Annali di Matematica pura ed applicata (IV), Vol. CLX (1991), pp. 193-206

~-Contractions and Attractors for Dissipative Semilinear Hyperbolic Equations

and Systems (*).

SUELY SIQUEIRA CERON - ORLANDO LOPES

S u m m a r y . - In this paper we discuss the existence of compact attractor for the abstract semili- near evolution equation it = Au + f (t, u); the results are applied to damped partial differen- tial equations of hyperbolic type. Our approach is a combination of Liapunov method with the theory of a-contractions.

1. - I n t r o d u c t i o n .

We consider an abstract semilinear evolution equation:

(1) it = A u + f ( t , u)

where A: ~(A) r X--~ X generates a strongly continuous C semigroup of linear opera- tors {T(t), t 1> 0} in a Banach space X and f satisfies the following basic hipotheses: fi R • X--~ X is continuous, maps bounded sets into bounded sets and is lipschitzian on u for (t, u) in bounded sets.

By a mild solution of (1) defined on J = [to, to + a] we mean a continuous function u: J ~ X that satisfies the integral equation

t

u(t) = T(t - to) u(to) + t T(t - s ) f ( s , u(s)) ds.

to

A strong (or strict) solution is a C 1 function u: J-->X such that u(t) e (~(A) and (1) is satisfied on J. Under the assumptions above we can show local existence, uniqueness, continuous dependence and continuation of the mild solution u ( t ) = ~(t, to;uo), t I> to, u(to ) = ~(to, to; uo) = uo. Moreover, ~(t, to; uo) is C 1 with respect to Uo i f f i s C 1 with re- spect to u; finally, u(t) is a strong solution if Uo is in 0~(A) and f(t , u) is C 1 in both variables.

In this paper we assumef is p-periodic in t and we are interested in the existence of compact invariant attractors and p-periodic solutions. The literature about the sub- ject is quite big; see for instance [1], [2], [3] and [4] for some accounts in the theory.

(*) Entrata in Redazione il 18 maggio 1989. Indirizzo degli AA.: S. SIQUEIRA CERON: IBILCE-UNESP, S. J. Rio Preto, SP-Brasil; O.

LOPES: IMECC-UNICAMP, Caixa Postal 6065, 13.100 Campinas, SP-Brasil.

194 S. S I Q U E I R A C E R O N - O . LOPES: a-Contractions and attractors etc.

The point if view adopted here is the same as in HALE and LOPES [ 5 ] and LOPES [6]; namely, we give conditions under which the Poincar~ map S(uo) = ~(p, 0; u0) is an a- contraction and uniform ultimately bounded. The uniform ultimate boundedness, as usual, is obtained through Liapunov functions. As far as we know, ~-contractions which have been encountered in the applications are contractions plus a compact map; here we have to enlarge that class because (at least apparently) it does not include the Poincar~ maps arising in the applications we are going to present. We also discuss the regularity of complete solutions lying in the attractor.

2. - The abstract theory.

Let X be a Banach space with norm I" I. Firstly we consider a discrete dynamical system, that is, a map S: X - . X.

DEFINITION. - For any bounded set A of X we define ~(A) as the infimum of the numbers d such that A can be written as a finite union A = F1 w ... u F~ of sets with diam F~ <~ d, i = 1, ..., n where diamF~ denotes the diameter of Fi.

D E F I N I T I O N . - A map S: X--~ X is an a-contraction if there is a number 0 <~ q < 1 such that ~(S(A))<. q~(A) for any bounded set A of X.

DEFINITION. - If a map S: X ~ X is given, we say A c X is a compact invariant at- tractor (under S) if A is compact, S(A) = A and, for any x c X, d(S ~ (x), A)) tends to zero as n--. + ~.

REMARK. - In [3] several concepts of attractors are discussed. Actually, in the ap- plications we will present the attractors attract bounded sets.

DEFINITION. - A map S: X-~ X is uniform ultimately bounded if there are fun- ctions a(R) and N(R) and a constant R0 such that for any u satisfying lul ~< R we have IS n (u) I <. a(R) for n/> O and }(S ~ (u) I <~ Ro for n <. N(R).

The proof of the assertions of the next theorem can de found in [3], [5] and [7].

2.1. THEOREM. - If S: X - - . X is uniform ultimately bounded, a-contraction and continuous then

a) there is a compact invariant attractor;

b) S has a fixed point.

The prototype of a-contraction is given by the next result whose proof can be found in[3] and [5]. First let us recall that the essential spectral radius re(U) of a bounded linear operator U e L(X) is the infimum of the numbers r > 0 such

S . S I Q U E I R A C E R O N - 0 . LOPES: a-Contract ions and at tractors etc. 195

that the part of the spectrum ~(U) of U which is outside the ball of radius r consists of a finite number of eigenvalues each one with finite multiplicity.

2.2. THEOREM. - If S = Q + K where Q is a contraction (that is, I Q ( x ) - - Q(y)I <~ ql x - yl, 0 < q < 1) and K is a compact map then S is an a-contraction (with the same q). Also, if S = U + K where K is compact and U is linear with re (U) < 1 then S is an a-contraction with respect to a norm II II in x which is equivalent to I" [.

Next we are going to enlarge that class of a-contractions because (at least appa- rently) it does not include the period maps arising in the applications we have in mind.

DEFINITION. - A pseudo-metric ~ in X is said to be precompact (with respect to the norm of X) if any bounded sequence (in norm) has a subsequence which is Cauchy with respect to p.

The proof of next lemma is standard.

2.3. LEMMA. - If p is a precompact pseudo metric and A c X is bounded then for any ~ > 0 there is a finite number of sets C1...C~ such that A = C1 u . . . w C~ and ~(x, y) < ~ if x, y e Ci.

2.4. THEOREM. - If S: X ~ X satisfies

IS(x) - S(y)I <~ qlx - Yl + ~(x, y),

where 0 ~< q is a constant and ~ is a precompact pseudo metric then ~(S(A)) <- qa(A) for any bounded set A. If particular, if 0 ~< q < 1 then S is an a-contraction.

P R O O F . - For any s > 0 there are sets F 1 , . . . , F ~ such that

A = F1 u . . . u F~ and diam Fi < a(A) + ~, i = 1, . . .n.

From Lemma 2.3 we know there are sets C1,.. . , C~ such that

A = C1 w... w C~ and p(x, y) < ~ if x, y e Ci �9

Since A = w (Fi • Cj ) we conclude S(A) = u S ( F i n Cj ). Moreover for x, y e F i u Cj we have

IS(x) - Z(y)l <- qlx - Yl + p(x, y) <~ q(a(A) + ~) +

and then diam S(Fi (~ Cj) <. qa(A) + ~(q + 1) and the theorem is proved.

REMARKS. - 1) From the proof of last theorem it is clear that it is not necessary to assume that ~ is globally defined; in other words, all we have to assume is for each R there is such a PR (X, y) defined for ]xl, lYl ~< R.

2) Theorem 2.2 is a consequence of Theorem 2.4 because the pseudo metric p(x, y) = IK(x) - K(y)I is precompact ff K is a compact map.

Now we go back to the equation (1) of the introduction and we consider the non autonomous (periodic) flow ~(t, t0;u0) defined by it.

196 S. S I Q U E I R A C E R O N - 0 . LoPEs: ~-Contractions and attractors etc.

D E F I N I T I O N I - We say equation (1) is uniform ultimately bounded if there are fun- ctions a(R), z(R) and a constant Ro such that if lu01< R then I~(t, t0;uo)l ~< a(R) for t >~ to and I~(t, to;Uo)l <Ro for t >~ to + z(R).

DEFINITION. - A set A is invariant under equation (1) if for any Uo in A and any to there is (mild) solution u(t) of (1) defined for t e ( - ~ , ~) such that u(to)= uo and u(t) e A for any t.

REMARK. - In the applications we are going to present the Cauchy problem is well posed also for negative time but the definition of invariant set does not required that.

DEFINITION. - A set A is an attractor for equation (1) if, for any to and Uo, d(~(t, to;Uo),A) tends to zero as t ~ + ~.

The next theorem is a version of Theorem 2.1 for flows (see [3], [5] for an indica- tion of the proof).

2.5. THEOREM. - Iff(t, �9 ) is p-periodic in t, equation (1) is uniform ultimately boun- ded and the Poincar~ map S(uo)= ~(p, 0;u0) is an ~-contraction then:

a) there is a compact invariant attractor;

b) there is a periodic solution with periodic p.

2.6. COROLLARY. - Iff(t, u) is a compact, p-periodic in t and re (T(p)) < 1 and equa- tion (1) is uniform ultimately bounded then the conclusions of Theorem 2.5 hold.

PROOF. - It is a consequence of the variation of the constants formula and Theorem 2.2.

3 . - A p p l i c a t i o n s .

We start with a transmission line problem. The differential equations for the cur- rent. The differentialequations for the current i(t, x) and the voltage v(t, x)are:

L ai = av Ri , caV =-)-~i _ G v , ~t ax at ax

with boundary conditions

E(t) = v(t, O) + Ro i(t, 0) at x = 0

and

d t)+g(v(t,t)) at x /. i(t, l) = C1 -~ v(t, =

Assuming E(t) is C 1 we can make the boundary condition at x = 0 homogeneous deft-

S. SIQUEIRA CERON - O. LOPES: ~-Contractions and at tractors etc. 197

ning I(t, x) = i(t, x) and V(t, x) = v(t, x) - E(t), and the differential equations beco- me

~_I_/= 1 aV R I ~_VV = 1 ~I G V + hi (t) ~t L 3x L ' ~t C ~x C

and boundary conditions

and

with

V(t, O) + Ro I(t, O) = 0 at x = O

I(t, l) = C 1 d V(t, l) -+- g(V(t , l) + E(t)) + h2 (t) at x = l

and

and

f ( t , u ) = ( ( O , h l ( t ) ) , - - ~ l g ( b + E ( t ) ) - ~ l h ~ ( t ) ) �9

It is not difficult to show that A is the infinitesimal generator of a strongly continuous Co semigroup T(t), t >t 0 in H. Moreover, if, E(t) and g(u) are C 1 functions then f is continuous in (t, u) and lipschitzian in u for (t, u) in bounded sets.

Let ~ and r be constants defined by:

e= 2v k- 5 -

~" = 2VR-G + - C L

G dE(t) h i ( t ) = - E ( t ) - dE(t) and h2(t)=C1 d---~

dt

The problem above can be viewed as an abstract evolution equation

(2) it = A u + f (t, u) ,

in the real Hilbert space H = Le [0, l] • L2 [0, l] • R with norm

l l

0 0

where A: 6)(A)r H--* H is given by

- -

(~(A) = {(r ~, b) e H: r ,~ e H 1 [0, l], ~(0) + Ro r = 0, b = ~(/)}

198 S. S I Q U E I R A C E R O N - O. LOPES: a-Contractions and attractors etc.

and

g(Y) m = lim i n f -

lyl~+~ Y

3.2. THEOREM. - Suppose the following conditions are satisfied:

(i) g and E are C 1 and E is p-periodic;

(ii) m > 2 V ~

~ > -2-~0 Z- +

�9 ,

g - c c

- - - - - I ] if ~ < R 0 < r or

if Ro ~ ~ or R0 >I ~'.

Then equation (2) has a compact invariant attractor and a p-periodic solution.

PROOF. - Let u(t) = (I(t, x), V(t, x), V(t, l)) be a strong solution of (2) for t >I to and let us consider the functional

l

CI -~11 a s c V S ( t , l ) + f [ C v s ( t , x ) + 2 a I ( t , x ) V ( t , x ) + l S ( t , x ) J d x . W(t) = --ff ~ - o

An elementary but tedious computation shows that, under (i) and (ii), the constant a can be chosen in such way that:

(i)

dW(t) (ii)

dt

ks Iu(t)L s <- w( t )k~ lu(t)Ls;

<~ -OW(t) + M,

where kl, ks, M, 0 are positive constants; from (ii) we get

and then

W(t) <~ e-~176 + M__ 0

kt -o(t- to) M lu(t)12 < --k2 e )u(t~ + ks 0 "

Since the set of the initial conditions for which u(t) is a strong solution is dense, we conclude the above estimate for l u(t) I holds for any mild solution and then (2) is uniform ultimately bounded. Furthermore, if g - 0 and E---0 we see that u(t) = T(t)uo. In this case the constant M above can be taken 0 and then

kl -or~2 IT(t) l <. k2 e

S. SIQUEIRA CERON - 0. LOPES: a-Contractions and a t t r a c t o r s e t c . 199

which implies

re (T(p))I <~ r(T(p)) < e-~ < 1

and the theorem follows from Corollary 2.6.

R E M A R K S . - 1) The essential spectral radius re (T(t)) is known exactly ([8]) and its value is e ~ot, where

R C + LG 1 In !Zo - Ro I ~o = 2LC 21~/--~ Zo + Ro

2) If ~ = i n f ( g ( y ) - g ( z ) ) / ( y - z) satisfies the same conditions as m in theorem 3.2 then the p-periodic solution is unique and globally asymptoticaly stable.

3) LOPES [6] has studied existence and stability of a periodic solution for the transmission line problem with R - - G = 0. In this case it is possible to reduce the hyperbolic system to a differential equation of neutral type. In the general case pre- sented here such a reduction is impossible.

As a second application we consider the second order equation

(3) utt - Cu + h(ut) + g(u) = e(t) ,

in a real Hilbert space H with scalar product <, > and norm I'l; together with (4) we consider also the equivalent first order system:

I n t = V ,

(4) [vt = Cu - h(v) - g(u) + e(t) .

We assume C: 0)(C) r H ~ H is a selfadjoint negative definite operator (in particular, <Cu, u> <~ -;(1 [u] 2 for some ~1 > 0 and any u in 0)(C)). We are going to look at (4) as an abstract equation

W = A w + f ( t , w), w = (u, v) ,

in the real Hilbert space

X = (P(-C) 1/2 x H ,

with norm I]wll 2 = I ( -c)V2ul + Iv[ 2, where

A w = (v, Cu),

6)(A) = ~ ( - C ) x (~(-C) 1/2

and

f (t, w) = (0, - h ( v ) - g(u) + e(t)) .

As a consequence of Stone's theorem A generates a Co unitary group T(t), - co < t < 0% because A is skew-adjoint. We assume h: H--) H and g: ( ~ ( - C ) 1/2 --> H are lipschitzian on bounded sets, map bounded sets into bounded sets and e: R--, H in continuous. These conditions guarantee the local existence and uniqueness of mild so- lutions for (4). It is important to notice that if w(t) = (u(t), v(t)) is a mild solution then u: R - o H is C 1 and du( t ) /d t=v( t ) .

200 S. S I Q U E I R A C E R O N - 0 . LOPES: ~-Contract ions and at tractors etc.

3.6. THEOREM. - Besides the basic assumptions made above suppose the following conditions are satisfied:

H1) there is a C 1 function G: (~(-C)1/2-~ ~ bounded below on 0)(-C) 1/2 and bounded above on bounded sets such that g(u) = grad G(u) with respect to the scalar product of H;

H2) (u ,g (u ) ) is bounded below on (~(-C)1/2;

Hs) h(O)=O and there are constants a , ~ > 0 such that ~lvl 2<< . ( v ,h (v ) ) and th(v)l <. ~lvl for any v in H;

H4) e: R ~ H is bounded.

Then system (4) is uniform ultimately bounded.

PROOF. - Consider the functional:

W(u, v) = + -~l(-C)l/2ul~ + 2b(u , v) + G(u) ,

where b > 0 will be chosen satisfying several conditions the first of which being b2< i(1/16; with this choice of b we see there are constants k2 > kl > 0 such that

I(_C)l/2 u[2 + 2b ( u, v } <<. k2 (Iv]2 + ](-C)1/2u[ 2) 1 kl<lvl ~ + I(-C)i/~ul 2) ~ +

for any (u, v) in X. The derivative of W along the solutions of (4) is given by

~V = - ( v, h(v) ) + 2b ( v, v) - 2b ( u, h(v) } - 2bl(-C)l/2 ul) 2 +

+(v, e(t)) - 2b(u ,g (u ) } + 2b<u, e(t)} <~ (25 - a)lv] ~ +

+2bfllu]lv I - 2bl(-C)~/2ul 2 + (v, e(t)} - 2 b ( u , g ( u ) ) + 2b(u , e(t)).

Next we choose b > 0 in such way that

(25 - ~)lvl ~ + 2b~lul Iv l - 2bl(-C)~'~ul ~ <- -O(tvl ~ + I(-0)1/27~[2 ) ,

for some 0 > 0; in order to achieve that we use

2bZ 2b~lullv ] ~ ~ ](-C)l/2u][vl

b 2 fl 2 and impose ~ < 2b(a- 2b), which is satisfied if

2a b < - - Z 2

4 + - -

Taking in account the hypotheses H1), H2) and //4) and using the elementary inequalities

(v, e(t))

S. SIQUEIRA CERON - 0. LOPES: a-Contractions and attractors etc. 201

and

m 2 2 2-~le(t)t2<~ m2 2 - ~ Ie(t)12 <v, e(t)> <~ y lU + l(-C)l/2ul 2 +

with m small we see that IV ~< -O/2(Ivl 2 + ](-C) 1/2 ul ~ ) + N where N is a suitable con- stant. Using H1 and the previous estimates we see that there are functions a(r), b(r), and c(r) that go to + ~ as r goes to + :~ such that a(r) <~ W(u, v) <. b(r) and W(u, v) ~< -c(r), r e= ]v12+ I(-C)l/2ul 2 and then, argueing as in [9], we conclude (4) is uniform ultimately bounded.

3.7. COROLLARY. - If Q: H---)H is a bounded positive definite selfadjoint operator in a Hilbert space H and C, h, g and e(t) satisfy the conditions in theorem 3.6 then the equation

(5) Quit + Cu + h(ut) + g(u) = e(t) ,

is uniform ultimately bounded.

PROOF. - Writing this last equation in the form

utt + Q-1 Cu + Q-1 h(ut ) + Q - l g (u ) = Q-1 e(t)

and defining a new (equivalent) scalar product in H by

[ul , u2 ] = ( Qul , u2 > = ( ul , Qu2 > it is easy to chech that Q-1C is selfadjoint negative with respect to [, ]. Moreover if g(u)=grad G(u) with respect to ( , > then Q - l g ( u ) = gradG(u) with respect to [, ] and

[u, Q-lg(u)] = <u, g(u)>;

finally [v, Q-lh(v)] = <v, h(v)> and the conclusion follows from Theorem 3.6. As an example of this type consider a system (see [10])

Ian utt + a12 Utt = -C1 u - hi (ut) - gl (u) + el (t) ,

(6) [a21utt+a22Utt _ C 2 U _ h z ( U t ) _ g z ( U ) + e 2 ( t ) '

in the product space H = H0 x Ho where the 2 x 2 matriz (aij) is selfadjoint positive definite, and C1 and C2, hi and h2, and h2, gl and g2, el and e2 satisfy the same assure- ptions as C, h, g and e in Theorem 3.6. In order to verify that (6) is in fact of type (5) it is enough to take in H = H0 x H0 the scalar product

(( (ul , u2), (Vl , u2 ) )) = < ul , > + < v l , u2 >.

What we are going to do next is to give conditions under which we can guarantee that the period map S is an a-contraction; we study two different situations.

3.8. THEOREM. - Besides hypotheses H 1 t o H 4 in Theorem 3.6 we suppose that the following conditions are satisfied:

Hs) e(t) is p-periodic;

202 S. SIQUEIRA CERON - O. LOPES: ~-Contractions and attractors etc.

and

H6) there are constants 0 < ~ < ~ such that

~IVl--V2}2~<Vl--V2, h (Vl ) -h(v2)>

Ih(v l ) -h(v2) l <<-~lvl-v21, for any vl , v2, in H;

HT) 0)(-C) 1/2 is compactly imbedded in H; Hs) g: 0~(-C) 1/2--* H is compact.

Then the period map S is ~-contraction; in particular there is a mild p-periodic sol- ution and a compact attractor.

PROOF. - Let w 1 = (Ul , V 1 ), W 2 = (U2~ V 2 ) be two points in X and wl (t), w2 (t), t >i O, be the solutions through them. From the uniform ultimate boundedness we know wl(t), w~(t) remain bounded by some constant R~ for all t~>0. Consider the functional

1 1 E = ~ l v ~ - v 2 1 + 2 b ( u ~ - u 2 , v l - v ~ > + -~[ ( -C) l /2 (u l -u2) l 2,

where b is chosen as in the proof of theorem 3.6. Using//6) it is easy to verify that the derivative of E(t) along the pair wl (t), w2 (t) of solutions satisfies

dE(t) d-----~ <" -OE(t) + ( vl (t) - v~ (t), g(u2 (t)) - g(u~ (t))> +

+2b<Ul (t) - u 2 (t), g(u2 (t) ) - g(u~ (t)) >,

where 0 > 0 is a constant and then

E(p) <~ e-VE(O) + K sup lu~ (t) - us (t)l + K sup Ig(u~ (t)) - g(u2 (t))l ~/2 , K constant, O<.t<.p O<.t<.p

and

IItS(w2 ) - S ( W l )]11 ~ e -0p/2 illw 2 _ Wl tll "~ ,~1 (w2, Wl ) -~ ~o2 (w2 , Wl )

where

and

1 iv l2+2b(u,v> + 1 1 ( _ C ) 1 / 2 u l 2 ILIwlll

~1 (W2, W l ) = K 1/2 sup tul (t) - u2 (t)l 1/2 �9 o~t<~p

p(w2, wl ) = K 1/2 sup Igl (ul (t)) - g(u2 (t))l TM �9 O<<.t<~p

Let (us, Vn) be a bounded sequence with respect to the norm �89 2 + �89 from 1 the uniform ultimate boundedness we conclude ~lv~(t)12+�89 2 remains

bounded by a fLxed constant for t in [0, p] and HT) implies us (t) has compact closure in H for each fixed t; moreover,

lUn( t l ) -un( t2) l <~ ]tl"---t21 sup ]vn(s)]

and Arzela-Ascoli theorem gives the precompactness of the pseudometric p with re-

S. SIQUEIRA CERON - 0. LOPES: ~-Contractions and attractors etc. 203

spect to the norm

1 Ivl +

and, hence, to the norm IIIwlll (because they are equivalent); with a similar argument we verify ~ is also precompact and the conclusion follows from Theorem 2.5.

3.9. COROLLARY. - Consider the semilinear wave equation

Utt = AU - h(u t ) - g(u) + e(t, x)

in a smooth bounded domain ~ r R n , n ~ 1, 2, 3, with Dirichlet boundary condition: u = 0 on ~ and assume the following conditions be satisfied:

U

H i ) the funct ions ' ]= f g ( s )ds is bounded below for u in R; o

Hi ) the functions ug(u) is bounded below for u in R;

H~ ) h(O) = O, h: R---~R is C 1 and 0 < ~ ~< h' (v) </~ for some constants a and ~;

H~ ) e: R ~ L2 (t~) is p-periodic and continuous;

H~ ) g:R--~ R is C 1 and satisfies the growth condition:

Pg' < a + bJ l

), < 2 ff n = 3, any ~, if n = 2, and no restriction on the growth if n = 1. Then the equa- tion above has a compact invariant attractor and a p-periodic solution.

PROOF. - It is not difficult to verify that the functional

G(u) = f '~(u(x)) dx , Q

is well defined on (~(-A)I /~=H~(t~) and satisfies H I ) in theorem 3.6; also h: L2 (t~)--~/-2 (t~) defined by h(v)(x) = h(v(x)) and g: Ho 1 (5) ~ L2 (~) defined by g(u)(x) = g(u(x)) are lipschitzian on bounded sets. The other hypotheses in theorem 3.8 are obviously satisfied and the conclusion follows.

REMARKS. - 1) NAKAO [11] has proved a result similar to the one given by Corol- lary 3.9 allowing ~ ~< 3 for n = 3, but requiring e(t, x) be bounded by a fixed con-

stant M0. The case g - 0 has been treated by Prodi and Prouse with less restriction on the

growth of h (see [9] for these references). In [12] Prodi studies the case of linear gro- wth in h and g ~ 0, but he puts a very strong restriction on the growth of g.

2) If the wave equation in Corollary 3.9 is taken with Neuman boundary condi- tions then the same conclusion holds if there is a constant c > 0 such that g(u) - cu sa- tisfies the hypotheses Hi' and Hi.

The next result deals with equations governing vibration of beams.

204 S. SIQUEIRA CERON - O. LOPES: ~-Contractions and attractors etc.

3.10. THEOREM. - Assume the term g(u) in system (5) has the form g(u)= = m(<Bu, u>)Bu where m: R--->R is a C 1 function and B: (~(-C)1/2-~ H is bounded non-negative selfadjoint with respect to <, >. Suppose also that the following condi- tions are satisfied:

?*

H1) the function M ( r ) = ~m(s)ds is bounded below for r i> 0; 0

H2) the function r re(r) is bounded below for r I> O;

H3 and H 4 a s H8 and H 4 in theorem 3.6. Then system 4 is uniform ultimately bounded. Moreover, if assumptions Hs, H6 and H7 of theorem 3.8 are satisfied then the period map S is an ~-contraction; in particular, there is a compact invariant at- tractor and a p-periodic solution.

PROOF. - Defining G(u) = �89 u>) we see that g(u) = grad G(u) with respect to the scalar product of H and assumptions H1 - H4 of Theorem 3.6 are satisfied and this gives the uniform ultimate boundedness. Let (ul, vl), (u2, v2) two initial data in X, (ul(t), vl (t)), (u2(t), v2(t)) the solutions through them and define the function

F(t) = E(t) - I m( < Bu~ (t), U 1 (t) > ) < U 1 (t) -- U 2 (t), BU 2 (t) - B u I (t) >,

where E(t) is as in theorem 3.8. Using the previous inequality for dE(t)/dt we see that

dF < _ o(r(t) + 2 m( <Bu~ (t), Ul (t) }) < u~ (t) - u2 (t), Bu2 (t) - B u I (t) >) -~- dt

+ (m(<Bu2 (t), u2 (t) >) - m(<Bu~ (t), u~ (t)))) <vl (t) - v2 (t), Bu2 (t) } +

+2b ( Ul (t) - u2 (t), g(u2 ( t) ) - g(u~ ( t) ) >

- m ' ( < Bu~ (t), ul (t) > )< Bu~ (t), vl (t)> <u~ (t) - u2 (t), Bu2 (t) - Bu l (t)>,

where 0 > 0 is a constant; using

<BUl, Ul > - <Bu2, u2 > -~ <B(u 1 -~ u 2 ) , U l - u 2 >,

the fact that m is C 1 and the uniform ultimate boundedness we get

F(p) <. e -VF(0) + K1 sup lul (t) - u2 (t)l, o<~t<~p

E(p) <~ e-VE(O) + K2 s u p t u l ( t ) - u 2 ( t ) [ , o<<.t<~p

and then IIiS(wl) - S(w2 )[11 ~ e-V/e IIIw~ - w2 III + p(wl, w2 ) where II1 II] is an in Theorem 3.8 and

~(Wl, W2) ----- V~2(O<~t~pSUp lu: ( t )-u2(t) l : /2) .

The precompacteness of ~ is proved in the same way as in theorem 3.8. The conclu- sions follow from theorems 2.4 and 2.5.

S. SIQUEIRA CERON - 0 . LOPES: a-Contractions and attractors etc. 205

REMARK. - Theorem 3.10 holds also for systems of type 3.9 provided g l ( U )

and g2 (u) are of the same type as g(u) in that theorem.

3.11. COROLLARY. - Consider the equation:

utt = -A~u - h(ut) - m(Igrad Ut2L~) Au + e(t, x) ,

in a bounded smooth domain ~9 r R 3 and Dirichlet boundary conditions

u = 0, 3__uu = 0, 3n

on at) and suppose that the following conditions are satisfied:

(i) h : R ~ R is C 1 and 0<a~<h ' (v)~<f l for some constants a, fl and any v in R;

?-

(ii) m: R - , R is C 1 and M(r)= fg(s)ds and rg(r) are bounded below in R+; 0

(iii) e: R -~ L2 (t)) is p-periodic and continuous.

Then the conclusion of previous theorem hold. An important question about attractors is whether they have finite topological di-

mension. If the damping term h(ut) is linear then the attractors whose existence we have proved have finite topological dimension (see [3], [13]). For nonlinear damping there is a technical difficulty because the operator h: L~(t))---~L2(t~) given by h(v)(x) = h(v(x)) is not Frechet differentiable.

A last word about regularity of solutions belonging to the attractor. There is a ge- neral result due to D. HENRY [14] about regularity of solutions defined on intervals of type ( - ~ , to]. That result can be applied to our examples provided the damping is li- near. Nevertheless, even in the case of nonlinear damping, we can give a direct proof for the regularity of solutions belonging to the attractor.

REFERENCES

[1] A. V. BABIN - M. I. VISHIK, Regular attractors ofsemigroups and evolution equations, J. Math. Pares Appl., 62 (1983), pp. 441.491.

[2] J. M. GHIDAGLIA- R. TEMAM, Attractors for damped nonlinear hyperbolic equations, J. Ma- th. Pares Appl., 66 (1987), pp. 273-319.

[3] J. HALE, Asymptotic Behavior of Dissipative Systems, to be published by AMS. [4] A. HARAUX, Two Remarks an Dissipative Hyperbolic Problems, Seminaire du College de

France, J. L. Lions, Ed. Pitman, Boston (1985). [5] J. HALE - O. LOPES, Fixed point theorem and dissipative processes, J. Diff. Eq., 13 (2)

(1973), pp. 391-402. [6] 0. LOPES, Stability and forced oscillations, J. Math. Anal. and Appl., 55 (3) (1976), pp.

868-698. [7] R. NUSSBAUM, Some asymptotic fixed point theorems, Trans. Amer. Math. Soc., 176 (1973),

pp. 349-375.

206 S. SIQUEIRA CERON - O. LOPES: ~-Contractions and attractors etc.

[8] A. F. NEVES - H. S. RIBEIRO - O. LOPES, On the spectrum of evolution operators generated by hyperbolic systems, J. Functional Analysis, 67 (1986), pp. 320-344.

[9] T. YOSHIZAWA, Stability theory by Liapunov's second method, Math. Soc. Japan, Tokyo (1966).

[10] N. G. ANDRADE, On a nonlinear system of partial differential equations, J. Math. Anal. and Appl., 91 (1) (1983), pp. 119-130.

[11] M. NAKAO, Bounded, periodic or almost periodic soluctions of nonlinear hyperboloic par- tial differential equations, J. Diff. Eq., 23 (3) (1977), pp. 368-386.

[12] G. PRODI, Soluzioni periodiche di equations a derivati parziali del tipo iperbolico nonlinea- r/, Ann. di Mat., 42 (1956), pp. 25-49.

[13] R. MAI~E, On the Dimension of the Compact Invariant Sets of Certain Nonlinear Maps, Lecture Notes in Math., Vol. 898, Springer-Vertag (1981), pp. 230-242.

[14] D. HENRY, Private communication.