Overdetermined systems of singular partial differential equations

Nonlinear

Analysis,

Theoty,

Pergamon

Methods&Applications, Vol. 30, No. 6. pp. 3855-3865, 1991 Proc. 2nd World Coongrm of Nonlinear Analysts 0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546x/97 $17.00 + 0.00

PII: SO362-546X(96)00235-0

OVERDETERMINED SYSTEMS OF SINGULAR DIFFERENTI:AL EQUATIONS

PARTIAL

MINOItU KOIKE Shibaura Institute of Technology, 307 Fukasaku, Omiya 330, Japan Key words and phrases: Overdetermined system, singular partial differential equation, complete integrability condition, Stolz’s path, Banach scale. 1. IXTRODCCTIOiX In this paper we consider equations of the form

an overdetermined

system of singular

y(t)au/at, = ji(t, z, 11,cl(f.)l)u) + feo(4x>,

nonlinear

i = l;.*

partial

,rn,

differential

(1.1)

where t = (1i;.. ,t,) E llV’&, x = (XI;.. ,z,) E V and L? = (a/&i,... ,i3/&,). Under the assumptions that p(1), p(t), J~(~,x,zL,<) and fie(t, 2 ) are continuous (not necessarily differentiable) in the time variable t and analytic in (x, u, <) (and y(t) > 0 for f $I II), we establish existence and uniqueness theorems for local solutions analytic in the space variable 2‘: differentiable in t # 0, and continuous up to the singular point t = 0 along Stolz’s paths (see Section 3) without initial condition. Since a complex variable is regarded as a two-dimensional vector! problems with complex time variables are not excluded. From our viewpoint we can characterize the case where existence and uniqueness results simultaneouly hold like as classical Fuchsian equations (e.g. [l, 21). Modifications of the Nirenberg-Nishida method [3, 4) were used in [5! 61. Here we need to modify it further. When 1)~ = 1, our problem is a first order version of the one treated in [Sj, but the condition (,44) in Section 3 is an improvement of (A-l) in [6]. For rrr. > l! we give a new fclrm of the complete integrability condition for the overdetermined system, which is stronger than the one used in [5] in a nonsingular case. We need to use it here because of the singularity.

2. THE

COhIPLETE

IXTEGRABILITY

CONDITION

The problem Let Q and R be open subsets of Iw” and @“, respectively. @I” valued functions v, = u(t,z) in t E &, z E a,,, with (~~.(t,1c),1-l(~)l),~(t,2)) (ih/&:,, . . * , a?l~/ax,,)). where so and &I are positive numbers, and CAY= {x E C”; /PI -~ y( < s for some

(1.1) concerns E f/,,, (Du =

y E 0},

lJH = ((u,E) E C:” x CnN; ITA/< R, \
Section

we assume

that

p(t),

smooth

in (2, ?I.,<). For z = (~1,. f. ,zl),

$5

(.fdt ,KC,%E) + fio(t,3.)) ‘~1 = (~1,.

3855

. . ,TQ) we denote

are continuous

in f and

by 8, Fur the quantity

Second World Congress of Nonlinear Analysts

3856

Furthermore,

we write

@(r; h, k) = i1 [c (t + et1+ k; h) - ~(t + eh; h) + 6’ ~(1 + eh + dk; h, k)d4] de, and! for y E (‘I( [0, l]‘, Q), ~(7)

= 1’ [Gene,

1); iw,

1)) - (=hvwi

+ L1 we, We use the condition (II) rem 1.1 and 1.2 in [5]).

THEOREM (I) @(t; (II) where

following

h, k) = @(t;k, h)

conditions if t + OIz+ &

It is obvious mollifiers.

that

Let x E C~(EP)

,$ we put

(I) implies

q&(t)

=

that for every

‘Pc(t;[h, qk) = /&+l’(t for small

(II) implies

(cf. theo-

= y($,B).

(I). I n order

to show that

and /x(t)dt

= /+--/~(t)dt~-..dt,

in Q

- m; Jh, rjk)da =

(I) implies

(II),

we employ

= 1. For E > 0

T)/E)+(r)dT = /x(cJ)$(t - Ea)da.

-

C”‘J,y((t

(t, h, k)

theorem

E [0, 112.

x

lP

x

R”

The

the equality

Jx(a)Q(t - &a;T/k,Jh)du = Qe(t;qk,[h)

6 2 0, q 2 0 and E > 0. Thus

13&\IIE(t;
4b+oe.

are equivalent.

E Q for (e,4)

by “7(0,4)

l]“, Q) is defined

446

0))

to prove the existence

(I) and (II)

‘7 E C’([O,

a function

holds

theorem

112,Q),

Friedrichs

condition

in the following

C@(T) = cP(~Y) for y E C’([O,

Proof.

and

2.1. The

4); i3ed44),

ad4

at < = 11 = 0,

is

&tc,(f~;h)k + I-l,(t; h, k) = &G,(t; k, h) + &(t; k, h).

Second World

Therefore,

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3857

for 7 E C2( [0, l]“, Q), we have

Integrating both sides of this equality by 0, 49 and letting E tend to zero! we get a(7) G(t7) for 7 E C2([0, 112,Q). For arbitrary 7 E C’([0,112,Q), we can find a sequence in C”([O, l]“, Q) which converges to 7 in C* as j + co. We have G(7) = /iir@(yj) lim @(t7j) j-cc

= a(“~).

This

completes

= 7j =

the proof.

3. R.ESULTS

In this Section we state the existence and uniqueness theorems for the system (1.1). Hereafter Q denotes a fixed open subset of &P, which is a cone with vertex at O! that is the in Q for t E Q. We write V E S(Q) if V is an open line segment {Bt; 0 < 0 < 1) is contained cone with vertex at 0, contained in Q, satisfying the following condition: If W is a bounded closed cone with vertex at 0 such that W C- Q U {0), then rW = (rt; t E W} is contained in VU (0) for some r > 0. Let X be a topological space. When V E S(Q), we write $ E SC(V, X) if $ is a map : V U (0) --t X such that the map : [0, l] x V 3 (0, t) H +(0t) E X is continuous. Obviously. G(V U {0),X) C SC(V,X) C C(V,X). A. map $ : V U (0) --t X belongs to SC(V, X) if and only if $ E C( V, X) and + is continuous at 0 along Stolz’s paths, that is $ E C’(W, X) for each closed cone W c V U (0) with its vertex at 0. We write Se(V) for SC(V, C). If V E S(Q), T/JE SC(V) and l+(O)] < c, then {t E V; I$(&)\ < c for 0 E [O,l]} E S(Q). Let us denote by A(W) (W = tit, or W = n,, x r/,,) the set of all cAV-valued functions bounded and analytic in W. We write 1121//,= sup{]?r(z)j;

J?9 = A(%), Then

Z3, forms a scale of Banach f&r 5) & ,

spaces? that

(Al) 9 and p belong to SC(Q), Uh)) for some so, & > 0. (-42)

~t;f,(O,z,O,O) i=l

is

II . Iis!F II . /Is

We write R.?(R) = {,u E B& j]ulls < R}. For each t we can regard the Ji(t,r,u,<) are as follows.

for

O
in (1.1) as a function

‘p > 0 on Q and ~(0)

= 0 for t = (tl;.

Z E a,}.

of (z,u.,<).

= 0. fi belongs

Our assumptions

to sC(Q,A(n,,

x

, 1,) E Qm and x E !%,,; where

Qm = {t E Q; v(O,t) = a~},

4Q, t) = I’

&c.

(3.1)

3858

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(A3)

The

ues crj(t,z)

inequalities (1 5 j

Congress

inf{Recrj(t,z)/ll/; 5 N)

of the

of Nonlinear

Analysts

t E Q, z E fl,,}

N :< N matrix

A(t,rc)

> 0 hold

for the eigenval-

= -~ti&ji(O,r,O,O) i- 1

(&ji

=

Hfil~~j)l,,,N). (A4)

l’#

da converges

locally

in t E Q as 0 \

uniformly

0.

(A5) Qoo is relatively closed in Q and v(B, t) converges locally uniformly in t E Q \ Qoo as 0. (A6) Let c > 0, t E Q and h E IF’” \ (0). If f 1 1s not parallel to t, there exists a p > 0 such that

0 \

‘+-“‘q$9(@) uniformly

+ qJ([(f + Oh))]Q3(<(t

---j 0

as E \

0

in 0 E (0, p]

We prove the following

THEOREM

two theorems

3.1. (Existence)

there

exist

positive

u E SC(Q(s),

B,(R))

in. Section

Assume

QM C 1 f? Q Then

+ oh))-”

5.

(Al)-(A6),

and

for some half line I with

end point

at 0.

(3.2)

numbers R, RI and a set Q(s) E S(Q) (0 < s < so) such that, if e complete integrability condition (I) holds, then there exists a ho E =(Qt 4, (RI 1) and th solution 1~ of (1.1) in t E Q(s). z E 0,: isatisfying n C’(Q(s),

THEOREM 3.2. (Uniqueness) Q,

B,),

Assume

> 1n Q

pLh

(Al)-(A4),

E SC(Q(s),

E S(Q),

for

0 < s < se.(3.3)

and

for some half line 1 with

Then, for some H > 0 and some Q(s) satisfying (3.3) is unique.

I~,(R)~)

th e solution

end point

at 0.

(3.4)

u of (1.1) in t E Q(s)

and x E R,

4. PRELIMINARIES Here we prove several lemmas (cf. Section 3, [S]). Hereafter hold and write Qe = 22’Q = {t/2; t E Q}. Then Qe E S(Q).

LEMMA 1. There (0, t) E (‘All

exists a function

x

&a>

w(t)

> Ip(

and

w E SC(Qe)

&w(@t)

such that

2 ltll~(Qt)l/v(~t)

we assume

that

(Al)-(A4)

w(O) = 0; &i~l(&)

is continuous

for

x Qo.

(0, t) E (O,l]

in

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Proof, We put

Then

w1 E SC(Qo),

B J

2t)-l em p1 (at)&

q(O)

is smooth

&ul(et) is cyntinuous

in (0,f)

I,

2 1:2 pt(o8t)d~:

= 0 and w](t)

I,

> 2 ,‘, Jp(t)lda

=

Ill(t)\.

Further:

wl(&)

=

in # E [0, l], so is wl(&)

=

in 8 E (0, l] and

= Q-1[2,u1(Qt) - p1(&/2)] - 282&,(rrt)drr 1’

E (0, l] x QO.

CL](&)

1s . nondecreasing

hence &UI (8t) 2 0. Therefore

w(t)= cdl(f)+ ItIJy y#k? has the required

properties.

LEMMA 2. The function v defined only if the condition (A5) holds.

in (3.1) is a continuous

map from

[0, l] x Q to [0, co] if and

Proof It is easy to see that the continuity of v yields (A5). Suppose, conversely, that (A5) holds. It is clear that I/ is continuous in ((0, l] x Q) U ((0) x (Q \ QOO)). Thus it suffices to verify that 11 is continuous at every point (0, to) in {0} x Qoo. For every A4 > 0 we can choose a d f (0,l) such that ~(n, to) > M. By the continuity of l/cp on Q, we can find a neighborhood W of to such that ~(6, t) > M for t E IV. Thus ~(e,t) 2 ~(6,t) > A4 for (8, t) E (O,f x W. Th ere f ore ~(6, t) tends to 00 = ~(0, to) as (0, t) + (0, to). We write .4(t)(z) (A3) implies that /lIC(6,t)vll,

w h ere A(t, z) is the matrix function as in (A3). = A(t,z), there exist positive constants C; and b such that 5 C~e(0,t)l/ull,9

for (0,t)

E [O,l]

The

condition

x Q and u E B, (0 c s 5 SO),

(4.1)

where E(6, t) = exp[-v(o,t)A(t)],

~(0, t) -= exp[-br/(8,

t)i!i]

for (0,t)

For each t E Q, ts(B, t) and ~$0, t) are smooth in H E (0, l] and I;:(O,t) = 0 and e(O,t) = 0 for t t Qco. If we assume (A5), then, e E C( [0, l] x Q, [0, 11) and E E C( [0, l] x IQ, Ug).

E [0,

continuous by lemma

l] x Q. at B = 0. 2, we have

LEMMA 3. Let Q1 E S(Q), s E (0, so], ‘u E SC(Q1, H,) and the map t H (h H I’(t)h) belong to SC(Q,, f,(lP, H,)), w h ere L(IkP, HB) dlenotes the set of all linear maps from IP to B,. Set A(0,t) = r(&)t + A(t)v(t?t). L e t u b e .a map from &I U (0) to H, such that U(O) = v(0). If

Second World Congress of Nonlinear Analysts

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p(ot)aJu.(et)

+ A(t)u(c?f)

= ll(0,t)

for (0,t)

E (0, 11 x QI,

(4.3)

t

(4.4)

then u(1) =

1 -f<(O, ” PC@)

I

’

r(O)t Conversely, Proof.

(4.4) implies Suppose

that

+l(O, = 0

that ,u(&)

l)fM + qo,

for t E Qoo

is smooth

(4.2) and (4.3) hold. 1 1 -----E(f), E p(e)

J

t)A(ft,

we obtain

(4.4).

Thus

for

E Q,,

(if Qoo # 0).

(4.5)

in B E (0, l] for f E Q1 and satisfies

(4.3).

For 0 < E < l! we have by (4.3) t)cl0 = j1 3@[1S(O, /)u,(et)]da = u;t,

Let E + 0. Then

l)u,(O)

- E(E, f)‘U(&t).

we have

u.(t) - u(O) = 71.j(I) + l@(l),

(J.6)

where f?(:(8.,/)l?(Of)lf10,

Ul(/)

= 1’

&j

dt)

= i1

-E(H, &)

I)A(t)w(Ht)dH + fqo, t)u(o) - u(0)

1

1 --E(B, =/ 0 P(W since ,u.(O) = ~(0).

t)A(t)[w(Bt)

- w(O)]dO

(4.1) implies

= c,(hjtj)-‘[l

- e(0,t)] .

sup

UjCf
I Cc, . sup

!!!h!t!)-‘A!t)[v(tlt)

IlA(f)

- v(O)]Ils

- @)](I,>

CJ
hence v2 is continuous

at 0 along Stolz’s ~~~ is continuous

Since

paths,

along

and u,z(O) = 0. Therefore

Stolz’s

and UI(O) = 0.

E(0, f) = 0 for t E Qm: we have

761 (t) - /l(t)-‘ryO)f = 6’ $$q”’ thus,

paths,

for c E (O,l);

I)[r(Ht) - r(o)]w,

(4.6) and (4.2) yield (4.~)

Second World Congress of Nonlinear Analysts

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Hence am - A(t)-‘l?(O)t converges to 0 as r~ \ 0 for each t E Qoo ((it belongs for small g > 0). Thus we have (4.5) by (4.7). C onversely, assume (4.4). Then (T E (0, 11 and t E Q,

to Q1 II Qm we have for

u(d)=r0LE(H/ aqqo, t)&l+E(0, at)u(0) dot) CT, =zqcr, t)-1[6”sir E(O,1)A(O, l)dO +E(0, t)u(0) I. Differentiating

the both

sides of this equality

by c: we have (4.3).

We write

LEMMA

4. Assume

that

Qo3 # 0 (that

is (3.4)).

(1) If (3.2) holds (that is QOO = 1 n Q), then for every R’ E (0, &J) there exists an R1 > 0 such that, for every fin E S(r(Q, &,(RI)), th ere exists a unique ‘u) t H,Y,,(H’) satisfying P’(O,w

t) = 0

for

f E Qco.

(2) For solutions mined

u of (1.1) satisfying (3.3). the initial and satisfies /\uII/,, 5 R and (4.8).

value

(4.8) u t = u/=~[) is uniquely

deter-

Proof (1) By the assumptions, Q, = {rt”; T > 0) for every fixed to t d),. The condition matrix. (1.8) is equivalent to ft’(O, 11:;to) =: 0. Since A(/“) = -c%,E’(O,o; to)! u 0 is a nonsingular the well-known implicit function theorem yields the conclusion. (2) We have ll~l)l,~ < R (0 < s < SO) by (3.3). Thus jl,~(~)i,,, 5 R. Since ‘(1 is a solution of (1.1). we have ~(ot)&~(Ot) + A(t),~(trl) = K(Ot, ~(Otj; t) for (0, t) E (0, 11 x Q(3), where Kjt,

U; hj =- E’(I, ‘~1;If,) + A(h)u.

putting 1: = II and l‘(f)/,, = k’(t, u,(,f); lb,) in lemma 71’ is unique since the solution of (4.8) is unique.

(4.9)

3. we see from

(,4.5) that

71’ satisfies

(1.8).

5 PROOFS Kow

we prove theorem

3.1.

!!1171,!!,~

wit,h some con&ant

(::.

We have by the Cauchy’s

5 C~!~l!,~/(s

- s’)

for

inequality

0 < s’ < s? 1~ E A,

(5.1)

We have by (5.1)

for f E Q U (0): IL E iRrmT II, ‘11 E Zl,(R). p(l)l)v, /~(1)1h 0 < I? < I&, where &(-, H) belongs to SC:(Q) (j = 1,2).

t I&(R), It follows

0 < S’ < s 5 SO and from the definitions of

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K and A that di(0, R) + 0 as R \ 0. di and d2 are independent of the choice of fa. If Qoo # 0, then we take the solution ?u E: B40(~&) of (4.8). Otherwise, we take w E B,(rR) arbitrarily. We put Kl(L)

=

y$,

and

ll~~~~~~~~;~~ll~,ll~l

--

n(t)

= Cub-‘[1

- e(O,t)][Ki(t)

+LJ(~)],

(5.3)

where Cs and b are the numbers as in (4.1). It follows from (A5) that 1 - e(0, ~2) - 0 as p \ 0 locally uniformly in t E Q \ Qm. tc,(pt) --) 0 as p \ 0 locally uniformly in t E Q,. Thus K E SC(Qs) and K,(O) = 0. We put

Ql

= {t E

Qo; 4(&N

da(t)f, R) < Cz, PC(&) < 7-R for

< &,

where C, = &(O, R) + 1 and T > 0 is a small number determined later. for small R > 0. We define a sequence up, p = 0, 1, . . * , recursively by

B E (0, l]

Q1 belongs

, (5.4) > to SC(Q)

uO(/) = w,

flt;u”(Ht); r)dB + I:(O, t)w

(t # 0)

and

“V+‘(O)

= 711

and put VP = ZI? ’ - YP. Since v”(l)

=

u’(f)

-

w

==

1 I0

1 --qe, P(W

t)l;(tQ,

w;

f)dtl,

we have by (4.1)

--e(t), ’ I Xl(f) L1 :;:;;i

f.)dO = Ki(l)C&‘[l

- e(0, /.)I < K(f),

s)

0 < s < so

thus the estimation

llvp(t)11.9< holds

~(t)~~A4~(t;

for

I E Qp(s),

(5.5)

for p = 0, where

UT,= a0 fi (1 + j-y, j-1

Qpb)

=

{f

E

Q,;

w(f)

<

up(.so

-

s))

E

S(Q),

.

In the same way of using w(f) as in [6] we can verify by induction on p that VP is well-defined, belonging to SC:(Q,(s), 1j,g(R)): and (5.5) holds for p > 0 when cl0 > 0 is small enough. 71. =

lim 72’ = II! + 3Eod

Qr; $y<

e(ss - s)}

u(t) =

converges

(U = jiz

in sCr(Q(s),

aP > 0). This

BJ(H)),

where

Q(s)

= 2 Q,(.s)

=

II. satisfies tt # 0)

and

u.(O) = ~1.

{t

E

Second World

By lemma

Congress

of Nonlinear

Analysts

3863

3 we have p(Ot)&u(Qt)

= P’(Ot,u(Bt):

t)

for t) E (0, 11, t E Q(s).

(5.6)

We show that 11.satisfies (1.1) by the same way as in [5]. In the complete integrability condition (II), putting ?(a,$) = it” + [c$(t + 01~) for /L E R”, p E (0, O], 6 E (0, I], where t” E Q(s) and 6 is small enough, we have

thus,

ZZ o1 aY(l, (6); i’ct + 11)) - m40,4); J[ putting r = 1 + h and changing variables,

6’ G(pP

+
Et) + 1’

H(y(~,

41;
we have

+ &-; T) - q/It0

+ f#d; t)]fi+

J J

’ dcj 09[fZ(pfo + q% + Oh; 1, h) - N(pt” + c#d + Bh; h, t)]dtl. 0 Here the left side of this equality converges as p \ 0, and the derivative of the right side with respect to [ converges locally uniformly in c E (0, l] as p \ 0. Thus we have +

‘c 4J0

G(@ + Oh; h)&3 = G(E‘T; T) - G(@; t) (5.7) +

J

$Z(
+ ML; t, h) - W(
andput Wewrite&).9(4cu, p(t)Dw)$= (&dAt,Z:,II,04 + p(f.)Q(t,1,l?lS)W) /ceep(l),k A(t, Substituting

IL) = U(T) - u(t) (v, Dw, D”v)

G(t; h)(.,u,j,X) hence it follows

from

- 1’

L(t + ‘Oh, u(T); h)dO,

for (u, <, A) in (2.1): = L(f,q

(5.7) that

h),

L(t,u;

h) = -b(t,u.; v(t)

h).

we have

fZ(t; h, k)(., v, <, A) = [f&,,f,(t,

II; h)]L(t,v;

k),

(5.3)

3864

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Further

Congress

of Nonlinear

Analysts

we put

a, t,h)= &K,6h)+ wr t,h). Then

we have by (5.9)

Therefore

we have (cf. lemma W,h)

3)

fd (El I,)I R tt , t, h)a(@,{h)

= /nl &

+ S(f, t, h)]d[ + E:(q, t)fl(qt, qh),

h) = P(<, t, h) + A(t). K ow let h be not parallel to t. Tha assumption (A6) converges to 0 as v \ 0. Thus we have by (5.10) (5.8) imply that IIWV, W(@, @)lIs

where

B([,t,

A(& h) = i1 We can show by (A6)

that

&

there

Wl, exists

llS(C, 4 h)lL

t)[W,

4 hP(@,

a constant

I Cd<,

I

llu(Et)ll.s

+ ll~(E~)ll,~

and

(5.11)

4 h)Pt.

C$ such that for every E > 0 the estimate

t)-1’21tl+lM(t,~,

holds for E E (O,l], t, T E Q(S) if 11L1 is small max{w(t),W(T)). Since (-46) and (5.8) imply e(E, t)““ll~(Et,
Eh) + W,

(5.10)

s)-”

enough!

where

+ t:mrst.e(C,

t)l’”

A~(~,T,s)

sup Ey(f(t OSff<

= a(sO - s) -

+ Oh))-’

1

< 2R + ccmst., repeating

use of (5.11)

yields

ll~(t,h)ll.~

I G(@

+ ~lhl~f-&Wf,;-,

p= 1,2;-.

s)-‘,

for some constant (;I%. Letting p tend to infinity, we see that if b is not parallel p-l &)lls converges to 0 as p \ 0, which means that + plro) is differentiable respect to p at p = 0 and

llA(t,

u(t

y(W,u(t In

the

(5.12)

ca,se

holds,

+ ~h~)j~=~

t,

-= WC u(t); ho)

for

t E

Q(s).

where ho is parallel to (5.6) implies that ~(t + /&J) is differentiable too. Thus 21.is a required so:lution of (1.1). This completes the proof.

to with

t:

(5.12) by /, and

If 71.is a solution of (l.l), then 11.satisfies (5.6). Here we give a short proof of theorem 3.2. It follows from lemma 4 that the initial value ‘~1 = U(O) is uniquely determined. Therefore the solution of (1.1) is unique since the solution u(Ot) of (5.6) with the initial condition

Second World

u(@)lOTO = ‘111is

unique for every

Congress

t E Q(s).

of Nonlinear

Analysts

3865

This completes the proof.

REFERENCES 1. BAOUENDI M. S. & GOULAOUIC C.: Cauchy problems with characteristic initial hypersurface, Comm. Pure Appl. Math. 26, 455.475 (1973). 2. Singular nonlinear Cauchy problems, J. Differential Equations 22, 268291 (1976). 3. G’BERG L., An abstract form of the nonlinear Cauchy-Kowalewski theorem, J. Diffemnti~l Germ. 6, 561-576 (1972). 4. XISHIDA T., A note on a theorem of n’irenberg. J. Differential Geom. 12. 629-633 (1977). 5. KOIKE M., An abstract nonlinear Cauchy problem with a vector valued time variable, finkcial. Ekvac. 32, l-22 (1989). 6. -i VoleviE systems of singular nonlinear partial differential equations, Nonlinear Anal. 24, 997.1009 (1995).