Dissipativity in the sense of Levinson for a class of second—order nonlinear evolution equations

Nonlinear Analysis, Theory, Printed in Great Britain.

Methodr

& Applications,

Vol. 6, No.

11, pp. 1207-1220,

1982.

0362-546&/82/l 11207-14 $03.00/O @ 1982 Pergamon Press Ltd.

DISSIPATIVITY IN THE SENSE OF LEVINSON FOR A CLASS OF SECOND-ORDER NONLINEAR EVOLUTION EQUATIONS ALAIN HARAUX Universitk Pierre & Marie Curie, Laboratoire d’Analyse NumCrique, (L.A. 189) Tour 55/65 5kme E., 4 Place Jussieu, 75230 Paris Cedex 05, France (Received

10 February 1982)

Key words and phrases: Levinson dissipativity, nonlinear evolution equations, second order. 0. INTRODUCTION LET V, H BE

AND NOTATIONS

two real Hilbert spaces such that

with continuous and dense embeddings. in V, H and V’. We set

We denote respectively by ( 1, 1111and 11]I* the norms

&=

Inf

U2

II II 2.

uEV,u#O 0u

The duality pairing in V’ x V will be represented by the symbol ( . ). Let Q, E C’(V, [w) and g : Iws x V-, V’ a ‘multivalued’ variable operator. We investigate boundedness properties of the eventual positive trajectories of u” + O’(u) + g(t, u’(t)) 3 0. In section 1 of this work, we prove that under some assumptions on @ and trajectories (defined in an adequate manner) are such that u E Y’(lR+, V) If in addition, s(t, . ) does not vanish too fast as t + t 00, the dynamical in V x H by (1) is dissipative in the sense of Levinson, which means there that for every solution u of (1) on IF!+,we have lim ]Iu(t>IIS R,lim

t--*+=J

(1) g, all the positive f7 W’*m(Iw+,H).

system generated exists R > 0 such

(u’(t) 1s R.

t++@J

This result is applicable to the class of purely dissipative systems considered in [4]. In section 2, we construct strong global solutions of U’I

-

Au + g(u) + P(u’) W(m)

on IO,T[ X Q on IO, T[ X aS21

u=o

(2)

where S2 is a bounded, open domain of Rn with smooth boundary, p is a maximal monotone graph in R x Iw and f, g satisfy some relevant hypotheses. The results of section 1 are then applied to the solutions of (2). In section 3, we show that if p-’ is uniformly continuous: IF!--) R and f, g satisfy additional 1207

1208

A. HARAUX

assumptions, the dynamical system generated by (2) has stronger dissipativity properties, which can be used for example to construct bounded solutions: R ---) V x H. We also prove an existence theorem of periodic solutions under suitable assumptions on f, g and P*

Notations Throughout the course of this paper, we will use the following conventions. The derivative with respect to t of a function u : IO, T[ + X will be denoted by u’(t) (even by u’ if the dependence on t is clear). with a vector function: IO, T[ + VJ’(S2) (m E Z), we If u:]O, T[ x Q + IT8is identified identify k&t and u’. A map between two metric spaces f:X * Y will be said to be ‘bounded’ if, for every B C X which is bounded, f(B) is bounded in Y. Let X be a (real) Banach space and J = IR’or J = [0, m[. We set t+l

Sp(J, X) e Sup

fE

tEJ

I

If(e)I”xde<

t

+m.

A function u : R + X is said to be S1-almost periodic if u E S’(R, X) and z--, u(t + z) is almost-periodic as a function R + s’(lR, X), endowed with the norm defined above. If p is a (possibly multivalued) maximal monotone graph in R x [w, we set

We use the usual terminology concerning Sobolev’s In section 3, the ‘energy’ space will be denoted by

spaces in S2 or IO, T[ x Q.

%Z=VxH. We assume that the reader is accustomed with the properties of the usual wave equation. For the monotonicity properties and weak solutions of purely dissipative systems, we refer systematically to H. B&is [5]. The development of sections 2 and 3 relies in an essential manner on some results of [2,6] and [ 111, respectively. 1. SOME

ABSTRACT

THEOREMS

In this section, we do not study the Cauchy problem associated to (1) but we prove boundedness results for eventual global solutions. We assume that, for a.e. t E lR+, a (possibly multivalued) operator g(t, . ) : V + V’ is defined and moreover that Q and g have the following properties

3fE

syR+), 3c

2

o(vu E

v, a.e. on R+

whenever

z(t) E g( t, u),

a.e. on R+

(5)

Dissipativity in the sense of Levinson for a class of second-order

3p > 0 such that #k(z)

3k E IT([w+, Iw+),

Ess inf k( 19)2 p Ess sup k( e>, [v+ 11 [t,t+ 11 and 3h ~s’(lR+),

nonlinear evolution equations

h > 0 such that for every

dz>

0,

a.e on II%+

[u, z(t)] like in (5)

(6)

we have

a.e on R+. Remark.

If k is continuous,

(6) is easily seen to imply,

k(t + 0) 3 pexp(-MB)k(t), while the above

inequality

implies

W3

(6) with p replaced

in turn

(7)

for some M related

Iftao,

1209

to p

0,

by p exp( - M).

THEOREM 1.1.Let u EVV/$‘(lR+, v) n N$$i([w+, V’) satisfy -u” - Q’(u) E g(t, u’(t)) a.e on R+. Then, u E L"([w+, V) and u’ E Lco(R+, H). Moreover, there exists K: IR’ x [w++ [w+ bounded

such that vt 2 0, IIu(t) II +

and K only depends The proof

upon

of theorem

Iu’(t) I s wll @) ILlu’60 I)

a, p, Cl, C2, p and the S1-norms 1.1. will be a consequence

(8)

of f, h and kh.

of the following

technical

lemma.

LEMMA 1.2.Let 1 =4&'/2d'2p? Assume

that,

for some t E R+, we have t+l (u” + W(u),

It Then,

0

we have sup {ilu’(t) tE[t,t+ l-j

where

u’) dza

P is bounded:

I2 + @(u(z))}

(R’)‘+

s flllcu,

l/B, C,

Cl,

c2,

lip,

a, b, 4

Iw+ and

t+l t+l a= SUP h(r) dz, b = Sup k( r)h( r) dz tao I t t*o I t t+z c = SUP f (9 dr tao I t Proof of lemma 1.2. We introduce p(r) It is clear that p(z) is differentiable

the numerical =

function

3)u’(z) I2+ @(u(r))*

a.e on It, t + I[ and

p’(z) = (u” + W(u), From

(7) and (9), we deduce,

24’)

a.e

with 0 = p? I(ilk(r)h(r) t

/+‘k(z)lu’(@12drs t + j-t+f lu’(z> t

I2dzs

;.

dz

(10)

1210

A. HARAUX

On the other

hand,

if we define 5( z= )

-(u”

+ W(u),

u’(z)) + k(z)h(z),

we have 5 2 0 and which implies t+l

t-f-1 (p’(z) ( dzs

I

((u” + W(u), u’)l dz=

2b.

t

For convenience

we introduce

Q = SUP b’(Z>I, [f,t +r] and P = sup P(T). [V + r] From

(11) we get:

From

(12), it is immediate

t+l

1

Ps2b+i

P(Z) dr*

It

to deduce 1 t-t1 P s 2b + Laa-ll-’ + Q>(u(z>) d?;. 2 1

Since ilu1l* G K’[@(u)

+ Cl], we have

vz E

+ Cl)“*,

1(u(z), u’(r)) 1s &“*Qa-“*(P

[t, t +

I].

As a consequence: t+l

I

t+l

@‘(u(r), u(z)) dz = -

t

I

(u” + z, u) dz

t

t+l =

ul)l:+‘+ [

-[(u,

lu’(z>

I* dz-

I”i(z(z); t

u(z)) dz

s 2&r l’* a- Ii2Q(P + C1)‘/* + aa-’ + d’*(P

+ Cl)‘/* \‘+f {C[ IL+) t

I2+ (z(z), u’(r))] +f(z)}

Or .. t+l

I t

W(u($), u(z>>dr s

+ aa-’ +

a+*(P

2&l/*a-

+ Cl)“*

“*Q(P

[Caa-’

+ C,) ‘I*

+ c].

dz.

Dissipativity in the sense of Levinson for a class of second-order

nonlinear evolution equations

1211

Since Qz G 2P, from (14) we deduce t+l

I

(@‘(U(Z)),

U(Z)) dzs

23’2Ai1’2#2[P(P

+ C#‘*

+ aa-’

t

+ From

(4)

+ c)(pl’* + cy’).

c@‘*(Ca6l

we find 1 t+l - j@(u(r)) 1 t

On combining

dzs

p-ll-’

the two lines above

l”i t

(@‘(U(Z)), u(r)> dr + C$-’

with (13)

we finally obtain:

P s 23i2&112~‘2~-11-1(P + ;Cl) + M(Z)P’” + N(1) M(I) = (Ca/j-1611-1

+ c~~1Z-1)a~1~2

N(l) = C2P_l + ad1Z-1($

+ /3-l) + 2b + d*C:/2(Caa-’

+ c) p-‘l-?

From (15), and since I >23’2&1’2$12p-1, we get a bound P for P, which has the expected This ends the proof of lemma 1.2. Proof of theorem 1.1. Let t 3 0. From lemma 1.2, it is clear that P(t + I) s Max@(t),

As an immediate

consequence,

G Max{&

(16)

THEOREM

(17) and (3)

there

(16)

1.3. In addition

+ b.

(17)

it is clear that (8) holds for some function to (3)-(7),

exists R > 0 such that,

+oO k(t) dt = +m.

for any function

lim ilu(t) 11s R, t--*+m

I (18)

u as in the statement

there

to+1 p’(z) dza

-

/“”

I*dz < la-’ + ad

k(z) dz

to +

to

1u’(r)

of theorem

lim 1u’(t) 1s R. t* +m

Proof of theorem 1.3. Since p(t) 3 0 and (18) holds,

I

K as stated.

assume

I0 Then,

Sup p(t)}. Ostsl

we have Sup p(t) q(O) Ostsl

From

P}.

we find Sup&t) tao

Since p’(t) s h(t)k(t),

form.

exists to s 0 such that

1.1.: (19)

1212

A. HARAUX

Proceeding

as in the proof

of lemma

1.2 with P=

SUP P(z)? [toJo+Il

we find that (15) holds with N(I) and M(f) slightly increased. right-hand side of (15) remains unchanged, we get clearly

SUP

[to,to+4 Then,

like in the proof

P(z>w*

of theorem

Remark.

of theorem

If k E L’(R+),

2. GLOBAL

SOLUTIONS

for some p* 2 I?

1.1, we conclude s Max{P,p*}

Sup&t) tat0 This ends the proof

= p”

1.3.

the conclusion AND

of theorem

1.3. may fail.

DISSIPATIVITY FOR A CLASS EQUATIONS

OF SEMILINEAR

Let Q be a bounded, monotone

graph

open subset of Rn with a smooth boundary in IF!x R, and a function g EVVt$(R). We set G(u) = Iug(B) 0

Throughout

this section,

THEOREM

2.1. We assume

I’ = &2, p a maximal

notation

H = L2(Q), V’ = H-l@),

% = {u E H, u(x) E o(p>

W = V n H@).

a.e in S2}.

exists (M, D) E [w+ x Iws such that:

that there

VUER,G(U)+M~U~~+DZO. If n > 1, we also assume

that there lg’(u)l

(21)

exists s E R+, (n - 2)s c 2 and K E IFi+such that: s K(l + luls>,

a.eon II%.

f E L1(O, T; H) and [uo, uo] E V x %, there E C(0, T; V) n C1(O, T; H) such that

Then, for any

(22)

exists one and only one u(t, x)

u(0) = u(), u’(0) = U(-J u is a weak solution

WAVE

Vu E R.

de,

we will use the following

V = H;(G),

of P in the

Since the coefficient

on [0, T

(23) ] in the sense of [6, theorem

III. 6,

p. III. 141 of the equation u” - Au + p(u’) 3

ft, n) - g(u(t, x)).

V X % X L’(0, T; H)+ Moreover, the map: (UO, ~0, f) -+ u(t) is continuous: Cl@, T; H), and we have the following additional regularity properties. If

(24) C(0, T; V) n

f E L1(O, T; V) and [uo, uo] E VV x V, then u E L”(0, T; w) n vlym(O, T; If)*

Dissipativity in the sense of Levinson for a class of second-order

nonlinear evolution equations

1213

of p If f E wyo, T; H), u. E V and [ -Au0 + &uo)] f7 H # 0, where b is the extension in the duality between V and V’, then u E W1y"(O, T; V) n W2yoc(0, T;H). If we assume that D(p> = R, then for any [uo, uo] E V x % andf E L1(O, T; H), the solution u of (23)-(24) is such that

p”(u’)

E

x Q), u” - Au E L’(]O, T[ x Q>

L'(]O, T[

and u” - Au + g(u) - f(t, x) E +(u’(t,

x)) a.e on 10, n x Sk

Proof of theorem 2.1. From (22) it is easy to deduce V - H and satisfies Vu E V, N.J E V,

that the map u + g(u) is bounded:

(25)

I&) - m I 6 fYll4~Ibll)llu- 41 where P: [w+x IRS*

IRS is bounded.

(a) Uniqueness and continuous dependence on the data Assume that u1 and u2 are two weak solution s in the sense of [6] of the respective

We set w = ui - u2. Then,

u;’ - Au1 + B(d) 3f1(t,4

- g(4)

u!i -

-

AU2

+

P(4)

3f2(u)

equations

g(uz)*

for all t E [0, 7’1:

{IIw(t)II2+ Iw’(t)12Y2 s {II40) II2+ Iw’(o) lY2 + I flfl(s) 0

-f2Ml

ds

+ I t fw1(s) ILIMS)II)lb449 IIh* 0

Thus,

with A =Su. P( llul(t) (1,/uz(t) II), we find 109

{IIw(t)II2+ Iw’(t)12P2 s exp(A mll w(O) II2+ I w’(O) w2 + I0 T Ifl(s> -

I w*

f2b)

(26)

This formula shows clearly that if fi = f2 and w(0) = w’(0) = 0, then w = 0. Also, if we prove that the map: (UO, ~0, f) -+ u is locally bounded: V x % x L1(O, T; H) + C(0, T; V) n C1(O, T; H), the continuity will follow easily from (26). (b) Solution for a regularized system In this paragraph, we assume that

P is

[uo, uo]E

Lipschi tzian,

x Kf

E

qo,

and TY V).

Then the operator w * p(w) is bounded: V- V and Lipschitz continuous: Ii+ Ii. On the other hand, the operator u ---) g(u) is bounded: VV--, V and Lipschitz continuous: V+ H in bounded subsets of MJ. Therefore, by using for example [lo, theorem 10, p. 121 we obtain at once the existence of a solution of (23), (24) in some subinterval [0, S], 6 G T, which is in the space C(0, 6; IV’) n C’(O, 6; V) n C2(0, 6; H).

1214

A. HARAUX

To prove that 6 = T, according to the standard procedure we need to find a priori bounds for u(t) in VVand u’(t) in V. As a first step, we multiply (24) by u’ and integrate over 52, thus obtaining

Let r(t) = IMt) II2+ I ~‘W I2+ 2{Wt)

2+

G(u(t)) I 52

dx + D IS2I} + 1.

Then, because of (21) (22) we have

1 + II40 112 + l u’(t)l2s 40 s Q( IIW 11, Iu’(t) I> with Q: lR+ x lR+ * IF!+ bounded. We set F =Sg If(t) 12dt. A s a consequence

of (27) we derive successively

r’(t) s 2 If(t) I Iu’(t) I + 4M(u(t),W)>

s [I + If(9 II (1+ Iw> 19+ 4M2I40 I2 s (1 + 4M2&l + If(t) I}r(t). Thus, for t E [O, 61, we have

r(t) s Q( Iluoll,Iuol)exp([l + 4M2AT1]T + F)

= R(llhIll,InIl,F)* Now we multiply (24) by -A& for all h > 0 ~$(IJ~~vu’~~ G

(28)

= -A(1 - AA)%‘,

h > 0. Setting JA = (I - AA)-‘, we obtain

+ IJ/~AU~~)

IlfOIIlb’II+ @A w> + lb IIl%w> I7vf E [OY 61

(because JA: H + H is a contraction). It is classical that (AAPV,pw) s 0, VW E H. Also:

IV(&)) I =

k’(4V~

I2 s C’ I Au I km)

In s c”l

Au 12,

where u = u(t), t E [O, 61 and C’ only depends on IIu& (ug( and F. (by (22), w + g’(w) is bounded: V + L”(S2)). Finally, with z(t) = /u’(t)ll’ + IAu(t) I2+ 1, we find z(t) s r(O) exP{C”(lluoll, (uol,F)T

+ j-*lV(9 IIdBJ,vi!E [O,q* 0

(29)

From (28) and (29), we deduce the solution u(t) is global and (28), (29) hold with 6 = T. (c) Solution

when [uo, uo] t

We consider a sequence /%, A > 0.

W x V n % and f E L’(0,

T; V)

fnE C(0, T; V) such that fn + f in L’(0, T; V) and replace p by

Dissipativity in the sense of Levinson for a class of second-order

As a consequence

nonlinear evolution equations

1215

UA,~of

of (b), we find solutions

p&l - AUA,rz + h(4,n) = fn- g(+z) It is clear by (28) and (29) that u A,nremains uniformly bounded in C(0, T; w> n C1(O, T; V). Let i& be a subsequence of the sequence u~i,,~ such that tik+ U in C(0, T; V). Then, clearly, g( &) * g(a) in C(0, T; H). A simple application of [5, theorem 3.16, p. III. 501, allows to conclude that U is a solution of (23), (24). We’ omit the details in checking the hypotheses of [5, theorem 3.161, which are very similar to [ 11, Lemma 34, p. 227-2291. Finally, because of the estimates (28) and (29) which are independent of n and A, we find that u E L”(0, T; W) n W1ym(O,T; V). This proves

the first ‘additional

regularity

property’.

(d) The General case and additional statements Finally, if [uo, uo] E V x %, we consider sequences [uom, ~0~1 E W X V n %, V) with uorn* uo in V, uom+ uo in H and fm-+f in L1(O, T; H). We use (c) to construct the solution urn(t) of

u;- bn + /q&n>3fmW) 1umc9= u()&&(O) = UOm.

lim urn(t) m_,+m

T;

- g(w?J

As a consequence of (26) and since by (27) u, is bounded in C(0, immediately that urn(t) is a Cauchy sequence in C(0, T; V). Let u(t) =

fmE L1(O,

T; V),

we can see

in C(0, T; V) strong.

It is clear that u satisfies (23) (24) since fm(V>- g(4&,x)>-+ f(t, x)- g(u
of section

Q(u) = +(I*

THEOREM 2.2.Let

f E S*([Wf, II).

1 to the solutions

+ 1 G(u) dx, Q We assume

3eO,g(u)o

W(u)

of (24). Then we have = -Au

that g satisfies

-(Ao-

(22) and

E)Iu~*- CJul

3S E IF!+,G(u) s S(1 + ilolul* + ug(u)). Moreover,

we assume

that p satisfies,

+ g(u).

(30)

for some q > 0

(31)

A.HARAUX and one of the following

conditions

(a >n

= 1 and Int(D(@)

3 0

wn

= 2 and ]@‘wl sC(l

+ 1~1’) for some r E Rs

(c) ns3and]/Pwl
there

holds

i,l”),r
exists R E [w+ such that,

for any solution

lim ]Iu(t) 11s R, t++m Remark 2.3. Conditions

(30) are fulfilled

q(u)

u E C([w+y V) n C1(Iw+, H) of (24)

lim Iu’(t)I s R. t+ +=J if, in addition

to (22)

g satisfies

s aluI’+* - E, lfu E R

the condition (32)

for some a > 0, and E E R+. An alternative sufficient condition is g(0) = 0 and g nondecreasing. Proof of theorem 2.2. Since (~0, ~0, f) + u(t) is a continuous map: V X % X L1(O, T; H) + C(0, T; V) f7 C1(O, T; H), it is enough to check the estimates proved in theorem 1.3 in the special case when [UO,V] E IV X V, p”( ~0) E H and f E C1([w+, V). Then, the compu-

tations made at theorems 1.1-1.3 are justified, provided we are able to check (3) (4) (5) (7) (with k(t) = 1). U n d er any of the conditions (b) or (c), the multivalued operator: g(t, u) = pu - f(t) maps V into V’ in a bounded manner for almost every t E R+. Just like in [4], we check for some CE IW’:ilh E Pu, IlhlI* aC(1 + (h, u)), VU E V. Thus, for all u E V and z(t) E g(t, u)

Ilmll* s Ilf(t)ll*+ cc1+ Wan4 + cm 4) s am 4 + IuI2+ Ilf(9II*+ C’lf(0 I* + c* In case (a), we first replace p by PA and make 3c-+ 0 (the associated to u). In both cases, (7) is an easy consequence of (31) since

solution

UAdoes converge

Finally, it is straightforward to check that (22) and (30) imply (3), (4) for some suitable values of a, p, Cl, C2. This ends the proof of theorem 2.2. Remark 2.4. Theorem 2.2. contains as a special case [4, theorem 1.1 p. 4251 when h = 0, and shows that the system considered in [4] is dissipative in the same of Levinson in HA(Q) x L*(Q). This result remains true if A > 0. Also, when A = 0, we may generally get the same results with f E S1(Rf, H) instead of S*, as shown by the following result. THEOREM

2.5.

In the

hypotheses (a) by

of theorem

2.2, let us replace fE S*(R+, H) by fE

t S’(R+, H) and condition

(a’ 1n

= 1 and either

D(p> = IF&

Then

the conclusion

of theorem

or -D(p) = [p, v] with ,u < 0 < v. 2.2 still holds.

Dissipativity in the sense of Levinson for a class of second-order

1217

then u’ is a priori bounded in 1 and D(p) is bounded, is enough to follow the steps of the proof of 0 E Int(D(P>) S1(R+, H). or (c) hold, we prove that for any E > 0, there exists C(E) such

Idea of the proof. If n = Locl(lR+ x B), and the condition theorems (l.l)-(1.3) when f E If n = 1 and D(p) = II%or (b) that Vv E Indeed,

nonlinear evolution equations

V, Vh E pU, llhll*s E< h, v > + C(E).

(33)

if n = 1 and D(p> = IR, we have

I

h(x) v(x) dx 2 f E

lup-l/E

lwl dx

Ijopll&

lb(x) 1dx s E 1 h(x) v(x) dx sz

3 lh(l s E< k v ’ + IQ1sup IP( (UI
Then (33) follows easily since L1(Q) G H-l@). If (b) or (c) holds, we know from [4] that

;+“-s llhll Since Ilh(l*sdlhlly ur + K(E) , (33) follows there

C(l + (h, v))

easily.

From

(33), we deduce

that for every

E > 0,

exists fE E S1([Wf) such that

Ilmll* s E(z(t)74 + E(f (t) 74

+ f E(t)*

(34)

Then, by following the lines of the proofs of theorems (l.l)-( 1.3), on using (34) with E small enough we can get an estimate on P with the weaker hypothesis f E S1(R+, IS). We omit the details which are tedious but very similar in essence to the proof of theorem 2.2. 3. COMPACTNESS

The following

AND EXISTENCE

result is an easy consequence

OF PERIODIC

of theorem

TRAJECTORIES

2.2 and the methods

4 .11. THEOREM 3.1. In addition

to the hypotheses (n - 2)s < 2,

We assume

moreover

p-l:

of Theorem na3,r<-

andif

R + IR is uniformly

2.2, we assume n+2 n - 2’

continuous

t+l

_I

t>O t SUP Then,

all solutions

If@+

h) -f(e>I

de-0

ash+O.

of (24) on lR+ are such that U(t) = [u(t), u’(t)] is uniform1 y continuous:

with a precompact

range

in X.

R+ + X

of [ 11, theorem

1218

A. HARAUX

Proof. Since u(t) is bounded in V and u’(t) is bounded in H, it is clear that g(u(t)) uniformly continuous IT%+ + H. The rest of the proof is a simple application of [ 111.

is

COROLLARY3.2: In addition to the hypotheses of theorem 3.1, assume that f is defined on II%and f: R + H is S1-almost periodic. Then, there exists at least a solution of (24) on IF!,with the range of [u(t), u’(t)] precompact in X. idea of the proof. Let ii(t) by any solution of (24) on lR+ and choose a sequence {tn}, tn + 00 such that f(tn + t) -+ f(t) in S1(R, H). Since ti: R + X is uniformly continuous and

lies in a coqpact

subset

of X, we may select a subsequence {&} of {tn} such that u’(t) respectively in C([a, b]; V) and C([a, b]; H) for all a, b with - 00 < a < b < + 00. Then u(t) is a solution of (24) on R, which is uniformly continuous with the range of [u(t), u’(t)] precompact in ‘X. ii(t + &,J + u(t) and S(t + &) *

Remark 3.3. The proof of corollary 3.2 shows actually that there exists a solution of (24) bounded on R in the ‘positive hull’ of any positive trajectory of (24). We have been until now unable to prove the existence of an almost periodic trajectory. THEOREM3.4. Assume that the hypotheses of either theorem 2.2 or theorem 2.5 are satisfied and fi R + H is T-periodic. If in addition (n - 2)s < 2, there exists at least a T-periodic curve U(t) = [u(t), u’(t)]: II%* X such that U(t) E C(R, ‘X), and u(t) is a weak solution in the sense of [6], of equation (24). Proof of theorem 3.4. We use a Schauder method and monotonicity For E > 0, A-E [0, l] we look for a eventual T-periodic solutions of the

combination of compactness arguments via the Leraytheory. uniform apriori bound in C(0, T; V) x C(0, T; H) for the system

p&A - V&J + EU&,A= 0 I v;3A - Au, ,A( + B(VE 9A>+ As a first step, for h E P(0, solution uE of

&&,A

3f(t)

- g(Au, ,A)*

T: H), we derive a priori estimates for the (unique)

(4 - VE + EU, = 0 c,d.- Au, + p(u,) + &vE= h.

(39 T-periodic

(36)

For E > 0 fixed, the map: h + [uE, u,] is Lipschitz-continuous L’(0, T; H) --) C(0, T; V) x C(0, T; H)* If h is replaced by h, E W’?‘(O, T; H), then as a consequence of [5, theorem 3.14, p. III 411, the solution uE of (36) is strong. Multiplying the second equation of (36) first by vE, then by

Dissipativity in the sense of Levinson for a class of second-order u,,

nonlinear evolution equations

1219

we obtain with zE(t) =h - u; + Au, - EU~ T

I0

W),

bW>

dt c

Tv@), I0

(37)

uxt)> d

T T( - A uE,uE) I0

dts

I0 +

(38)

Tllr,(911*IIUEWIIdt I0

- g[&(t)],

If in (37), we plug in h(t) =f(t)

(h(t), u&Q)>dt + 0TlUr(f)12df I

we obtain

IT (z,(t), M)) dt s IT If(t>I lvxt>I a + cE(I + I,‘luXtV

d)

0

0

(since (21) and (30) imply -ug(u) (22) imply T (g(&), I0

s C(l + u2)) because u, EC(O, T; V) n C1(O, T; H)

u;) dt = ; j=- G(&(t,

x)) d,]’

and

= 0. 0

Then, by the same method as for the proo on

of lemma 1.2, we can use (38) to get an estimate

au,), UJ dt, hence on

IT {; llud12 + ; j-- G(W b} dt* 0

We end up with an estimate of the form

sup tqo, sup ~lluE,*(t)l12 + lUE,A(t> I21< + O” q

O<&Q&o

OsAsl

Since the operator T(u) =f(t) - g(u(t)) is completely continuous: C(0, T; V) n Cl(O, T; H) +P(O, T; H) , we conclude from Leray-Schauder’s theory that (35) has at least a solution uE,~for 0 < E G ~0 and h E IO, 11. If we keep now A = 1 and make E+ 0, we may assume that for some sequence E, + 0, we have u,,~ =i&--,u in C(0, T;H) aswt-, +m and [ u,, z&] remains bounded in C(0, T; X). Then, by compactness, g(z&) converges in C(0, T; H) to g(u) as vz ---) + 00. As a consequence of [5, theorem 3.16, p. III 501, it is clear that u is a weak solution of (24) on [0, 7’j. This ends the proof of theorem 3.4. REFERENCES 1. AMERIO L. & PROUSEG., Abstract Almost Periodic Functions and Functional Equations, Van Nostrand-Reinhold (1971). I 2. BENILANPH & BRI~ZISH., Solutions faibles d’kquations d’kvolution dans les espaces de Hilbert, An&. Inst. Fourier Univ. Grenoble 22, 311-329 (1972). 3. BIROLI M., Bounded or almost periodic solutions of the nonlinear vibrating membrane equation. Ricer&e 22, 19&202 (1973).

Mat.

1220

A. HARAUX

4. BIRIXI M. & HARAUX A. Asymptotic behavior for an almost periodic, strongly dissipative wave equation, J. dif$ Eqns. 38, 422-440 (1980). 5. BRBZISH., Opkateurs Maximaux Monotones et Semi-groupes de Contractions dam les Espaces de Hilbert, North-Holland, Amsterdam/London (1973). 6. BRI?ZIS H., Probltmes unilatCraux, J. Math. pures appl. 51, 1-168 (1972). 7. BRBZISH. & LIONS J. L., Sur certains problkmes unilatCraux hyperboliques, Cr. Acad. Sci. Paris, Ser. A 264, 928-931 (1967). 8. BR~ZIS H. & NIRENBERG L., Forced vibrations for a nonlinear wave equation, Communs pure appl. Math. 31, l-30 (1978). 9. FICKEN F. A. & FLEISHMAN B. A., Initial value problems and time-periodic solutions for a nonlinear wave equation, Communs. pure appl. Math. 10, 331-356 (1957). 10. HARAUX A., Nonlinear evolution equations: global behavior of solutions, Lecture Notes in Mathematics 841, Springer, Berlin (1981). 11. HARAUX A., Almost periodic forcing for a wave equation with a nonlinear, local damping term, to appear. 12. LERAY J. & SCHAUDER J., Topologie et Cquations fonctionnelles, Annls scient. Ecole norm. sup. Paris 51, 4578 (1934). 13. LIONS J. L. & STRAUSS W. A., Some nonlinear evolution equations, Bull Sot. math. France 93, 43-96 (1965). 14. PRODI G., Soluzioni periodiche della equazione delle onde con termine dissipativo non lineare, Rc. Semin. mat. Uniu. Padova 33 (1966). 15. PROUSE G., Soluzioni quasi periodiche della equazione delle onde con termine dissipativo non lineare, I, II, III, IV, Atti Accad. naz. Lincei Rc. 38-39 (1965). 16. STRAUSS W. A., On continuity of functions with values in various Banach spaces, Pacific .I. Math. 19 543-551 (1966).