BOL. SOC. BRAS. MAT., VOl_ 19, N ~ 1 (1988), 141-150 141

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF A NONLINEAR WAVE EQUATION

ORLANDO LOPES

Introduction and Statement of the Main Result.

In this paper we deal with the damped nonlinear wave

equation:

( l ) u t t -Au + cu t + f ( u ) = h ( t , = ) , c > O,

We assume u ( t , x ) and h ( t , x ) are defined for a l l

x = ( x l , x z , x 3) in ~3 and are 2~-periodic in each x i ; in

other words, we take 2~-periodici ty in the spatial variables as

boundary condition. For each non-negative integer k and

l ~ p ~ ~, Hk,p(~ 3) (Hk(~3) for p = 2) denotes the usual 27 3) (H~(~ 3) for Sobolev spaces with the usual norm; Hk,p(~

p = 2) denotes the Sobolev spaces of functions which are

2~-periodic in each variable (of course, the integrals defining

the norm are taken over the fundamental cube ~,2~x ~,2~ • ~,2~).

Equation (1) can be viewed as a system

(2) u t = v

v t = Au-cu t ~ f(u) - h(t,=)

or, more compactly,

dw (3) ~ = Aw + G(t ,w), where

r

w = (u,v), Aw = (v,au), ~(t,w) = ~ I_ov o I -f(u) +h ( t , . )

Recebido em 01/08/88.

142 ORLANDO LOPES

I t is very well known that A generates a strongly

continuous semigroup (actua l ly a group) of l inear operators in 2~ H~ ~," for each integer k > 0 As phase the space X k = Hk+ I x _ .

space for equation (3) we take the space X I = H~ ~ • H~. Since 2= L~ ~ ~ H z continuously, i t is eas~ to see that w G X ~ G(t,w)EX~

is l i psch i t z ian on bounded sets provided f is C z. So, i f

th is is the case and h:.~+ § Hz is continuous, i t fo l lows that

local existence and uniqueness of mild solutions of (3) i s

guaranted in the space X~; moreover, global existence in time

can also be gu~ranted provided we get an a p r i o r i e s t i m a t e for

the norm of the mild solutions in the space X~. For t ~ t o > 0

and w o belonging to X~, we denote by w( t , t o ;wo) the solution

sat is fy ihg w( to , to ;uo ) = wo.

Bef in i t ion . Equation (3) is uniform ul t imately bounded in the

space XI i f there are functions a(R) and T(R) and a constant

H such that lw01X ~ R implies l w ( t , t o ; w o ) I X I ~ a(R) for

t > t o and lw ( t , t o ;w '~ ) IX l -< R o for t _> to + T(R).

Our main resul t is the fol lowing:

Theorem A. Equation (3) is uniform ul t imate ly bounded in the

space X I provided the following conditions are s a t i s f i e d .

( i ) fCu) is a cZ-function;

( i i ) there are constants k I > 0 and k~ such tha t

uf(u) ~ klu 2 + k z;

( i i i ) such t h a t

( i v )

bounded.

Moreover, i f the map t § h i t ) .

then the Poincar~ map ~o§ ~{p,0;uo)

there are posi t ive constants

If'(u)l ~ ~(I + I IB) 2~

the map t G ~+ ~ h(~) G ~i is continuous and

k 3 and B, 0 < B < 4

is periodic of period p > 0,

is the sum of a l i n e a r map

NONUNEAR WAVE EQUA~ON 143

with spectral radius s t r i c t l y less than one and a compact

d i f fe ren t iab le map.

Before getting into the proof, we believe i t is

interest ing to make the following remarks about Theorem A and

i t s consequences:

l ) I f we consider the problem of global existence of

solutions of the equation u~t ~ u + luIBu = O, with i n i t i a l

condition (uo,vo) in H2(~ ) H I ~ ) , then the exponent

B = 4 is c r i t i c a l ; in other words, for B < 4 solutions are

g lobal ly defined, and for B = 4 they are global ly defined

for small i n i t i a l data ( ~ ] , ~2]). Assumption ( i i i ) in

Theorem A is related to this fact .

2) Some authors have studied the existence of a p-periodic

solution and of f i n i t e dimensional attractors for equation (3)

in the case h(t) is p-periodic in t and the phase space is 2 ~ H~• ( [3 ] , E4], ~5], ~ ] , E7]) (here, f i n i t e dimension is

understood in the sense of Hausdorff dimension). For that phase

space, the c r i t i c a l exponent is B = 2. For the case we are

t reat ing, the existence of a p-periodic solution and of a

f i n i t e dimensional at t ractor follows immediately from the

decomposition of the Poincar~ map given by Theorem A (see ~ ] ,

~9]).

3) I f the wave equation is given in a bounded open set

~ ~ with smooth boundary and we impose, say, D i r i ch le t

boundary, then everything we have done works, except the proof

of Lemma 3. This lemma gives an Lp - Cq estimate for the

l inear wave equation and i t s proof follows from a s imi lar resul t

for the wave equation i n Almost surely i t also holds in

the case of D i r i ch le t boundary; so while we wait for such a

proof, we r e s t r i c t ourselves to the case where per iod ic i ty in

x is taken as boundary condition. Lp - Lq estimates have been

used by several authors to t reat the Cauchy problem for nonlinear

wave equations in ~n (see, for instance, ~ l ] , ~ and the

references therein) .

144 ORLANDO LOPES

Proof of Theorem A. As a preparation for the proof of Theorem A

we s t a r t w i t h a L - L e s t i m a t e f o r t h e p r o b l e m P q

3

(4) u t t - Au = 0 in ~ , u(0,z) = 0, u t ( O , z ) = f ( z ) .

temma 1. The map f § S ( t ) f = u ( t , . ) sa t i s f ies

Is(t)fl L <_ k~t 3~-llflml oo ,p

l ( ~ ) , t >o, p - - l ~ - ~, ~ _>o.

P r o o f . See DO]-

Next lemma deals with equation (4) in the case p e r i o d i c i t y

is takenas.boundary condit ion.

Lemma 2. ]S(t)flL~ < ks(t3~-I + t3~-l+3/P)Jfl ~ > o

- HI, p P r o o f . F i r s t o f - a l l we have t o n o t i c e t h a t , due t o t h e w a v e

p r o p a g a t i o n p r o p e r t y , Lemma 1 can be r e s t a t e d as Is( t) f lL (~) kst3~-I I f l H l , p ( ~ ( t ) ) , where ~ is any subset of ~3 and

R(t) is the domain of dependence of R. I f 0 < t ~ l and

is the fundamental cube [0,2~3 x [0,2~] x ~ , 2 = ] , then the

number of cubes necessary to cover R(t) is less or equal to

some f ixed integer No; also, i f t is large, the number of

such a cubes grows as fast as t 3, and th is proves the lemma.

2 0

Lemma 3. Consider the equation u t t - Au + cu t + 7 u = 0

u ( O , z ) = O, u t ( O , = ) = f ( x ) , a n d s u p p o s e 2 ~ - p e r i o d i c i t y i n

is taken as boundary condit ions. Then f § ~ ( t ) t = u ( t , . )

sa t i s f i es :

= I l~(t)fl~ _<~-~t/z(t3u-I +t3~'1+31p)Jad_z~r, t > o, p I~'~* ~l,p

NONLINEAR WAVE EQUATION 145

Proof. Defining U(t) = eOt/2u( t ) , we see that Utt - kU = O, U(O,x) = O, Ut(O,x ) = f ( x ) , and the conclusion follows from the

previous lemma.

Lemma 4. Under the assumptions of Theorem A, equation (3) is 2~ 2~

uniform ul t imately bounded in the space X o = Hz x L 2 ( for 2 2rr i n i t i a l conditions in H ~ xH~ , as long as solution exists in

this space).

Proof. Recall that the norm in E 2~

i 2 2 T

(Ig rad u-I 2~ + lul L 2 L~ ~') "

1 2 O + 7 Ig earl ul + -~ <u'v> L

2

E(u) = f( s) ds, 0

2

- c 2 - c l g r a d U.lL2~ t ir = ~ l v l r 2 ~ 2 2

is defined by

2 I

Defining ~l(u,v) = -g IVlL 2 + 2

L2 + P(u(x))dx where Q

an easy computation s'hows that

uf{u)d=+<v,h( t )>+<u, h ( t ) >

and using assumptions ( i i ) and ( iv) we get

2 12 #(u,v) <_ -Y([VlL2~a + [grad u L~Z~ + I"12L2~) + M,

where y > 0 depends on e and k z and M depends on Y and

t_>oSUp lh(t)IL22x. Let RI > 0 be defined by -yR 2z + M = -l and

le t us cal l w(t) = ( u ( t ) , v ( t ) ) . Notice that assumptions ( i i )

and ( i i i ) imply F(u) is bounded below and I~Cu)l<kT(l+lu.IB+l);

in par t i cu la r , since L 2w ~H 2~ continuously, we conclude f o r

each R there is a constant CI{R ) such that I F(u(x))dx <Oz(H ) Q

provided l ul .2~< H. Moreover, i f for some t a we have H z

IW(t 1)IXo = R I and ]W(t)Ixo > R z for t > t z then necessarily

146 ORLANDO LOPES

(5) w(u(.e),v(t)) ~ w(u(t~),v(~)) - (t - t~).

This i n e q u a l i t y together wi th the previous remarks proof

the lemma.

Lemma 5. Let u : Eto,+=) § be a non negative cont inuous func t io r

s a t i s f y i n g

-c(t-t )/2 ~ -c(t-*)/Z u(t) < ~ e + M +~M31t e k(t-8)u(8)de, t o <_ t,

0

are non negative constants and k(s) is a where MI, M 2, M 3

I+= -eS/4k(e) nonegative funct ionosuch that EM 3 e de < ~. t - to)~4 0

~(~) < ' l u ( t ) = 2Mze + 3M 2 , t ~ t o .

Then

Proof. Suppose u(s) < u (s ) , r < s < t ; then 0 -

- c ( t - t o ) / 2 + u(t) ~ M1e

I "e (s - to ) /4 + 3Mz]ds t + cM 3 e ' O ( t - s ) I Z k ( t - e ) 2MIeF

+ M z to

-~ I t e-e( t -s - c (e . t ~ )/2 + M2+2MIc~ 3 e ) /4~(t_8)ds = M1e t o

I -c(t-t o )/4 s ~(t) t _c(t_s)/2k(t.s)d s ~ 2M1e + 3Mz~M 3 e + M2 <

to

and th i s proves the lemma.

Proof of Theorem A. We .start by rewr i t i ng system (2) as

ut =V

v t = ~ u - e u t . - 4 - u - g ( u ) + h C t ) ,

~2 where g ( u ) = f ( u ) - - 4 - u ' C e r t a i n l y , gCu) a l s o s a t i s f i e s

> O, a and b, assumption { i l l ) ; so, we can f i nd constants k 8 .

NONUNF_..AR WAVE EQUATION 147

with 0 < a < 3, 0 < b< I , such that a + b = 8 and

Ig ' (u ) I < ks( l + lulb). Then, using that L ~-H 2~ continuously l + I ~ I ~ -

6 and Lemma 5 with p = ~ > l we get, by the va r ia t i on of

constants formula:

- o C t - t o )/2 I ~ ( t ) l ~ < K~ Iw(to l lx~ +

I t e_C(t_s)ig "YI + K [(t-~) + (t-~) ~2] Ig(~(.))l 2= a. to HI ,p

0 < Y~ < l , Y2 > O, w(to) = (uCto),V(to)). Using H~Ider

- ~-iau IL2= -< i~ , , (u)l ,,au 6 .2xl~-~T. 1.2~ ' q = E' and i nequa l l t y Ig' (u) At "D I#. 2

P q the obvious estimate

11+ I ul a c2~ - I +1~1 ~ Lq q

< ksC1 + I~1 b _ L 11(1 + l u l a ) l 2x q

we get

-cCt-to)/2 I~(t) Ic| <- Ke Iw( to ) l x~ +

K it -o( t-8)/2[(t_8)-Y1+C t-~) ~2] Fksl graduC~)IL2Jl+ lu(~)I~)2~r u(~)l ~) I t o q

Now, i f I ~ { t o ) l x i

c2(R ) such t h a t

I1 + l u ( 8 ) l a l L 2 ~ 2 ~ (R) q

and so

+ I r (~(~) ) l L2~7 a~. P

= R then, according to Lemma 4 there is a

lu(s)IH~ ~ ~ c2(R), s ~ to, in p a r t i c u l a r ,

(we have used the inclusion L~ ~ c H ~ ) ,

148 ORLANDO LOPES

-c(t-to)/2 l~(t)lc ~ K~ lW(to)Ix~ + ~(R) +

it e_C~t_s)/2 -YI + ~, (~) [Ct-~) + ( t -~) I t o

b d

Since 0 < b < l , for any c > 0, there is a K(e)

lul b ~ K(c) + ~ lu l , and then

- c ( t - t o ) / 2 l~( t ) l c _< Mie lW(to)IX + M(R ,~ ) +

such that

I t t-s)12 -YI ( t -s ) Y2] u(s) ds + ~M, CR) e - ~ E(t-~) + I Is �9 t o

I f E(H) fS chosen to sa t i s fy eM3(R ) +s )d8 <_ T"

then Lemma 5 gives l u ( t ) l L < o~(R), t > to, for some convenient

au ( t , = ) v ( ~ , = ) = av c~(H). Defining U(t ,x) = ~ . ~-~. ( t , x ) ,

i = l . . . . . n, we see ( u ( t ) , y ( t ) ) is a mild solution ( in the space

X o = H 2~ x 2~ i C 2 ) of the system

U t = V

02 f' t) ~ c 2 au ~h (t) v t = ~ u - c v + - T u + Cu( ) ( t ) - T ~ T . ( t ) + ~ "

Since the semigroup generated by the l inear part goes to zerc.

exponential ly (in the space Xo) and

@ t > t au t) o = au t) + - - h ( t ) l~2x, _ o, If' c=(t))~C - -T ~ ( a=~ -~

is bounded above by some constant Cs(R ) (as a consequence of

Lemma 4, the previous estimate for l u ( t ) I L = and assumption ( i v ) ) ,

we conclude l(u(t),v(t))IXo, t -> to, is bounded above by some

constant c6(R); hence, l ( u ( t ) , v ( t ) ) I X l , ~ ~ t o , is bounded

above by some cT(R ). Furthermore, using Lemma 4 again and

NONLINEAR WAVE EQUATION 149

previous remarks, we know there are os(Ro) and T(R) such

that

so for

ig rad u(s ) lS2 ~ ~ cs(Ro),

Ig(~(~))IL2 w ~ o~(R ) for

P t ~ tz(R) we have

I1 + lu(8) lalc2x ~ o~(R0) q

-o(t-t1(R))12 l u ( t ) l c ~ Ke jwCt~(R))l x + eg(R o)

1

and this implies l u ( t ) IL m ~ 2e~(R0) provided t ~ t o + Tz{R),

for a convenient T~(R). Using the functions U(t) and V(t) defined above and argueing exactly as before we can see that

are elo(Ho) and T2(R ) such that l (u( t ) iv( t ) ) IX2 ~ olo(Ro)

for t ~ t o + T2(R) and this proves the uniform ultimate

boundedness. The f inal assertCon follows imediately from the

2~ 2~ fact that the map u G H 2 + g(u) @ H~ is compact and

di f ferent iable, and this proves the theorem,

D]

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150 ORLANDO LOPES

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