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On the Cauchy problem for finitely degenerate hyperbolic equations of second order

Ferruccio Colombini, Haruhisa Ishida and Nicola Orrfi

A b s t r a c t . This paper is devoted to the study of the Cauchy problem in C ~ and in the Gevrey classes for some second order degenerate hyperbolic equations with time dependent coef- ficients and lower order terms satisfying a suitable Levi condition.

(1)

1. I n t r o d u c t i o n In this paper we shall consider the Cauchy problem

f i(t, cot,co~)u(t,x)=O,

,[ u(o, x) = ~o(x),

|

( o,~(o, x) = ~ (z),

on [0, T] x R~, where

L(t, COt, COx) : coy-L2(t, COx)- L1 ( t, COx), L2(t, cO~) ~

= a~j(t)O;,x~,

2

i , j = l

n

Ll (t, cO=) = E bj(t)cO=~,

j = l

under the weak hyperbolicity condition

(2)

~ a~j(t)~j >0

for all (t,~) c R • "-1

i , j = l

(2)

224 Ferruccio Colombini, Haruhisa Ishida and Nicola Orrd

Let us define

(3) a(t, () = a~5 t~) T~F'

i,5=1

(4) b(t,() = ~ ' b s ( t .

j = l

We shall assume from now on t h a t

a~seC~(R) and •eC~ It is well known

t h a t the Canchy problem (1) can fail to be C~-well posed, even if

bj-O,

due to too fast oscillating coefficients (see [CS]); or, on the other hand, when the Levi condition is not satisfied by L1, even if the coefficients ai5 are constants (see, e.g., [M]).

On the contrary, if L2 is effectively hyperbolic, then (1) is C~-well posed for any choice of L1 (see [N2] and its bibliography). We observe t h a t in this simple case the effective hyperbolicity of L2 means t h a t if for some (t-, ~)E [0, T] x S '~- 1 we have a(f, ~)=0, then

(51 o~ar ~) > 0.

The aim of this paper is to study the Cauchy problem (1) when the condition (5) is weakened to an assumption of finite degeneracy, and under a very precise

Levi condition

on the lower order term L1. More precisely, we shall prove the following theorem.

T h e o r e m 1.

Assume that

(6)

~ l ~ a ( t , ~ ) l y ~ O for all (t,~)e[O,T]xS n-1.

5=0

Let k be the minimal integer satisfying

k

(7) ~ I~a(t,~)l r for all (t,~) 9 [0,T]

x S n - ' .

5=0

Suppose that there exist

C > 0

and 7 9

[0, 89

such that

Ib(t,~)l < Ca(t,~)~ for all (t,~) 9 [0, T] x S "-1.

(8) Then, if

(9)

' 7 + l / k < 89

the Cauchy problem

(1)

is well posed in .y(8) for

(10) s ~_~ 1 1-'T

~ - ('1,+ l / k ) "

(3)

On the contrary, if

(11)

~/+ l / k > 89

the Cauchy problem

(1)

is C~176 posed.

We can easily show t h a t under the assumption (6), some k exists for which (7) is satisfied, thanks to the regularity of

a(t,

~) and the compactness of [0, T] • S n-1.

We denote by 7(s) the (projective) Gevrey class with exponent s (_>1), t h a t is, the set of all functions f c C ~ ( R ~) such t h a t for any r > 0 there is a constant C r > 0 fulfilling

sup

]O~f(x)l < Crrl~l(~!) ~

x E R n

for every multi-index c~E Z+.

We remark t h a t some hyperbolic second order equations finitely degenerating at a point are studied in [K] and in [IO], who consider the coefficients depending also on x, but under more restricted conditions. More precisely, in [IO] it is proved t h a t if

(12)

a(t, ~) _> 6t 21,

(13) [b(t, <_ c(t + )

for some positive constants 6 and C, and for l and v with 0 < u < l - 1 , then the Cauchy problem (1) is "y(~) well posed for

s < s o = ( 2 l - v ) / ( l - u - 1 ) .

It is easy to see t h a t if (12) and (13) are satisfied, then we can apply Theorem 1 with

k=21 and

"y=v/21,

obtaining the same Gevrey exponent; but, conversely, the assumptions of Theorem 1 are more general. In fact, under the hypothesis (7) an inequality like (12) is not true in general, even for t near 0; moreover the condition (13) is more restricted t h a n (8), as shown by the following examples.

Example

1. Let us consider, in the case n = 2 , the following coefficients:

a l l ( t ) = t 6, a12(t)=a21(t)=O, a22(t)=t 2,

bl (t) = t, b2(t) =

t 1/3.

Owing to Theorem 1 with k = 6 , ~ = ~ , we know t h a t the Cauchy problem (1) is well posed in -/(') for s_<5, meanwhile by Theorem 1.2 of [IO] we get s < ~ .

Example

2. Let us now consider:

a11(t)=t 4, a12(t)=a21(t)=O, a22(t)=t 2, bl(t)

= t , b2(t) = t 1/2.

In this case (11) is fulfilled and so (1) is C~-well posed.

(4)

226 Ferruccio Colombini, Haruhisa Ishida and Nicola Orrd

Finally we remark that in the case of one space variable, if (12) is satisfied, then the Gevrey exponent given by Theorem 1 coincides with the one of [IO], see the example below, studied in [If.

Example

3. For the operator

it is proved in [If that the Cauchy problem (1) is well posed in 7 (8) if and only if

S < S o = ( 2 l - v ) / ( l - v - 1 ) .

The techniques used in the present paper are in part similar to those of [CDS]

and [CJS], but we also require the following precise estimates; their proofs are inspired by [N1].

L e m m a 1.

Let us consider a(t,~) defined by

(3),

satisfying

(7).

Then there exist M and ~o positive such that for any

eC(0,e0]

we have

(14)

fo T IOta( t, ~)1 1

a(t, ~)+e dt <_ M

log -'e

L e m m a 2.

Let us consider a(t, ~) defined by

(3)

and let k be given by

(7).

Then for any ~?>_0 there exist M~ and Eo positive such that for any

eE(0,e0]

we have

fo T (a(t,~)+z)'71 { Mu, 1 if~?<l/k, dt <

U n l o g ,

if n= l/k, Mne 1~k-n, if 77 > 1/k.

2. P r o o f s o f L e m m a s 1 a n d 2

Proof of Lemma

1. Let us fix (t,~)C[0, T ] x S n-1 and let k ~ k be an even integer such that

(15) ~ a ( t - , ~ ) = O , j = 0 , . . . , k - 1 , and Otka(t,~)#O.

Then, by virtue of the Malgrange preparation theorem (see, for instance, [HI, The- orem 7.5.5), we can write

(16)

a(t,~) =e(t,~)[(t-~)k+bl(~)(t-~)k-l+...+b~(~)] =e(t,~)p(t,~)

(5)

for

(t,~)EU={(t,~)ERI+":It-~[<5 ,

where

e(t,~) and bj(~) axe C ~

functions with e({, ~ ) # 0 and bj(~)=0, j = l , ..., k, respectively. Further, due to (2), we may suppose that

e(t, ~)

is positive in U and so that

p(t, ~)

is nonnegative in U.

For (t, ~)EU we can factorize

(17)

p(t, ~) = ( t - t l (~) )(t-t2(~) )

...

(t-tk(~) ).

Let Co, C1, C2, C3 and Ca be positive constants satisfying

Co<e(t,~)<C1, IOte(t,~)l<_C2,

max_ltj(~)l_<C4

j = l , . . . , k

~=~ i#j

in V. Then we have, for 1~-~1<5_ (< ~T) 1

(18)

Here, noting that

[t+z IOta(t,~)l I t + ' ]Otelp d t + / t : 6 e[Otp[ dt Jr--5 a(t,~)+edt<--J~-~ ep+e - ep+e

C2 C1 [~+~ IOtpl

<- -Coo T + -Coo j {_,~ p + E / Co k

O t p ( t , =

II(t-

j=l iCj

and taking

5<Ca and eo<CoCa(T+2Ca)<l/2eo,

we find

J { - z

p+z/---~o dt <

j = l t--

- It-tj(~)l+r dt

- - dt.

T+5

3=1J-~ It-Ret~l+e/CoC3 dt

(19)

fT+2C4 1

< k --/-2c4

[tl +e/CoC3 dt

]fT+2C"

1 dt = 2k log ( 1 +

COC3(T+2C4))

-<2ks0

t+~/CoC3 c

< 4k log - 1

for cE (0, E0]. Therefore, by repeating the calculations in (18) and (19), thanks to the compactness of

S n-l,

we obtain

[~+~ IOta(t,

~)1 1 t_j~_~ a ( t , ~ ) + r dt < M l o g -r

for some 5>0, M > 0 , all ~ E S '~-1 and r162 (r is retaken small enough, if necessary). Finally we conclude (14) due to the compactness of [0, T]. []

(6)

228 Ferruccio Colombini, Haruhisa Ishida and Nicola Orrd

Proof of Lemma

2. Let ~/>0 be fixed. We also fix (t, ~) e [0, T] x S n-1 and, with the same notation as in the proof of Lemma 1, we can deduce

[

~+~ 1

.~_~ (a(t,,~)+~)',

(20)

1 ~+~ 1

dt<-~o ~_ - ~ (p(t,~)+e/Co)" dt

1 ~ / ~ + 6 1

dt

< ~

j=l

J~-~ (It-~tjl~+~lCo),

1 fT+C4 1

< ~ookJ_c, (itl~+e/Co)n dt

< _2 k(/a

1 )

- c~' \ J o

(tk+e/Co)" dt+T+C4

2 / / --C \l/k--rt

f l

1 dt+T+C4)

Hence, by using a compactness argument as at the end of the proof of Lemma 1, Lemma 2 immediately follows from (20). []

3. P r o o f o f T h e o r e m 1

First of all, if k = 0 , then L is strictly hyperbolic; moreover obviously k is even and so we may assume that k > 2 . Since the case

s=l

is well known (see [CDS]), we suppose t h a t s > 1 and u0 and ul are compactly supported. Then the Cauchy problem (1) has a unique solution

uEC2([O,T];'D (s)')

for l < s < 2 (see [CJS]). Here :D (s)' is defined as the dual space of :D (s). Thus we need only check the regularity of the solution with respect to x variables. For this purpose, denoting the partial Fourier transform of u in x by

v(t, ~) = / R - u(t, x)

exp(--vrL--f x.~)

dx,

it will be sufficient to estimate the growth order of

v(t, ~)

with respect to ~. The function

v(t,~)

solves the ordinary differential equations in t, depending on the parameter ~,

(21)

02v+a(t, ~)1~12v

+ v/E-f

b(t, ~)I~l v = O.

With the same method in [CJS], we define a,(t,~) = a ( t , ~ ) + E

(7)

and introduce the e-approximate energy

(22)

E,(t, ~) = a,(t,

01~121vl2 + 10~vl 2.

Differentiating

E~(t, ~)

in t and taking (21) into account, we enjoy

d E,(t,~) < ( IOta(t'~)l EI~I Ib(t,~)l ~E,(t,~)

Ja(t, ~)+e ]

and, Gronwall's inequality and (8) yield

(23)

( f o

T lata(t,5)l

dt+elSI fo T 1

Ee(t,~)<Ee(O,~)exp a(t,~)+e ~/a(t,~)+e

T C dt)

+~o

( ~ ( t , ~ ) + e ) u ~ - ~ "

dt

Here, putting ]~l=e-~, we distinguish two cases.

(i) If

~/+l/k> 1,

choosing

a=(k+2)/2k,

we obtain by Lemmas 1 and 2

(24)

E~ (t, ~) < E~ (0, ~) exp(C log I~1) for some C > 0 and for

I~l

large enough.

(ii) If ~ / + l / k < 1, then we select a = l - 7 and hence we get by Lemmas 1 and 2

(25) E~(t, ~) <

E~(0, ~) exp(C log

I~I +CISI [1/2-('r+l/k)J/(1-'r))

for some C > 0 and for

I~1

large enough.

Thus we arrive at the conclusion from (24) and (25) by using arguments similar to the ones in [CDS] and [CJS]. []

Acknowledgment.

This work was carried out during the stay of the first and third authors at the Institute of Mathematics, University of Tsukuba, during Febru- ary and March 1999. They would like to express their sincere gratitude to Professor K. Kajitani for his hospitality and kindness.

R e f e r e n c e s

[CDS] COLOMBINI, F., DE GIORGI, E. and SPAGNOLO, S., Sur les ~quations hyperboliques avec des coefficients qui ne d~pendent que du temps,

Ann. Scuola Norm. Sup.

Pisa Cl. Sci.

6 (1979), 511-559.

(8)

230 Ferruccio Colombini, Haruhisa Ishida and Nicola Orrti:

On the Cauchy problem for finitely degenerate hyperbolic equations of second order

[cJs]

COLOMBINI, F., JANNELLI, n. and SPAGNOLO, S., Well-posedncss in Gevrey classes of the Cauchy problem for a non-strictly hyperbolic equation with coefficients depending on time, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10 (1983), 291-312.

[CS] COLOMBINI, F. and SPAGNOLO, S., An example of a weakly hyperbolic Cauchy problem not well posed in C ~176 Acta Math. 148 (1982), 243-253.

[H] H()RMANDER, L., The Analysis of Linear Partial Differential Operators I, 2nd ed., Grundlehren Math. Wiss. 256, Springer-Verlag, Berlin-Heidelberg, 1990.

[IO] ISHIDA, H. and ODAI, H., The initial value problem for some degenerate hyperbolic equations of second order in Gevrey classes, Funkcial. Ekvac. 43 (2000), 71- 85.

[I] IvRII, V. JA., Cauchy problem conditions for hyperbolic operators with character- istics of variable multiplicity for Gevrey classes, Sibirsk. Mat. Zh. 17 (1976), 1256-1270, 1437 (Russian). English transl.: Siberian Math. J. 17 (1976), 921- 931.

[K] KINOSHITA, T., On the wellposedness in the Gevrey classes of the Cauchy problem for weakly hyperbolic equations whose coefficients are H61der continuous in t and degenerate in t-~T, Rend. Sere. Mat. Univ. Padova 100 (1998), 81-96.

[M] MIZOHATA, S., On the Cauchy Problem, Notes and Reports in Math. in Sci. and Eng., Academic Press, Orlando, Fla.; Science Press, Beijing, 1985.

[N1] NISHITANI, T., The Cauchy problem for weakly hyperbolic equations of second order, Comm. Partial Differential Equations 5 (1980), 1273-1296.

[N2] NISHITANI, T., The effectively hyperbolic Cauehy problem, in The Hyperbolic Cauchy Problem (by Kajitani, K. and Nishitani, T.), Lecture Notes in Math.

1505, pp. 71-167, Springer-Verlag, Berlin-Heidelberg, 1991.

Received May 10, 1999 Ferruccio Colombini

Dipartimento di Matematica Universit~ di Pisa

IT-56127 Pisa Italy

Haruhisa Ishida

Institute of Mathematics University of Tsukuba Ibaraki 305-8571 Japan

Nicola Orrd Via Genova 5 IT-07100 Sassari Italy

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