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Branching of singularities for a degeneratehyperbolic systemKeneo Shinkai aa Depertment of Mathematics , University of Osaka Prefecture Sakai , Osaka, 591,JapanPublished online: 08 May 2007.

To cite this article: Keneo Shinkai (1982) Branching of singularities for a degenerate hyperbolic system,Communications in Partial Differential Equations, 7:5, 581-607, DOI: 10.1080/03605308208820233

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CoMM IN PARTIAL DIFFERENTIAL EQUATIONS, 7(5), 581-607 (1982)

BRANCHING OF SINGULARITIES FOR A

DEGENERATE HYPERBOLIC SYSTEM

Keneo Shinkai

Depertment of Mathematics

University of Osaka Prefecture

Sakai, Osaka 591, Japan

Introduction. The branching of singularities of

a solution for a weakly hyperbolic operator D 2 t21 2 t- J=x+

lower order terms was studied by Alinhac [l], Hanges

[5], Taniguch~ and Tozaki [13] and ~akane[l~]. Ichi-

nose and Kumano-go [ 6 ] treated a modified type of the

above operator. Amano [2] and Amano and Nakamura [3]

treated higher order operators that have solutions

with branching singularities. In the present paper we

treat a degenerate hyperbolic system of the following

type:

Copyright O 1982 by Mdrcel Dekker, l n c

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where h.(t,x,C), j=l,..,,m satisfy J

hj(t9x,c)E ~[1,e], real valued, (0.2) e

I~~(t,x,~)-~~(t,x,t)l z c It I ( c } (jf*)

for a positive integer .! and a positive constant c, and

symbols b . k(t,x,c) of (j9k)-elements of B satisfy J ,

Here S[a,b] is a variant symbol class of Boutet de Mon-

vel [4] (see Definition 1.1). It was proved in the

previous paper [12] that operators treated by the above

authors [1][2][3][5][6][10][13] are able to reduce to

systems of type (0.1) with (0.2) and (0.3). In [12] we

constructed the fundamental solution E(t,s) of the

Cauchy problem

when s 2 0.

In the present paper we construct E(t,s) for -T < 0'

s i t $0 and -To (' s $0 i_t &To for a small constant T 0

(Theorem 2.1). For the last case we have

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BRANCHING OF SINGULARITIES

(0.5) ~ ( t , s ) = N ( ~ ) E , ( ~ , ~ ) C ( X , D ~ ) E ~ ( O , ~ ) N - ~ ( ~ )

+ "smoothing operator",

where

N(~)Es[o,o] is the perfect diagonalizer of L,

E2(t,s) is a diagonal matrix of -Fourier integral

operators of single phase,

and

c(x,5) e so .

Thus the branching of singularities is known by using

c(x,<) and the expansion formula of Fourier integral

operator with double phase in [ 9 ] . The principal part

co(x,<) of c(x,<), that is, ~ ( x , S ) - c ~ ( x , ~ ) € S-O, is

given by solving a system LO of ordinary differential operators, where the coefficients of Lo coincide with

the principal symbols of L (Theorem 3.2). We note that

CO(x,F,) is the Stokes coefficient of Lo ema ark 3.3).

1 , Diagonalization. I n this section we diagon-

alize L and LO to Ll and Lo,, respectively and show

that the coefficients of & coincide with the prin- 0,1

cipal symbols of L1.

Definition 1.1. Let 1 be a positive integer and

03 =li(e+l).

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584 SHINKAI

i) For real a and b we denote by S[a,b] the space

of all ~ @ - s ~ m b o l s p(t,x, 5 ) on [-T,T]XR~XR~ ( 0 ( T jl)

such that for any non-negatke integer j and any multi-

indices d , p we have

2 1 / 2 where D =-pla/at, p"' = # ~ $ p and (5) =(l+ /ti ) .

t ( P I t; ii) For real a, b and c we denote by S[a,b,c] the

n space of all cm- symbols p(t,s,x,C) on [-T,T]X[-T,T]~R~

XR; such that for any non-negative integers J , k and

any multi-indices d , (j we have

iii) we set wa= n~[a-&v,-v] , q=(-)~[a-dv,-v ,o] v >O v 70

iv) A pseudo-differential operator P(t) with the

symbol ~(~(t))=p(t,x,c) is defined by

where X(Y,c) E $(R**) (the set of rapidly decreasing

functions of Schwartz) such that X(0,0)=1 and 3 is the

set of all c*-functions whose derivatives are all

bounded.

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BRANCHING OF SINGULARITIES 585

For the fundamental properties of the pseudo-diffe-

rential operators with above symbols, we refer to 91

of [12].

Proposition 1.2(perfect diagonalization of L). Let

L be the operator given by (0.1) which.satisfies (0.2)

and (0.3). Then, there exists ~ ( t ) such that ~ ( t ) has

the inverse N-l(t), N(o)=I, the symbol o(N(~))(x,~) E

s[o,O] and we can write

for some L1 of the form

where

and

(1.7) u(R(~)) e 74:

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Proof, Let d (j=l, ..., rn) be real constants such j

that d.fdk if jfk, and set J

where

Then, there exists a constant C 0 > O such that

N

+ (5) -')'(c) if jfk.

Set

and

where b(.') is the (j,k)-element of B ( v) and J ,k

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BRANCHING OF SINGULARITIES

Then we have

and by Proposition 1.5 ii) of [12] there exist ~ ( t ) and

~ ( t ) such that

(1.11) N(~)-JN ( ) + N + mod 731'

and

(1.12) F ( ~ ) - - F ('I + F(') +.,. mod ]$.

v Here, 'tAdA(o)+A(l)+. . . mod H~ means that A- c A(p)

u=o

g~[a-v,-v(l+l)] for any V. The existence of the inv-

erse N-l(t) of ~ ( t ) is shown by using Theorem 1.4 of

Kumano-go [ 7 ] . ~ ( t ) is given by

let ao(t,x,() and Bo(t,x,c) be the "principalv

symbols of L, that is,

where h J .(t,x,c)-ho , J .(t,x,<) E ~[1,$+1]

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Then we have the following

Theorem 1.3. There exists the perfect diagonalizer

NO(t,x,E) o f LO such that

and we can write

(1.18) N ~ ( ~ , ~ , c ) - ~ ~ ~ N ~ ( ~ , ~ , ~ ) = 'Lo,, for some , of the form

and

(1.21) R ~ ( t,x,S) E H:

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BRANCHING OF SINGUIARITIES

Furthermore we have

and

Proof. By using the symbol product instead of the

operator product used in the proof of the previous pro-

position, we get No(t,x,5), Fo(t,x,S) and Ro(t,x,S)

which satisfy (1.17) and (1.23). Set R(~)(t,x,~) =

Since

we have (1.24).

52. Fundamental solution E(t,s) of L for s g O l t .

When O S s d t s T o ( T ~ > O is a small constant), we const-

ructed in [12] the fundamental solution E(t,s) of L in

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SHINKAI I

I

the form

where ~(~(t,s))(x,c) E s[o,o,o], Q(S,S)=O and E2(t,s) is

the fundamental solution of L2=Dt-a(t)+~(t). E2(t,s)

has the form .

where +jj(t,s,x,5) is the solution of the eiconal equ-

ation

and a.(t,s,x,<) is the symbol that is given by solving J

transport equations. By Theorem 3.1 of [12] there ex-

ist constants M and M! such that j J

The operator e a .(t,s,X,Dx) is a Fourier J

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BRANCHING OF SINGULARITIES

integral operator defined by

for u(x) ~ ( R Z ) , We also have by Theorem 1.11 of [12]

When -Togs I_tI_O or -Toss &O&tSTo, E(t,s) is

given by the following

Theorem 2.1. Let L be the operator given by (0.1)

which satisfies (0.2) and (0.3). Then the fundamental

solution E(t,s) of the Cauchy problem (0.4) can be

found in the form

where

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592 SHINKAI

and Q (s,s)=O. 5

Furthermore, if we fix exists

c(x,#) G SO such that

Here sa is the usual symbol class of pseudo-differential

operators of order a.

0 Proof. 1. Assume that -To< s d t l O . First we -

note that, in view of the equation E (t,5)E2(0,s) = 2

E2(t,s), E2(t,s) satisfies

If we get the solution E (t,s) of 1

then, since it also satisfies E (t,a)E (C7,s)=El(t,s), I 1

we have

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BRANCHING OF SINGUIARITIES 593

Set

(2.14)+ R+(t,s)= fi~~(s.t)~(t)~~(t,s) for O$sLtiTO,

(2.14)- R - (t,s)= ~~~(t.s)R(s)E~(s, t) for -TOLs$tlO,

and let Q (t,s) be the solution of -

Then El(t,s)=(~+~-(t,s))~2(t,s) satisfies (2.12)

and

is the fundamental solution of (0.4). So, we shall

solve (2.15). But this can be done by the same prose-

dure to Lemma 4.1 of [l2]. We also have

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594 SBINKAI

24 When -To s 2 0 5 t ST0, using the results of

Kumano-go, Taniguchi and Tozaki [ g ] and Kurnano-go and

Taniguchi [ 8 ] , we can define the product E(t,s)=~(t,O)

~ ( 0 , s ) ~ where E(t,O) has the form (2.1) and E(0,s) has

the form (2.16). The (j,k)-element E (t,s) of El(t, j ,k

s) is represented as a Fourier integral operator with

a double phase and for u C B ( R ~ ) we have

where

1 1 y = (x-x ). <+(x -x2)* <1+(X2-X3).C2

and

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BRANCHING OF SINGULARITIES

3: We prove t h e l a s t s t a t e m e n t of the theorem. B y

( 2 . 1 5 ) we have

The f i r s t i n t e g r a l of t h e r i g h t hand s i d e belongs t o S 0

*o and t h e second t o . Thus, t h e r e e x i s t s C - ( x , ~ ) € s 0

such t h a t

I n t h e same manner we can prove t h a t t h e r e e x i s t s

C + ( x , z ) such t h a t

Simce we f i x e d s < 0 , I + G ( Q - ( O , s ) ) ( x , ~ ) - C - ( x , t;)E S - a .

S e t C(X,<)=~(C+(X,D~)C-(X',D~,)), and we have ( 2 . 1 0 ) .

Q.E.D.

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596 SHINKAI

Now, we consider the propagation of singularities

for the Cauchy problem (0.4). We fix s < O and assume

0 0 with i=j, O = s and the Cauchy data (~,v)=(x ,< ), and

let

where (qj,k(t),pj,k(t)) is the solution of (2.21) with

i=k, O=O and the Cauchy data (y,q)=(qj(~),pj(0)). Let

~ ( t ) be the solution of (0.4). Then we have WF(U(~))

m C U rj,k(t) when t > 0 (see Theorem 3.4 of [ B ] ) .

j,k=l

Furthermore we have

Corollary 2.2. Let c (x,F,) be the (j,k)-element j ,k

of C(x,E). Then we have

i) if there exist a conic neighbourhood K of

(qj(0),pj(O)) such that c (X,E)~O on K, then j ,k

(2.2)) WF(U(t))nc,k(t) = a for 0 < t STo .

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BRANCHING OF SINGULARITIES 597

ii) if there exist a conic neighbourhood K 1 of

(qj(0),pj(O)) such that c (x,~) never vanisn i n K t , j,k

then

93. The principal symbol of c ( x , ~ ) . We quote from

[12] a -ernma e em ma 1-13) that is important in the proof of Theorem 3.2. For the simplicity we assume

0 f_s &tI_To, When -To$s$t10, the lemma is also va-

lid with slight modifications.

Lemma 3.1. Let OSsSt&To, p(t,s,x,C)€~[a,b,c],

r(t,s,x,C)€ gk, $(t,s,x,~) be real valued and ~-~(~,s,x,c)-x-c es[l,e+l,o]. set

and ~=r(t,s,X,D~).

Then both PjR and RP+ are pseudo-differential op-

erators. Setting rl(t,s,x,C) =O(P+R) and r2(t,s,x,c) = CV k+a

o(RP+). we have r . (t, s , x , 5)s 74 . Furthermore we 3

have

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Theorem 3.2. Let Lo be the system of ord inary

d i f f e r e n t i a l opera tors given by (1 .16) . Then the fund-

mental s o l u t i o n E o ( t , s , x , E ) of , t h a t i s , E 0 ( t , s , x ,

5 ) s a t i s f i e s

i s given i n the form

where E ( t , s , x , < ) i s the fundamental s o l u t i o n of 092

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BRANCHING OF SINGULARITIES 599

% =D,- 80(t,x,c)+~o(t,x,c), No(t,x,c) is the perfect 0,2

diagonalizer of given by Theorem 1.3 and Q (t,s,x, 0, +

Furthermore, if we fix s < O[ then there exists

c,(x,c) E so such that

and

Proof. The products except that has the form ~ ( A B )

in the follwing proof are all symbol products. So for

the simplicity we omit to describe the variablesx and

. The proofs of (3.2) and (3.3) are similar to that

of Theorem 2.1. So we shall prove only (3.4) by three

steps, Let R (t) be the symbol defined by (1.19) and 0

set

o,+

for 0 i s 2 t iTo,

and

3 5 Ro,- ( t , ~ ) = PE 0,2 (t,s)~~(s)~~,~(s,t)

for -To $ s i t L O .

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600 SHINKAI

1 Let R+ be the operator defined by (2.14)+. We

first prove

The proof of

is similar to that of (3.6) + ' Let r , J 9 k(t), ro, j,k(t)r r+, 3, k(t9s) Or r ~ , + , j,k (tt

s) be the (j,k)-element of G(R(~)), Ro(t), ff(R+(t,s))

or R (t,sf, respectively. Let a .(t,s) be the symbol o,+ J

given by ( 2 . 2 ) , and set

and

Then by Lemme 3.1 we have

and W - 1 (3.8) r +, J, k(t,s)-r

(t,s)~ Jd . +,j,k

BY using (1.14), (1.23), (1.24), Lemma 3.1 and the re-

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BRANCHING OF SINGULARITIES 60 1

we have

and

BY (3.8) and using Lemma 3.1 we have

For the simplicity we write r(t,s)=~(R (t,s)) and + r (t,s)=~ (t,s). We set

0 o.+

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and

Then we have

Since we have by Lemma 4.1 of [12]

we have (3.12)+ if we proved

and

Thus we have

Now we assume that

(3.191,

(1,(t.s)-I 0 ,v (t,s)\ < (2~)~(t-s)~(5)-~(t+(5)-*)-"/v: Dow

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BRANCHING OF SINGULARITIES 603

h o l d f o r v = l , 2 , . . . , p . Then, s i n c e we have by ( 4 . 2 0 ) of

L e m m a 4 . 1 o f [ 3 2 ]

Thus we have ( 3 . 1 9 ) f o r any v L 1 , and v

D i f f e r e n t i a t e bo th s i d e s o f ( 3 . 1 3 ) and ( 3 . 1 4 ) and e s t i -

mate s i m i l a r l y . Then w e have ( 3 . 1 8 ) .

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3'. Similarly we have

(3.12) - Q-(t,s)-so,-(t,s)~ s[-~,o,o] for -T,~s i t $0.

Set

and

R e m a r k 3.3, Let t ' and s t be constants such that

0 (t' < -s' $To. We set two solutions of L0u=0 by

and

Z ,4

Then there exists a matrix C 0 such that v(~)=u(~)c~.

Since Qo -(t,st) E Dt(s-@) when s' A t A-t' , we have by 9

( 3 . 2 )

(3.22) v(~)-N~(~)E~,~(~,O) E Bt(s-@) for s t & tJ.-t'.

Similarly w e have

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BRANCHING OF SINGULARITIES

N These mean that C is the Stokes coefficient matrix of

0

the lateral connection problem for Lo (See [ll]). When t f $ t A T o , by (3,21), (3.2), (3.3) and (3.22) we

have

Since there exists the inverse matrix of ~ ( t ) , we have

If d; have a typical form: 0

1 when It( (cy22, where p .(x,C)€ s ( j = l , . . . ,m) and

J rV

B(x,<)EsO, then, setting z=WtLtl(~}, the equation

L 0 u = 0 is reduced to

where AO, Ale SO, and A1 is diagonal. This is one of

the equationstreated by Okubo and Kohno [ll].

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REFERENCES

SHINKAI

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