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ANALYTIC AND GEVREY SOLUTIONS OF NON-LINEAR PARTIAL DIFFERENTIAL

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					Far East J. Appl. Math.




    ANALYTIC AND GEVREY SOLUTIONS OF
NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS

G. DE DONNO, A. OLIARO and L. RODINO

Dipartimento Di Matematica, Università Di Torino
Via Carlo Alberto 10, 10123 Torino, Italy
e-mail: dedonno@dm.unito.it
        oliaro@dm.unito.it
        rodino@dm.unito.it

                                         Abstract

         We prove some results of existence and regularity in analitic-Gevrey
         spaces for solutions of linear and non-linear partial differential
         equations. In particular we consider Navier-Stokes equations and
         perturbations of powers of the Schrödinger operator.


                                    1. Introduction

    An elementary, but instructive, example of the use of the Gevrey
classes for the study of the non-linear equations of the Applied
Mathematics is given by the Cauchy problem:
                      2
                    ∂ t u = F (t, x , u, ∂ t u, ∂ x1 u, ..., ∂ xn u ),

                    u(0, x ) = u0 (x ), (∂ t u ) (0, x ) = u1 (x ),                    (1.1)

where x = (x1 , ..., x n ) are the space variables in a domain of R n .
Assume for simplicity that the non-linear function F is analytic, but allow
2000 Mathematics Subject Classification: Primary 35S05.
Key words and phrases: Please provide.

Work supported by NATO grant PST. CLG. 979347.

Communicated by Guidorzi Marcello

Received February 19, 2004
                                                               2004 Pushpa Publishing House
2              G. DE DONNO, A. OLIARO and L. RODINO

possibly that dx u = (∂ x1 u, ..., ∂ x n u ) is replaced in (1.1) by Adx u, where
A is a matrix of classical pseudo-differential operators of order 0.
Problems of the type (1.1) appear often in the applications, related to
equations of fluids, or with different Physical meanings, according to the
choice of F. It is well known that (1.1) is well-posed in the classical sense,
i.e., for initial data in Sobolev spaces or other standard function or
distribution spaces, only if restrictive assumptions are satisfied by the
right-hand side F. In other relevant cases, despite the solution is
expected by Physical intuition, a theorem of existence is not possible in
this frame.

    Gevrey classes provide a different functional frame, allowing to satisfy
the natural hope, that (1.1) should be always solvable, independently of
F. Namely, if the initial data u0 (x ), u1 (x ) belong to the Gevrey class G s ,
with index 1 ≤ s ≤ 2, than (1.1) admits a solution u(t, x ), t ≤ T , in the

same Gevrey class. Note that, if u0 (x ), u1 (x ) belong to G1 , i.e., are
analytic functions, the existence of an analytic solution u(t, x ) is already
granted by the Cauchy-Kowalevsky theorem.

    To be more definite, let us recall here the definition of the Gevrey
classes, which play the role of intermediate spaces between the spaces of
the C ∞ and analytic functions. Given Ω, open subset of R n , we say that

u ∈ G s (Ω ), s ≥ 1, if u ∈ C ∞ (Ω ) and for every compact subset K of Ω we
have
                        sup ∂ α u(x ) ≤ C α + (α!)s ,
                                             1
                                                                            (1.2)
                       x ∈K

for a constant C depending only on u and K. When s = 1 we recapture
the analytic case, whereas for s > 1 we obtain larger spaces, containing
functions with compact support. It is interesting to observe that if
     s
u ∈ G0 (R n ), i.e., u ∈ G s (R n ) has compact support, then its Fourier
transform u(ξ ) satisfies the estimates
          ˆ

                                        ξ1s
                       u(ξ ) ≤ Ce − ε
                       ˆ                      , ξ ∈ Rn ,                    (1.3)

for some positive constants C and ε. Actually, the estimates (1.3)
      ANALYTIC AND GEVREY SOLUTIONS OF NON-LINEAR …                                   3

characterize the G s -regularity of a Fourier transformable function, or
distribution. Observe that (1.3) implies

                       ξ1s                              ξ1s
                  eτ
                                             ∫   e 2τ
                                     2
                             u(ξ )
                             ˆ           =                        u(ξ ) 2 dξ < ∞
                                                                  ˆ                (1.4)

for a sufficiently small τ > 0. In the sequel we shall also consider Gevrey
classes on the n-torus T n . Given a function on T n

                        u( x ) =         ∑u en
                                                 α
                                                     iαx
                                                           , uα ∈ C,               (1.5)
                                     α∈Z


we have u ∈ G s (T n ), s ≥ 1, if and only if


                               ∑e    n
                                         2τ α 1 s
                                                        uα    2
                                                                  < ∞              (1.6)
                              α∈Z


for some τ > 0. The problem (1.1) can be obviously reset for x ∈ T n ,
with identical results of existence in G s (T n ), 1 ≤ s ≤ 2. For all the
related proofs we address the reader to classical works, see Kajitani [10]
and the references there; for the linear case see also the monographies of
Rodino [19], Mascarello and Rodino [14]. In the sequel of the paper we
want to show the effectiveness of the Gevrey functional frame in the
study of some other linear and non-linear partial differential equations,
with emphasis on equations of the Mathematical Physics. Namely, in the
next Section 2 we review a result of Foias and Temam [6] concerning
Navier-Stokes equations. In that paper existence of solutions is assumed
already achieved in a certain Sobolev space by other methods, let us refer
to Solonnikov [20, 21], and the aim is to show the Gevrey regularity of
such solutions.

    The other Sections of the present article are devoted to prove new
results for perturbations of powers of the Schrödinger operator (without
potential), in one space variable:
                                         S = i∂ t − ∂ 2 .
                                                      x                            (1.7)

In fact, the local properties of
                           Su = F (t, x , u, ∂ x u )                               (1.8)
4                G. DE DONNO, A. OLIARO and L. RODINO

do not differ from those of the non-perturbed operator S, see for example
Messina and Rodino [15]. On the contrary, for the equations
                                            l j
                         S 2u = F (t, x , ∂ t ∂ x u )2l + j ≤ 3                  (1.9)

we have completely different phenomena, depending on the lower order
terms in the (linear or non-linear) expression of F. So for example we
prove that
                                 S 2u = i∂ t ∂ x u                              (1.10)

is hypoelliptic, i.e., all the solutions are smooth, whereas the solutions of
S 2u = 0 admit obviously singularities.

    Gevrey classes provide a frame for the unified treatment of (1.9):
leaving a more detailed analysis to the future, we begin to give in the last
part of this paper some simple but representative results of Gevrey
solvability. The proofs rely on the techniques introduced in De Donno and
Oliaro [4].

                         2. Navier-Stokes Equations

     In the following we report the results of Foias and Temam [6] in a
lightly more general form, as suggested by the final Section of that paper.
We do not give proofs, since they are essentially a repetition of those in
[6].

    We consider the Navier-Stokes equations of viscous incompressible
fluids with space periodicity boundary conditions, in space dimension
n = 2 or n = 3. Setting then the problem in [0, T ] × Tn we obtain for the

unknown function u = (u (1) , u (2 ) ) for n = 2, or u = (u (1) , u (2 ) , u (3 ) ) for
n = 3, the abstract evolution equation

                        du
                           + νAu + B (u ) = f , u(0 ) = u0 ,                     (2.1)
                        dt

where ν is the kinematic viscosity and A, B are defined as standard.
Namely, we refer to the Hilbert space H defined as the closed subspace of
L2 (T n )n of all functions
     ANALYTIC AND GEVREY SOLUTIONS OF NON-LINEAR …                                             5

             u =   ∑u e  n
                              α
                                  iαx
                                        , uα ∈ Cn , u− α = uα , u0 = 0,                     (2.2)
                   α∈Z


such that for every α ∈ Z n
                                                    n
                                  α ⋅ uα =        ∑ α u( ) = 0
                                                  j =1
                                                             j α
                                                                j
                                                                                            (2.3)


(i.e., ∇u = 0). The operator A in (2.1) is the Stokes operator with space
periodicity boundary conditions. It is a self-adjoint unbounded positive
operator in H. The domain D ( A ) ⊂ H and more generally the domain of

the positive power D ( A r 2 ), r > 0, is the set of all u ∈ H such that


                                     ∑     α     2r
                                                        uα   2
                                                                    < ∞.                    (2.4)
                                   α∈Z n


We then consider (G s (T n ))n , the space of all the n-tuple of functions
satisfying (1.6). More generally, given s ≥ 1, τ > 0 and r ≥ 0 we define
           s
the space Gτ, r (Tn ) of all the (n-tuples of) functions satisfying


                             ∑       α   2r 2 τ α 1 s
                                            e                    uα    2
                                                                            < ∞.            (2.5)
                             α∈Z n


Note that Gτ, r (T n ) ⊂ (G s (T n ))n and for every fixed r ≥ 0
           s



                                  ∪G
                                  τ >0
                                           s      n
                                           τ, r (T )       = (G s (T n ))n .


The spaces G1, r (T n ), τ > 0, r ≥ 0 are then subsets of the analytic class in
            τ

T n . Finally, concerning B in (2.1), we have B (u ) = b(u, u ) where b(u, v )
is defined as usual by
                                                  n
                                                                            ∂v (k ) (k )
                    (b(u, v ), w ) =            ∑∫
                                                j , k =1
                                                             T   n
                                                                   u( j )
                                                                             ∂x j
                                                                                   w dx ,


which one can easily re-interpret in terms of the Fourier coefficients in
(2.2).
6                G. DE DONNO, A. OLIARO and L. RODINO

    Let us now state the result of Gevrey regularity for the solutions of
(2.1). We shall prescribe initial data u0 in D ( A1 2 ) ⊂ H , cf. (2.4). It is
then well known that in dimension n = 2 the solution u = u(t ) exists

and remains bounded in D ( A1 2 ) for all positive times, whereas for
n = 3 this is granted only in a finite interval [0, T ]. For all t in the

domain of D ( A1 2 ) existence, we have:

    Theorem 2.1. Let the right-hand side f in (2.1) be given in (G s (Tn ))n
∩ H , s ≥ 1. Then the above solution u(t ) is an analytic function of t, with

values in (G s (T n ))n .

                                s
    More precisely, assume f ∈ Gτ, 0 (Tn ) ∩ H , s ≥ 1, τ > 0. For small

values of t > 0 we have u(t ) ∈ Gts, 1 (T n ) ∩ H , whereas for large values of
                  s
t we get u(t ) ∈ Gσ, 1 (T n ) ∩ H , for a suitable constant σ, σ ≤ τ.


    Let us observe that for f ≡ 0, or f ∈ (G1 (T n ))n ∩ H , from Theorem
2.1 we deduce analyticity of the solution with respect to space and time
variables, for t > 0.

    3. Gevrey Hypoellipticity of Perturbations of Powers of the
                             Schrödinger Operator

    This Section deals with the properties of hypoellipticity and Gevrey
hypoellipticity of perturbations of powers of the Schrödinger operator.
The main properties of the operators depend heavily on the lower order
terms of their symbol. In particular we consider the following class of
linear differential operators:

           P (x , y, Dx , D y ) = (D y − Dx ) p +
                                     2
                                                     ∑ b ( x , y )D D ,
                                                    ( )
                                                    l, j ∈I
                                                              lj
                                                                   l
                                                                   x
                                                                       j
                                                                       y   (3.1)


                   ∂             ∂
where Dx = −i        and D y = −i ; with p, l, j ∈ Z + , p ≥ 2, and blj : Ω → C
                  ∂x             ∂y
are smooth functions. The subset of the indices I corresponding to lower
       ANALYTIC AND GEVREY SOLUTIONS OF NON-LINEAR …                                              7

order terms will be described in the sequel. The respective class of
symbols is the following:

                p(x , y, ξ, η) = (η2 − ξ ) p +        ∑ b (x , y)ξ η ,
                                                     ( )
                                                     l, j ∈I
                                                               lj
                                                                              l j
                                                                                             (3.2)


obtained from (3.1) by replacing the couple (ξ, η) to the couple of
derivatives (Dx , D y ) . We denote by z = (x , y ) the real variables in Ω,

open subset in R 2 , neighborhood of a point z0 ; ζ = (ξ, η) the dual
variables of z.

    To investigate (3.1), we follow closely the arguments in De Donno and
Oliaro [4], where the class of operators

                  D m − a(x , y ) Dx +
                    y
                                   d
                                          ∑ b (x , y)D D ,
                                         ( )
                                         l , j ∈I
                                                    lj
                                                                    l
                                                                    x
                                                                        j
                                                                        y     d <m


is studied in the anisotropic frame; also see De Donno and Rodino [5] about
isotropic case and pseudo-differential coefficients. Hypoellipticity and
solvability in [4], [5] depend only on lower order terms corresponding to a
                                             m ∗               1
unique couple (l ∗ , j ∗ ) ∈ Z 2 with k ∗ :=
                               +               l + j∗ > m −        ( k ∗ > m − 1,
                                             d                 2
in the case of constant coefficients), such that Im b ∗ ∗ (x , y ) ≠ 0. We also
                                                                            l j

                                                                                         1
obtained a sufficient condition for G s hypoellipticity, s ≥                                      in
                                                                                     ∗
                                                                                    k − (m − 1)
                md
[4], s ≥    ∗
                         in [5] (cf. the definition of Gevrey spaces in (1.2)). On
           k − (m − 1)
the contrary, for P in (3.1), terms of much lower order may play a role,
see the next Theorem 3.1. It is also natural to allow more couples
  ∗    ∗
(li , ji ), i = 1, ..., n, to give the order k ∗ = 2li∗ + ji∗ . The case of p-powers
                                      d
of generic operators (D m − a(x , y )Dx ) shall be detailed in a future paper.
                        y

      Let us observe that for operators of the form
                                                         ∗   ∗
                            (D y − Dx )p + iDx D y ,
                               2             l   j
                                                                                             (3.3)

we shall prove in this Section that k ∗ := 2l ∗ + j ∗ varying in the interval
8                     G. DE DONNO, A. OLIARO and L. RODINO

( p, 2 p) is allowed if p ≥ 2; while in the case p = 1 we obtained in [4],
[5] and [15] that the lower order terms have no influence.

    We define the following sets for k ∈ Q + , 0 < k < 2 p :

                                 I k = {(l, j ) ∈ Z + × Z + : 2l + j = k}

and fix k = k ∗ such that p < k ∗ < 2p. We use the notations k − for all
k < k ∗ , k + for all k > k ∗ . We define I = I − ∪ I                                   ∪ I + , with I − = ∪ I
                                                                                   k∗                            k−
and I + = ∪ I             .
                     k+

                              m
    In this Section we prove Sρ, δ estimates of (3.2) in order to obtain
hypoellipticity and Gevrey hypoellipticity of the class of operators (3.1).

    In the plane (ξ, η) we consider the conic neighborhood Λ = { η < Cξ}
of the semi-axis ξ > 0, for a suitable constant C; we observe that the

intersection of the half-plane ξ < 0 with the curve η2 − ξ = 0 is empty.

    We denote with Γ the set Ω × Λ, and we state the following:

    Theorem 3.1. Assume that I                                    consists of couples (li∗ , ji∗ ), i = 1, ..., n,
                                                             k∗

k ∗ = 2li∗ + ji∗ with p + 1 ≤ k ∗ < 2 p such that:

             n                              ∗       ∗
    (i)   ∑i =1 Im bli∗ ji∗ (x , y)ξli η ji                 ≠ 0, for all (z , ζ ) ∈ Γ, η ≠ 0;


    (ii) blj (x , y ) ≡ 0 for all (l, j ) ∈ I                          , (2l + j > k ∗ );
                                                                  k+

                 n                              ∗       ∗
    (iii)   ∑i =1 Re bli∗ ji∗ (x , y)ξli η ji                ≥ 0, for even p,

                                 n                                 ∗     ∗
                              ∑i =1 Re bli∗ ji∗ (x , y)ξli η ji              ≡ 0, for odd p.

Then
                                                                        1 ∗
                                                                          k
                                 p(x , y, ξ, η) ≥ b ζ                   2      in Ω × R 2 ,

for a suitable constant b > 0, and for all α = (α1 , α 2 ) ∈ Z 2 , β = (β1 , β2 )
                                                               +
        ANALYTIC AND GEVREY SOLUTIONS OF NON-LINEAR …                                  9

∈ Z 2 and for all K ⊂⊂ Ω we have with suitable constants Lα, β and B
    +

that:
         α     α      β                      ρ α −δ β
        Dx 1 D y 2 Dξ 1 p(x , y, ξ, η) ζ
                                                        ≤ Lα, β ,   ξ + η > B,      (3.4)
                          p(x , y, ξ, η)

where
                                           k∗ − p
                                    ρ =           , δ ≥ 0,                          (3.5)
                                             2p

for any δ < ρ. Since we have assumed                         p + 1 ≤ k ∗ < 2 p, we have
        1       1
0 <        ≤ ρ < .
        2p      2

      Remark 3.2. By formula (3.4) and by Mascarello and Rodino [14,
Theorem 3.3.6], we have that an operator P (z , Dz ), associated to the

symbol       p(z , ζ ) in (3.2), is C ∞ -microlocally hypoelliptic in Γ; i.e.,
Γ ∩ WFPu = Γ ∩ WFu, for all u ∈ D ′(Ω ), where W F u is the Hörmander
wave front set. A microhypoelliptic operator is hypoelliptic too.

      Remark 3.3. If the coefficients are analytic, formula (3.4) holds for
Lαβ = L α     + β +1
                       α! β! , so by Kajitani and Wakabayashi [11, Theorem 1.9],

we have that an operator P (z , Dz ), associated to the symbol p(z , ζ ) in
                                                           1  1  1
(3.2) is G s -microlocally hypoelliptic in Γ for s ≥ max  ,
                                                                  = ,
                                                         ρ 1 − δ  ρ
                  2p
that is, s ≥                .
                  ∗
                k −p

    As examples of operators satisfying Theorem 3.1, beside (3.3), we can
consider the following:

      Example.

              P (x , y, Dx , D y ) = (D y − Dx )p + i( y 2 + 1) Dx D y + h − 2l ,
                                        2                        l p


having k∗ = p + h, 1 ≤ h ≤ p − 1. They are hypoelliptic and G s hypoelliptic
           2p
for s ≥       in a neighborhood of the origin.
           h
10               G. DE DONNO, A. OLIARO and L. RODINO

     Example.

P (x , y, Dx , D y ) = (D y − Dx )4 + iy 2 Dx D y + (x 2 + i( y + 1)) Dx D 2 + (x + 1) Dx ,
                          2                     4                      2
                                                                           y
                                                                                        3



where k ∗ = 6, the imaginary part is given by ξη2 ( y 2 η2 + ( y + 1)ξ ) ≠ 0 in
Ω × Λ, η = 0, with Ω neighborhood of the origin and ξ > 0; and we also
         /

have    ξ2 (x 2η2 + (x + 1)ξ ) ≥ 0.                    We     obtain           hypoellipticity              and   Gs
hypoellipticity for s ≥ 4.

     Remark 3.4. When ρ < 1, and δ > 0, one can prove by means of
interpolation theory as in Wakabayashi [22, Theorem 2.6] that (3.4) is
valid for any (α, β) ∈ Z 4 , if (3.4) holds for
                         +                                                     α + β = 1. Hence it is
sufficient to verify (3.4) for α + β = 1 since we can consider δ > 0 in
Theorem 3.1.

     Proof of Theorem 3.1. First we estimate the numerator of (3.4) and
then we give some lemmas to estimate the denominator, see Lemma 3.5,
3.7 and 3.8. In view of hypothesis (ii) in Theorem 3.1 we have I + ≡ 0, so
for α1 = 1, we get

                                                                                                        
                                                                                                        
 Dx p(x , y, ζ ) ζ   −δ
                          =
                              (l , j ∈I
                                  )
                                    ∑          Dx blj (x , y )ξl η j ζ    −δ
                                                                               ≤ L1            ∑ ξ l η j ζ
                                                                                     (l, j )∈I ∗ ∪ I −  
                                                                                                                  −δ
                                                                                                                       ;
                                          k   ∗ ∪I−                                           k         

and similarly for α 2 = 1

                                                                                           
                                                                                          j
               D y p(x , y, ζ ) ζ                −δ
                                                      ≤ L2        ∑      ξ l η
                                                            (l, j )∈I ∗ ∪ I −
                                                                                             ζ
                                                                                            
                                                                                                   −δ
                                                                                                        ,
                                                                     k                     

for suitable constants L1 , L2 . If β1 = 1,

                                                                                                       
                                                                                                    j
        Dξ p(z , ξ, η) ζ         ρ
                                     ≤ L3  η2 − ξ
                                          
                                                            p −1
                                                                   +
                                                                            )
                                                                              ∑ξ
                                                                       (l , j ∈ I
                                                                                            l −1

                                                                                         ∪ I−
                                                                                                   η  ζ ρ;
                                                                                                     
                                                                                   k∗               

and for β2 = 1
     ANALYTIC AND GEVREY SOLUTIONS OF NON-LINEAR …                                                                               11

                                                                                                               
                                                                                                         j −1 
     Dη p(z , ξ, η) ζ        ρ
                                  ≤ L4  η2 − ξ
                                       
                                                        p −1
                                                                η +
                                                                             )
                                                                               ∑
                                                                         (l , j ∈I      ∪ I−
                                                                                             ξ   l
                                                                                                     η           ζ
                                                                                                                
                                                                                                                         ρ
                                                                                                                             ,
                                                                                    k∗                         

with suitable constants L3 , L4 .

    To prove (3.4), it will be then sufficient to show the boundedness, for
 ζ > B, of the functions

                                                               j
                     
                        ∑(      l, j )∈I ∗ ∪ I −
                                                    ξ   l
                                                            η    
                                                                 
       Q1 (z , ζ ) =                    k                       ,
                                        p(z , ζ )

                      2                                                                             j
                     η −ξ
                     
                                       p −1
                                              η +       ∑(    l, j )∈I ∗ ∪ I −
                                                                                     ξ   l
                                                                                             η        
                                                                                                       ζ
                                                                                                                ρ

       Q2 (z , ζ ) =                                                 k                                            ,
                                                            p(z , ζ )

                      2                                                                         j
                     η −ξ
                     
                                       p −1
                                              +   ∑(    l , j )∈I ∗ ∪ I −
                                                                            ξ    l −1
                                                                                             η    
                                                                                                      ζ
                                                                                                            ρ

       Q3 (z , ζ ) =                                            k                                             .
                                                            p(z , ζ )

First we introduce three regions in the plane (ξ, η) :

                                  R1 : cξ ≤ η2 ≤ Cξ

                                  R2 : η2 ≥ Cξ

                                  R3 : η2 ≤ cξ,                                                                              (3.6)

                                                                                                                        1
for suitable constants c, C satisfying the inequalities c <<                                                              , and
                                                                                                                        2
C >> 2, cf. [4, 5].

    The following estimates then hold:

                                        const. η − 2δ ,                ζ ∈ R1
                                 −δ     
                                        
                         ζ            ≤ const. η − δ ,                 ζ ∈ R2                                               (3.7)
                                        const.ξ − δ ,                  ζ ∈ R3 ;
                                        
                                        
12                       G. DE DONNO, A. OLIARO and L. RODINO

and
                                             const. η 2ρ ,            ζ ∈ R1
                                     ρ       
                                ζ          ≤ const. η 2ρ ,            ζ ∈ R2                                        (3.8)
                                             const.ξρ ,               ζ ∈ R3 .
                                             
                                             

By abuse of notation, in the following we shall also denote by R1 , R2 , R3
the sets Ω × R1 , Ω × R2 , Ω × R3 ; recall that Γ = Ω × Λ.

      The following three lemmas give us some relevant estimates from the
below of p(z , ζ ) in (3.2).

      Lemma 3.5. Let p(z , ζ ) be the function (3.2), such that (i), (ii) and (iii)
in Theorem 3.1 hold. Then there are positive constants K1 < 1, B, such
that, for (z , ζ ) ∈ R1 , ζ > B.

                                                                                                        1
                                                                      2 2
                          2           n                           ∗
                                                                li ji  
                                                            ∑
                                                                 ∗
          p(z , ζ ) ≥ K1  (η − ξ ) + 
                                   2p
                                             Im b ∗ ∗ (x , y ) ξ η          .                                        (3.9)
                                               li ji                 
                                      i =1                           

Proof. We have that

                                                                                                  2
                                                                                          l j
     p(z , ζ )   2
                     =  (η2 − ξ )p +
                       
                       
                                                ∑(  l, j )∈I ∗ ∪ I −
                                                            k
                                                                       Re blj (x , y ) ξ η 
                                                                                           
                                                                                              


                                                                                                                2
                 ∑                                            ∑(
                     n                              ∗   ∗
          +
                           Im b ∗ ∗ (x , y ) ξli η ji +                            Im blj (x , y ) ξl η j  .
                                                                                                                   (3.10)
                    i =1          li ji                               l , j )∈ I −                        

By developing (3.10) and removing the terms

                                                        2                                                       2
                                                                                              
                                               
                    ∑    Re blj (x , y )ξl η j 
             (l, j )∈I ∗ ∪ I −                 
                                                            and 
                                                                          ∑
                                                                           Im blj (x , y )ξl η j 
                                                                                                 
                      k                                        (l, j )∈I −                    

respectively from the real and imaginary part of p(z , ζ ), we can write

                                                                                           2
                                                  n                             ∗
                                                                                                   3

                                                   ∑                                             ∑ J (z, ζ ),
                                                                            ∗
        p(z , ζ )    2         2
                         ≥ (η − ξ )        2p
                                                +      Im b ∗ ∗ (x , y ) ξli η ji  +                      i
                                                           li ji                  
                                                  i =1                                         i =1
       ANALYTIC AND GEVREY SOLUTIONS OF NON-LINEAR …                                                                      13

where


                                                 ∑
                                                                                             ∗       ∗
        J1 (z , ζ ) = 2(η2 − ξ ) p                            Re b ∗         ∗
                                                                       li , ji
                                                                                  (x , y )ξli η ji                   (3.11)
                                          (li∗ , ji∗ )∈I ∗
                                                        k


                          n

                         ∑                                              ∑ Im b (x , y)ξ η ,
                                                          ∗        ∗
        J 2 (z , ζ ) =          Im b ∗ ∗ (x , y ) ξli η ji                              lj
                                                                                                         l j
                                                                                                                     (3.12)
                                      li ji
                         i =1                                          ( )
                                                                       l, j ∈I −


        J 3 (z , ζ ) = 2(η2 − ξ ) p        ∑ Re b (x , y)ξ η .
                                          ( ) l, j ∈I −
                                                              lj
                                                                                 l j
                                                                                                                     (3.13)


The function (3.11) is non-negative by hypothesis (iii). Let us fix attention
on J 2 (z , ζ ) and J 3 (z , ζ ) defined respectively by (3.12) and (3.13). We
have for all ε > 0

     n                                   2                                                                                2
1                             ∗                             n                           ∗
   ∑                                                 1 − ε 
                                                                                        ∑
                            ∗                                                         ∗
                          li ji 
        Im b ∗ ∗ (x , y ) ξ η      + J 2 (z , ζ ) ≥            Im b ∗ ∗ (x , y ) ξli η ji  ,
2          li ji                                  2      i =1     li ji                  
  i =1                                                                                    

in R1 , ζ > B. In fact by (3.6) in R1 and hypothesis (i) in Theorem 3.1,
for all ε > 0 we get for B sufficiently large

                                                                                           k∗ + 2l + j
                 J 2 (z , ζ )                                                          η
                                                 2
                                                     ≤ const.             ∑                      ∗
                                                                                                         < ε,    ζ > B.
       ∑                                                                                   η2k
        n                            ∗  ∗
            Im b ∗ ∗ (x , y ) ξ η 
                                   
                                     li ji                             (l , j )∈ I −
       i =1     li ji             

We remark that k ∗ = 2l ∗ + j ∗ > k − = 2l + j. Concerning (3.13) we also
have that:

                                n                                             2
                                             ∗
  1 2
                                ∑
    (η − ξ)2 p + 1  Im bl∗ j∗ (x , y )ξli η ji  + J 3 (z, ζ )
                                         ∗

  2              2       i i                   
                    i =1                       

                                                                                                                2
                                              n                             ∗
                        1
                                                                   ∑
                                                                        ∗
                    ≥  − ε  (η2 − ξ )2 p + 
                                                  Im b ∗ ∗ (x , y ) ξli η ji                                  
                      2                              li ji                                                   
                                              i =1                                                             
                                                                                                                 

in R1 , ζ > B, since
14                         G. DE DONNO, A. OLIARO and L. RODINO

                                            J 3 (z , ζ )
                                                                                            < ε,       ζ > B.   (3.14)
                                                                                        2
                                     ∑
                                            n                                ∗     ∗
            2
         (η − ξ)         2p
                                +
                                                Im b ∗ ∗ (x , y ) ξ η 
                                                                       
                                                                            li    ji
                                           i =1     li ji             

In fact, considering the curves ξ = (1 − r ) η2 in the plane (ξ, η), r < 1,
we obtain by (3.6) in R1 that the left part of (3.14) is estimated by

                             2 p + 2l + j                             2 p + 2l + j −2k∗
                  rp η                                      tp η
     ∑
  (l, j )∈I − r
                  2p
                       η   4p
                                 + η2k
                                            ∗
                                                ≤     ∑
                                                    (l, j )∈I − η
                                                                    4 p − 2k ∗
                                                                                 + t2 p
                                                                                            < ε, t + η > B,     (3.15)


         1                    ∗
where t = , by dividing for ηk ; since k − < k ∗ . Recall that:
         r

      For all ε > 0 there exists Bε > 0 such that for x + y > Bε ,

                           x α yβ
                                            < ε if and only if (2γ − α ) (2ν − β) > αβ.
                       x 2 γ + y 2ν

For r ≥ 1, in R1 , we have

                                 2 p + 2l + j − 2k∗
                       tp η
        ∑
       ( )
        l, j ∈I −        η    4 p − 2k∗
                                                +t   2p
                                                          ≤ η       − 2 p + 2l + j
                                                                                       < ε, since k − < 2p.     (3.16)


Then
                                                                                                   1
                                                              2 2
                  2           n                           ∗
                                                        li ji  
                                                       ∑
                                                         ∗
                           2p 
  p(z , ζ ) ≥ K1  (η − ξ ) +        Im b ∗ ∗ (x , y ) ξ η          , ζ > B.                                    (3.17)
                                       li ji                 
                              i =1                           

for a suitable positive constant K1 .

     Remark 3.6. From the previous estimate (3.17) follows easily that
                           1 ∗
                            k
 p(z , ζ ) ≥ ζ             2 .

     Lemma 3.7. Let p(z , ζ ) be the function (3.2). Then there are positive
constants K 2 < 1, B, such that:

                                                          2p
                           p(z , ζ ) ≥ K 2 η                   , (z , ζ ) ∈ R2 , ζ > B.                         (3.18)
         ANALYTIC AND GEVREY SOLUTIONS OF NON-LINEAR …                             15

     Lemma 3.8. Let p(z , ζ ) be the function (3.2). Then there are positive
constants K 3 < 1, B, such that:

                                        p
                  p(z , ζ ) ≥ K 3 ξ         , (z , ζ ) ∈ R3 , ζ > B.            (3.19)

     For the proofs of Lemma 3.7 and Lemma 3.8 see De Donno and
Rodino [5] concerning similar operators, the present case actually does
not involve more complications.

     By the relations (3.6), (3.7) and (3.8), and the inequality η2 − ξ ≤
     2
 η       + ξ , we easily estimate the numerators N i , i = 1, 2, 3, of Qi , in
the regions R2 and R3 in the following way:

                           const. η     2l + j − δ
                                                     ,           ζ ∈ R2
                 N1 (ζ ) ≤                                                     (3.20)
                           const. ξ     l +1 j − δ
                                           2       ,             ζ ∈ R3 ;

                            const. η    2 p −1 + 2ρ
                                                         ,        ζ ∈ R2
                 N 2 (ζ ) ≤                                                    (3.21)
                            const. ξ    p −1 + ρ
                                           2 ,                    ζ ∈ R3 ;

                                        2 p − 2(1 − ρ )
                            const. η                         ,      ζ ∈ R2
                 N 3 (ζ ) ≤             p −1 + ρ
                                                                                (3.22)
                            const. ξ
                                                   ,                ζ ∈ R3 .

We have N 3 (z , ζ ) ≤ N 2 (z , ζ ) in R2 and R3 , so we can just consider the
functions N1 (z , ζ ) and N 2 (z , ζ ) in those regions. Now Lemma 3.7,
Lemma 3.8 and the estimates (3.20) and (3.21) show the boundedness of
Q1 (z , ζ ) and Q2 (z , ζ ) in the regions R2 and R3 since in (3.5) always it
          1
is ρ <      ; so for Q3 (z , ζ ), too. For Q1 (z , ζ ) in R1 we have immediately
          2
boundedness since we apply Remark 3.6, formula (3.8) in R1 , recalling
that in Q1 , (l, j ) ∈ I        ∪ I − with δ ≥ 0.
                           k∗


     Regarding Q2 and Q3 in the region R1 , we observe that η2 − ξ
vanishes in it, so estimates of the previous type are not optimal.

     For Q2 , by (3.6), (3.8) and (3.9) we get easily:
16                 G. DE DONNO, A. OLIARO and L. RODINO

                                                                       p −1         1 + 2ρ
                                                             η2 − ξ             η
     Q2 (z , ζ ) ≤ const.                                                                                   1
                               2                                                                        2 2
                                                             ∑
                                                                  n                               ∗  ∗
                               (η − ξ )2 p + 
                                                                      Im b ∗ ∗ (x , y ) ξ η 
                                                                                             
                                                                                                 li ji      
                                                                i =1     li ji                           
                                                                                                           

                                                     2l + j −1 + 2ρ
                                                η
                   +
                              )
                                ∑
                        (l , j ∈I        ∪ I−            η   k∗
                                                                           ≤ L, (x , y ) ∈ Ω, ζ > B.
                                    k∗

                                                       1
The second term in the right-hand side is bounded since ρ <
                                                          by (3.5).
                                                       2
About the first term we can argue in the same way we have done in
                                                                                                                 k∗ − p
Lemma 3.5, see formulas (3.14), (3.15) and (3.16), obtaining ρ ≤                                                        .
                                                                                                                   2p

      The study of the boundedness of the functions Q3 (z , ζ ) in the region
R1 actually does not involve further complicated statements, so arguing
like the previous step and using the estimates (3.5) on ρ, we have proved
that Qi (z , ζ ), i = 1, 2, 3 is also bounded in R1 .

      The following Remark ends the proof:

      Remark 3.9. By formulas (3.9), (3.18) and (3.19), we obtain that
                       1 ∗
 p(z , ζ ) ≥ b ζ       2k ,   b > 0, ζ > B. In fact we obtain that                                               p(z , ζ ) ≥
                                                                                                                       1 ∗
            k∗                                      k∗       1        k∗       1     k∗                     k∗         2k )
const. η         in R1 , then η                          =     η           +     η           ≥ const. ( η        + ξ
                                                             2                 2
                 1 ∗          1 ∗                             1 ∗                                   1 ∗
≥ const. ( η     2k    + ξ    2k )        ~ const. ζ          2k
                                                                    , so p(z , ζ ) ≥ b ζ            2k .    In the same
                                                p
way we get        p(z , ζ ) ≥ b ζ                    in R2 and R3 , the result follows since we

have k ∗ < 2p.

     4. Gevrey Local Solvability for the Square of the Schrödinger
               Operator with a Nonlinear Perturbation

      In this Section we consider the following equation:

              (D y − Dx )2 u + F (x , y; ∂ lx ∂ yu ) |2l + j < 4 = µf (x , y ),
                 2                              j
                                                                                                                       (4.1)
      ANALYTIC AND GEVREY SOLUTIONS OF NON-LINEAR …                                             17

where F (x , y; z ), ∈ C6 , is a nonlinear function. Before giving the main

result let us recall that, given an open set Ω ⊂ R 2 , the Gevrey
anisotropic space G (λ1 , λ 2 ) (Ω ), λ1 ≥ 1, λ 2 ≥ 1, is the set of all C ∞
functions such that for every compact set K ⊂ Ω there exists a positive
constant C K            satisfying sup K ∂ l ∂ y f (x , y ) ≤ C K j +1l!λ1 j!λ 2
                                           x
                                               j                l+
                                                                                        for every
                          (        )
l, j ∈ Z + . As standard G0λ1 , λ 2 (Ω ), λ1 , λ 2 > 1, is the set of all the

functions in G (λ1 , λ 2 ) (Ω ) with compact support in Ω.

    We shall prove the following result.
    Theorem 4.1. Let us fix λ = (λ1 , λ 2 ), 1 < λ1 < 4, λ 2 > 1. We suppose
                                 λ
that the datum f in (4.1) is in G0 (Ω δ ), Ω δ := {(x , y ) ∈ R 2 : x 2 + y2 < δ}.
Moreover, rewriting F (x , y; z ) as G (x , y; ℜz; ℑz ) and setting w = (ℜz , ℑz )
we suppose that:
    G (x , y; 0 ) = 0;

    G (x , y; w0 ) ∈ G λ (Ω δ ) for every w0 ∈ R12 ;

    G (x 0 , y0 ; w ) ∈ G σ (R12 ), for every (x 0 , y0 ) ∈ Ω δ , σ < 4.

    Then the semilinear equation (4.1) admits a classical solution in Ω δ ,
for δ and µ sufficiently small.
    Remark 4.2. Arguing in the isotropic case we have from the previous
theorem the G 4 − ε local solvability of (4.1) for every ε > 0.

   We want now to recall the definition of a class of Gevrey-Sobolev
spaces on the strip R × (− δ, δ ), δ > 0. These spaces have been studied in
the isotropic form by Gramchev and Rodino [9] and then generalized to
the anisotropic case in Marcolongo and Oliaro [13], see also De Donno and
Oliaro [4]. To start with, let us recall that the space H s (R × (− δ, δ )),
                                                          p

p < 1, s integer, is the set of the L2 functions satisfying

                              s     δ    ∞
     f (x , y )   2
                  Hsp
                        :=   ∑∫ ∫
                             k =0
                                    −δ   −∞
                                              (1 + ξ   2p s−k
                                                         )        k~
                                                                D y f ( y, ξ ) 2 dξ dy < ∞,   (4.2)
18                    G. DE DONNO, A. OLIARO and L. RODINO
      ~                          − ixξ
where f ( y, ξ ) =          ∫e           f (x , y )dx is the Fourier transform of f with

respect to x. This definition extends to every s > 0 by interpolation. We
                                                           1
fix now s > 0, p < 1, τ > 0 and r =                          + ε, for a fixed (arbitrarily small)
                                                           2
                     1
ε; we set q =          .
                     p

     Definition 4.3. Let us suppose that ψ( y, ξ ) is a positive function
                                   pr
belonging to the Hörmander class S1, 0 ((− δ, δ ) × R ). We define the Gevrey-
                   ψ
Sobolev space H s, q , r (R × (− δ, δ )) as the set of all functions f ∈ L2 ( R ×
                τ,

(−δ, δ)) such that f                s, q   is finite, where


                        f   s, q   := e τψ ( y, Dx ) f (x , y )    H s ( R × ( − δ , δ )) ;                     (4.3)
                                                                     p


the operator e τψ( y, Dx ) acts on the function f in the following way:

                                                      1         ixξ τψ ( y, ξ ) ~
                        e τψ ( y, Dx ) f (x , y ) =
                                                      2π   ∫e      e             f ( y, ξ )dξ.


     Observe that the operator e τψ( y, Dx ) and its inverse e − τψ( y, Dx )
establish an isometry between Hilbert spaces:

           e τψ ( y, Dx ) : H s, q , r (R × (− δ, δ )) → H s (R × (− δ, δ ))
                              τ,
                                 ψ
                                                           p                                                    (4.4)

           e − τψ ( y, Dx ) : H p (R × (− δ, δ )) → H s, q , r (R × (− δ, δ )) .
                                s
                                                      τ,
                                                         ψ
                                                                                                                (4.5)

     In the next theorem we summarize the most important properties of
     ψ
H s, q , r (R
  τ,            × (− δ, δ )).

     Theorem 4.4. Let us suppose that the function ψ( y, ξ ) satisfies the
condition

                                                                                         1q              1q r
  ψ( y, ξ1 ) − ψ( y, ξ1 − ξ2 ) − ψ( y, ξ2 ) ≤ −b min{1 + ξ1 − ξ2                              , 1 + ξ2     } (4.6)

for a constant b > 0. Then there exists salg > 0 such that, for every s > salg ,
     ψ
H s, q , r is an algebra.
  τ,
       ANALYTIC AND GEVREY SOLUTIONS OF NON-LINEAR …                                                       19

                      q                     (λ , λ )
     Let us fix 1 < λ1 <, λ 2 > 1. Let f ∈ G0 1 2 (R × (− δ, δ )). Then, for
                      r
every function ψ satisfying the assumptions of Definition 4.3 for every
                                      ψ
τ > 0 and s ≥ 0 we have that f ∈ H s, q , r (R × (− δ, δ )).
                                   τ,


     Let us consider now the linear part of the equation (4.1) P = (D 2 +
                                                                      y

Dx )2 : we want to analyze the following operator:

                                 ~
                                 P = e τψ( y, Dx ) Pe − τψ( y, Dx ) ,                                  (4.7)

                                             y
where we choose from now on ψ( y, ξ ) = 1 +  ϕ (ξ ) ξ
                                              
                                                                                  r 2
                                                                                        , ϕ being a C ∞
                                           2δ 
                                                                  1
function satisfying 0 ≤ ϕ (ξ ) ≤ 1 for every ξ, ϕ (ξ ) ≡ 0 for ξ ≤ , ϕ (ξ ) ≡ 1
                                                                  2
for ξ ≥ 1 (observe that ψ( y, ξ ) satisfies (4.6)). The next proposition
then follows from the results of De Donno and Oliaro [4].

     Proposition 4.5. The symbol of the operator                                        e τψ ( y, Dx ) (D 2 −
                                                                                                          y

Dx ) e − τψ ( y,   Dx )
                          is given by:

                                  τ            r 2
                    (η2 − ξ) + i ϕ (ξ) ξ             η + p1 + r − ν (x , y, ξ, η),                     (4.8)
                                  δ

where p1+r −ν (x , y, ξ, η) satisfies the following estimates in a neighborhood

Γ of the set ∑ = {(ξ, η) ∈ R 2\ (0, 0 ) : η2 − ξ = 0}, Γ being of the form Γ =
                          −
{(ξ, η) ∈ R 2\ (0, 0 ) : c0 1η2 ≤ ξ ≤ c0 η2 }, c0 > 1 :
      l j k h
     Dx D y Dξ Dη p1 + r − ν (x , y, ξ, η) ≤ Cljhk (1 + ξ               12
                                                                             + η )(1 + r − ν )− 2k − h ,

for every l, j, k, h ∈ Z + .

     The results in De Donno and Oliaro [4] assure us that the operator
e τψ ( y, Dx ) (D y − Dx ) e − τψ ( y, Dx ) is C ∞ microlocally hypoelliptic in Γ, in
                  2

particular, denoting its symbol by q(x , y, ξ, η), we have:

                                q (x , y, ξ, η) ≥ c( ξ       12
                                                                  + η )1 + r ,
20                 G. DE DONNO, A. OLIARO and L. RODINO

       Dx D y Dξ Dη q (x , y, ξ, η) ( ξ 1 2 + η )r (2k + h )− (1 − r )(2l + j )
        l j k h
                                                                                ≤ Lljkh ,
                                 q(x , y, ξ, η)

for every l, j, k, h ∈ Z + and ξ + η >> 0.

Now we observe that, since e τψ ( y, Dx ) e − τψ ( y, Dx ) = e − τψ ( y, Dx ) e τψ ( y, Dx ) = Id
             ~
we can write P , cf. (4.7), in the following way:

 ~
 P = (e τψ( y, Dx ) (D y − Dx ) e − τψ( y, Dx ) ) (e τψ( y, Dx ) (D y − Dx ) e − τψ( y, Dx ) ). (4.9)
                       2                                            2


    So we can apply Proposition 4.5 to the two factors in the right hand
                                                         ~
side of (4.9), obtaining that the symbol of the operator P equals, modulo
lower order terms, the square of the (microlocally hypoelliptic in Γ )
                                                                  ~
symbol (4.8). Then we can construct a microlocal parametrix of P in Γ.
                             ~
Since outside Γ the operator P is microlocally quasi-elliptic, a standard
technique of patching together the microlocal parametrices in Γ and
outside Γ (cf. Gramchev and Rodino [9], Marcolongo and Oliaro [13]),
                      ~    ~                s
gives us a parametrix E of P in the space H1 2 (R × (− δ, δ )) :

                 ~
                 E : H1 2 (R × (− δ, δ )) → H1 + 2(1 + r ) (R × (− δ, δ )),
                      s                      s
                                               2

with
                                   ~~                 ~
                                   PEu = ~(x , y )u + Ru,
                                         χ

                                                     ~
where ~(x , y ) ≡ 1 in a neighborhood of (0, 0 ) and R is a regularizing
      χ
             s
operator in H1 2 (R × (− δ, δ )). Then by (4.4), (4.5) and (4.7) we have the

following result.

     Proposition 4.6. There exists a linear map

                E : H s, 2, r (R × (− δ, δ )) → H s,+22(r + r ), ψ (R × (− δ, δ ))
                      τ
                        ,ψ
                                                  τ ,
                                                        1


such that
                                   PEu = χ(x , y )u + Ru,
       ANALYTIC AND GEVREY SOLUTIONS OF NON-LINEAR …                                                                  21

                   (        )                2
where χ(x , y ) ∈ G0λ1 , λ 2 (Ω δ ), 1 < λ1 < , λ 2 > 1, χ(x , y ) ≡ 1 in a neighborhood
                                             r
                                                                 ,ψ
of the origin, and R is a regularizing operator in the space H s, 2, r .
                                                               τ

    The nonlinearity can be treated with the technique developed by
Bourdaud et al. [1]. The following proposition holds, cf. De Donno and
Oliaro [4], Oliaro and Rodino [16].

     Proposition 4.7. Let us consider the nonlinearity in (4.1); we write
                                         j             j
for shortness J (u ) = G (x , y, ℜ(∂ l ∂ yu ), ℑ(∂ l ∂ yu ))|2l + j <4 . We take u(x ) ∈ B ⊂
                                     x             x
     ψ                                               ,ψ
H s, 2, r (R × (− δ, δ )), where B is bounded in H s, 2, r . Then we can find a
  τ,                                               τ

continuous non-decreasing function Φ : (0, + ∞ ) → (0, + ∞ ), Φ(0 ) = 0, such
that
                                    J (u )   s, 2   ≤ Φ( u       s + 3, 2 );                                      (4.10)

moreover, if u, v ∈ B we have:

                            J (u ) − J (v )      s, 2   ≤ CB u − v        s + 3, 2                                (4.11)

for every s > salg .

     We can now prove the solvability of (4.1).

                                                        ,ψ
     Proof of Theorem 4.1. Let us fix the datum f ∈ H s, 2, r . Using
                                                      τ

Proposition 4.6 and arguments as in Gramchev and Popivanov [7], see
also Gramchev and Rodino [9], Marcolongo and Oliaro [13], De Donno
and Oliaro [4], we can find a positive, continuous, non-decreasing
function L : [0, δ0 ] → [0, + ∞ ), L(0 ) = 0 such that, defining

                                          Ev     s + 3, 2                                            Rv    s, 2
       A s (δ ) :=        sup                               ,   B s (δ ) :=          sup                          ,
                           v≠0               v   s, 2                                v≠0             v    s, 2
                            ψ                                                          ψ
                     v∈H s, 2, r (Ω δ )
                         τ,                                                    v∈H s , 2, r (Ω δ )
                                                                                   τ,


we have A s (δ ) ≤ L(δ ), B s (δ ) ≤ L(δ ). We are looking for a solution of the
form u = Ev, so the equation (4.1) becomes v(x ) = Q (v(x , y )) + µf (x , y ),
                                                        j                         j
where Q (v(x , y )) := − Rv(x , y ) − G (x , y; ℜ(∂ l ∂ y (Ev )(x , y )), ℑ(∂ l ∂ y (Ev )(x , y ))|2l + j <4 .
                                                    x                         x
22                  G. DE DONNO, A. OLIARO and L. RODINO

We then have to find a fixed point of the operator Q (⋅) + f ; we choose δ
and µ such that the following conditions are satisfied:

              B s (δ ) (1 + µ f   s, 2 )   + Φ(A s (δ ) (1 + µ f   s, 2 ))   ≤1       (4.12)

              B s (δ ) + A s (δ )C B < 1,                                             (4.13)

                                                                  ,ψ
where Φ(⋅) and C B are the ones of Proposition 4.7, B := {w ∈ H s, 2, r (Ω δ )
                                                                τ

: w − µf    s, 2   ≤ 1}. Now, by (4.10) and (4.12) we have that Q (⋅) + f : B → B;

moreover (4.11) and (4.13) imply that Q (⋅) + f is a contraction. We then
obtain a solution as an application of the Fixed Point Theorem in the
Banach space B. Taking s sufficiently large the solution is classical. By
                                      1
Theorem 4.4 and since r =               + ε, with ε arbitrarily small, we obtain the
                                      2
                              λ
solvability of (4.1) for f ∈ G0 (Ω δ ).


                                            References

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[19]   L. Rodino, Linear partial differential operators in Gevrey spaces, World Scientific,
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[20]   V. Solonnikov, On estimates of the tensor Green’s function for some boundary value
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[22]   S. Wakabayashi, Singularities of solution of the cauchy problem for hyperbolic
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                                                                                              g
24          G. DE DONNO, A. OLIARO and L. RODINO

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