On a nonlinear evolution equation and its applications

NonlinearAnalysis, Theory,Methods &Applications, Vol. 24, No. 8, pp. 1221-1234, 1995 ElsevierScienceLtd Printed in Great Britain

Pergamon

0362-546X(94)00193-6

ON A NONLINEAR EVOLUTION EQUATION AND ITS APPLICATIONS? ANGELA C. BIAZUTTI Instituto de Matemfitica, UFRJ, CP 68530-Cep 21944, Rio de Janeiro, Brazil

(Received 8 August 1993; received for publication 17 August 1994) Key words and phrases: Asymptotic behavior, global weak solutions, nonlinear evolution equations, nonlinear hyperbolic problems.

1. I N T R O D U C T I O N

We are concerned with global existence and asymptotic behavior for weak solutions of the abstract Cauchy problem u"(t) + A u ( t ) + B ( t ) u ' ( t ) + Gu'(t) = f ( t )

(1.1)

u(O) = Uo u'(O) = u I

(u"(t) = d Z u ( t ) / d t 2, u ' ( t ) = d u ( t ) / d t ) , where A and G are nonlinear operators. Various

examples of equations of type (1.1) arise in physics; for instance, if 3

Au=- ~ ~

<(Ux)

and

B(t)u' = -Au',

(1.2)

i=1

the equation represents a longitudinal motion of a viscoelastic bar obeying the nonlinear Voight model. Existence theorems concerning global solutions of (1.1) have been given by many authors, under some appropriate additional conditions (see, e.g. Crippa and Biazutti [1], Tsutsumi [2] and Yamada [3]). For the asymptotic behavior of solutions of (1.1), Nakao [4, 5], Yamada [6] and Zuazua [7] have results under the appropriate assumptions on A and B(t). In all of the results above G - 0 and the operator A is the Fr6chet derivative of a convex functional. Dang Dinh Ang and Pham Ngoc Dinh [8] have studied the existence and uniqueness of solutions of (1.1) in the special case when A and B(t) are like (1.2) and G is the monotone operator Gu' = lu'l~sgn(u'), 0 < ,~ <_1. In the present work we obtain global solutions of the problem (1.1) and we study its asymptotic behavior, assuming that the operators A and G are not (necessarily) monotone. We also prove that there is uniqueness of solutions in a special case. The content of this paper is as follows. In Section 2 we introduce some notation, give some assumptions on A , B ( t ) and G and prove the existence and uniqueness theorems. In Section 3 we study the asymptotic behavior of the solutions obtained in Section 2, with f = 0 to avoid technicalities, following the 1-This work was partially supported by CNPq/process no: 201657/91-0. 1221

1222

A.C. BIAZUTI'I

ideas of Nakao [4]. Finally, in the last section, we give some examples to which our results can be applied. 2. G L O B A L

SOLUTIONS:

EXISTENCE

AND UNIQUENESS

Let V and H be real separable Hilbert spaces with Hilbert structures given, respectively, by ((.,.)), I1.11, (.,.),1.1 and X and Y two real separable reflexive Banach spaces. We assume X c V c H and X c Y c H where the inclusion mappings from V into H and from X into Y are compact and all the others are continuous. Moreover, we assume that X is dense in V, H and Y. We identify H with its own dual H', so that the following inclusion relation holds: X c V c H c V' c X' and X c Y c H c Y' c X ' . The natural pairing between u ~ V' (resp. X', Y') and v ~ V (resp. X , Y ) is denoted by ( u , v ) . Besides these spaces we are given two nonlinear operators A and G and a linear operator B(t) which satisfy the following assumptions.

Assumption 1. A: X ~ X' is an operator that takes bounded sets into bounded sets, such that, for all T > 0, if u n ~ u in L2(0, T; X) weakly, Au~ ~ X in L2(0, T; X ' ) weakly and lim sup p~

oc

f0

(Aum (t),Um (t))dt<

f0

(X(t),u(t))dt

then g =Au, where (urn), is a subsequence of (Urn) m. There exist three real positive functions tr(.), ~" and ~, such that ~- and ~ are continuous and lims_~+® z ( s ) = + ~ . For each u ~ C I ( R + ; X), tr(u(t)) is differentiable on t, (Au(t), u'(t)) > (d/dt)cr(u(t)) and, for each v ~ X , ~0(Ivlx) >_ o-(v) >_ r(lvlx).

Assumption 2. For each 0_ b e [[u[[2, for u, v ~ V, t ~ R +, where b I and b 2 are positive constants. For each u, v ~ V, the mapping t--* b(t, u, v) is in Cl(g~ +) and, if vn ~ v weakly in V, then, for each T > 0, lim sup n..-~oc

f0Tb ' ( t , v , , v n ) d t <_ f0Tb ' ( t , v , v ) d t .

Assumption 3. G: H ~ Y' is an operator which takes bounded sets into bounded sets, such that (Gu, u) > - K ( 1 + lu12), for u ~ Y, where K > 0. For all T > 0, there is a p > 1, such that, if Wm ~ W strongly in L2(0, T; H), then there exists a subsequence of(win)m, (Wmv)v, such that GWm~~ Gw in L P(0, T; Y') weakly. THEOREM 2.1. Under assumptions (A.1)-(A.3), if u 0 ~ X , u 1 ~ H and f ~ L : ( 0 , T; V'), then there exists u: [0, T] ~ X , 0 < T < 0% satisfying the conditions

u~L~(O,T;X),

u' ~ L ° ~ ( O , T ; H ) A L 2 ( O , T ; V )

u" ~ L2(O, T; X ' ) d (u'(t),v) + (Au(t),v) +b(t,u'(t),v) + (Gu'(t),v) = (f(t),v) dt

(2.1)

On a nonlinear evolution equation and its applications

1223

in D'(0, T), for all v ~ X. We say u is a weak solution of (1.1). u(0) = u 0, u'(0) = u 1. Proof. The proof consists of several steps. Step (i) is devoted to constructing Galerkin approximations and establishing a priori estimates. Step (ii) is concerned with a limit process of the approximate solutions. Step (iii) will settle that the functions obtained in Step (ii) are really solutions of the problem (1.1), in a weak sense.

(i) Approximate problem. According to a result of Browder and Bui An Ton [9],since X is a separable reflexive Banach space, then, there exists a separable Hilbert space H, dense and continuously embedded in X, therefore with compact embedding into H. Consider a special basis of H: w I . . . . . w m . . . . formed by the eigenvectors that solve the spectral problem ((v,w))t7 = )flu, w) for all v ~ H. Denote by H m the subspace of H generated by the first m eigenvectors above. Let Um(t) ~ iCIm be defined by (u"(t),v)

+ (Aum(t),v)

+b(t,u'(t),v)

+ (Gu'(t),v)

= (f(t),v)

(2.2)

for all v ~ Hm" urn(O) = Uom ~ Uo strongly in X

(2.3)

U'm(O) -= Ulm ~ Ul strongly in H.

(2.4)

Applying the Caratheodory's theorem, the system (2.2) with initial conditions (2.3) and (2.4), has solution on a sufficiently small interval [0, tin[, which can be extended to the interval [0, T], as a consequence of the a priori estimates below. This happens because this system can be put in the following form y'(t) = F ( t , y ) y(0) =Y0, where F(t, y) is measurable on the variable t, for each y fixed, since t ~ b(t,.,.) is continuous in ~+ and f ~ L2(0, T; V'); F(t, y) is continuous on the variable y for each t fixed, because b(t,.,.) is bilinear, A is continuous from the strong topology of X into the weak topology of X', due to the assumption (A1) and G is continuous from the strong topology of H into the weak topology of Y' due to the assumption (A3); F(t, y) is bounded since G and A takes bounded sets into bounded sets and [b(t, u, vl < b~llull Ilvll for each u, v ~ V due to (A2). First a priori estimate. Taking v = urn(t) in (2.2), applying Cauchy-Schwarz and Young inequalities, integrating from 0 to t and observing the assumptions (A.1)-(A.3) it follows from Gronwall's lemma that lU'm(t)l z + O'(Um(t)) +

f0t IIU;,(S)II 2 ds < cT,

(2.5)

for all t c [0, T], where c r is a constant independent from m, but depending on T. Observing again (A.1) and (A.3) we obtain also from (2.5) that lUm( t )lx + IAum( t )lx' + I a u " ( t )ly, < c T.

(2.6)

1224

A.C. BIAZUTTI

Second a priori estimate. From the approximate equation (2.2) we can deduce that, for each V ~ ./'t,

I(Um(t), v ) l - c { I f ( t ) l v ' + I A u m ( t ) l x , + I G u ' ( t )ly, + Ilu'rn(t)ll}lvln.

By (2.5) and (2.6) it follows that

i

r lu"(t)J 2' d t <

(2.7)

cT•

(ii) The limiting process. From (2.5), (2.6) and (2.7), we can extract a subsequence of (Um)rn, still denoted by (Urn)m, such that Um --4 U in L~(0, T; X ) weak *

(2.8)

u~, ~ u' in L 2 (0, T; V) weak

(2.9)

U~n -~ u' in L °~ (0, T; H ) weak *

(2.10)

U "m

(2.11)

---,u" in L2(0, T ; H ' ) weak

Aum -~X1

in L ~ (0, T; X ' ) weak *

G u " ~ X2 in L~(0, T; Y') weak * u(Z)

in X weak

Um(T)

~

u'(T)

~ u ' ( T ) in H weak.

(2.12) (2.13) (2.14) (2.15)

Using Aubin-Lions compactness lemma [10] applied to (2.8) and (2.10), on one hand, and to (2.9) and (2.11) on the other hand, we can extract from (Urn) m a subsequence still denoted by (urn) m , such that u m --, u strongly in L 2 (0, T; Y)

(2.16)

u " ~ u' strongly in L 2 (0, T; H ) .

(2.17)

It follows from (2.16) and (2.17) that, for each t ~ [0, T], urn(t) ~ u ( t ) strongly in H.

(2.18)

From (2.17), recalling (A.3), it follows from (2.13) that there is a subsequence of u~,, still denoted by u ' , such that G u " ---, Gu' in L ® (0, T ; Y ' ) weak *.

(2.19)

Letting n ~ +oo in (2.2) we find from (2.9), (2.12) and (2.19) that u satisfies the equation d (u'(t),v) + (x1(t),v) dt = (f(t),v)

+b(t,u'(t),v)

+ (Gu'(t),v)

(2.20)

in L2(0, T), for all v ~ X .

The initial conditions are satisfied since, by (2.18), urn(0)= U0rn --~ U(0) strongly in H, then, from (2.3), it follows that u(0) = u 0 and, from (2.10) and (2.11) we obtain ( u ' ( 0 ) - u'(0), v) ~ 0 for each m ~ ~ and v ~ / 4 , which, together with (2.4), implies that u'(0) ----u 1.

On a nonlinear evolution equation and its applications

1225

So, in order to complete our proof of the existence of weak solutions of (1.1), we only need to verify that X1 = Au. (iii) Verification that XI =Au. Observing (A.1), (2.8) and (2.12) it is sufficient to verify that

f0

lira sup m

..-. o c

(Aum(t),um(t)) dt <

f0

(Xl(t),u(t))dt.

(2.21)

Returning to the approximate equation (2.2) and taking v = Urn(t), from (2.3), (2.4), (2.8), (2.14) to (2.19) and (A.2), observing that ( O/ Ot) b( t, u(t ), u( t )) = b'( t, u, u) + 2b( t, u'( t ), u'(t )), the norm in V is given by Ilull = (b(t, u, u)) 1/2 and using the Uniform Boundness Principle, we obtain lim sup m

~.~ ~

f0

(AUm(t),u,,(t))dt <-

- ( u ' ( r ) , u ( r ) ) + (Ul, u0) +

(f(t),u(t))dt lu'(t)l 2 d t

if _ l b ( T , u ( T ) , u ( T ) ) + lb(O,uo,Uo) + -~

-

(Gu'(t),u(t))dt

(2.22)

r ~0 b(t,u(t),u(t))dt.

Taking v--wj, j fixed, in the approximate equation (2.2), multiplying it by a function 0 C~[0, T], integrating by parts and passing to the limit it follows that

(u'(r),o(r)wj)-(Ul,O(o)wj)+

+

foT (X~(t), O(t)wj) dt + (Gu'(t), O(t) N ) dt =

(u'(t),o'(t)wj)dt b(t, u'(t), O(t)wj) dt

fo

(2.23)

( f ( t ) , O(t)•) dt.

We define • = {4'lOeU(o,r; X), q,' ~L2(0,T; H)}. Since the class of 4,-- E~=I O(t) N is dense in ~ , (2.23) remains true using 4, in place of O(t)wj. In particular, taking 4, = u we have ( xl(t),u(t))dt

=

(f(t),u(t))dt + (ul, uo) +

- (u'(r),

lu'(t)l 2 dt -

u(r))

(Gu'(t), u(t)) dt

(2.24)

- ½b(T,u(T),u(T)) + ½b(O, uo,uo) + gi f r ~O b(t,u(t),u(t))dt. By (2.24) and (2.22) we obtain (2.21) and it follows from (A.1) that X1 =Au.

•

1226

A.C. BIAZUT/'I

COROt~Y 2.1. Under the hypothesis of theorem 2.1, if f = 0 and G satisfies the following assumption.

Assumption 4. (Gu, u) > 0, for each u ~ Y then, there exists a function u:R+--*X, satisfying the conditions u~L°~(R+;X),

u'~L~(~+;H)NLZ(~+;V),

u" ~ Lz(R+; X ' )

u" + Au + B ( t ) u ' + Gu' = 0 weakly as (2.1) u(0)

= u 0,

u'(0)

= u 1.

Remark 2.1. In the particular case where X = V, in order to establish global weak solutions for (1.1), the existence of ~-: ~ ~ R, defined at (A.1) is not necessary, since we can obtain the same boundness for lUm(t)lx = Ilum(t)ll observing that, by (2.5), fdllu'(s)ll 2 ds < c. We shall consider now the problem of uniqueness of weak solutions. To this end, we shall have to strengthen conditions on the space X and on the operators A, B and G. THEOREM 2.2. Let us assume that X (U.1)

y0

V and

I(Au(t) - A v ( t ) , u ' ( t ) - v ' ( t ) ) l d t < k I

f0

Ilu(t) - v ( t ) l l l l u ' ( t ) - v ' ( t ) l l d t

for each T > 0, u, v, u', v' ~ L z (0, T; V). Either

f0

P(Gu'(t) - a v ' ( t ) , u'(t) - v'(t))l d t < k z

f0

tu'(t) - v'(t)l z dt

or G is monotone.

l ab

--~ (t, w, w) < k 3 Ilwllz,

for all t ~ [0, T], w ~ V,

where kl, k2, k 3 are positive constants

If u, v are two functions satisfying 2.1 then u = v.

Proof. From (2.1), taking account that X -= V, if w = u - v we obtain w"+B(t)w+B(t)w'+Au-Av+Gu'-Gv'=B(t)w

in L2(O,T;V').

Since w' ~ L2(0, T; V) we have

f0t ( w " ( s ) , w ' ( s ) ) d s + f0tb ( s , w ( s ) , w ' ( s ) ) d s + f0tb ( s , w ' ( s ) , w ' ( s ) ) d s = -

f/

(Au(s)-Av(s),u'(s)-v'(s))ds+

- f t ( G u ' ( s ) - Gv'(s), u'(s) - v ' ( s ) ) ds.

.Io

Yo

b(s,w(s),w'(s))ds

On a nonlinear evolution equation and its applications

1227

From (A.2) and (U.1), observing that w(0) = w'(0) = 0 and applying Young's inequality we get

½Iw'(t)l 2 + -~- IIw(t)ll 2 +

IIw'(s)ll 2 ds < c

y0

(llw(s)ll 2 + [w'(s)l 2} ds.

By Gronwall's lemma we conclude that w - 0 and the proof is completed.

•

The next step of our work is dedicated to obtaining information on the behavior of the energy associated to (2.1), Section 2, when t goes to infinite. 3. A S Y M P T O T I C

BEHAVIOR

To avoid technicalities we are going to assume f - - 0. We also assume that Y--- Y' - H and make the following assumption.

Assumption 5. ~([Vlx) >

a 2 Ivl p°,

(Av, v)>aj~r(v),

for each v e X

d (Aw(t),w'(t)) = -~ tr(w(t)), IGul< E~=1 ctlul pt, for each u ~ H ,

for each w ~ C1(1~+; X )

where al, az, P0, c l , . . . , ck, p~,..., Pk are positive constants. THEOREM 3.1. Under the hypothesis (A.1)-(A.5), with Y - Y ' = H solution obtained by corollary 2.1, then: (i) if Pz > 1 for each l = 1. . . . . k and 0
and f = 0 ,

if u is the

E(t) < ce -~t, for all t > 1; and (3.1) (ii) if 0 < mint= 1..... k {1, Pt} < 1 and 0 2, E(t) < c(1 + t ) - ~, where /3'=

for each t > 0,

(3.2)

(2 - mint= 1..... k {1, Pl}) max{2, P0} -- 2 max {2,p 0} mini= 1..... k {1, Pt}

c, o" positive constants and

E(t) = 1 lu,(t)12 + a2 lu(t)lPx0.

We use, in the proof, the method of Nakao [4]. This method depends on an inequality, which is isolated in the following lemma. LEMMA 3.1 (Nakao). Let 4~: ~ + ~ R be a bounded nonnegative function satisfying sup t
[oh(s)] 1+~' < c0[4,(t) - 4~(t+ 1)],

for t > 0 ,

(3.3)

1

where c o > 0 and /3' > 0 constants. Then we get: (i) if/3' = 0, there exist c and 6 positive constants, such that ¢k(t) < ce -~t, for each t > 1; and (ii) if/3' > 0, there exists a positive constant c, such that ~b(t) < c(1 + t ) - ~, for each t > 0.

Proof of theorem 3.1. It is not possible to work directly with the energy E(t), because it lacks sufficient regularity to the derivative of the solution u'(t). It is sufficient to obtain (3.1) and

1228

A.C. BIAZUTTI

(3.2) for the approximated solutions u m of (2.2). The convergences obtained in the proof of theorem 2.1 imply the inequalities (3.1) and (3.2) for the limit u. By this reason, we write u instead u,, and E(t) instead of Era(t). We define the auxiliary e n e r ~ / ~ ( t ) by/~(t) = ½1u'(t)l 2 + o-(u(t)). As we can observe, by (A.1) and (A.5) we have E(t) < E(t), so we can work with /~(t) instead of E(t). We have from the approximate equation (2.2), taking f - 0 and v = urn(t) and using (A.5) that

l~'(t) + b(t, u'(t), u'(t)) + (Gu'(t), u'(t)) = 0.

(3.4)

Integrating (3.4) from 0 to t, it follows from (A.1), (A.2), (A.4), (2.3) and (2.4) that 0
for each t > 0.

(3.5)

Integrating now (3.4) from t to t + 1, we have [ F ( t ) l 2 def=/~(t) --/~(t + 1)

=

f

t+ 1 b ( s , u ' ( s ) , u ' ( s ) ) d s

+ ftt+ 1 ( G u ' ( s ) , u ' ( s ) ) d s ,

for each t > 0. It follows from (A.2) and the continuity of the inclusion from V into H that

ft

lu'(s)l 2 ds < c

r

Ilu'(s)ll 2 as

~t

C f t + 1 b(s, u'(s), u'(s)) ds < c[F(t)l 2. ~E ~t

(3.6)

We also have

1 ftt t+ (Gu'(s),u'(s)) ds <_c[F(t)] 2.

(3.7)

The continuity of u', the Mean Value Theorem for integrals and (3.6) imply that there exist tl~[t,t+j]and t 2 ~ [ t + 7 , t3+ l ] s u c h t h a t

lu'(ti)l < 2 F ( t ) ,

i = 1,2.

(3.8)

Taking v = u(t) in the approximate equation (2.2), integrating from t 1 to t 2 and observing (A.2), (A.5), (3.6) and (3.8) and the continuous inclusion mapping from X into V and H we obtain

~

t2 ( A u ( s ) , u ( s ) ) d s

<4F(t)c

+c

sup lu(s)lx+ [ F ( t ) ] 2 t
t2

fli

k ~tltz Ilu'(s)ll lu(s)lx ds + ~_, c lu'(s)l a, lu(s)lx ds. 1=1

On a nonlinear evolution equation and its applications

1229

It follows from (A.1) and (A.5) that

i

t2 o ' ( u ( s ) ) d s < c { F ( t ) s u p t ~ , ~ t + l [or(u(s))] ~° + [F(t)] 2

(3.9)

+

Ilu'(s)ll(~r(u(s)))~ds+ ~_,

[u'(s)[m[o'u(s))]'°ds

.

I=1 If we assume 0 < P0 < 2, it follows from (3.5), (3.6) and Young's and H61der's inequalities, since /~(t) is decreasing, that t~ c r ( u ( s ) ) d s < c F ( t ) [ E ( t ) ] ~ + [ f ( t ) ] ~ + Y'~ [ F ( t ) ] 2minll,o~} .

(3.10)

l=1 Let us consider now P0 > 2. If p~ is the conjugate of Po, going back to (3.9), applying Young's and H61der's inequalities and using (3.6) and (3.7) we have t2 o - ( u ( s ) ) d s < c

F ( t ) ( E ( t ) ) ~ + [F(t)] 2 + [F(t)] p; + ~

( f ( t ) ) mi~{°'p;'21 .

1=1 Comparing with (3.10) we get

~

t2 o ' ( u ( s ) ) d s _< c { F ( t ) ( E , ( t ) ) ~ + ( F ( t ) ) 2 + H ( t ) } defl = ~ ( L ( t ) ) 2,

(3.11)

where k

E ( F ( t ) ) 2min{l'0/}, H(t) =

if0
I= 1 k

( F ( t ) ) p'° + Y'~

(F(t)) min{°tp'°'2},

if Po > 2.

I=1 By the definition of/~(t), and observing (3.6) and (3.11) we finally have i t2 E ( s ) d s <_ ~1 [F(t)] 2 + ~1 [L(t)] 2. It follows from the Mean Value Theorem for integrals that there exists t* ~ [t 1, t 2] such that /~(t*) _< [ f ( t ) ] 2 + [L(t)] 2.

(3.12)

Integrating (3.4) from t to t*, observing (3.12), (3.6), (3.7) and using Young's inequality we obtain /~(t) < {[F(t)] 2 + H(t)}.

(3.13)

If O < p o < 2 and Pt >-1 for each 1 = 1..... k, remembering that /~(t) is decreasing and bounded, we get sup t<_s
E ( s ) _< c{E(t) - E ( t + 1)}.

1230

A.C. B I A Z U T r I

This inequality, together with (3.5) allows us to apply lemma 3.1 to obtain (3.1), since

E(t) < E(t). Returning to (3.13), if 0 - 1 for each l, imply that sup [j~(S)](1/rninl= 1..... ktl,p,~) < c(E(t) - ff~(t + 1)). (3.14) t 2 and 0 < mint= l ..... k {1, Pt} < 1, it follows again from (3.13) and (3.5) that sup [F_~(s)] (2/p'°minl=l ..... k{l'Pl}) <_c(E(t) -ff~(t + 1)). (3.15) t<_s~t+ l Finally, if P0 > 2 and mint= 1..... k {1, Pl} = 1, the same arguments used above imply that 2

sup [/~(s)] ~ < c(ff~(t) - ff~(t + 1)). t<_s~t+l Combining (3.14), (3.15) and (3.16) and applying lemma 3.1, we find (3.2).

(3.16) •

4. A P P L I C A T I O N S

Let 11 be an open bounded set of R n, with a sufficiently smooth boundary F, /3 E CI(R+), with /30) >/30 > 0 in ~+ and decreasing. 4.1. Example 1 We consider the following system

lion? on)

~ ~/~U~

vtt - ~

~

~

-~

~ ~l~1 ~U~ ~1 sgn(ut) = fl

- fl(t)Ao t + lu,I ~ Iv, I ~sgn(v,) =f2

i=1

U(x,O) = Uo(X) , Ut(x,O) = Ul(X),

v(x,O) =Vo(X),Vt(x,O)= vl(x), u=v=-O, o n F × l ~ +,

in II

where p > 2, 0 < oti < 1

and

0
i = 1,2.

Let us take

X---[Wd'P(II)] 2,

V = [ H ~ ( I I ) ] z,

H=-Y=-Y'=[L2(II)] z

and

2 U-- (u, v),

with

Iglx =

)l/p

~ lujl~,d.p¢~)

,

j=l 2 ((UI,U2))v = ~, ((uj,vj))no,(n) j=l

and

2 (U1,U2)I4 = ~ (uj,vi)t~(a). j=l

(4.1)

On a nonlinear evolution equation and its applications

1231

We put A U = -

,

i=1

B( t )U = - /3( t )( Au, Av ) and G U = ( G1U, G 2U ) , where Gi(s 1, s 2) = Isil~"lsil ~i sgn(si), i = 1,2,j 4: i. Then, all the assumptions (A.1)-(A.5)will be satisfied with ¢r(U(t))=lU(t))l~v, ~'(s)= ~O(s)=s p, p o = p and IGUIH
Uo,V o ~ W01'P(fD, Ul,V I ~ L 2 ( f D

and

f l , f 2 ~ L2(II~+; H - I ( ~ ) ) ,

we can apply theorem 2.1 and corollary 2.1 to obtain the existence of weak solutions

u,v: ~ × R +--) R o f ( 4 . 1 ) . If fl ----f2 - 0, it follows from theorem 3.1 that the energy of the system satisfies: (i) i f p = 2 a n d a~+/3i=l,i=l,2then E ( t ) < c e -~', f o r e a c h t>_>_l;and (ii) i f p > 2 a n d ot~ + /3i < 1 , i = 1 , 2 or p > 2 , al +/31< 1 and a 2 +/32 < 1 or p > 2, al + / 3 1 < 1 and a 2 + / 3 2 < 1 , we have E(t) < c(1 + t)- ~', for each t > 0, 2(P - 1) - p m i n i = 1,2( _1~ ' ~ , } where/3'=

c, 6 positive constants.

Remark 4.1. The same results as above can be obtained if we consider in (4.1) more general G~(ut, vt). As for the existence of weak solutions we can a s s u m e Gi: [ L 2 ( ~ ) ] 2 ~ L P(~),

where

'

p > max 1 , n p _ n + p

taking bounded sets into bounded sets, with Gi:[~ 2 ~ R, continuous and satisfying

siGi(si,s 2)> - c i ( l + s 2 ) , i = l , 2 , s i ~ E ,

wherec,>0.

To apply the corollary 2.1 it is sufficient to consider c i - 0, i = 1, 2. We only have to take Y = [LZ(l))] 2, where p' is the conjugate of p, if p < 2. Finally, to obtain the results of theorem 3.1, we need also to assume

Iai(sl,s2)l<_cilsll'~'ls21 t3'

i = 1,2.

In the next examples we are going to assume Y = Y' = H = L2(fD, Gu = lul"sgn(u), 0 < a _< 1

1232

A.C. BIAZUTH

and b(t, u, v) = fn fl(t)Vu. Vv dx. Then, the assumption (A2) is satisfied, since flo Ilullz -< b(t, u, v) < fl(O) [[u[[2 because /3 is monotone decreasing, /3 >/30; t ~ b(t, u, v) is in CI(R +) because/3 ~ CI(R + ) and, if v, ---)v in V, we have lim sup, _++~ for b'(t, o~, v~)dt < for b'(t, u, v ) d t , applying the Uniform Boundness Principle, since for /3'(t) dt < 0. 4.2. Example 2 Let J:R n -+ R be a convex and continuously differentiable function such that: (i) Otxl~lP 2 , 0/1' 0/2' Or3 > 0 and 1~12 = Eni=1 /z2 bl" Let us take a~(~:) = ( 3 J ( ~ ) / 3 ~ i ) , i = 1,..., n. We choose X = Wol,P(O), V = H i ( o ) and define A u -~- - Eni= l ( °3// cOxi) (ai(Vu)). Taking o ( u ) = fa J ( V u ) d x , z(s) = al sp, @(s) = ot2(mes(Fl) + sP), u o ~ Wd.P(I~), u 1 L2(f/) and f E L 2 ( R + ; H-I(~)) it follows from the theorem 2.1 and corollary 2.1 that there exists u: f / × R+-~ R global weak solution of the problem Ut,

--

iE

(ai(~TU))--/3(I)Au t + Gu t = f ,

in f / × R +

-

(4.2)

] u ( x , O ) = u o ( x ) , u ~ ( x , O ) = u l ( x ) , in l~ l u ( x , t ) = 0 , in F × R +. If we also assume En,=l (c~J(¢)/c~:~) ~ _>c I J( ¢ ), for each ¢ ~ R ~, i = 1. . . . . n, where Cl > 0, then, as (A.5) is satisfied with Po =P, we obtain the asymptotic behavior of the solutions of (4.2) with f = 0. We can observe that the example 2 includes the case in which

[l ur

A U = - ~/=1 " ~ / ~ [ ' ~ / [

"~/

'

p > 2. It is sufficient to take

1 ~ i~ilP.

J ( ~ ) = P i=1 4.3. Example 3

Let us consider A u = A(IAuIP-2Au), p >_ 2, X = Wo2'p (1~) and V the same as in example 2. We choose tr(u) -- ( l / p ) [u[~ and ~(s) = ~'(s) = ( l / p ) s p. Then the problem

utt + A(IAuIp-2Au) - / 3 ( t ) A u t + Gu t = f ,

l

u(x,O) = U o ( X ) , U t ( x , O ) = u l ( x ) , u(x,t) = 0 in F × R + Vu(x,t)=O in F × R +

in I~

in f~ × R +

On a nonlinear evolutionequation and its applications

1233

has global weak solutions and we also can apply theorem 3.1 to obtain the asymptotic behavior of these solutions. It is sufficient to assume u o ~ W o 2"p ([l), u l ~ L 2 (12) and f ~ L2(R+; H - 1(~'~)). 4.4. Example 4 We consider the nonlinear problem in D, X R + (4.3)

u(x,O) = uo(x),ut(x,O) = ul(x), in a , u ( x , t ) = O i n F X N ÷, where M: R + ~ R +, M ( s ) > 0, M S 0 and

(4.4)

M is a nondecreasing continuous function. We choose X = V = H ~ ( 1 2 ) . Defining A u = - M ( f ~ l V u L 2 d x ) Au, ~ t ( r ) = / ~ M(s)ds, o-(u) = ½ ~ (fnlVul 2 dx), ~-(s) = qJ(s) = ½~(s 2) and taking G -= 0, it follows from the theorem 2.1 and corollary 2.1 that there exist global weak solutions for (4.3) if we take u 0 ~ H0~([l), u 1 ~ L2(II) and f ~ L2(R+; H-~(12)). If we also consider the condition M is Lipschitz continuous,

(4.5)

the theorem 2.2 implies that there is uniqueness of solutions. We can also observe that, if we have assumption (4.4) and also there exists 3' > 0

such that At(s) > cs ~,

Vs > 0

(4.6)

and choose f = - 0 , the energy associated to the problem satisfies E(t)<_Cl e-6t, Vt > 1, if 0 < 3' _< 1, and E(t) <_c2(1 + t ) - ~, Vt >_ O, if 3' > 1, where /3' = ((3' - 1)/3').

Remark 4.2. We can observe that, if M(s) > m 0 > 0, we have exponential decay, since 3' = 1 and if M(s) = s °, 0 > 0, we obtain E(t) < c2(1 + t ) - 1- ',. Matos and Pereira [11] have obtained the same algebraic decay in the particular case 0 = 1 and Munoz Rivera [12] has obtained existence and algebraic decay with M(s) satisfying another condition instead of (4.6), which is not satisfied by M(s) = s °, if 0 > 1 and considering u 0 ~ H~(II) n n 1+ "(1"~), U1 ~ n ' ( I I ) , e > 0. Medeiros and Milla Miranda [13] have obtained the existence and exponential decay for the weak solutions of the equation u , - M (fn IVul 2 d x ) Au + ( - A ) ~ u t = 0, where 0 < v < 1 and M(s) > m 0 > 0, assuming u 0 ~ H01(tD n D(A ") and u I ~ L2(II), with M not necessarily monotone. They also have obtained uniqueness of solutions, if 51 < v ___1, in the case which M ~ CI(~+). 4.5. Example 5 Let o-i, i = 1. . . . . n be real valued functions satisfying the following: o-i in C(R,•), nondecreasing, tri(0) = 0, each tri: L2(fD ~ L2(II) takes bounded sets into bounded sets and

1234

A.C. BIAZUTrI

is locally Lipschitzian. We consider the following initial and boundary value problem

I

~=1 ~--~icri(Ux)-fl(t)Aut

u,- i ' lu(x,O) =Uo(X),Ut(x,O)= ul(x), ~u(x,t)=O inF×R +

'~

+ lu,I sgn(u,) = f , (4.7) in 11,

where 0 < a < 1. Taking X, V and H the same as in last example, defining n

0

hu--- - i=~1 ~ tri(Ux')'

n f

o'(u)

=

E )1) °i(Ux')dx i=1

and

~b(s)=cl $2,

where d-i(s) = f~ o-i(¢)d ~: and c I is the maximum among the n lipschitz constants it follows from the theorem 2.1, corollary 2.1 and theorem 2.2, remembering remark 2.1, that there exists an unique global weak solution for (4.7). If we also assume d-i(s)> c2 s2, Vs ~ •, choosing r ( s ) = c2 s2 and f = 0, the energy associated to the system decays exponentially if t ~ = l and satisfies E ( t ) < c ( l + t ) - ~ , 'tit_>0, if 0 < a < l , where f l ' = ( ( 1 - a ) / a ) . The existence and uniqueness of weak solutions for (4.7) have been proved by Dang Dinh Ang and Ngoc Dinh in [8]. Acknowledgements--We wish to thank Prof. Luiz Adauto Medeiros and Prof. Jamie Munoz Rivera for their useful remarks on this work. We also appreciate very much the suggestions presented by the referee. REFERENCES 1. CRIPPA H. R. & BIAZUTTI A. C., Sobre uma classe de equaq6es de evolu~o nao lineares, Atas do 23° Sere. Bras. Amilise 139-145 (1986). 2. TSUTSUMI M., Some nonlinear evolution equations of second order, Proc. Japan Acad. 47, 950-955 (1971). 3. YAMADA Y., Note on certain nonlinear evolution equation of second order, Proc. Japan Acad. Ser A, 55(5), 167-171 (1979). 4. NAKAO M., A Difference inequality and its applications to nonlinear evolution equations, J. math. Soc. Japan 31R4), 747-762 (1978). 5. NAKAO M., Decay of solutions of some nonlinear evolution equations, J. math. Analysis Applic. 60, 542-549 (1977). 6. YAMADA Y., On the decay of solutions for some nonlinear evolution equations of second order, Nagoya Math. J. 73, 69-98 (1979). 7. ZUAZUA E., Stability and decay for a class of nonlinear hyperbolic problems, Asymptotic Analysis 1, 161-185 (1988). 8. DANG DING ANG & NGOC DINH A. P., Strong solution of a quasilinear wave equation with nonlinear damping, SIAMJ. math. Analysis 19(2), 337-347 (1988). 9. BROWDER & BUI AN TON, Nonlinear functional equation in banach spaces and elliptic super regularization, Math. Z. 105, 177-195 (1968). 10. LIONS J.-L., Quelques m~thodes de r~solutions des probl~mes aux limites non lin~aires. Dunod, Paris (1969). 11. MATOS M. P. & PEREIRA D. C., On a hyperbolic equation with strong damping, Funkcialaj Ekvacioj 34, 303-311 (1991). 12. MUNOZ RIVERA J. E., Smoothness effect and stability of nonlinear evolution equation (to appear). 13. MEDEIROS L. A., & MILLA MIRANDA, M. On a nonlinear wave equation with damping, Revta Mat. Unit). Comp. Madrid 3(2,3), (1990).