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A mixed problem for weakly hyperbolic equations ofsecond orderKimura Kyoko aa department of Mathematics , University of Toronto , Toronto, Ontario, CanadaPublished online: 08 May 2007.

To cite this article: Kimura Kyoko (1981) A mixed problem for weakly hyperbolic equations of second order, Communicationsin Partial Differential Equations, 6:12, 1335-1361, DOI: 10.1080/03605308108820213

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A M I X E D PROGLEM FOR WEAKLY HYPERBOLIC EQUATIONS

Kyoko Klrnura > e 2 z r t ~ e n t nf M a t h e m a t i c s

i _ i n i v n r c i nf T o r o n t o

T o r o n t o , O n t a r i z , Canada

$1 I n t r o d u c t i o n

I n t h i s p a p e r we s t u d y a mixed p r o b l e m f o r h y p e r -

b o l i c e q u a t i o n s o f s e c o n d o r d e r w i t h d e g e n e r a c y on t h e

i n i t i a l s u r f a c e .

L e t S? be a domain i n R" w i t h compac t smoo th

b o u n d a r y a R = S . W e c o n s i d e r a m i x e d p r o b l e m f o r t h e

e q u a t i o n

Copyright O 198 1 by Marcel Dekker, Inc.

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d e f i n i t e quadratic fo rm w i t h r e a l c o e f f i c i e n t s i n

The Cauchy p r o b l e m f o r e q u a t i o n s o f t h e t y p e ( 0 . 1 )

h a s b e e n s t u d i e d by many a u t h o r s . I n p a r t i c u l a r

O l e i n i k i7] showed t h a t t h e C a u c h y p r o b l e m f o r e q u a t i o n s

w i t h more g e n e r a l d e g e n e r a c y i n t i s w e l l - p o s e d u n d e r

c e r t a i n c o n d i t i o n s on t h e c o e f f i c i e n t s . The mixed

p r o b l e m f o r d e g e n e r a t e e q u a t i o n s h a s b e e n a l s o

c o n s i d e r e d , f a r e x r m p l e , by K r a s n = v [ 4 ] a n d G l e i n i k M, - a n d t h e y o b t a i n e d t h e e x i s t e n c e t h e o r e m s a n d t h e

u n i q u e n e s s t h e c r e m s i n t h e s e n s e o f "a g e n e r a i i z e d

s o l u t i o n " u n d e r some c o n d i t i o n s on t h e coefficients

a n d t h e d a t a . I t i s o u r p u r p o s e h e r e t o f i n d a

s o l u t i o n u ( t , x ) o f t h e m i x e d p r o b l e m ( 0 . 1 ) , (0.2) , a n d

( 0 . 3 ) i n a b e t t e r s e n s e , n a m e l y s o t h a t

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HYFERSOfTC EQUATIONS OF SECOND ORDER

Our aim is to obtain a unique solution

n i t , x ) E Too

rt in 9 of the problem

that u(t,x) is k-times continuously differentiable

conditions in Oleinik [71 for the Cauchy problem it

seems to be naturai to impose a t K-l t e r m on the

coefficients of the first derivatives in x in case

the coefficients of the second derivatives in x have

t2K term of degeneracy. Furthermore, we shall

impose a compatibility condition of infinite order upon

the data.

The essential point of the problem is to establish

rhe energy estimate. T h e device u s e d for this purpose

is that of ~ z r d i n g [21, but modifications are necessary

to get the estimate near the boundary. (See SakamotO

[8].) In fact, both u and its boundary values can be - estimated using integration by parts.

This paper consists of two sections. In the first

section the existence and uniqueness theorem is derived

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from an energy i n e q u a l i r y . The succeeding sect$on is

devoted to a proof of this inequality.

51 Statement of Results

with the conditions

I1.2) B[UJ = o on [O,T] x S,

(1.3) u(0,x) = Dtu(O,x) = 0 in R,

where 0 5 E < 1. L is identical with L when E

E = 0; otherwise LE is regularly hyperbolic because

of the assumption (0.4).

The following lemma will be established in the

latter section.

Lemma. Let m be a non-negative integer. Then there

exists some integer N(>m) such that for any solution

u (t,r) s E : ( H ' ~ + ~ (a)) n.. .n E:'~ (IL2 (5-2)) of the nixed E

problem (1.1) , (1.21, and (1.3) the estimate

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h o l d s

o f i l

f [ c ; %

R e r c

- - w e now g i v e t h e d e f i n i t i o n o f the compatibility

c o n d i t i o n o f B r d e r m ( 2 O ) f o r mixed p r o b l e m i P ! .

L e t u s d e f i n e u < x > 9 , - recursively by ~ + 2

t h e f o r m u l a

w h e r e

a (t,x;Dj i s a d i f f e r e n t i a l = p e r a t o r of order j with j

r e s p e c t to x [ j=1,2) , and

W e s a y t h e d a t a [uo (x) ,ul ( x ) ; f (t , X I 1 satisfy t h e

compatibility condition of order rn when

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From the above lemma we can prove:

Proposition- Let m be a non-negative integer.

- v ~ n e n we can find some integer N ! > m ] such that for sny

Proof. Choose N so that for a solution u of the E

problem (1,l) , (1,2) and (1.3) the estimate

hoids for t E [ O , T j when f(t,x) e E~[H-(R))

satisfies D F ~ ( 0 , X I = 0 !p=O,1,. . . ,N) . Here we note - that the right hand side of (1.7) is independent of E .

When E > 0, the existence of the solution

E0 y, m+3 sE (t , x ) . E is) 1 r ~ . . . n E:+~ ( L ~ i n , ) is assured,

since L E is regularly hyperbolic and the compatibility

condition is satisfied. (See Ikawa [31) .

Thus, integrating (1.7) over [O,T], we have a

bounded set {uE (t ,x) }E,O in Eim-t3 ( ( 0 , ~ ) x R ) . That is,

for some constant M > O

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- . < ... . - - - .

i l { t i u ! 6 B ( i o , T i x G j such t h a t ii- + ti w e a K l y L > - . 2.

"' 9- i R-" - ( ( 0 , ~ j x 2 j w h e n k + = -

it i s e a s y t o s e e t h a t

( 1 . 9 ) ~ [ u ] ( t , x ) = f [ t , x j i n t h e s e n s e o f P' ( ( 0 , ~ ) " n j r

( 1 . 1 0 ) ~ [ u j ( t , s i = 0 i n (0 ,TI x S ,

In f a c t , f r o m L - [u 1 = f we c a n w r i t e ik Ck

P a s s i n g t o t h e l i m i t a s k + m , we o b t a i n

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Uere n R: = { : x x I E R ; > o j .

n

It suffices to prove that for a11 j

or, equivalently,

where we use m7 i t p y ) = T u ( t ,mi (xi) to denote i 3

-4 c P j u ( t , x ) for x 6 U . n S L . We set v = @ . u and take

3 m

3

X { X ] E 2 ( g l j which is equai t o 1 in a n e i g h b ~ r = n

hood of y,= 0 and v a n i s h e s wher? y,>1 .

Since

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P a s s i n g to t h e limit a s 6 + 0 shows that

r 1 pi ( t r y 1 ) = - / D ~ ~ X V ) (t,y:y,j%

- -

J 0

Similarly, if we set v = -

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Thsref ore

Since (1.11) follows from the same argument as that of

Mizohata [41, pp. 332-334, we omit the proof. We note

that

D'U ( 0 , ~ ) = 0 (p=O,l, . . . ,m+2) imples that D'U ( 0 , ~ ) = 0 &k

t

(p=O,. , . . . ,m+2) because D'U + DPu in D ' ( ( o , T ) x R) Ek

t

when k -t !p=O ,I,. . - ,m+3) .

Next, by changing the value of u(t,x) on a sat

with measure O in IO,T1 r R if necessary, we shall

see that u(t,x: is the solution of iP,i in u

E ( R ) ) . . . E ( L ) . (See Mizohata 1 4 1 1 .

Let P + jal < m + 2 . Then we can write

J t '

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i n Hm+ 2 -p

D P ~ ( T , X ) = f i m ~ : u ( t , x ) t

(R) . t'T

Then t h e mapping t + ~ ' u ! t , x ) t H m+2-P , ,n\ i s s t r o n g l y t

c o n t i n u o u s on [O,T] , w h i c h p r o v e s

O ( R ~ + ~ (0.1 ] n . . . n E ~ + ~ ( L ~ ( 5 2 ) ) . The e s t i m a t e U ! t ,X) 6 ft \-- c

( 1 . 4 ) when E = G a s s u r e s t h e u n i q u e n e s s o f t h e

s o l u t i o n o f ( P o ) . Q . E . D .

Theorem. F o r g i v e n d a t a

mixed p rob len ; (P) p r o v i d e d t h a t t h e d a t a s a t i s f y a

c o m p a t i b i l i t y c o n d i t i o n o f i n f i n i t e o r d e r .

P r o o f . FOX any f i x e d n o n - n e g a t i v e i n t e g e r m , t a k e

an N a s i n the p r o p o s i t i o n . L e t u (XI (p== ,.,..., N) P+ 2

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5 2 Energy inequality; Proof of Lemma.

We shall convert our problem into the one in a

n half space R+ = {x= (x l,...,~n); x > 01 by means of a

partition of unity and a local coordinate change. W e

begin by considering the mixed problem with zero

initial data:

n where V is a small neighbsrhood of X = 0 in R .

Let A and be positive pgrameters. We write

t = t + E and define the operators E P E and QE by

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Then ( s e e S a k a m o t o [81)

where a c t u a l c o m . p u t a t i o n g i v e s

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2348

and

The constant Ci depends only on the supremum of

a 1 , a 1 , iokaij (i.j.k=l.--- .n). ! b 1 . ! c ! A J 0

and 2 depends only on the supremem of 2

1b.l [j=18--.,nl. 1

We shall denote by c and C (j=O,l,. - - 1 j j

positive constants depending only on the supremum of

n the coefficients of L in [ O , T ] x v n R + .

It folLows from the assumption (0.2) in

Io,T] x V n R~ that for a sufficiently large ?, , + -

G t i u , u ) is equivalent to the term Dow

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for a suf f i c ien t iy large h . 14oreover,

- . -,- , - c '>'-!n !II oi2 !

Now we i n t r o d u c e t h e n o t a t i o n

where 8 i s a p o s i t i v e p a r a m e t e r .

m L e t 4 E C O ( V ) and l e t E be a s o l u t i o n of (2.1).

We s e t v = $uE. We s h a l l f i r s t o b t a i n t h e e s t i m a t e E

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t i f n N

G ~ \ D ; D ; V ~ . D > ) ~ ) ( t . ~ : o ) . - ~ ~ ~ dx'

j vn Rn-I

w h e r e R = R1.+ R 2 + R 3 a F i r s t o f a l l we w a n t t o

e s t i m a t e a l l t h e b o u n d a r y t e r m s w h i c h c o n t a i n t h e

a n o r m a l d e r i v a t i v e D n ( a =1, . . . , k+l) . To d o t h i s we n n

c o n s i d e r t h e f o r m

cinr,. 1 7 0 \ with p o s i t i v e unkown constants .M: ,,,,,, , L . U , ci n

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2-3 ( 2 . l! i n p l i e s v t u = 0 , we have

e : : 1 . 1 sc that the ~ 3 5 0 ~ ~ a j u l , i ~ i s u i ; r ---,- .- . = -. - 2 rnF--

i ( 2 - 6 j and ;;. ; ; a r e u u r n v ; l ~ ~ = . A A A - A ~ - - -

,- ,,ing ,- t h e inequalities ( - 2 . 5 1 ( 2 - 6 ) , and ( 2 . 7 ) for

= D F c Z - J ~ , we obtain

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-.L--- -.- 3. - - L n r l l r r . . - r . c . r , '-zr -...a,--- d J c,;~- ( ~ 1 1 a form estimated as

C 3

fo l ;ows:

On the o t h e r hacd, y e "ave

Hence (2.10) and ( 2 . 1 2 ) y i e l d

(2.13) ~ ~ G ~ f u ~ ( t ) l E -

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- * _ . - - w, .... ,,* - - - l i = , & j - j [ k = Q r l , > * r ' x & : i . i a A l u - k

=,,- nn the estimates ( 2 . 1 3 ) f o r k = O , . , . . . , m + l . ?has - -... -.=

we get f o r any lJ > O

I n a d d i t i o n , w e have

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C o I i e c t i n g the ii.l4j a n d (2.i51, we c a n

= a a + b = + + = ?.*... +.,L.., ::-- - - - m u - - - - - - - - - - - . - - - = = LGRI i i : ~ : 2 a i f i n i e L e r s 6 arid u i a r g e

w e c a n n e g l e c t t h e l a s t t w o t e r m s i n t h e r i g h t h a n d

s i d e o f (2.14). T h e r e f o r e , t a k i n g a c c o u n t o f t h e

estimate

We o b t a i n t h e e s t i m a t e f o r v = $ u E : E

i2.i6; " "t"Lq,uE:t) P ' A . . I $ + C S I l l i I i Y *,2 (+, ' - , I l l ' 1 ' 8 *

f o r 8 s u f f i c i e n t l y l a r g e . H e r e

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I n t a g r a t i n g (2-16) o v e r [O:t] y i e l d s

T o j u s t i f y t h i s i n t e g r a t i o n e v e n i n t h e case E = 0 ,

r e must assume t h a t

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L e t ! $ . ( X I > j=l, be a p a r t i t i o n o f u n i t y 3 . . . ,J

s u b o r d i n a t e t o a c o v e r i n g { u j ) o f S . W e may

assume t h a t t h e r e exists a s m o o t h t r a n s f o r m a t i o n

Y . x , . . ) f r o m U o n t o V i n Rn I j 1 J n 3 j

s u c h t h a t

Then t h e p r ~ b l e m ( P o l E i s converted i n t o t h e p r o b i e m

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where

Then it i s e a s i l y s e e n t h a t

and

- A p s r j - =.c r i 2 - T h e z e f o r e , we can apply the p r e v i o u s e s t i m a t e (2.18)

ta t h e problem ( P ~ ) ~ ~ ~ (j=1,2,...,J) . S i n c e t h e

- - C I - . - - - ~ . , i O q d nnlv harmless t e r m s , we h a v e C 0 0 r u ~ I l c i L ~ b,,Cz"y,kL 1 ---- ----'

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we set m o = 1 - mj . Since @3 vanishes in a j=1

neighborhood of S , we can find

under the same assumption as in (2.19). Thus

collecting the estimates (2.19) and (2.201, we

obtain

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xhen we take 9 sufficiently large, we can neglect the

i a s t - term in the right-hand side of (2.21). Hence we

nave

sufficiently large N . To complete the proof we note that D'£ t (0.x) = 0

ip=O,l, ..., N) implies

by virtue of Taylor's formula. Using Schwarz's

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which c o n c l u d e s t h e p r o o f .

ACKNOWLEDGEMENTS

I s h o u l d l i k e t o e x p r e s s my g r a t i t u d e t o P r o f e s s o r

T . X a k i t a f o r p r o p o s i n g t h i s p r o b l e m and f o r s o many

v a l u a b l e s u g g e s t i o n s .

REFERENCES

[ I ] G . Darboux , "Lecons s u r l a t h g o r i e g g n g r a l e d e s s u r f a c e s " , T . 11, G a u t h i e r - V i l l a r s , 1 9 1 5 .

1 2 1 L . ~ Z r d i n ~ , " S o l u t i o n d i r e c t e du p r o b l s m e d e Cauchy p o u r l e s i q u a t i o n s h y p e r - boliques", Colloques i n t e r n a t i e n a u x du C . N . R . S . ( 1 9 5 6 ; , p p . 70-90 .

[33 M. I k a w a , mu: -..A - - - % , - - - r * r * s u p ~ u ~ ~ e m b f o r h y p e r b o l i c

e q u a t i o n s o f s e c o n d o r d e r " , J. Math. Soc . J a p a n , 20 ( 1 9 6 8 1 , pp . 580-608.

[ 4 ] M.L. K r a s n o v , "The mixed b o u n d a r y p r o b l e m f o r d e g e n e r a t i n g l i n e a r d i f f e r e n t i a l e q u a t i o n s o f s e c o n d o r d e r " , Mat. S b . , 49 (91) ( 1 9 5 9 1 , pp . 29- 84. ( R u s s i a n )

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" Z e probline de C s u c h y 5 caracteristiques multiples", Ann. Scuola Norm. Sup. Pisa 4 - 4 (1977!, E D . 7 5 7 - 8 0 5 . -

"On the C a u c h y problem for weakly L ilyrsrbslic equations", Comm- P u r e

~ p p l . Math. 23 (1970f, p p - 5 6 9 - 586.

. P I A P ~ ~ . p L V d I C , -,.C, =3.2 for hyperb~?lic

equations I", J, Math. Kyoto U n i ~ . 10-2 (19721, pp. 349-373.

Rece ived A p r i l 1961

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