(we do not write out the dependence of the norm on ~o and t o ) . We distinguish separately the case o = 0, i.e. , t h e e a s e i n w h i c h f o r a n y t , It--tol~<8, the function u(t, z) as a function of z belongs to the same space E-~m,r (~o; nC~) a s at "time" t = t 0. We denote this space by ~Y(6; ~ m , ~ ( ~ ; C~)), 3.
Cauchy Problem for P/D Equations with Variable Symbols We consider the Cauchy problem for the system of equations
u ' - - A ( z , t , D ) u = h ( t , z ) (u'=--Ou/Ot), tt (to, z) = ~ (z), zECn,
/ /
.
.
.
.
.
.
.
.
.
.
(3.1)
(3.2)
.
where u = (ul,. ...?uN)A h - (hl,...,hN) are vector-valued functions of the variables t~C' and z6C=; ~(z)=i~1(z ).....~v(z)), A(z, t, D ) = ( A u ( z , t, D))N• is a matrix of p/d operators with Symbols Aij(z, t, ~) analytic in C ~ X V , where V is a domain in the space of the variables t, ~. We denote by ~t the section of the domain V by the plane t = const, i.e., f2t-----Vn{t-~--const}. We further minimal type in i the operators the question of
suppose that for each t = const the functions Aij(z, t, r) are functions of z and satisfy inequality (1.2), In correspondence with Assertion 1 of Sec. Cn Aij(z, t , D) are then defined in the spaces ExPet(z), and we are able to pose the solvability of problem (3.1), (3.2) in classes of exponential functions.
Definition 3.1. We say that the Cauchy problem (3.1), (3.2) is locally well posed in the i n classes of exponential functions E-~m,r+altl~Cz), if for any point,it0,~0)EV there are munbers R > 0 and o > 0 such that for any r < R for any
~(z)6E-~m,r
(~; Czn)
and
h(t, Z)6(-](7R -;r Expm§ l,r+cqt-tol
(~; C2))there exists a unique solution
it, z)ce ( R--r. - ? - , Expm,,+~,,-,o,(;o; of this problem.
C z) ~)
[Here m + I = (m I + 1 .... ,mN + i).]
THEOREM 3.1. The Cauchy problem (3.1), (3.2) is locally well posed in the classes of exponential functions Expm,:+el~l(C~)if and only if the symbols Aij(z, t, ~) are polynomials in z, and the degrees mij of these polynomials satisfy the inequalities
mr
(i,
j----1 . . . . . N )
[if Aij(z, t, ~) - O, then by definition mij = -~].
(3.3)
Moreover~
II it, z)11 ~U~:m'"~ ''< m ,(11W(z) e-=, z It m,, + !1h (t, z)II ~;m+,,~,o)'
(3.4)
/
where M > 0 is a constant Proof. Sufficiencl: the class of solutions. closed polycylinder
n o t d e p e n d i n g on r < R. - We f i r s t o f a l l , i n d i c a t e t h e n u m b e r s R > 0 and a > 0 d e t e r m i n i n g Namely, f o r R and .~ i t i s p o s s i b l e t o t a k e a n y n u m b e r s s u c h t h a t t h e
U R,m~(~o, ~" t o ) = { l ~j--~oyl~
inside
the original
It is clear that for any (~0$t0). Moreover, we note that large values of o the domains U.e,~/~(~0~to)(this is essential
i t-toI
domain V. r < R the domains {l~]--~o]l
It may henceforth be assumed with no loss of generality that ~0 = 0, t o = 0 [the change u +-+ uexp(-r t +-+ t - to]. Problem (3.1), (3.2) is then equivalent to the integrodifferential equation t
t
(t, z) = l A iz, ~, D) u (~, z) d~ + 9 (z)+ f h (~, z) d~. 0
We s h a l l
show t h a t
0
the operator
!
Bu (t, z)-- I A (z, ~, D) u (~, z) d~ 0
2766
acts in the space G ( R---r" --lCn~\ ~ , Expm.r+~ill k zT] and is a contradiction if o > 0 is chosen sufficiently large. LEMMA 3.1.
If g(z)~.Ex-p~,r(C~), then for lu]>~m] (j = 1 .... ,N) there is the estimate
I]D~ltj (Z) [Imi,r < M (f -{-I (z I )my (~1. . . (Zn)It2ri~l-m]]Ib:i(Z)11mi,r' where M > 0 i s a c o n s t a n t Proof.
( 3.5 )
( n o t d e p e n d i n g on r ) .
From Cauchy's formula for any z6C n we have
IO ~ui (z) [ -<- ~-al (1 + I east+ Iz I )~1 e~(l~i+i,o IIuj (z) I1~j,,, where the vector a=(a~, .... an), a t > 0 .....an>O, is arbitrary. Setting a = = / r [in the case r = 0 the estimate (3.5) is obvious] and using Stirling's formula, we find that
D~uj (z) i < M (r + j a I-l- r ] z I)~Y ( e l . . . a~)II2e'i'Irl~'-~i l[ u1 (z)[[~ i,,,, whence, obviously,
I D~u(z) l~<M(r-~-I ~l )rn] (r which gives the estimate (3.5).
G~n)l/2i( 1 +[z[)
'niertzirl~l-mj [] tt:(z)II mi,~,
The lemma is proved.
We continue the proof of the theorem. Let u(-c,.)EEx--pm,r§ Then by Lenmm 3.I for ]TI~<(R--r)/o and for all a such that Ic,l>j~n~ (I~<j~
it D~u] (t, z) ][ mi, ~+~l*i~< A4 (r + g t t I +1 a I )~Y (a~.. 9a,) ;/2 ~4 • ( r + (r I 9 l )~1-'~ II uj (% z)!l ~,,+~l~. < M (~+1 ~ i )'~; ( ~ . . . ~ A ~ R I ~ ' - ' ~ I] ~ (% z)U ,n~,,+ol~l, since r-~- c~] x [ ~ ]~. It is also clear that for lal < mj
i] D=u~ (t, z)il =i,,+~I,:l <M l[ u~ ('L z)It ,,..,§ where M = M(R~ m) is a constant. Further, by hypothesis the functions Aij(z , t, ~) have the form
A,~(z, t, ;)~
~
Igl<m~7
z~A~(t, ;),
(3.6)
where A~(I, ;)are analytic in the domain V and, in pariicular, for ltl~
' i (~,z)J! iI A ~ t' t , D)u~(t, z)[<Mtltt mi,r+(rl~I (l+tzl)~e where M > O i s a c o n s t a n t
('+~I'*l)l'l,
n o t d e p e n d i n g on r < R and o > O.
From the representation ( 3 . 6 )
we now see that forltl..<(R--r)/~
and all
z~C n
! A u (z, t, O)u~ (~, z)[ ~ 0 is a constant.
Hence,
! i A~l(z' ~'D)"i(t' z)d~l~
u
l~i< t)
"
< M_~maxl,
mi~-.<m~--mj+l.
From this we immediately obtain the inequality
:!
j~ A~j (z, t, D ) u] (z, z) d t
0
ll
mi ,r+~ft!
< M~ -~ max ~j (~, z)lbjo,+~l~l. ~l< ti
2767
Thus, for B u (t, z) ------((Bu)I . . . . .
ll(Btt)il],,t,r+ol~t < ~ j=l
(Ba)N) we have the estimate
A u (z, x, D) tt i (~, z) dz 0
mi .r+oltl
and hence the estimate [in the norm of the space
II Bu (t, z)lb-~ --;m,t,G
max ~ .
"< M -8
i=1 .....
N,
"=
R--r. Exp,t.r+~itt(Cz)~ 0,(.._.7_, n)]
M
< ~- il u (t, z)/b_,.
;m,tj~
.
Since the constant M > 0 does not depend on o > 0, by taking o sufficiently large, we
find
that
the operator
To c o m p l e t e
Bu i s a c o n t r a c t i o n
t h e proo~f i t
remains
in t h e s p a c e 0
to note
that
;Exp,~,r+
f o r any f u n c t i o n
h(t, z)fiO 'R--r. ---7- ' Expm+~,r+ot/I
R - r " E--~m,r+zl/l(C:)) (C,~)) its primitive with respect to t belongs to the space 0 ~ I--7,
R - r . Expm,r+
It is also
In summary, we find that the operator t
(u) = Bu + ~ (z) + S/* (% z) d~ 0
acts in the space (Y (R--r. --7-" E-~m.~+glt,(C~))and point
is a contraction.
It
therefore
has a unique
fixed
it(t,z)EO( R-r -n) --$--;Expm,~+zm(Cf),which is the solution of the original problem (3.1), (3.2).
The estimate (3.4) is obvious.
The existence of a solution has thus been proved.
Necessity. We seth(t,z)------0,q0(z)~-(O..... I..... 0), i.e.,~y(z)=1, ~(z)--=0 for i ~ j, where j is some fixed index. For the corresponding solution of problem (3.1), (3.2)~(t,z)6(Y(R/a; ExPzltI(cn))for t = 0 we then have
UI(0, z ) - - A ~ i ( z , 0 , 0 ) - - 0 ,
z6C n,
(3.7)
Since by hypothesis [u~(t, z)l
0 is a constant (depending, generally speaking, on R and o). Thus, from (3.7) it follows that the entire function Aij(z, 0, 0) satisfies the inequality
I A,j(z, O, O)] ..< M (1-+- I z l) m*+' and hence is a polynomial of degree no greater than my + i. Since t h e case of the Cauchy problem in a neighborhood of a point'(~0, t0)6V reduces, as already noted, to the considered case by the change u ++ uexpp(-~0, z), t +-+ t - t o , it follows that the functions Aij(z, t, ~) are polynomials in z of degree no greater than m i + i with analytic coefficients depending on (~, t)6V~ From this it follows that all the derivatives (with respect to ~) 0~Ao(z, t, ~) are also polynomials of degree no greater than m i + i. We now set (again fixing the index j ) ~ ( z ) = & , k = l ..... n; e#~(z)-UO,i=/=]. Then for the corresponding solution u(t, z) (we do not write the dependence of the solution on the chosen values of j and k) we have for all z6C ~
tti (0, z)-- A u (z, O~O) z~ --3~o Au(z ' O, 0)~0. By what has been said above, from this it follows that Aij(z, 0, 0)z k are polynomials of degree no greater than mi + i, and hence the functions Aij(z, 0, 0) themselves [and hence Aij(z, t, ~) for all(t, ~)6V ] are polynomials of degree no greater than m i. Repeating these arguments for initial data zY for I~I = 2,...,mj, we find finally that the degrees of the polynomials Aij(z, t, ~) in z are no greater than m i - m j + i. This is what is required. Necessity and together with it the theorem have been completely proved. Remark 3.1. As is evident, in the proof of necessity solvability of the Cauchy problem only for polynomial initial data [in other words q~(Z)~.'Expm, r(C n) for r = 0] was usually used. Thus, well-posedness of the Cauchy problem (3.1), (3.2) for polynomial initial data implies 2768
the required conditions on the degrees of the polynomials Aij(z, t, ~) and hence the wellposedness of the Cauchy problem for the exponential classes of data~Expm, r(C~) for 0 ~ r < ~ . Example.
We consider the Cauchy problem for a single equation of order s ~ I u(~) - - A~_~ (z, t, D ) u( ~-I~ - - . . .
u (0, z)
=
% (z), . . . . u(~-~) (0, z)
where hi(z, t, 5) are functions analytic in C ~ X V The change u I = u, u 2
u'
~,..~U
9
/0
s
- - Ao (z, t, D ) u = h
= u(S-i) 1
0...
0
1
=
(3.8) (3.9)
(t, z),
%_~ (z),
and having minimal type in z. leads to the system
\No A1 A2
0)
0 0
0
0
1 Ju+~)
A,_2 A,-1/
\h
.
In this case conditions (3.3) give the inequalities 0~mi--mi+12~l (i=l ..... s--I); m,--nzi§ whence we immediately find that d e g A ~ < s - - L
degA i
Conversely, if deg A~-.<s--i~ then on going over to the system it suffices to set m i = k + i - i, where k is an arbitrary natural number. Thus, for a single equation the Cauchy problem is well posed in the classes of exponential functions Expm, r (,) if and only if the C~ characteristic polynomial of equation (3.8) is a Kovalevskaya polynomial in the variable z.
4.
The Case of Evolution of a Solution in the Class of Initial Data
In Sec. 3 we established the existence of a solution of the Cauchy problem (3.1), (3.2) in the spaces E-~m,r+ 0 and 6 > 0 such that for any r < R and any functions ~(z)CExpm, r(~) C ~' and h(t, z)~9 -Expm+~, r (z)) C ~ there exists a unique solution e(t,z)E9 s 0f problem (3.i), (3.2). THEOREM 4.1. The Cauchy problem is locally solvable (in a well-posed manner) in the spaces Expm, r(Cz) if and only if the symbols Aij(z , t, ~) are polynomials in z, and their degrees mij satisfy the inequalities
(i.j = l
m~]~mi--m]
. . . . . N)
(4.1)
(as before, if Aij(z , t, ~) e 0, then mij = -~)~ The proof is altogether similar to the proof of Theorem 3.1, and we omit it. Examplee We consider the Cauchy problem (3.8), (3.9) for a single equation of o r d e r s ~ i. Arguing as in the example of See. 3, we find that for a single equation the Cauchy problem is locally well posed in classes of initial data Exp~,~(C~) if and only if degAi-%0 (i = 0, 1 ..... s - I), i.e., the symbols Ai(z , t, r ~ Ai(t, ~) do not depend on z. Remark 4.1. It is clear that if some (3.3) or (4.1), then any index of the form satisfies these same conditions. Thus, if m = (m I ..... mN), then it is well posed for
multiindex m = (m ,...,mN) satisfies conditions ~ = (m I + k .... ,mN + k), where k is an integer, the problem is well posed for some one multiindex all multiindices
l(ff' (m) -=-{l = (ml + k . . . . .
.zN + k), k~Z}.
If we set def
Exp----r(m),r+altt(C~)= U Exp-'-~,r+
F~r(,n),,+~it!(C~)if
and only if the degrees mij Of the
2769
