Mathematische Zeitschrift

Math. Z. 191, 449-465 (1985)

9 Springer-Verlag 1986

Existence of Regular Solutions for a Quasilinear Wave Equation with the Third Boundary Condition* Peter Weidemaier Universit~it Bayreuth, Fakult~it fiir Mathematik und Physik, Postfach 3008, D-8580 Bayreuth, Federal Republic of Germany

w 1. Introduction The aim of this paper is to establish local (in time) existence of classical solutions to the initial-boundary value problem 3

u,(t,x)-

~ Oi(aij(t,x,u(t,x))8ju(t,x)) i,j=l

=f(t,x,u(t,x), Vxu(t,x)),

(t,x)e]O,T] x 9 c N + xlR 3,

3

a,j( t, 4, u(t, 4)) v ,( ~) Oj u(t, 4) + a(t, 4) u(t, 4) =0,

(BVP)

i,j=l

(t, ~)z]0, T] x 89,

u(O,x)=uo(X),

u,(O,x)=ul(x), xeg.

Here f a , Uo,U ~ are given data with certain regularity properties, 9 is a bounded domain in Na with a smooth boundary, v=(vl,...,v3) denotes the outer unit normal to 9, and the spatial part of the differential operator is assumed to be uniformly elliptic (the module of ellipticity may depend on sup lu(t, x)l). x6~Q

Our functional analytic approach to (BVP) will establish a local (unique) solution in the class {us C4([0, T], L2(9))lu(k)(t)~H4-kg), t~[0, T],

u(k)(')eC~

H4-k(9)), k=O ..... 3}

(1.1)

under natural compatibility conditions between the initial values of time derivatives and the right hand side of the equation (cf. Theorem 3.1 for a precise statement of these compatibility conditions). They are natural in the sense that * This work was part of my doctoral dissertation completed under the direction of Prof. Dr. W. yon Wahl

450

P. Weidemaier

they are conserved along the solution in the class (1.1); thus, global existence of the solution will follow, if a suitable a priori estimate (cf. Remark 3.9) can be obtained. In general, hyperbolic initial-boundary-value problems are more difficult to treat than the Cauchy problem (cf. the preface in [Ka]). In the treatment of (BVP) in particular, difficulties arise from the time-dependence of the boundary condition. In a possible abstract formulation as an equation of evolution in Lz(~2), this results in operators A(t) with time dependent domains of definition

D(A(t))={ u~H2((2) i,j=~ aij(t'u)vic~Ju+a(t)u=O ~ ~0} and the abstract theory for evolution equations ([Ta], 4.4) seems to be not applicable. A possible way out was given recently by Yagi ([Ya]); he observes that at least D(A1/2(t)) is independent of time, proves the differentiability (in t) of A1/Z(t), and obtains, in the frame of an LZ-theory, a strong solution to the corresponding linear (BVP) (a~j=aij(t,x), f=f(t,x)). Earlier, Ikawa treated (BVP) in the linear case as an evolution equation in the dual space of Ht(~?), which has the advantage of time-independent domains of definition; but Ikawa's approach to regularity (cf. [Ik], Theorem 4) requires too much regularity of the coefficients a~j(t,x) and is no longer practicable in our quasilinear case. Instead, we follow the basic ideas of the regularity method in [C/vW], which uses the inverse, in some sense, of the elliptic part in the wave equation (cf. (Pwv) in Sect. 3.2). As Prof. T. Kato communicated to Prof. v. Wahl, he has recently obtained a local existence result for (BVP), too. This paper is organized as follows. In Sect. 2 we give some (essentially wellknown) abstract existence theorems and a priori estimates for certain linear equations. Then, in Sect. 3, we study the map S, which carries w (from a suitable function space) into the solution u (in the same function space) of the linear problem

u,(t,x)-c~i(ai~(t,x, w(t,x))c3ju(t,x))= f (t,x, w(t,x), Vw(t,x)),

(Pwu)

%(t, ~, w(t, 4)) v~(~)c~ u (t, ~)+ ~(t, ~)u (t, ~)= O, (t,x)e] 0, T] x ~2, (t, ~)~]0, T] x ~2; we estimate Sw in terms of w (Sect. 3.3) and construct a fixed point of S in the class (1.1) (Sect. 3.4). Finally, in the Appendix, some elliptic regularity results are proyed. A few words about the notation: (HS'P(~2), [I 1Is,p), 0 < s e N , l < p < o e are the usual Sobolev spaces ([Tr]); abbreviating, we write (Hs, I[ IIs) for (HS'2(~2), II IIs.2); ( ' , ' ) denotes the scalar product in Lz(f2); ( ' , . ) is the duality bracket between Ha(~?)* (the dual space of Hi(f2)) and HI(~); I[" 11-1 is the operator norm in HI((~)*; i: L2(~?)-~HI(~?) * denotes the canonical inclusion (gu,~5): =(u,q~) (q~eL2 (~2)). Bck(~?) is the set of all functions f: f2~@, whose derivatives up to the k-th order exist and are continuous and bounded on f~, BC(~): =BC~ C~ ~([O,T], X) is the set of Lipschitz-continuous X-valued functions; L(X, Y) is the set of bounded linear operators X ~ Y. A(. )s ck([O, T], Z, Y) means that Z c D(A) ~ Y and that A(. ) xe Ck([O, r], Y)

Regular Solutions for Quasilinear Wave Equations

451

for each fixed xEZ. ( A i h q ) ) ( x ) : = h - l { ( p ( . . . , x i + h

.... )-q)(...,x

i....

)} (0@h~]R)

denotes a difference quotient in the i-th coordinate direction, gl (l=i__
w2. Linear Equation: Abstract Existence Theorem and A-Priori-Estimates

First we will give some well-known existence theorems for abstract evolution equations using variable norm technique. Assume that the Banach space (B, [[ [[) is equipped with a family of equivalent norms (ll I[,)t>=o, which depend on t Lipschitz-continuously: 1

-Ilxl[t<=llxll<=cllxl[t, c

IIIxll~-Ilxllsl<=clt-slllxll.

(x6B; t,s>=O)

Denote by B t the space (B, 11 Elt). Theorem2.1. Let (A(t))t>__0 be a family of densely defined closed operators with time-independent domains of definition D(A(t))=D(A(O)) in the reflexive Banach space B which satisfy the following properties: (a) There exists a 6 > 0 such that ]6, o o E c p ( - A ( t ) ) f o r all t>=Oand

1I(2+4(t))_XlILr

<

1

=2-6

for 2>6.

(2.1)

(b) 4(" ) is strongly continuously differentiable on D(A(0)). Assume that _PeC~ T],B); gzC ~ o([0, T],B); ~(t)sD(4(0)), A(. ) ~(. )eD([0, T], B). Then for each (beD(4(0)) the problem ,i'(t) + 4(t) fi(t) = Y(t) + ~(t),

tz[0, T],

(te [0, T])

a (o) = ~, has a unique solution fi in the class

Ca(E0, T],B)c~ (fil~(t)~D(4(0)), re[0, T]; A( .)fi(" )~ C~

T], B)}.

Proof Assumptions (a), (b) imply ([Ta], Corollary to Thm. 4.4.2, p. 102; Prop. 4.3.2) that the family (-4(t))t~to ' rl generates an evolution operator [7(.,-) with the usual properties (cf. [vW2], p. 164); then, in the well-known representation formula for a(.), the F-term is integrated by parts as in [vW2], p. 172/173. Since 4(t)O(t, s)(4(s)+ 2o)-1 (20 > 6) is well defined and strongly continuous on B in O<_s<_t<_T, the ~-term can be treated by obvious modifications of [Pal, Chap. 4, Theorem 2.4, Corollary 2.6. [] We are going to apply Theorem 2.1 to certain second order evolution equations. Consider the time-dependent sesquilinear form a(t,-,.) defined on

452

P. Weidemaier

Hi(Q) x/~1(~2):

a(t,u,~).'= ~ f%(t,x)~ju(x)6a~)dx i,j=l

f2

+ I a(t, ~)u({)~(Odo({)+fi fu(x)~(x)dx, OYl

(2.2)

Yl

where O is a bounded domain in ~ " with a sufficiently smooth boundary,

aij, ~t aijEBC([O, T] x f2, ~); aij=aji ,

a, 0 t aeBC([O, T] x ~t'/, IR),

~ aij(t,x)r162

]I~H2, r

(2.3) (2.4)

i,j= 1

with a positive constant c o independent of t; by (2.4) fi can be chosen so big that a(t,u,u)>=~ I[ull2 for all ueHi(O). We associate with a(t,.,') an isomorphism A(t): Hl(t-2)~Hl(O) * in the usual way: (A(t) u, ~)." =a(t, u, q~), (u, ~peHi(O)) and equip HI(~/) * with the time dependent scalar product

(f,h)t:=a(t,A-i(t)f,A-l(t)h) Writing H* for (Hi(Q) *, I]" H t = ~ ,

(f, h aHi(t'/)*).

we then have

Theorem 2.2. Let Fe C ~ T], Hi(t/)*); g(t)eL2(t-2), tel0, T], g(-)eLl([0, T], L2(O)), ~'geC~ T], Hi(O)*). Then for each (Uo,Ul)eHi(O) x L2(O) the problem

u"(t) + A(t) u(t) = F(t) + i g (t), (0 <-t <=T)

u(0)=Uo, u'(0)=ul,

(2.5)

has a unique solution u in C2 ([0, T], Hi(O) *) c~ Cl([0, T], L2(O))c~ C~

T], H~ (t~)).

(2.6)

Proof. Apply Theorem 2.1 with S = L2(O) • H~ (Y2)*,

(0 (o)

~(t) = A(t)

if(t)= F(0 '

0

' ~(t)=

St=L2(t'2) • H*,

D(A(t)) = Hi(f2) x (~ L2(O)),

(0)

'g(t) '

(~=(u~

and take as u(t) the first component of the solution a(t) obtained there. The resolvent estimate (2.1) is proved in [Ik], Lemma 3.4. []

Regular Solutions for Quasilinear Wave Equations

453

A-Priori-Estimates For the convenience of the reader, we list some standard a-priori-estimates for equations of the type (2.5), with detailed dependence of various constants on the coefficients ai~. (The dependence on o- is of no interest and will be suppressed).

Theorem 2.3. Let A(t) be associated with a(t, ", "), a(t,., .) as in (2.2)--*(2.4). a) A solution u of (2.5) with f e C ~ r ] , H1((2)*), g(t)~L2(f2), t ~ [ o , r ] , g( ")eLl(J0, r l , L2(f2)) in the class (2.6) satisfies Ilu(t)[I 1 + Ilu'(t)l[ + Ilu'(t)ll-a

( O < t < T)

__
(2.7)

t

+ S { IIV'(s)ll-a + Ilg(s)[I } ds), 0

where c (T, aij ) = c (T, II% I[BOO(E0,r] • ~), ilOt % II,cO(r0, T1 • ~))" b) Assume that in addition to (2.3) 32talj, 02ta are in B C ( [ O , T ] x f 2 , G), B C([0, T] x ~2, IR) respectively. If, moreover, F~ cz([0, T], Ha(~?)*), ge C a([0, T], LE(g2)) and u solves (2.5) in Ca([0, T], Hl(f2))c~C2([0, T],LZ(f2)), the following estimate holds for tel0, T] Ilu'(t)lla + Ilu'(t)ll < c(r, aO(l[Uo[I ~ + Ilu~ Ila + Ilu'(0)ll + IIf'(0)tl_ t

+ S { IIF'(s)II - 1 + IIg'(s)II } ds),

(2.8)

0 2

with c (T, ai~) = c (T, ~ II~k al j [IBc(Lo,rl • ~))" k=O

For a formal proof of (2.7), cf. [L/M], Vol. I, p. 266, and [Ik], Proposition 3.1, for the necessary regularization procedure. (2.8) is proved in I-L/M], Vol. II, p. 96; the details of the regularization can be found in [We], Appendix, Theorem A.2.

w3. The Quasilinear Equation 3.1. Assumptions (A1)

~2 is a bounded domain in N3 with C~~

(A2)

aij(t , x,

rl)~ C4([0, T] x f2~ x N2,112)1 and alj = aji , 1 <=i,j <--_3.

3

(13)

~

alj(t,x,~)~i~j>co(l~ll)ll~ll 2, ~eI~ 3,

i,j= 1

1 Here 112~t1 is identified with (Re q, Im t/)s]Rz.

P. W e i d e m a i e r

454

for (t,x,t/)~[0, TJ • (2~ • ~ with Co(')~C~ (A4) a~ C4([0, T] • O, ~). (A5) f 6 C3([0, T] • (2~ x lR 2 • R6, ~ ) , f ( . , . , 0,0)-0. Here (2~ denotes the &neighbourhood of (2.

3.2. The Solution of the Linearized Problem

Let K(4): = {uc C4([0, T], L2(Q))lu(k)(t)~H~-k(~), t~[0, T]; u(k)( ")e C~

T], H4-k((2), k =0,..., 3},

and let 4

w(0)=Uo, w'(0)=u. w"(0) = u2 : = f (O, Uo, Vuo) + ~i(alj(O, Uo) ~ Uo)}. Theorem3.L Assume that (A1) through (A5) hold. Let w6MT(uo,ul); assume further that (Uo,UOEH4((2) x H3((2) satisfy the (nonlinear) compatibility conditions (cf. [Ik], (3.44)): (C1) B(O)uo =0, (C2) B'(0) u o + B(0) u~ = 0, 2 (C3) B"(O)uo+2B'(O)u 1 +B(0)u 2 =0. Then the problem u!'(t) - 6i(alj(t, w(t)) 0 r u(t)) = f (t, w(t), Vw(t)), B~(t) u(t):-- alj(t, w(t)) v i 3 ju(t) + a(t) u(t)[e~ =0,

(0 <=t <=T)

(Pwu)

u(0)=uo, u'(0)=u. has a unique solution u in the class K (4) defined above. Remark3.2. Due to the assumption f(t,x,O,O)=-O, the set of admissible initial values Uo, ul is not empty (since it contains C~((2)).

The plan for proving Theorem 3.1 is the following: If u e K ~4) solves (Pwu), then v(t): = - c~i(a,j(t, w(t)) Oj u(t)) - f (t, w(t), V w(t)) + flu (t),

(3.1)

(which equals - u " ( t ) + fl u(t)), obeys the inhomogeneous boundary condition t Ut (t) + Bw(t) H S w(t) v(t) = 2 Bw(t) u (t)

2

B'(O)u o m e a n s c~t[t=o(Bw(t))Uo; B,~(t)u o as e x p l a i n e d in (Pwu) b e l o w

Regular Solutionsfor Quasilinear Wave Equations

455

and, using the relation u ( t ) = A ~ 1 ( t ) ( i v ( t ) + i f w ( t ) ) (which follows from (3.1)), one easily shows that the equation for v(t), formulated as an evolution equation in H i ( O ) *, is

i v"(t) + A w (t) v (t) = A~ (t) A ~ 1 (t) (i v (t) + ifw (t)) + 2 A ' ( t ) {A w-1 (t)(~9v(t)+~fw(t)) 9 } , +*'fi(v(t)+ fw(t)) - ~ f.' ( t,,) , 3

(Pw v)

v(0), v'(0) as implied by (3.1). Here A~)(t), 0= 1), whereas f~k)(t) stands for c ~ { f ( t , ' , w ( t , . ) ) , Vw(t,'))}. Henceforth, the index w is suppressed where no confusion will arise. We aim at establishing a solution to (Pwv) in K(2): = C~

T ] , H 2 ) c~ C1([0, T],UX)c~ C2([0, T], L2);

to accomplish this, we first consider the problem obtained formally from (Pw v) t

by differentiation with respect to t (v(t) = v(O) + ~ y(s) ds): 0

y"(t) + A(t) y(t) = (G(t, v(t), y(t)))' - A ' ( t ) v(t),

y(O)=v'(o),

(py)

y'(O) = h + fl(v(O) + f(0)) - f"(0), where G(t,v(t),y(t))-G(t,v(t),v'(t)) is the right hand side of the equation in (P~ v), and hEL2(y2) is chosen such that i h = - A (0) v (0) + A"(0) z (0) + 2 A'(0) z'(0), (z (t)."= A - - 1 (t) (i V (t) + i f (t))) ;

(3.2)

(this is possible by virtue of the compatibility conditions (C1) through (C3) and will be proved later). Before we establish the existence of a solution to (Py) with the help of Theorem 2.2, we need some regularity results for A - ~ ( t ) and its time derivatives. Lemma 3.3. Let A ( t ) = A w ( t ) with w ~ M T. Then (i) A-~(-)EC~

T],iH~,H3), ][A- l(t) i v H3 =
Bc2(~)) [[vii1 ;

(ii) A-~(')~Ca([O, T ] , i U , H2),

II(A- l(t))' i u[12 < c~

ttBc~(~), lla'ij(t)llBco(a)) [lutl ;4

(iii) A-1(')~C3([0, T], iLZ, H~),

I]( A - l(t))'"g u [la =
In the proof of these facts, the following lemma turns out to be useful.

3 4

i: L2(f2)~H~(O) * is the canonicalinclusionvia L2-scalarproduct a~j(t)('):=3~{a~i(t, ", w(t, .))} etc.

456

P. Weidemaier

Lemma 3.4. Let f2 be as in (A1). T], Lp) c~ Cl([0, T], LP),

al~)( 9)e C~ v(-)~ C~

A(k)(') v(')e C1([0, r ] , Hi*),

T ] , / ~ ' 2 ) c~ C1([0, T], Hm'2), 3

provided that (m, p), (nS,p) satisfy m _>- + I, if 2 < p < co, m > 3 + 1,/f 2 = p, P Proof. By standard Sobolev imbeddings (e.g. [Ad]), one finds I[al~)(t)Ojv(t)]]L2
[]

Proof of Lemma 3.3. The regularity of w6M T implies (by suitable Sobolev imbeddings):

a~j(t, x, w(t, x))eBCZ([O, T] x (2),

(cf. [vW1], Hilfssatz 8)

(3.3)

a'~)(")e C1([0, T], L3),

(3.4)

a'~)'( 9)e C~([0, W], L2).

(3.5)

(i) is a standard elliptic regularity result (cf. [W/3, Satz 20.4). (ii) By (3.3) one has A ( ' ) e C2([0, T], L(H 1, Hi*)) and therefore also A - I ( . ) E C2([0, T],L(HI*,Ht)), (A- l(t))' = - A - l(t) A'(t) A - ~(t),

(3.6)

(cf. [Kr], Introd., Lemma 3.8). (3.6), Lemma A.2 and Remark A.3 yield:

(A-l(t))','ueC~

2)

for ueL 2

and IIA - '(t) A'(t) A - 1(0 ~ u JI2

<=c([laij(t) LIBcl)(lla'ij(t)0j A- l(t), u][ 1 + tla'(t)HRco [IA-l(t)iuH 1) <=c([[aij(t)lIBr I[a'i2(t)Ilsc,)I[A-l(t)~iull2
(3.7)

II{A'"(t) A - 1(t)}',' v II- 1
(3.8)

Regular Solutions for Quasilinear Wave Equations

457

(3.7) is a consequence of L e m m a 3.4 (with p = 3 , m = 2 , p = 2 , n5=3 by (3.4), (3.5), and L e m m a 3.3(i), (ii)); (3.8) is obtained from the estimates in L e m m a 3.3(i), (ii).

Proof of Theorem 3.1. As motivated above, we implement an iteration to solve (Py) in g(1): -= C~ T],H~)c~ C~([0, r ] , L2) c~ C2([0, T], Hi*): Yn'+ 1(t) + A (t) y, +1 (t) = (G(t, v, (t), Yn(t)))' - A'(t) v, (t), (3.9)

Yn+ 1(0) = V'(0) e H 1 ( ( 2 ) '

Yl,+1 (0) = h + fl (v (0) + f (0)) - f " ( 0 ) E L2((2), where

G(t, v.(t), y.(t)):

=

A"(t)A-l(t)rn(t)+2A'(t){A-l(t)r,(t)}'+,'firn(t)-if"(t),

(3.10)

t

with

r,(t):=g(v,(t)+ f(t)); v,(t)'.=v(O)+ f y,(s)ds; v(O), v'(O) as in (Pwv). Starting 0

with yo(')-=v'(0), one proceeds from y, to Y,+I by applying T h e o r e m 2.2 in each step; note that for y,~K (1) the Hl*-valued part of the right hand side in (3.9) is still in CI([O,T],HI*); (cf. (3.7), the remaining terms can be treated similarly, with the help of L e m m a t a 3.3, 3.4). The a-priori-estimate (2.7) and the estimates in L e m m a 3.3 and (3.8) yield:

E,(t)
(n~N)

(3.11)

00<-z<-s 2

where E,(t)." = ~

. (k) Y, + 1 (t) -- y~k)(t)l] 1 - k ; (3.11) implies that (y,), =>o is a C a u c h y

k=0

sequence in K(I); its limit

y e K (a) clearly is a solution to (Py). Defining v(t)."

t

= v(O) + ~ y(s) ds, we see that v is a solution of (Pwv) (this is due to the choice of 0

h, cf. (3.2)). By elliptic regularity (Lemma A.2), one also finds from the equation in (P,~v) that v~ C~ T],H 2) and therefore vEK (2). Next, we define

u(t):=A-~(t)r(t),

r(t)'.=iv(t)+if(t);

then u(.)eC2([O, T],L2); u(0)=Uo, u'(0)=ua (cf. (3.14)) and from the equation in (Pw v) we easily deduce

gu"(t)+A(t)u(t)=gf(t)+gflu(t).

(O<=t<=T)

(3.12)

definition of u(t) it also follows, by elliptic regularity, that (cf. [ W / ] , Satz 20.4; note that this theorem remains valid if one only assumes "aifiH3(f2)c~BC2(f2) '' instead o f " a~fiBC 3(O)). ,, Moreover, From

the

u(')~C~

u'(t) = --A- 1(0 A'(t) A - 1(0 r(t) + A - a(t) r'(t)6 C~

T], H3),

458

P. Weidemaier

(Lemma A.1, Remark A.3), and from the Eq. (3.12), written out pointwise as in (Pwu), one deduces successively

u"(.)~C~

u"(.)~C2([O,T],L2),

u'"(.)e C~

T], Ul),

whence uffK (4). To complete the proof, only (3.2) remains to be shown. Integrating by parts and using the relation - v ( 0 ) = u 2 - f i u o , we find

(r.h. side of (3.2), ~ ) = (h, ~o)+ ~ (B(0) u 2 + 2B'(0) z'(0) + B"(0) z(0)) ~b of2 - f l ~" (B(0)Uo) @ of2

(~oeHl(f2))

(3.13)

for a h~L2(f2); the boundary integrals clearly vanish (by (C1), (C3)), if we verify z(O)=uo

and

z'(O)=u 1,

(z(t):=A-X(t)(~'v(t)+;f(t)));

(3.14)

but the first relation is a consequence of (C1), and the second one follows since Z'(0) = -- A - 1(0) A' (0) A - 1(0) (~"v (0) -1-i f (0)) + A - ~((3)(i v' (0) + gf'(0)) = A - 1(0)( - A'(0) u o +g v'(0) + / i f (0)), and since the definition of v'(0) (cf. (3.1)) implies ( i if(0) +~'f'(0), q)) = (A'(0) u o + A(0) ul, qg) - ~ (B'(0) u o + B(0) ul) ~o

= (A'(O)uo+A(O)Ul,fp).

(c2)

(goeHl(f2)). []

Remark3.5. Theorem 3.1 remains valid under the following weaker assumptions for w:

w(.)eC3([O,T],L2), w'"(t)~H 1 V tel0, T], w(k)(')~C~

w"'sC~ w'"(')~LI([O,T],H1), k = 0 , 1,2,3, 0 < e < 8 9

In this case, by imbeddings for non-integer Sobolev spaces (el. [Tr], 4.6.1, Thm.), (3.3) and (3.4) still hold; moreover A'"( 9) A - l ( ' ) e C O,1([0, T],g H 1, Hi*),

if"~CI([O,T],HI*),

(if")'(t)=gg(t) with a g(t)eL2(f2),

g(" ) e D ([0, T], L2 (f2)), so that Theorem 2.2 is still applicable in the iteration (3.9).

Regular Solutions for Quasilinear Wave Equations

459

3.3. Estimates for u = S w

Denote S the map which carries weMTc into the unique solution u of (Pwu) in K (~) and define M~a:= {w~Mffl IIw'"(0)ll 0 such that

(i) Mc,T* ~,(Uo,Ul),es, (ii) SMc* T*a*(Uo,ul) ~ Mc.r*,a,(uo, ul). The second assertion in Theorem 3.6 follows from Theorem 3.7. for every d>0, where tc(...) depends on ~ only through expressions of the form c~(~) T ", c~> 0, and is an increasing function in each such expression. Proof of Theorem3.6. Extend Uo,U1 to ~oe/~4(f2~), file/t3(~20) respectively (cf. [WI ], p. 105). The auxiliary problem

~"-0~(a~j(0,fio)Oj~)=f(0,a o, V~o) in [0, 1] x f20, ~=0 on ~(t?~),

~(0)=~o,

~'(0)=~,

has a unique solution ~EC4([0,1], L/(f2~)); ~(k)(')eC0([0,1], /~4-k(f2~)), k = 0 .... ,3 (cf. [C/vW]). Defining v:=~lo, we see that v~M~.a(uo,ul) for ~, sufficiently large. To prove (ii) we choose d*__-d so big (depending only on Uo, ul in view of the equation in (Pwu)) that II(Sw)"'(0)ll__d* for all we ~ Mr~(Uo,Ul);then T~]0, 1],c>~

we set c*."= m a x {x(d*,uo,u ~, 1),~} and choose T * = T*(c*)e]0,1] so small that c~(c*)T *~<=1. [] Proof of Theorem 3.7. The a priori estimate (2.8) applied to the equation g u'"'(t) + A(t) u"(t) = F(t) + g f " ( t ) + i fl u"(t), F(t).. = - A " ( t ) u (t) - 2A'(t) u'(t),

(3.15)

(obtained by taking the second derivative of the equation in (3.12) with respect to t) yields: Ilu"(t)ll 1 + Hu'"'(t)H

< c(T, %) t,~(Uo, ul) + ]lF'(0)]I _, +

i (llf'(s)ll_ 1 + Ilu'"(s)ll + IIf'"(s)ll)ds} 0

< c(T, aij) t)~ (Uo, ul, d) +const(e) ~(llul]3+ Ilu'112+ Ilu'ltl+ Ilu"rl~)(s) ds + 0

I[f'"(s)[I ds , 0

(3.16)

460

P. Weidemaier

(with c(T, ai) as in (2.8)); here we have already used the fact that the higher time derivatives u'"(0), u""(0) can be expressed in terms of Uo, u 1 (through the equation in (PwU)). By elliptic estimates (Lemma A.2) we further get from (3.15)

Ilu"(t)tl 2
< c(alj(t)){ Ilu(t)[is + ]lu'(t)]l 2 + Ilu"(t)ll + Ilu""(t)ll + Ilf"(t)ll}, where c(ai~(t))=c(Haij(t)l[Bcl, L e m m a A.1 applied to

(3.17)

Lla'ij(t)LlRcl, Ila'i~(t)[[Bco, ]la'i~(t)[10. Analogously,

g u'"(t) + A(t) u'(t) = -A'(t) u (t) + i f'(t) +,'fi u'(t) yields [lu'(t) ll a <=c(llaij(t)l[Bc2){l[u'"(t)]ll + I]u'(t)l[ 1 + Hf'(t)l[ 1

+ IIa'ij (t) ~j u (t)II 2 + IIu (t)I[ 2} <=c(aij(t)){llu(t)ll3+][u'(t)Hl +l[u'"(t)]ll +]lf'(t)H1},

(3.18)

where c(aij(t))=c(Ha~(t)l]Bc2, I[a'ij(t)l[ 2); from (3.12) we conclude (cf. our remarks following (3.12))

[[u(t)ll4<=c(llaij(t)llBc ~, Halj(t)l[3){llu"(t)l[2+ I[u(t)H 2 + [[f(t)ll 2}.

(3.19)

Collecting (3.16) through (3.19), we end up with E(t):=k=O ~

[lu(k'(t)l[4-kZC(T'aiJ){~(d'uO'ul)-[-]lu(t)[13

+ Ilu'(t)ll 2+ Ilu"(t)l[ + Ilu'"(t)]l x + ][u""(t)ll +

[If(k)(t)llZ_k+COnSt(g) ~.E(s)ds+ ~ Ilf'"(s)[[ ds , k=0

0

(3.20)

0

c(T, a~j) depending on T and sup

ts[o, T] kk = 0

Ilai~l(t)HBc. . . .

)

~ IIa~)(t)]/s_k.

(3.21)

k= 0

Inserting (3.16) again in the right hand side of (3.20) and making use of the t

relation Ilu(t)ll s --< Ilu(0)ll 3 + ~ llu'(s)]l ads etc., we see that 0

E(t)<=c(T'ai'){ 2(d'u~176

oi E(s)ds .%

+ i (]lf'(s)ll2 + IIf"(s)lll + IIf'"(s)ll)ds~ 9 ) 0

(3.22)

Regular Solutions for Quasilinear Wave Equations

461

Estimating the last integral by const(~). T and applying Gronwall, we get

E(t) < c (T, aig) {2(d, Uo, u 1) + const (~) T} e c(T'"'~)c~

r

(O<_t<=T). To complete the proof, we note that by suitable Sobolev imbeddings ([Tr], 4.6.1 Thm.) all the norms in (3.21) can be estimated in terms of IIw(k)(t)ll4_k_~

with 0 < e < 1,

(kE{0,1,2})

and thus, using the interpolation inequality ([Tr], 4.3.1, Theorem 1, formula (11) in 2.4.2, formula (3) in 1.9.3), in terms of expressions of the type 2(uo,u 0 +c~(~) T ~ for suitable c~>0, e.g. IIw'(t)ll 2_~ < Ilw'(0)ll 2_~+ IIw'(t)-w"(0)ll ~-~ IIw'(t)- w'(0)ll?-

< 2(Uo,Ul)+ 2Eff/2.

[]

3.4. Existence of a fixed Point of S T* Taking an arbitrary wosMc.,a.(Uo,Ul) ( which is not empty according to Theorem 3.6) and defining w . : = S w . _ 1 recursively, one obtains a sequence (w.).ao in Me.T*,a, which satisfies

g w~ (t) + A (t, w._l (t)) w. (t) = i f (t, w.-1 (t), V w._ 1(t)) + g fl w. (t), w'.(O)=Ul,

w.(0)=u 0,

(O<_t<_T*)

and which will be shown to converge; namely, v,+ ~: =w,+ 1 - w , fulfills

; v~+ 1(0 + A(t, w.(t)) v.+ l(t) =F(t) + i g(t),

(3.23)

"On+ 1(0) = Vt.+ 1(0) =/);t. t_ 1(0) = 0 ,

where F(t), = {A(t,

Wn_

l(t)) --A(t, w.(t))} Wn(t),

g (t)-"= f (t, w. (t), V w. (t)) - f (t, w n-1 (t), 17W._l (t))+ flv. +a (t). From (3.23) the a priori estimates in Theorem 2.3 yield E,+ a(t)." = [Iv,+ 1(0II1 + I]v',+ 1(0111+ I/v;+ 1(011

(3.24)

t

< const (T*, c*) S (ll F'(s)l1-1 + HF"(s)ll_ 1 + IIg'(s)Dds; 0

using suitable Sobolev imbeddings -aij(t, w,_ l(t))}, we find that

and

]lF'(t)ll_, < It{...}'Ojw,(t)ll + II{...}

writing

~jW'n(t)ll

_
{...}

for

{aij (t, Wn(t))

462

P. Weidemaier

[IF"(t)[[_ 1 < LI{...}"LI USjw,(t)UL~ +2 U{...}' tl ILS~w',(t)lLL~ + [l{...} HL4Hajw'~(t)]lL4 < const (c*) { I1v~'(t)I[+ It~;(t)II + IIv,(t) 111}, IIg'(t)tl <___const (c*) { IIv',(t)[11 + I[v,(t)II1 + IIv'+l(t) ll}. Inserting these estimates in (3.24) and using a version of Gronwall's inequality (cf. [Ik], Lemma 2.3), we arrive at t

E,+ l(t) < const (T*, c*) ~ E,(s) ds,

(3.25)

0

from which we conclude w,--* :=w in C1([0, T*],H1)~ C2([0, T*],LZ). Also, since (w,)~ Me., r* we get by weak convergence for k = 0, 1, 2

w(k)(t)eH4-k(f2),

t~[0, T*],

sup [Iw(k)(t)[14_g<=C*.

(3.26)

rE[O, T*]

Further, an Arzela-Ascoli type argument for weak convergence (cf. [Br-l) shows the existence of a subsequence (w,~) c (w,) such that w','~(t)--* :=v(t) weakly in L2(Q) for all t~[0, T ' l , which, together with the estimate I(w~"(t)-w~'(s), h)[ < c* It-s[ IIh]l (h~L2), easily implies w(')~Ca([O,T*],L2), w'"(')-v(')~C~ 2) (3.27) and the validity of (3.26) for k = 3 also. Clearly, w satisfies the equation

i w"(t) + A(t, w(t)) w(t) =if(t, w(t), Vw(t)) +i fl w(t)

(3.28)

for tel0, T*-I. Moreover, it has all the weakened regularity properties listed in Remark 3.5 (for all e~]0,89 by (3.26), (3.27) and the interpolation inequality), so that we even get

weC4([O,T*],LZ),

w(k)(.)eC~

k = 0 ..... 3.

Uniqueness of w follows from (3.25) by Gronwall's Lemma. Summarizing, we have proved

satisfy (A1) through (A5) and assume that (C1)--*(C3) in Theorem 3.1. Then there is a T * = T*(uo,Ul)>O such that the problem

Theorem3.8.

Let f2, aij, a, f

(Uo,Ut)EH4(f2)xH3(~2) satisfy the compatibility conditions u"(t) -Si(ai~(t, u(t))) ~~u(t))= f (t, u(t), Vu(t)),

aij(t,u(t))vi~ju(t)+a(t)u(t)lo~=O , u(0)=u0,

u'(0)=ul,

(O<_tNT*)

Regular Solutionsfor Quasilinear Wave Equations

has a unique solution ueC4([0, T*],LZ(Y2)); u(k)(')e C~

463

u(k)(t)~H4--k((2),

T*], H4-k(Q)),

t~[0, T*],

(k = 0 .... ,3).

In particular u is in BC2([0, T*] x ~) (cf. [vW1], Hilfssatz 8). Remark3.9. Since the compatibility conditions, which are natural in some sense, are conserved along the solution constructed above, global existence of this solution will follow, if an a-priori estimate for [Iu(t) l[4 + l[u'(t) ll3 is available. Further calculations show that the estimate

IlullBc2
Nu(t)ll3 +

I[u'(t)ll2
with h ( . ) s C O([0, oe [, IR +) is already sufficient.

Appendix: Some Elliptic Regularity Lemmata We give some regularity results for weak solutions of elliptic equations which allow boundary integrals on the right hand side of the Eq. (A.1); since we have been unable to locate these results explicitly in the literature, we have decided to include a brief proof. Let 3

a(u,v):= ~

~%(~)~ju(x)O~v(x)dx+ ~ o(Ou(O~(~)do(O

be Hi-coercive (i.e. there are fl, Co>0 such that the inequality Rea(u,u) +/~ Uul[2>Co Ilu[[2 holds for all ueHl(f2)).

LemmaA.1. Let f2cP, 3 be as in w (A1). Assume that alj, o-~BC2(~'2) and that ueHl(f2) is a solution of a(u,q~)=~fCp+~gOkcP+~hCp

Vq~6H~(~?)

(A.1)

for some k6{1 .... ,3} with feHa(f2), g, h6H2((2). Then u~H3(f2) and Ilul[3
(W3:={(xl,x2,.c)ER3lxi, zE]--l,l[}),

~?f2-- {(xl,xz, O)[xie ] - 1 , 1[}. By standard manipulations we see that for a tangential derivative ~?k, a tangential difference quotient A~ (e.g. k, ie{1,2}), )~eC~(W3), and for small h the identity i a(Ah~3k(ZU),q~)=a(u, zc2k(AZ_hq~))+I(u, qg) (peHa(W+3)) (A.2)

464

P. Weidemaier

holds, where ]I(u,q~)[
(h'z)(x',O)C~k(A~_h (O)(x',O)dx '

(dx'=dxl dx2)

w 2

1 = - ~ O ~ ~ (h" )O(x', v) Ok(A~_h(o)(x', z)dx ' & 0

W 2

~_ _ ~ Cq~(h"x)(X', Z)Cqk(Ai-h ~O)(X', Z) dx' dz - ~ (h'z)(X',Z)O~Ok(Ai-h(O)(X',Z) dx'dz. w? Now, integrating by parts, we take away Ok from ~ in the first term and both Ok and Ai_h in the second one; the resulting expressions can be stimated as desired and (A.3) is proved. Combining (A.2), (A.3) and inserting ~0: = A~ Ok()~u)eHl(W3+) (which is admissible), we get in a standard way

O~Ok()~u)eHl(W3+)

(i,k~{1,2});

then, exploiting the equation in (A.1), the desired regularity of the remaining (normal) derivatives follows as usual (cf. [ W / I , p. 314). [] Following the same lines, one shows LemmaA.2. If in Lemma A.1 we only assume that aq, a~BCl(~), f~I~((2), g, heHl(f2), then u~H2(O) and liu]l 2<=const([laij[]Bc~)(llfll + Ilgll 1 + llhll 0.

Remark A.3. In the case of time-dependent coefficients a~), a and data f, g, h, the estimates derived above show: If all the norms mentioned in the assumptions of Lemma A.1 (Lemma A.2) depend continuously on t and if u ( - ) s C~ ([0, T], H1), then we also have u(. )e C~ ([0, T], H3), (u(,)E C~ r ] , H 2) respectively).

References lAd] [Br]

Adams,R.A.: SobolevSpaces. New York-San Franc.-London: Academic Press 1975 Briill,L.: On the existence theory for nonlinear Schr6dinger and wave equations without Lipschitz estimates. Nonlinear Anal. 7, 1193-1208 (1983) [C/vW] Chen, C., v. Wahl, W.: Das Rand-Anfangswertproblem fiir quasilineare Wellengleichungen in Soholevr~iumenniedriger Ordnung. J. Reine Angew. Math. 337, 77-119 (1982)

Regular Solutions for Quasilinear Wave Equations [Ik] [Ka]

[Kr] [L/M] [Pa] [Ta] [Tr] [vW1] [vW2]

[We] [Wl] [Ya]

465

Ikawa, M. : Mixed problems for hyperbolic equations of second order. J. Math. Soc. Japan 20, 580-608 (1968) Kato, T.: Linear and quasilinear equations of evolution of hyperbolic type. In: Hyperbolicity, Edited by G. DaPrato/G. Geymonat, Centro Internazionale Matematico Estivo, II ciclo, Cortona, pp. 125-191 (1976) Krein, S.G.: Linear differential equations in Banach space. Transl. Math. Monogr. 29 (1971) Lions, J.L., Magenes, E.: Nonhomogeneous boundary value problems and applications II. Berlin-Heidelberg-New York: Springer 1972 Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Berlin-Heidelberg-New York: Springer 1983 Tanabe, H.: Equations of evolution. London: Pitman 1979 Triebel, H.: Interpolation theory, function spaces, differential operators. Amsterdam: North-Holland PuN. Comp. 1978 v. Wahl, W.: Klassische L/Ssungen nichtlinearer Wellengleichungen im GroBen. Math. Z. 112, 241-279 (1969) v. Wahl, W.: Analytische Abbildungen und semilineare Differentialgleichungen in Banachr~iumen. Nachr. Akad. Wiss. G6ttingen, Math.-Phys. K1. II, 153-200 (1979) Weidemaier, P.: Ober lokale und globale Existenz klassiscber LSsungen flit Randwertprobleme quasilinearer hyperbolischer Gleichungen 2. Ordnung. Dissertation, Bayreuth 1985 Wloka, J.: Partielle Differentialgleichungen. Stuttgart: Teubner i982 Yagi, A.: Differentiability of families of the fractional powers of self-adjoint operators associated with sesquilinear forms. Osaka J. Math. 20, 265-284 (1983)

Received June 28, 1985