k1 +1 ∂t u +

+

  





O (t ) t k1 +1 ∂t

( j ,α )∈Λ\I1

+



j





u

1m−1

( j ,0)∈I1

+



O t k1 (m−) t k1 +1 ∂t



O t p 1, j,α



u+





O t k1 ( j −) t k1 +1 ∂t



 u

1 j −1



t k1

σ |α |

t k1 +1 ∂t



j α  ∂x u





O t p 1, j,α +k1 ( j −) t k1 σ |α | t k1 +1 ∂t

 α  ∂x u = t h1 F (t , x).

( j ,α )∈Λ\I1 1 j −1

This is nothing but the equality (Eq)1 with q1, j ,α (( j , α ) ∈ Λ0 ) defined in the following way: under the setting

B 1, j ,α =

min

(d,α )∈Λ,d−1 j





p 1,d,α + k1 (d − j ) ,

if j = 0 and (0, α ) ∈ Λ then q1,0,α = p 1,0,α ; if j  1, |α | > 0 and ( j , α ) ∈ Λ then q1, j ,α = min{ p 1, j ,α , B 1, j ,α }; if j  1, |α | > 0 and ( j , α ) ∈ Λ0 \ Λ0 then q1, j ,α = B 1, j ,α ; if j  1 and ( j , 0) ∈ I1 then q1, j ,0 = min{1, B 1, j ,0 }; if j  m and ( j , 0) ∈ Λ \ I1 then q1, j ,0 = min{ p 1, j ,0 , B 1, j ,0 }; if 1  j  m − 1 and ( j , 0) ∈ Λ \ I1 then q1, j ,0 = min{ p 1, j ,0 , B 1, j ,0 , k1 (m − j )}; if j  m and ( j , 0) ∈ Λ0 \ Λ then q1, j ,0 = B 1, j ,0 ; if 1  j  m − 1 and ( j , 0) ∈ Λ0 \ Λ then q1, j ,0 = min{ B 1, j ,0 , k1 (m − j )}. Let us show (8.5). If p ∗ = 1 we have k2 = ∞: in this case by Lemma 4.4 we see that the set {( j , α ) ∈ Λ0 ; j + σ |α | > l1 } is empty and so (8.5) for i = 1 is clear. Let us consider the case p ∗  2. Since h1 + ordt (a j ,α ) − j − k1 l1 = p 1, j ,α + k1 ( j + σ |α | − l1 ) holds, by (5) of Lemma 4.5 we have

H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

k2 =

p 1, j ,α + k1 ( j + σ |α | − l1 )

min

j + σ |α | − l 1

( j ,α )∈Λ, j +σ |α |>l1

3631

(8.8)

.

If ( j , α ) ∈ Λ \ I1 , 1    j − 1 and  + σ |α | > l1 hold, we have j + σ |α | > l1 and

( p 1, j ,α + k1 ( j − )) + k1 ( + σ |α | − l1 ) p 1, j ,α + k1 ( j + σ |α | − l1 ) =  + σ |α | − l 1  + σ |α | − l 1 >

p 1, j ,α + k1 ( j + σ |α | − l1 ) j + σ |α | − l 1

 k2 .

(8.9)

These two conditions (8.8) and (8.9) lead us to (8.5) for i = 1. Thus, the case i = 1 is proved. The cases i = 2, . . . , p ∗ can be proved in the same way. The assertion (2) is clear. 2 8.2. Proof of (2) of Theorem 3.7 Let Uˆ (t , x) be as in (8.3); by Proposition 8.1 we see that Uˆ (t , x) is a formal solution of (Eq)1 and we have the formal equality







P 1 t k1 +1 ∂t Uˆ = t h1 F (t , x) +



j

A1, j ,α (t ) t k1 σ |α | t k1 +1 ∂t ∂xα Uˆ .

(8.10)

( j ,α )∈Λ0

We note that each term in the formula (8.10) belongs to the class









t h1 −m O C N [[t ]] + t h1 O C N [[t ]] +







t h1 − j +ordt (a j,α ) O C N [[t ]],

( j ,α )∈Λ

and we know h1 − m = k1 m and the following: by (4) of Lemma 4.5 we have h1 − j + ordt (a j ,0 ) = k1 j if ( j , 0) ∈ I1 , and h1 − j + ordt (a j ,α ) > k1 ( j + σ |α |) if ( j , α ) ∈ Λ \ I1 . We set





w 1 (ξ, x) = Bˆk1 [Uˆ ](ξ, x) ∈ ξ μ−k1 O C N [[ξ ]] which is the formal k1 -Borel transform of Uˆ (t , x). Since the coefficients un (x) (n = 0, 1, 2, . . .) of uˆ (t , x) satisfy the estimates (5.3) (in Theorem 5.1), the series ξ k1 −μ w 1 (ξ, x) is convergent in D δ × C N for some δ > 0 and so w 1 (ξ, x) defines a holomorphic function on R( D δ \ {0}) × C N : moreover we have the estimate

 



 w 1 (ξ, x)  M |ξ |μ−k1 exp b|x|γ on R D δ \ {0} × C N for some M > 0 and b > 0. For ( j , α ) ∈ N × N N we write

 M1, j ,α [ W ] =

ξ k1 σ |α |−k1 Γ (σ |α |)

∗k1 ((k1 ξ k1 ) j W ), if |α | > 0,

(k1 ξ k1 ) j W ,

By applying the formal k1 -Borel transform Bˆk1 to (8.10) we have

if |α | = 0.

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H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

Proposition 8.2. The above w 1 (ξ, x) satisfies the following convolution partial differential equation





P 1 k1 ξ k1 w 1 = f 1 (ξ, x) +







a1, j ,α (ξ ) ∗k1 M1, j ,α ∂xα w 1



(8.11)

( j ,α )∈Λ0

on R( D δ \ {0}) × C N for some δ > 0, where f 1 (ξ, x) = Bk1 [t h1 F ](ξ, x), and a1, j ,α (ξ ) = Bk1 [Ai , j ,α ](ξ ) (( j , α ) ∈ Λ0 ). Moreover we have the following: (1) f 1 (ξ, x) is a holomorphic function on R( D δ \ {0}) × C N , and a1, j ,α (ξ ) (( j , α ) ∈ Λ0 ) are holomorphic functions on R( D δ \ {0}). (2) There are F 1 > 0, c > 0, b > 0 and A 1, j ,α > 0 (( j , α ) ∈ Λ0 ) such that the following estimates hold:

 



 f 1 (ξ, x)  F 1 |ξ |h1 −k1 exp c |ξ |k1 + b|x|γ on R D δ \ {0} × C N ,  



a1, j ,α (ξ )  A 1, j ,α |ξ |q1, j,α −k1 exp c |ξ |k1 on R D δ \ {0} ( j , α ) ∈ Λ0 . Proof. By applying the formal k1 -Borel transform Bˆk1 to each term of the formal equation (8.10) we have



   Bˆk1 P 1 t k1 +1 ∂t Uˆ = Bˆk1 t h1 F +





j

 Bˆk1 A1, j ,α (t ) t k1 σ |α | t k1 +1 ∂t ∂xα Uˆ .

( j ,α )∈Λ0

Since t h1 F (t , x) and A1, j ,α (t ) (( j , α ) ∈ Λ0 ) are convergent, we have Bˆk1 [t h1 F ] = Bk1 [t h1 F ] and

Bˆk1 [A1, j,α ] = Bk1 [A1, j,α ] (( j , α ) ∈ Λ0 ). Since a calculation yields



j 

j Bˆk1 t k1 +1 ∂t Uˆ (ξ, x) = k1 ξ k1 w 1 (ξ, x), and 

j    Bˆk1 t k1 σ |α | t k1 +1 ∂t ∂xα Uˆ (ξ, x) = M1, j ,α ∂xα w 1 (ξ, x), we have the equality (8.11). The estimates in (2) are clear.

2

We note: since 0 < μ  k1 m + m = h1 is supposed, the first estimate in the above (2) can be replaced by | f 1 (ξ, x)|  F 1 |ξ |μ−k1 exp(c |ξ |k1 + b|x|γ ) on R( D δ \ {0}) × C N . Since I 1  Z1 holds, we have 0 < | I 1 | < 2π /k1 and

λ1,1 , . . . , λ1,l1 −m ∈ C \ π ( S k1 I 1 ). We note that P 1 (λ) = p1 (λ) × λm holds and so the roots of P 1 (λ) = 0 are λ = 0 and λ1,1 , . . . , λ1,l1 −m . By (8.5) with i = 1 we have

 k2 = min ∞,

 min

q1, j ,α + k1 ( j + σ |α | − l1 )

( j ,α )∈Λ0 , j +σ |α |>l1

Hence, we can apply Theorem 6.1 (with m, k, k1 and respectively) to (8.11) and we have

j + σ |α | − l 1

 .

κ0,1 replaced by l1 , k1 , k2 and κ1,2 = 1/k1 − 1/k2 ,

Lemma 8.3. The function w 1 (ξ, x) has an analytic continuation w ∗1 (ξ, x) on S I 1 × C N satisfying the following estimate: for any sufficiently small > 0 there are M 1 > 0, c 1 > 0 and b1 > 0 such that μ−k1   ∗

 w (ξ, x)  M 1 |ξ | exp c 1 |ξ |κ1,2 + b1 |x|γ on S I 1 −[ ] × C N . 1 k l 1 1 (|ξ | + 1)

(8.12)

H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

3633

In the case p ∗ = 1, we have I 1 = J , k2 = ∞ and κ1,2 = k1 : by taking > 0 sufficiently small we have I ⊂ J − [ ]. Thus, in the case p ∗ = 1 the assertion of Lemma 8.3 leads us to the conclusion that Uˆ (t , x) is (k1 )-summable in I -direction. In the case p ∗  2 we continue our argument. By (8.11) and the unique continuation property for holomorphic functions we have





P 1 k1 ξ k1 w ∗1 = f 1 (ξ, x) +







a1, j ,α (ξ ) ∗k1 M1, j ,α ∂xα w ∗1



(8.13)

( j ,α )∈Λ0

on S I 1 × C N . We set





w 2 (ξ, x) = Ak2 ,k1 w ∗1 (ξ, x), f 2 (ξ, x) = Bk2 [t h2 F ](ξ, x), and a2, j ,α (ξ ) = Bk2 [A2, j ,α ](ξ ) (( j , α ) ∈ Λ0 ). For ( j , α ) ∈ N × N N , we write

 M2, j ,α [ W ] =

ξ k2 σ |α |−k2 Γ (σ |α |)

∗k2 ((k2 ξ k2 ) j W ), if |α | > 0, if |α | = 0.

(k2 ξ k2 ) j W ,

Then we have: Proposition 8.4. The function w 2 (ξ, x) is a holomorphic function on S I 2 (δ2 ) × C N for some δ2 > 0 and satisfies

 

 w 2 (ξ, x)  A |ξ |μ−k2 exp b1 |x|γ on S I (δ2 ) × C N 2

(8.14)

for some A > 0. Moreover, it satisfies the following equation





P 2 k2 ξ k2 w 2 = f 2 (ξ, x) +







a2, j ,α (ξ ) ∗k2 M2, j ,α ∂xα w 2



(8.15)

( j ,α )∈Λ0

on S I 2 (δ2 ) × C N , that is just the equation obtained by applying k2 -Borel transform to the equality (Eq)2 . Proof. Since (8.12) and I 2  I 1 + [π /2κ1,2 ] hold, by Proposition 2.2 we see that w 2 (ξ, x) is welldefined as a holomorphic function on S I 2 (δ2 ) × C N and satisfies the estimate (8.14) for some A > 0 and δ2 > 0. {γ } Let us show (8.15). We denote by E k ( S I 1 −[ ] × C N ) the set of all holomorphic functions W (ξ, x) on S I 1 −[ ] × C N satisfying

1

 

 W (ξ, x)  M 1 |ξ |μ−k1 exp c 1 |ξ |k1 + b1 |x|γ on S I −[ ] × C N 1 for some M 1 > 0, c 1 > 0 and b1 > 0. {γ } In the case w ∗1 (ξ, x) ∈ E k ( S I 1 −[ ] × C N ), the equality (8.15) is verified in the following way. In 1 this case we can define U 1 = Lk1 [ w ∗1 ] and so we have w 2 = Bk2 ◦ Lk1 [ w ∗1 ] = Bk2 [U 1 ]. Since (8.13) is known, by applying Lk1 to Eq. (8.13) and by using the relation



j 

j  

j Lk1 k1 ξ k1 w ∗1 = t k1 +1 ∂t Lk1 w ∗1 = t k1 +1 ∂t U 1

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H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

we have the equality (8.10) (with Uˆ replaced by U 1 ), that is, the equality (Eq)1 (with u replaced by U 1 ) on W I 1 −[ ],k1 (1/c 1 ) × C N . Then, by multiplying this by t h2 −h1 and by (8.6) we have the equality (Eq)2 , that is,







P 2 t k2 +1 ∂t U 1 = t h2 F (t , x) +



j

A2, j ,α (t ) t k2 σ |α | t k2 +1 ∂t ∂xα U 1

( j ,α )∈Λ0

on W I 1 −[ ],k1 (1/c 1 ) × C N . Since I 2  I 1 + [π /2κ1,2 ] holds, by applying Bk2 to this formula and by using the relation



j 

j

j Bk2 t k2 +1 ∂t U 1 = k2 ξ k2 Bk2 [U 1 ] = k2 ξ k2 w 2 we have the equality (8.15). To prove the general case, it is enough to show

(8.15) =

  ξ (h2 −h1 )−k2 ∗k2 Ak2 ,k1 (8.13) . Γ ((h2 − h1 )/k2 )

(8.16)

For simplicity, we set









G 1 [ w ] = P 1 k1 ξ k1 w − f 1 (ξ, x) −

 ( j ,α )∈Λ0

G 2 [ w ] = P 2 k2 ξ k2 w − f 2 (ξ, x) −















a1, j ,α (ξ ) ∗k1 M1, j ,α ∂xα w , a2, j ,α (ξ ) ∗k2 M2, j ,α ∂xα w .

( j ,α )∈Λ0

Then, (8.13) is nothing but G 1 [ w ∗1 ] = 0, and (8.15) is nothing but G 2 [ w 2 ] = G 2 [Ak2 ,k1 [ w ∗1 ]] = 0. Therefore, the result (8.16) follows from Lemma 8.5. Let W (ξ, x) be a holomorphic function on S I 1 × C N satisfying the estimate (8.12) (with w ∗1 replaced by W ) for some M 1 > 0, c 1 > 0, b1 > 0 and > 0 with I 2  I 1 + [π /2κ1,2 − ]. Then, we have





G 2 Ak2 ,k1 [ W ] (ξ, x) =

  ξ (h2 −h1 )−k2 ∗k2 Ak2 ,k1 G 1 [ W ] (ξ, x) Γ ((h2 − h1 )/k2 )

(8.17)

on S I 2 × C N . Proof. We will show this by approximations. Let σ > κ1,2 , let θ be the middle point of I 1 , and let 0 < φ < min{| I 1 |/2 − , π /2σ }. Set I 1,0 = (θ − φ, θ + φ)  I 1 − [ ] and α = e −i θ σ . Then, for any η > 0 we have





exp −ηα ξ σ   exp −η cos(σ φ)|ξ |σ , For

ξ ∈ S I 1,0 .

η > 0 we set



W η (ξ, x) = W (ξ, x) × exp −ηα ξ σ ,

(ξ, x) ∈ S I 1,0 × C N

which is regarded as a holomorphic function on S I 1,0 × C N . Since {γ }

σ > κ1,2 and cos(σ φ) > 0 hold,

we have W η (ξ, x) ∈ E k ( S I 1,0 × C N ). Therefore, by the same argument as in the case w ∗1 (ξ, x) ∈ 1 {γ }

E k ( S I 1 −[ ] × C N ) we can show the equality 1

H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637





G 2 Ak2 ,k1 [ W η ] (ξ, x) =

3635

  ξ (h2 −h1 )−k2 ∗k Ak ,k G 1 [ W η ] (ξ, x) Γ ((h2 − h1 )/k2 ) 2 2 1

(8.18)

on S I 1,0 +[π /2κ1,2 ] × C N . Here, we note that W η (ξ, x) converges pointwise to W (ξ, x) (as η → 0) and we have | W η (ξ, x)|  | W (ξ, x)| on S I 1,0 × C N . Since Ak2 ,k1 is an integral operator defined by (2.4), we can express the both sides of (8.18) to the form of integration. Therefore, by letting η → 0 in (8.18) and by using Lebesgue’s convergence theorem we have the equality (8.17) on S I 1,0 +[π /2κ1,2 ] × C N . Thus, by the unique continuation property for holomorphic functions we have (8.17) on S I 1 +[π /2κ1,2 − ] × C N . This proves Lemma 8.5. 2 This completes the proof of Proposition 8.4.

2

By Proposition 8.4 we have seen that w 2 (ξ, x) is a holomorphic function on S I 2 (δ2 ) × C N with | w 2 (ξ, x)|  A |ξ |μ−k2 exp(b1 |x|γ ) on S I 2 (δ2 )×C N for some A > 0 and δ2 > 0, and it satisfies Eq. (8.15). Therefore, by the same argument as in Lemma 8.3 we have Lemma 8.6. The function w 2 (ξ, x) has an analytic continuation w ∗2 (ξ, x) on S I 2 × C N satisfying the following estimate: for any sufficiently small > 0 there are M 2 > 0, c 2 > 0 and b2 > 0 such that μ−k2  ∗ 

 w (ξ, x)  M 2 |ξ | exp c 2 |ξ |κ2,3 + b2 |x|γ on S I 2 −[ ] × C N . 2 (|ξ |k2 + 1)l2

In this way, we can show inductively on i (1  i  p ∗ ) that the functions w 1 (ξ, x), . . . , w i (ξ, x) are well-defined by

w 1 (ξ, x) = Bˆk1 [Uˆ ](ξ, x),





w 2 (ξ, x) = Ak2 ,k1 w ∗1 (ξ, x),

.. . 



w i (ξ, x) = Aki ,ki−1 w ∗i −1 (ξ, x) and that w i (ξ, x) has an analytic continuation w ∗i (ξ, x) on S I i × C N satisfying the following estimate: for any sufficiently small > 0 there are M i > 0, c i > 0 and b i > 0 such that μ−ki  ∗ 

 w (ξ, x)  M i |ξ | exp c i |ξ |κi,i+1 + b i |x|γ on S I i −[ ] × C N . i (|ξ |ki + 1)li

Moreover, we see that w ∗i (ξ, x) satisfies the equality





P i ki ξ ki w ∗i = f i (ξ, x) +







ai , j ,α (ξ ) ∗ki Mi , j ,α ∂xα w ∗i



( j ,α )∈Λ0

on S I i × C N , that is just the equation obtained by applying ki -Borel transform to the equality (Eq)i , where f i (ξ, x) = Bki [t hi F ](ξ, x), ai , j ,α (ξ ) = Bki [Ai , j ,α ](ξ ) (( j , α ) ∈ Λ0 ), and Mi , j ,α is defined in the same way as M1, j ,α and M2, j ,α . Finally, in the case i = p ∗ we have

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H. Tahara, H. Yamazawa / J. Differential Equations 255 (2013) 3592–3637

Lemma 8.7. The function w p ∗ (ξ, x) has an analytic continuation w ∗p ∗ (ξ, x) on S I p∗ × C N satisfying the following estimate: for any sufficiently small > 0 there are M p ∗ > 0, c p ∗ > 0 and b p ∗ > 0 such that μ−k p ∗   ∗

∗  w ∗ (ξ, x)  M p |ξ | exp c p ∗ |ξ |κ p∗ , p∗ +1 + b p ∗ |x|γ on S I p∗ −[ ] × C N . p k p∗ l p∗ (|ξ | + 1)

Now, let us recall that J and I p ∗ are taken so that I  J = I p ∗ holds: we have I ⊂ I p ∗ − [ ] for a sufficiently small > 0. Since κ p ∗ , p ∗ +1 = k p ∗ holds, the assertion of Lemma 8.7 concludes that our formal solution Uˆ (t , x) in (8.3) is (k p ∗ , . . . , k1 )-summable in the I -direction. Since

uˆ (t , x) = Uˆ (t , x) +



un (x)t n ,

0nμ−1

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  N −1   

 ∗  un (x)t n   Ch N Γ ( N /k1 )|t | N exp b|x|γ u (t , x) −   n =0

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