On the existence of solutions of quasilinear wave equations with viscosity

J. Korean Math. Soc. 37 (2000), No. 3, pp. 339-358

ON THE EXlSTENCE OF SOLUTIONS OF

QUASILINEAR WAVE EQUATIONS WITH VISCOSITY

JONG YEOUL PARK AND JEONG JA BAE

ABSTRACT. Letnbe a bounded domain inJRNwith smooth bound- aryan. In this paper, we consIder the existence of solutions of the following problem:

(1.1)

Utt(t, x) - div{(7(1Vu(t,x)1²)Vu(t, x)} - Au(t, x) - AUt(t, x)

+<>lut(t,x)IP-1Ut(t,x)=J.'lu(t,x)IQ-1 u(t,x), xEn, tE[O,Tj,

u(t,x)lan=0,

u(O, x) =uo(x), Ut(O,x)=U1(X), x En,

where q > 1, p ;::: 1, <> > 0, J.' E JR, A the Laplacian in JRN and

( 2) . ·t· funct· lik 1

(7= (7 V 15a POSllve Ion e as (1+,,2)1/2.

1. Introduction

Many authors have been studied about the existence and uniqueness of solutions of (1.1) by using various methods. Equation (1.1) with

<5 = JL = 0 was introduced by Greenberg and Maccamy ([5]) as a model of a modified quasilinear wave equation which has global smooth solutions for large data. Since then, many authors have investigated the global existence as well as the asymptotic behavior of solutions to this and related equations (see Alikakos and Rostamin [2], Engler [4]' Kawashima and Shibata [7], Mizohata and Ukai [9] and Nakao [10] and the references cited in these papers). When u = JL = 0 and

<5 = 1, Nakao ([12]) has considered the existence and decay properties

Received July 24, 1998. Revised January 3, 2000.

2000 Mathematics Subject Classification: 35L70, 35L15, 74S05.

Key words and phrases: quasilinear wave equation, energy identity, Galerkin method, exist~nceand uniqueness.

of solution of (1.1). On the other hand, Matsuyama and Ikehata ([8]) have considered the following quasilinear wave equation

(1.2) Utt -

M(llvu

dX)

mo > 0 with a constant moo ^{In fact,} Equ~tion (1.2) has its origin in the mathematical description of small amplitude vibrations of an elastic string whose ends are. held a fixed distance apart, hinged or clamped and is either elastic or compressed by an axial force. Infact, a mathematical model for (1.2) is an initial boundary value problem for the nonlinear hyperbolic equation

(1.3)

tPu {

(ou)

ph&t2 = _^Po+ ^2L 10 ^{ox _ dx} O~2 +f for 0<x <L, t ~ 0,

where u is the lateral deflection, x the space coordinate, t the time, E the Young modulus, p the mass density, h the cross section area, L the length, Po the initial axial- tension and f the external force. See Narasimha ([13]).

Kirchhofffirstintroduced (1.3) in the study of oscillations of elastic strings and plates, so that (1.3) is called the wave equation of Kirchhoff type after his name.

Inthe present paper, we will study the existence and uniqueness of solutions of (1.1) with 8>0 andJL=F 0 by using Galerkin method. Our paper is organized as follows: Insection 2, we give lemmas and state the main result. Insection 3, we study the existence and uniqueness ofweaksolution of (1.1).

2. Preliminaries

In this section we present some lemmas that will be necessary throughout this· paper.

LEMMA 2.1 (Sobolev-Poincare [1]). Ifeither 1 ~ q < +00 (N =

1,2) or 1$ q~ Zi:~ (N;::: 3)issatisfjed~then thereis a constantC*

such that .

lIullq+l ~C*IIVuIl2 for uE HJ(n).

Quasilinear wave equation with viscosity 341

LEMMA 2.2 (Gagliaxdo-Nirenberg [1]). Let 1 ::; r < q ::; +00 ~d

p ::; q. Then the inequality

holds with some C >0 and ()=

(~

~

~

(~

~

~

that 0< () ::; 1 (we assume 0 < () <1ifq= +00).

For further a priori estimates, we need the generalized Gronwall's inequality which is due to Bihaxi and Langenhop.

LEMMA 2.3. ([3]). H k > 0 and c ~ 0 are constants and g(s) is positive, nondecreasing for s _~ 0, then the inequality

(2.1) 4>(t) ::; k+^C

it

implies that 4>(t) ::; G-1(c

I;

I:

k > o. Ifg(s) = s, then the inequality (2.1) isthe usual Gronwall's inequality and Lemma 2.3 reads as follows:

4>(t) ::; k+c

lot

4>(t)::; k exp

(c it

~

Lastly, weindicate the following two propositions which are necessary to obtain convergence results.

PROPOSITION 2.4. (Temam [14]). Let X and Y be two Banach spaces such that X c Y with a continuous injection. H a function 4> belongs to D)O(O,T;X) and is weakly continuous with valuesinY, then 4>is weakly continuous with valuesin X.

PROPOSITION 2.5 (Lions [6]). Let X be a Banach space. If! ^E

V(O,T; X) andf' EV(O,T;X) (1 ~ p ~ 00), then!, possiblyafter redefinition on a set of measure zero, is continuousfrom [0,T] toX.

Nowweconsider the following boundary value problem:

Utt(t,x) - div{u(lV"u(t, x)12)V"u(t, x)} - Dou(t, x) - Dout(t, x)

+<5lut(t, x)IP-1ut(t, x) = J.Llu(t, x)lq-1u(t, x),

(2.2) x En, t E [0,T],

u(t,x)lao = 0,

u(O, x) =Uo(x), Ut (0, x) =UI(X), x En.

We define the potential J(u(t)), energy E(u(t)) and I-positive set W associated with.equation (2.2) by

J(u(t)) = ~1IV"u(t)lI~ ^-

q:

1 1

r

E(u(t)) = 211^{Ut (t)\12}+210 10 U(TJ) dTJdx+J(u(t)) and W = {u(t) E HJ(n)nH2(n) II(u(t)) >O}U{O},

where I(u(t))= \IV"u(t)lI~- J.Lllu(t)lI:t~.

To obtain our theorem, we shall make the following assumptions on u:

u(·) belongs to 0²([0,00)) and satisfies (HI)

°

(H2) there existsr > 0such that u(v2) - 21u'(v2)lv22: k2{U(V2)+

2Iu'(v2)lv2y for some k_b k2 >O.

REMARK. Without loss of generality, we may assume r > 2 in (H2)of the above hypotheSis. When u(v2)= VI~V2' we see u(v2) - 2Iu'(v²)lv² = (1 +v2)-i and u(v2)+2Iu'(v2)lv2 _~ 2(1+v2)-i and hence we can taker =^3.

Our results read as follows.

Quasilinear wave equation with viscosity 343 THEOREM 2.6. Let N be a positive integer. Under assumptions (HI) and (H_{2 ),} suppose that 8 > 0, It> 0 andp <min{q, N+4~-Nq}

issuch that

(i) 1~p <⁰⁰ (N = 1,2),

(ii) 1 ~p ~3, 1<q~ 5 (N= 3), (iii) 1

~

~ N~

N . {N+2 N-2}

N - 2 ~ ^q ~^{mIn N _ 2' [N _ 4]+ (N} ~ ^4).

HUQ E WnH 2(0), U1 E HJ-(O) and

ItC;+l(2~:11))~² E(UQ)~ ^{< 1,}

then the problem (2.2) has a solution U = u(t,x) satisfying U E

LOO (O,T; HJ- (0) n H2(0)), u' E Loo (O,T;HJ(O)) and u" E Loo (0, T; L²(0)).

3. Proof of Theorem 2.6

We now consider the problem (2.2). Throughout this section, we always assume that UQ E W nH2(0) and U1 E HJ(O). We employ the Galerkin method to construct a solution. We present by (Wj)jEN be a basis in HJ(O) n H2(0) which is orthonormal in L2(0) and by V_m the subspace ofHJ(O) n H²(0) generated by the m-first vectors WI,'" ,Wm ^{and define} um(t) = 2:;:1gjm(t)Wj, where um(t) is ·the solution of the following problem:

(U~(t)- div{u(IVum(t)1²)Vum(t)} - ~um(t) - ~u~(t),w)

+8Iu~(t)IP-1(U~(t),w) = Itlum(t)/q-1(um(t),^{w),wE V}m,

(3.1) um(O) =UQm =I)UQ,Wj)Wj ^{- t}UQ in HJ(O) nH²(0),

j=I

U~(O)⁼ U1m = 2)Ul'Wj)Wj ^{- t} U1 in HJ(O).

j=l

Note that we can solve the system (3.1) by a Picard's iteration method.

Hence the system (3.1)hasa local solution in [0,Tm )with 0<Tm ~ T.

The extension of the solution to the whole interval [0,T] is a conse- quence of priori estimates we are going to obtain below.

A Priori Estimates I

Multiplying the equation (3.1) by u~{t),then we have (3.2)

~ (~IIU~{t)lI~

~IIVUm(t)II~

~ l

~ ll1um{t)lI:t~)

+IIVu~{t)lI~+8I1u~{t)lI~t~ ^{= 0,}

wherer{t) = fJVum^{(t)12u{ry) dry.}

Integrating (3.2) from 0 tot, then we have the energy identity

where E{um{t)) = !lIu~{t)lI~ + !IIVum{t)lI~ + ! fn r{t)dx - ~

lIum(t)lI:t~·

Next to bbtain a priori bound, we need the following results.

LEMMA 3.1. Assume that either 1 ~ q < +00 (N = 1,2) or 1 ~ q ~ ~!~ (N 2:: 3) is satisfied. Let um{t) be the solution of{3.1) with

UOm E WnH²(O) andUl_m E HJ{O). If .

then um{t) EWon[0,T], that is, IIVumll~-JLIIUmll~t~ >

°

Proof Since I{uom} > 0, it follows from the continuity of um{t) that

(3.5) I{um{t)) 2::

°

Quasilinear wave equation with viscosity 345

Lettmax be a maximal time (possiblytmax ^{= Tm )} when (3.5) holds on

[0,t_max). Note that

(3.6)

1 q-1 2

J(um(t)) = q+1I(um(t)) +2(q+1)IIVum(t)lb

q-1 2

2:: 2(q+1)IIVum(t)lb on [0,tmax).

By the energy identity (3.3) and (3.6), we have

(3.7)

IIVUm(t)lI~ ~ 2~q~^11)J(um(t))

~ ^{2(q+11)E(um(t))}

q-

< ^2(q+l)E( )

- 1 UOm on

q-

It follows from the Sobolev-Poincare's inequality, (3.4) and (3.7) that /LIIUm(t)lI~ti ~ /LC2+11IVum(t)II~+l

.t.!.

~ ^{/LC2+ 1}(2~q~^{11)E(Uom)) 2} IIVUm(t)lI~

< IIVum(t)ll~ on [0,^tmax ).

Therefore, we get I(um(t)) >

°

Note that Lemma 3.1 implies that

(3.8)

Thus, (3.3) and (3.8) imply

(3.9)

1 1 2 q-1 2 1 (

"2l1um (t)lb+2(q+1)IIVum(t)112+"210.r(t)dx

+

l

8I1u~(s)lI~t~

l

IIVu:n(s)lI~

~ E(uom).

A Priori Estimates 11

Multiplying the equation (3.1) by -~um(t),then we have (3.10)

d Id

dt (u:n(t),~um(t)) -IIVu~(t)II~+ II~Um(t)lI~+"2^dtlI~um(t)lI~

+

L

- 8

L

= -1£

L

Since P 2': 1, we can use the fact that L²P(0) ^{' - t} L²(0) and so (3.9) implies

(3.11)

18

L lu~(t)IP-lU:n(t)~Um(t)

~ 8I1u~(t)II~plI~um(t)1I2

~ 8C~lIu~(t)II~II~um(t)1I2

~ 8C~(2E(uom))~ lI~um(t)1I2

1 2

~Cl +211~um(t)1I2'

where Cl = ~82C~P(2E(Uom))P.

Quasilineax wave equation with viscosity 347

(3.12)

On the other hand, Schwarz's inequality and (3.9) imply

IlIUm(t)lq-1um(t)~Um(t)

~ qllum(t)lIg+~IIVum(t)II~+l :s; qC2+lIlVum(t)II~-111~um(t)ll~

!l.::l

:s; qC2+l(2~q~11) ^{E(Uom )) 2} lI~um(t)II~.

From (3.10)-(3.12), we get (3.13)

~(u~(t), ~um(t)) ⁺~11~Um(t)lI~⁺~^:tII~um(t)lI~

+

l

!l.::l

:s; Cl +IIVu~(t)lI~^+qj.LC2+l(2~q~^{11)E(uom)) 2} II~Um(t)II~.

On the other hand, integration by parts gives (3.14)

l div{u(IVUmI2)VUm}~Umdx

=

~ ~

L- L-In ^ox-ax- OXk OX-OXk ox-

i,j=l k=l t J t t

02Um dx

aXiaXj

+(N - 1) ( (u(IVum

_lan

a:

~ ^{l (u(IVumI}2) - 2Iu'(IVumI²)1 IVumI2)ID2Um(t)12 dx

+(N -1) ( (u(IVum

_lan

a:

whereH(x) denotes the mean curvature ofan^{at x}with respect to the outward normal and ID2Um \2 ⁼ 2:';J-1_{(0, ux?,uX:J}!2v;,m1^2• By assumption (H1) of Hypothesis and a standard trace theorem, we see

(3.15)

I(N

fan

H(x)dBI

~

~ C3I1Vumll;(l-82)II.6.umll~82

_ 1 _

~ (1-(2)C;-62I1Vumll~+0211.6.umll~

~ ^{C;!82 2(q}+_1)(~- (²⁾E(uom)+0211.6.umll~

q-

~ C4 +0211.6.umll~,

h C _C1!622(q+1)(1-02)E( )

were 4 - 3 UOm ·

q-1 Thus (3.13)-(3.15) give

(3.16)

~ (u~(t),^{.6.um(t» +}~1I.6.um(t)lI~⁺~ ~ lI.6.um(t)lI~

+

l

!L=!:

~ ^Cl+IIVu~(t)"~⁺qj.LC2+l (2(q_{q - l}+1)E(Uom

»)

+^{I(N -1) (} (u(lVumI2)+2u'(IVUmI2)IVUmI2)188Um 1²H(x)dBI

Jao n

~ Cl+^C4+IIVu~(t)lI~+C611.6.um(t)II~,

!l.=.!.

where C₆ = O2+qj.LC2+l (2(q+1)E(uom») ^{2 •} q-1

Therefore, (3.17)

:t

(u~(t),

~ ~ lI.6.um(t)lI~

+

l

Quasilinear wave equation with viscosity 349

::; C7+IIVu~(t)lI~+C8I1Llum(t)II~, whereC7 =Cl+C4 andC8=IC6 - ~I.

Integrating (3.17), from (3.9) we get

Lt L

1 2

+21lLlum(t)lb

::; lIu~(t)1I2I1Llum(t)1I2 ⁺IIVuIil211Vuoll2+~IILluoll~

+

Lt

IIVu~(s)lI~

C8I1Llum(s)II~)

::; C9I1u~(t)lI~ ⁺EIILlum(t)1I2+IIVuIi/2II Vuo112+~IILluoll~^+E(uom) +

it

C8I1LlUm(s)II~)

::; EIILlum(t)lb+ClO+Cn

Lt IILlum(s)lI~

where 0 ::;E ::; ~. Thus, we have (3.18)

it L

1 2

r

+(2 - €)IILlum(t)1I2 ::;ClO+^Cnlo IILlum(s)lb ds.

Next, multiplying the equation (3.1) by -Llu~(t),then we have

(3.19)

~ ~ (IIvu~(t)ll~

IILlum(t)II~)

+po

L

+IILlu~(s)lI~⁺

L

=^-JL

L

Integrating (3.19) from 0 tot, then we have

(3.20)

l I t

2I1Vu~{t)lI~+2I1Aum(t)II~+10 IIAu~{s)lI~ds

+

it L diV{CT{IVUm{s)12)VUm{s)}AU~{s)

1 2 1 2

~ 211Vu1lb+211 Auo lb

+

JL'Lt L IUm{s)lq-lUm{s)AU~{s)

where we have used the fact thatp8

I;

O.

Note that

JL it ^L

(3.21)

~

Lt L IUm{s)lq-lIVUm{s)IIVu~{s)1

~

Lt IIlum{s)lq-lVum{s)1I2I1Vu~{s)1I2

Sobolev-Poincare's inequality implies

(3.22) IIlum{s)lq-1

Vum{s)1I2 ~C12 I1um{s)IIf;!l)NIIVum(s)1IJ!!2

~ ^C13I1Um(s)lIf;!1)NIIAum{s)112' Now, in the case ~2 ~ ^q ~ min{_Z!~,[%-4]+} (N ;::: 4), we observe from Gagliardo-Nirenberg's inequality that

(3.23)

lIum(s)!If;!l)N

~C14l1um{s)!I(~;1)(1-93) /IAum{s)^lI~q-l)93

N-2

~ C15I1Vum{s)/I~q-l)(1-93)/IAum(s)^lI~q-l)93

Quasilinear wave equation with viscosity 351

(3.25)

1 (q-1)(1- 8 a)

~ (1- (q -1)8₃)C₁¹₅^{(q 1)83}IIVu_m(s)112^{1 (q 1)83}

+(q-1)8311~um(s)1I2

(q-1)(1-8j!)

~ ^{(1 _ (q _} 1)83)C;5 (/1)83 (2~q~¹¹⁾_E(Uom))^{2(l (q 1) 3)} +(q-1)8311~Um(s)lb

N-2 1

with 83 = - - - --1« 1).

2 q-

Thus, (3.22) and (3.23) imply (3.24)

11lum(s)IQ-1Vum(s)112

~C13(C16+(q-1)8311~um(s)1I2)II~um(s)1I2

~ C17(II~um(s)1I2 +II~Um(s)II~),

(q-1)(1-8a)

1

)2(1

where C16 = (1- (q -1)83)C;5(q 1}83 q-1 E(uam) . Therefore, from (3.21) and (3.24), we have

J.L

l

llum(s)IQ-1um(S)~U~(S)

~

(11~um(s)1I2

lI~um(s)II~)IIVu~(s)1I2

we get

(3.27)

/I

div{o-(IVum(s)12)Vu.n(s)}~U~(S)

::; ^~ I

L

)+2o-'(IVUm(SW)IVum(s)/2)² ID2um(s)12dxds

+

~ I

ll~u~(sWdXdS.

Now, assuinption (H2 ) implies that (3.28)

%

I

L

=

~ I

L

)+2o-'(IVum(s)12

)IVum(s)12}2

2 ⁴ 2 .^2(r-2)

ID um(s)P=ID um(s)1 ^r dxds

::; ^~ (I

L

)IVum(sWY

2 r-2

ID2Um(S) 12dxdS) r

(I

L

::; C;8

(I

2 r-2

ID2um(S)12dXdS) r

(I

^.ln

2 ^r-2

::; C;8 ( C^lO+Cn

I

"~um(s)ll~dS)

(s~p lI~um(t)II~)

::; C19+C20

it "~Um(S)"~

€1I~um(t)II~,

where we have used (3.17) and the Young's inequality at the last stage.

Quasilinear wave equation with viscosity 353

(3.30)

Thus, from (3.27) and (3.28), we have (3.29)

IL

l div{(T(I\7Um(s)12)\7Um(S)}~U~(S)dxdsl

::; GI⁹+

EII~um(t)ll~

Lt II~Um(s)lI~

~ Lt lI~u~(s)lI~

Hence (3.20), (3.25) and (3.29) imply

1 1 1

r

211\7u~(t)lI~ +(2 - E)II~um(t)ll~ +2lo lI~u~(s)ll~ ^ds

1 2 1 2

::; 211\7U1112+_211~uolb

+qJLGI7it

(lI~um(s)112

II~Um(s)II~)II\7u~(s)112

+GI9+G20

it II~Um(s)lI~

Therefore we have

II\7u~(t)lI~

lI~um(t)ll~

Lt II~u~(s)lI~

(3.31) ::;C2I +G22L

t

((II~um(s)1I2

lI~um(s)II~)II\7u~(s)1I2

+

lI~um(s)II~)

whereG2I ^and G22 are some constants. Therefore, we obtain EI (t) ::; G2I+^G23LtEI (s) +EI (s)! ds, where EI (t) ⁼ II\7u~(t)ll§+lI~um(t)II§·

Now we setg(s) = s+s! ons~ O. Then we have

Note thatg(s) iscontinuous and nondecreasing on s~ O. By applying Bihari-Langenhop's inequality, we get

I

E1(t) ~ G-1(C23t), where G(s) = - ()dr, s>C21 > 0

C₂₁ 9 r and this estimate imply

(3.32) E1(t) = lIV'u~(t)lI~+lI~um(t)lI~ ~ M1(T) for some constant M1 >0independent ofm.

In particular, (3.31) and (3.32) imply

(3.33)

Lt lt~u~(s)lt~ds ~

A Priori Estimates III

Finally, multiplying the equation (3.1) by _u~(t),then we have (3.34)

ltu~(t)lI~^-

i

- (~um(t), u~(t))- (~u~(t), u~(t))+6Iu~(t)IP-l(U~(t), u~(t))

=JLlum(t)IQ-¹(um(t),u~(t)).

Note that from L²p(0) ^<--t L²(0) and (3.9),

6Iu~(t)IP-1(u~(t),u~(t») ~ ⁶

i

(3.35) ~ 8I1u~(t)lI~pllu~(t)1I2

~ 8Cl'lIu~(t)II~lIu~(t)112

~ C27I1u~(t)1I2'

Now, it follows from (3.9) the Gagliardo-Nirenberg's inequality and our assumptions on q that

(3.36)

JLlum(t)IQ-1(um(t),u~(t)) ~ JLllum(t)II~Qltu~(t)1I2

~ C2811V'um(t)II~4I1um(t)II~-o4I1u~(t)1I2

~ C29I1u~(t)lI§ ^{with 0}4 = (q -l)N 2q

Quasilinear wave equation with viscosity

for some constants C₂₈ andC_29• Now, we have

355

(3.37)

Thus, our assumption(HI) implies that (3.38)

Il div{(7(IVurn(t)12)VUm(t)}u~(t)

~ CsolI.6.urn(t)1121Iu~(t)"2'

Thus (3.34)-(3.38) imply

(3.39)

lIu~(t)lI~ ~

(1

lI.6.u~(t)1I2) lIu~(t)1I2

Thus, from (3.32), (3.33) and (3.39) we have (3.40)

lIu~(t)1I2 ~ Ms(T) for some constant Ms> 0 independent ofm.

Limiting process

By above estimates, {um}has a subsequence still denoted by {Urn}

such that

(3.42) u~ ^{- t}u' in LOO(0,T; HJ(O» weak*,

(3.44) (3.45) (3.46)

U~ ~ u' in L2(0,TiHJ(fl» weak, U~ ~u' in LP+l(O, Ti LP+l(O» weak,

div{u(IVumI2)Vum } ~ ~ in LOO(0,TiL2(0» weak*,

(3.51)

(3.47) lu~IP-lU~ ~^{<P in} L7((0,T) x 0) weak,

lumlq-1um^{~'l/J in L~((O,T)x 0) weak.}

It follows from a classical compactness argument (cf. Lions [6]) that

asm~oo

(3.48) lu~IP-lu~ ~ ^{lu'lp-1u' in L}7 ((0,T) x 0) weak, (3.49) lumlq-1u=~ ^{lulq-1u in} L~((O,T)^{x 0) weak.}

We shall show that ~ =div{u(lVuI 2)Vu} is satisfied. For any w E

Co(O,OOiHJ(O», we have

iT ^(~-

l

(~-div{u(lVUmI2)VUm},

+

l

It follows from (3.46) that the first term of the right hand sides of (3.50) tends to zero. Also, (3.41) and mean value theorem imply

l

=

I

::; C241T

+C2S max:{U'(S)}

iT

s 0

~Oasm~oo

Quasilinear wave equation with viscosity 357

and hence we conclude~ = div {a(lV"uI²)V"u}. On the other hand, we can use Aubin-Lions' compactness lemma (cf. proposition 2.4 and 2.5) and so we can extract from {um } subsequence still denoted by {um }

such that for eacht E (0,T]

(3.52) Um(t) -+ u(t) strongly in HJ(O).

By letting m -+ 00 in (3.1), we can find thatusatisfies the equation:

(3.53)

(u"(t) - div{a(lV"u(t)1²)V"u(t)} - .6.u(t) - .6.u'(t),w)

+8!u' (t)I^{P - 1}(u' (t),w) = JLlu(t)l^{q- 1}(u(t) ,w) for all w E HJ(0).

Now, the above result (3.52) implies

(3.54) um(O) = UOm^-+ u(O) strongly in HJ(O).

Thus, from (3.1) and (3.54), u(O) = uo. Also, from (3.42) we obtain (3.55) _(u~(0) - u'(0), w) -+ 0 as m -+ 00 for each w E HJ(0).

Thus, (3.1) and (3.55) imply u'(O) = Ul. This completes the proof of

Theorem 2.6. 0

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Jong Yeoul Park

Department of Mathematics Pusan National University Pusan 609-735, Korea

E-mail: [email protected] Jeong JaBae

Department of Mathematics Pusan National University Pusan 609-735, Korea

E-mail: [email protected]