Asymptotic behavior of solutions of a generalized Boussinesq type equation

Pergamon NonlinearAnal~sts, Theory, Methods & Applications, Vol. 25. No.l 1, pp. 1147-1158, (1995)

Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved

0362-546X/95 $9.50 + .00

0 3 6 2 - 5 4 6 X ( 9 4 ) 0 0 2 3 6 - 3

A S Y M P T O T I C B E H A V I O R O F S O L U T I O N S O F A

G E N E R A L I Z E D B O U S S I N E S Q T Y P E E Q U A T I O N

FELIPE LINARESt and MARCIA SCIALOM$§

t lnst i tuto de Matematica Pura e Aplicada, Estrada Dona Castorina 110, 22460 Rio de Janeiro R J, Brazil; and :~Departamento de Matematica, Universidade Estadual de Campinas, 13081-970 Campinas SP, Brazil

(Received 16 Not)ember 1993: received in revised fi)rrn 9 June 1994; received for publication 26 August 1994)

Key words and phrases: Asymptotic behavior, Boussinesq equation, initial value problem.

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

Consider the initial value problem (IVP) for the generalized Boussinesq type equation

l u , -u~ ,+u ...... = - ( ~ ( u ) ) .... x ~ , t > 0 ,

u(x,O) = f ( x ) (1.1)

't u,(x,O) =g(x),

where xlt(u)= ]ul ~- 1u and c~ > 1. The equation above arises in the modelling of nonlinear strings. This equation is a

generalization of the classical Boussinesq equation which describes in the continuous limit the propagation of waves in a one-dimensional nonlinear lattice, and the propagation of waves in shallow water (see [1,2]).

Local and global results have been obtained for the IVP (1.1). Bona and Sachs [3] showed local well-posedness for (1.1) for smooth solutions using Kato's abstract theory for quasilinear evolution equations. Tsutsumi and Matahashi [4] proved that for f ~ H1(R) and g = X", X e H1(~) the IVP (1.1) is locally well-posed. In [5] Linares established local well-posedness of (1.1) for data f ~ HI(~) and g = h' e L2([~). Moreover, assuming smallness in the data, it was proved that these solutions can be extended globally in H*(~).

We shall notice that the equation in (1.1) admits solitary-wave solutions. As it is known the existence of solitary-wave solutions shows the perfect balance between the dispersion and the nonlinearity of the equation in (1.1). In particular, for ~ integer, these solutions are given explicitly by

U~:( ~: ) = A sech :/'~ i ( Bs~ ), (1.2)

where

A =A(c, ce) = { ( a + 1 ) ( 1 - c 2)]1 ..... 1

J 2 B=B(c,a)= (1 - c2 )1 /2 (0~ - 1)

§Work partially supported by CNPq and FAPESP, Brazil.

1147

1148 F. LINARES and M. SCIALOM

~ = x - ct, and c is the wave speed satisfying c 2 < 1 (see [3]). Bona and Sachs [3] showed, under some restrictrions on the wave speed, that these solitary-wave solutions are stable. Moreover, using this result, they established global existence of solutions corresponding to data closed to stable solitary waves. We also notice that in order to satisfy the IVP (1.1), the speed u t has to be the derivative of a certain function, i.e. (1.2) implies that u t = U~. For more details see [5].

The purpose of this paper is the study of the asymptotic behavior of solutions of the IVP (1.1). In this direction, we first obtain a decay property exhibited by small solutions of (1.1). More precisely, we have the following theorem.

THEOREM 1.1. Let f e H~(N) n Lq/2(N), g = h', h ~ L2([~) OLq(~), and a > (4 - 3y - y 2 ) / y . If Irlfllr + Ill glJl---Ilflll.2 + IlfJJ~/2,q + Ilhll2 + Ilhilq < 6 small. Then there exists C > 0 such that the solution u of the IVP (1.1) satisfies

]lUllp <_ C ( l + t ) , t > 0 ,

where p = 2/(1 - 3;), q = 2 / ( l + 2"),) and y ~ (0, 1/2). This result was motivated by the work of Tsutsumi and Matahashi [4] where it was

established that for f ~ H~(~) ~ L~(/t~) and g = )f", g ~ H~(~) A La(N), and a > 4, the solution u e H l ( l ~ ) of the IVP (1.1) satisfies

lu(x,t)l_<c(1 ~-[t[) 1,'2 (1.3)

for all x and t. They obtained this result as a direct consequence of the decay property presented in solutions of the nonlinear Schr6dinger equation. To exploit the decay of solutions of the nonlinear Schr6dinger equation they had to neglect the term Uxx. In this work we look for conditions on the initial data for which decay of solution of the 1VP (1.1) can be obtained when the complete structure of the equation in (1.1) is considered, i.e. without neglecting the term uxx. Besides the results in theorem 1.1, that deals with this concern, we can establish a similar result as in (1.3) under some weak conditions on g and using our method of proof. We do not require g to be a second derivative of a Ha-function (see remarks in Section 3).

Under some suitable conditions of the initial data and the nonlinearity we obtain a result which ensures that small solutions of the IVP (1.1) behave asymptotically like solutions of the associated linear problem, that is, we have the following theorem.

THEOREM 1.2. Let ( f , g) = (D3"/4f~,DJ + y 4 h) c H l([~) X L2(~) with IIf[ll,2 + Ilgll < 6 small, c~ = 4 / 7 , y ~ (0, 4 /5 ) and u bc the solution of the IVP (1.1). Then there exist unique solutions u _ of the linear problem associated with (1.1) such that

I l u ( t ) - u ± ( t ) l l j : - - ' O a s t - * ___~c. (1.4)

In the proof of this theorem we follow ideas used by Strauss [6], Pecher [7], and Ponce and Vega [8] to establish similar results for the nonlinear Schr6dinger equation, nonlinear wave equation, Kle in-Gordon equation and Korteweg-de Vries equation, respectively.

Affirmative results on scattering for small solutions are interpreted as the nonexistence of solitary-wave solutions of arbitrary small amplitude. In this case, we notice that a simple calculation shows that the solitary-wave solutions U,(~ c) in (1.2) satisfy IIg,.(')ff2 > e > 0 for ce > 5. So according to the statement above we should expect to have scattering as in (1.4)

Asymptotic behavior of solutions of a generalized Boussinesq type equation 1149

when the nonlineariW power c~ satisfies the previous constraint. The results in theorem 1.2 show that scattering for small solutions of (1.1) occurs when the restriction on a mentioned above is satisfied. In this sense we could say that the scattering results presented here are optimal.

This paper is organized as follows. In Section 2, we prove some estimates for solutions of the linear problem associated to (1.1). The proof of theorem 1.1 and some remarks concerning it are provided in Section 3. Finally, the result related to scattering for small solutions of (1.1) is given in Section 4.

1.1 Notation

The notation II'llr p is used to denote the norm in L i' - ( 1 - A)r/2Lp. A l s o , II'llp instead of II'll0,p denotes the norm in L p and H ' is used instead of Lz,.

We use c to denote a constant that may change from line to line. The Fourier transform of a function f is denoted by f (~ ) , and )~(x) denotes the inverse

Fourier transform of f. We use D ' = ( - c ? ~ z ) ~':. For l < p , q _ < z a n d f :N×[R-~ [R

2. PRELIMINARY RESULTS

We begin this section with the statements of some results that are used to establish estimates for solutions of the linear problem associated with IVP (1.1).

The following results were proved by Kenig et al. in [9]. Define for y>O, ( x , t ) ~ ~2 and 0 as in [9]

THEOREM 2.1. Let ~h(~) and W~(t) be defined as above. Then for any ",/e [0, 1].

IIW~(t)uolli~ <_ cltl ~ ':Hu0llq,

where p = 2 / ( 1 - 3'), q = 2 / ( 1 + y).

Proof See Kenig et al. [9, theorem 2.2]. •

THEOREM 2.2. Let ~b(~) and Wr(t) be defined as above. Then for any ~/~ [0, 1]

HW~/e(t )u,~Hl,,~:t p~)) <_ cHuoll2,

where (q, p) = 4/3,, 2/(1 y).


Proof See Kenig et al. [9, theorem 2.1]. •

The next result provides the existence of a global solution in H~(R) for the IVP (1.1).

THEOREM 2.3. Let f ( x ) • H 1(~) and g(x) = h ' (x ) ~ L2(~). If IIf[ll,2 + Ilgll2 << 1, then for any T > 0 and a > 1 the solution u of the IVP (1.1) satisfies

u(x , t) • C([0, T ] : H I ( ~ ) ) .

Moreover,

sup Ilu(t)l[l,2 ~ c( Ilfll,,2 + Ilgl12). 10.7]

Proof See Linares [5]. •

The second part of this section is devoted to establishing the linear estimates needed in the proofs of theorems 1.1 and 1.2.

We begin by considering the linear problem

l u . - u ~ + u ~ , ~ = O , x ~ , t > O ,

u ( x , O) = f( x ) (2 .1 )

u,(x,O) = g ( x ) ,

where g(x) = h ' (x ) and whose solution is given by

with

u ( x , t ) = U ( t ) f ( x ) + V ( t ) g ( x ) ,

U ( t ) f ( x ) = 2 ~ r - "

(2.2)

and

V ( t ) g ( x ) = ~ (1 + ~2) ~/2 d ~,

where ~b(s c) = (~c z(1 + ~z))l/2. The following result is concerning the decay presented in solutions of the linear problem

(2.1).

PROPOSITION 2.4. (i) Let g = h' • L2(ff~) (~ Lq(~) Then

IPV(t)g[rp <_ c(1 + t) ~/2(llhlt2 + Ilhllq), where p = 2/(1 - y), q = 2/(1 + 23,) and 3 , • ((), 1/2).

(ii) Let f E H 1(~) n Lq/2(~). Then

Hg(t)fllp <_c(l + t) ' - ( l l f lh,2+llf l ly/Z,q) ,

where p = 2/(1 - y), q = 2/(1 + 2y) and y ~ (0, 1/2).


Proof.

I I r ( t ) g l l p <_ IIV(t)glD,2 _< Ilhl12,

where p is as above. For t > e > O, we write

l f /e ,< ,~<~,+,~ , l~ , , / ' ( ~ ) V ( t ) g ( x ) = ~ ( ~ )1~, '214/ '( s c )~/:(1 + ~ 2) 1/2

where ~b"( s c ) = I ~!12~ -~ -~ 3)/11 + ~: z)3/: Since qS( s c) satisfies the condit ions in t heo rem 2.1 (see [5, 9]), we have that

I I r ( t )g l l p < ct '!:IIGII2/~I . ~>,

here G ( x ) = ( h ( ~ ) / l ~ l ~ -~Xx). Applying the H a r d y - L i t t l e w o o d - S o b o l e v theorem (see [10]) we find that

IIGI[_~,,,1 • , ,-< cllhllq,

for q = 2 / (1 + 2-/) and ` / ~ (0, 1/2) . Thus

IIV(t)gllp <_ ct ~/211hllq.

There fore , combining (2.3) and (2.4) we obtain the desired result. To prove (ii) we can use a similar argument , hence it will be omit ted, •

To show (i) we use Sobolev embedd ing theorem for t small to obta in

d~:,

(2.3)

(2.4)

the linear p rob lem 12.1) satisfies

Ilullp = I IV( t )g l l f , <_ ct ~ -'llhliij

where p = 2/11 - y), p ' = 2 / (1 + 3,) and y ~ [0, 1].

Proof. We write V ( t ) g ( x ) as

1 ( 2 l / ( t ) g ( x ) =

then applying t heo rem 2.1 the result follows. •

LEMMA 2.7. (i) Let , [~ L-'(R). Then

IIU(t)fil~ _< Ilfll:

d~"(~:)lv/2(1 + ~2) 1/2 dsC

for t > 0 ,

COROLLARY 2.5. Unde r the assumpt ions on f and g in proposi t ion 2.4. The solution u of the linear p rob lem (2.1) satisfies

liullp _< c( l + t) ~ 2(Nf!l~ , - I l f l l~ ,Z,q + Ilhll2 + Ilhllq),

where p = 2/11 - y). q = 2/(1 + 23,) and y ~ (0, 1/2).

Proof. A combinat ion of (i) and (ii) gives the result. •

LEMMA 2.6. Let f=- 1) and g ( x ) = h" (x ) , h ~ L2(R) ~ L ( 2 / ( 1 + `/)XIR). Then the solution u of

1152

(ii) Let g = h' E Lz(R). Then

F. LINARES and M. SC1ALOM

tlV(t)gll2 <~llhll 1,2

Proof. It follows directly from the definition of U ( t ) f ( x ) and V( t )g ( x ) . •

PROPOSITION 2.8. (i) Let g = D ~j + z')/4h E L2(I~). Then

IIV(t )glIL~¢~:z.~¢R,, <-- cllhll-1 +(y/4),2"

(ii) Let f = D ~ / Z f j ~ H ~ ( ~ ) . T h e n

IIU(t)fjlL~:L,',~l~ <-- cllflllv/4.2,

where p = 2 / (1 - 3'), q = 4 / - / and 3 '~ [0.1].

Proof. To prove (i) we write V ( t ) g ( x ) as

V ( t J g ( x ) = ~ ~ 14¢'(~)1r/4(1 q- ~2)1/2 d~:.

Thus from theorem 2.2 it follows that

c J~clY/4/~(') ) [ I IV( t )g l l t -~:cPm'<- iq~,, ( ~: )1-//4(1 + ~ 2)1/2

<_ cllhll i +,~4,.2.

This shows (i). The proof of (ii) is similar, hence it is omitted. •

COROLt.ARV 2.9. Under the assumptions on f and g in proposi t ion 2.8. The solution u of the linear problem (2.1) satisfies

IlulIL~:~C~R), -< c(llflill ~(T/4),2 A-Ilhll~r/4j,2),

where p = 2 / (1 - 3'), q = 4 /3 , and y ~ [0, 1].

Proof. The result follows as a consequence of (i) and (ii) in proposi t ion 2.8. •

From lemma 2.7 and corollary 2.9 we can conclude that a solution u of the linear problem (2.1) satisfies

u ~ L ~ ( ~ : H I ( ~ ) ) A L q ( ~ : L P ( O ~ ) ) ,

where p, q and the initial data satisfy the conditions in those results.

3. DECAY OF SOLUTIONS

In this section we prove theorem 1.1 concerning decay of solutions of the IVP (1.1) and give some remarks.


Proof of theorem 1.1. The solution of the IVP (1.1) is writ ten as

u(t) = U( t ) f ( x ) + r ( t ) g ( x ) - r ( t +-)(lul +' lu)x+(r) dr , (3.1)

U( t ) - and V(t). are defined as in (2.2). F rom (3.1) it follows that

Ilu(t)llv <_llU(t)fllp +llr(t)gllt, + I IV( t - ' c ) ( lu l" -~u) , ( r ) l lp d~ ",

then use of proposi t ion 2.4 and l emma 2.6 leads to

f Ilu(t)llf,<_ctl+tl ~ 2 ( l l l f l l i÷ l l l g l l l )+c ( t - ~ ) "/:lllul'~-~u('r)lle/u+~>d'c. (3.2)

On the other hand, G a g l i a r d o - N i r e n b e r g interpolat ion yields the inequality

Ilull~,~ 'u+ ~,-<cllu,ll~ ~ ~ " ~ + Ilullf: ~:~,

Hence the integral in (3.2) can be bounded as follows

where a = ( a ( 1 - , / ) - Next we define

) ' ¢ t - ~/211u(~')lLi;~ d'r, u( r ) l l e /u ~ , ,d r_< c( sup ilu(t)[[t.: r ) " [I), T] '.'t~

(3.3)

1 + ` / ) ) / ( 2 - 3,).

M ( T ) = sup(1 +t)TJ:llu(t)llp. [,. r l

Therefore , combining (3.2) and (3.3), and the definition of M(T) we obtain

M ( T ~ _< c< IIfH + Ilgll)

a fl + c sup!lutt)]i~,: (l +t)~': M ( T ) ' (t 7) +'/2(l +'r)-+ "+" d~c.

'[0+T]

Now using the hypothesis c~ > (4 - 3 , / - 3,: )/3, we have that

)° MiT~<_c(l l f l+l lgl l )+c'{ sup Ilu(t)lL+ : MCT) +++:~; ~'

1, !+~ o r

M ( T ) -<c~ + c'6 ~ M ( T ) ~+-':~'.

Therefore , for ~ sufficiently small we will havc

M ( T ) <_C

for any T > 0, where C is the smallest positive zero of the function f ( x ) = c 6 + c'6x +~;' - x . Thus we obtain the desired result. •


Some remarks are listed.

Remark 1. If we assume in theorem 1.1 f = D ~/2 f l • H i(N), f l • H1 + ~,/2(N)(3 Lq/2([~) ' and g = D 1 + ~,/2 h • L2(~), h • Hz ' /2(~) (3 Lq(l~), we obtain

Ilu(t)llp <_c(l + t ) -y'2 t > 0 ,

where p = 2 / (1 - y), q = 2 / (1 + y), and ~ > max{(1 + y ) / ( l - ,/), (4 - 3 y - y 2 ) / y } , for y • (0, 1).

Remark 2. Moreover , if wc assume f = DJ /2 f l • H i ( N ) and g = D3/2h • L2(R), where f l ~ H3/2(N) n L~j/2(~) and h • HW2(N) c~ L~(ff~), it can be proven that the solutions of the IVP (1.1) satisfy for any x • JR, t > 0 and ~ > 4

lu( x, t IF <_ c(1 + t ) I

The proof of this follows an argument similar to the p roof of theorem 1.1 but easier.

4 SCATYERING

In this section we establish theorem 1.2 concerning nonl inear scattering for small data for solutions of the initial value problem (1.1) under some restrictions on the nonlinearity power.

To prove theorem 1.2 we first need to establish an existence result for the IVP (1.1) under some additional conditions on the initial data. In the p roof of this result we will use the integral equat ion form of the IVP (1.1) (see (4.1) below), where the linear part is written as in (2.2).

THEOREM 4.1. Let ( f , g ) = {D~:,"4f l , D I" ~ 4h) • H ~ ( R ) XL2(~) with [Iflll,2 + IlgJ[2 < 8 sufficiently small and c~ = 4 / y , y • (0 ,4 /5) . Let g, denote the solution in H i ( N ) of the linear problem (2.1). Then the integral equat ion

f(i l , l ( I } = t ~ ( t ) - - l / ' ( t - - T)(I/.,/I a l/.,/)rx(T) dr (4.1)

has a unique solution in X = L" (E :LP (N) ) ~ L~(~:HI(~) ) , where p = 2 / (1 - y).

Proof To establish this result we will usc the contract ion mapping principle. Define

and

f0 • ~1,~(u)( t ) = ~ ( u ) ( t ) = O( t ) - V( t - r)(luf~-O~u)xx('r) d'c

X, = {v • L"(IR:LP(N))( '1 L ~ ( N : H I ( N ) ) :

where A(v) = max{sup, II v(t)[I i. 2, JI v r[ L~(~ :l ('~ ~ ,,}.

A(v) < a},

(4.2)


We first shall show that (I):X, ~ X , and next that qb is a contract ion in X~.

qb:X, --, X, .

Notice that the hypothesis o ~ = 4 / - / with y ~ ( 0 , 4 / 5 ) implies o~> 1 / ( 1 - y). With this observat ion we can begin the p roof of the previous s ta tement .

Use of the definition (4.2), l emma 2.7, H61der 's inequality in the x-variable and the fact that L ~([R) c L2(,, I)/~ (JR) for c~ > 1/(1 - -/), y ~ (0, 4 /5 ) , leads to

f ~ l t II~(u)(g)ll , .2 ~ il~(t)ll~ 2 ÷ c (11 lul" ~u(~)ll2+llulul~-2u~(~)ll2)d~

f'( ) <!l~/~( t ) l l , ._~-~c Ilu(r)ll '~:,. ',llu('r)llz, ~-llu(~r)ll~:~,llu~('~)llp dr • r "

<ll4,(t)li, ~.+c. f Ilu(~)ll'~,, dr. (4 .3 )

On the o ther hand. definition (4.2), l emma 2.(5. H61der 's inequality in the x-variable and the embedd ing L('([R) c L 2~'' ~)'Y (JR) for c~ > 1/ l 1 - y), y ~ (0, 4 /5 ) , yield

f ,,

II~(u)it)li~,<~c'llO(t)ll~,~-(" It r)

f _< ~' ]~b(t)l]t, + c I t r )

f < , ' l O ( t ) l l p + , ( t - - r )

~": II lul ~ I u(T)llq d r

- Ilu¢ ~')11 ~%,,_L,Ilu(~')l12 dr r

Hu(~') l p Ilu(r)ll2 d r

setting y / 2 = 2/c~ and applying the H a r d y - I . i t t l e w o o d - S o b o l e v t heo rem (see [10]) we obtain

[l~(u)!l..ivl;<~l,<_cll&(t)Ht,l~t,.<r~,, c supl lu( t ) i : ~ 1 + ]]UhIL,'~:L((~)>. (4.4) t

A similar a rgument leads to

( l l~ (u ) ) , l l ¢ . ,< :~ L;(~,)_< c i lO, ( t ) ! l~ . . , : ; , , ¢ ;(~), ~-c" supllu(t)lll.2llu[lg,,(~:Lt'(~>. (4.5) I

Therefore , a combina t ion of (4.3), (4.4) and (4.5) gives

~%('b(u)) _< c~X(~) + 2 c ( A ( u ) ) '~.

So if Ilflll.2 + I[gll_" is sufficiently small such that c'~( O)_< a/2 with 2ca'*-t < ¼, we obtain that

A(qblu)) <_a, this shows that q):A'~ -, X~.

The next step is to prove that '1) is in fact a contract ion. We consider u and v in X~, thus

( ~ ( u ) ¢l)(t~)) lt) = V(z r I ( lu l" ' u - l v l ~ ' v ) .... ( r ) dr .


To es t imate sup~[l(qb(u) - q~(v)Xt)[l~,2 we use l emma 2.7 to have

II(O~(u)-~(v))(t)ll~,2<_c II(lul = 1 n t - l / ) l a - l ) ( / A - - / ) ) ( 7 " ) [ [ 2 d~"

+ / l ( [ u l ' ~ -~u - lu l ~ ~v)~(~-)ll2 d r

= I~ + 12 .

Using a similar a rgument as in (4.3) it follows that

l, _<c ( l lu(r) l lC~ ' + l lv(~- ) l lF .~ ' ) l l (u- v ) ( r ) l lp d r

c~ 1 o~ 1 ~ - I I V l I L , , ~ : L ~ , ) I l u - VlIL"~:L~m) dr . ( 4 . 6 ) <- c(I lulILO~:L~)~

The last inequality follows f rom H61der 's inequality in t. On the o ther hand, we have that

i :<_c II(lul':' = + l v l ~ ' ~ ) ( u - v ) u ~ ( r ) l l ~ d ' c + c I l v l v l " - 2 ( u ~ - v x ) ( ' c ) l l e d ' r = I ~ +I~2.

The a rgument in (4.3) leads to

c~ 1 1~ __< cllvllc°c~:t.C(~)) IIu -- UIII.,~I~:L¢(~ D. (4.7)

To bound I 1, we first apply the general ized H61der 's inequality to obtain

II(lul" : + i v l " - Z ) ( u - v)uJ~-) l l2 _< (llu/l~. ~" + Ilvll~',-Z)llu - vllp,llu~llp, (4.8)

where p ' = ( 2 p ( c ~ - 1 ) ) / ( p - 2 ) , p = 2 / ( 1 - Y). Then using the fact that for c~ > 1/(1 - y), - /~ (0 ,4 /5 ) , and H61der 's inequality in t it follows that

1)~ _<c (llu(~-)11 ?. p' + IIv(~-)ll ~'--',~ Ilu(r )11, . p)ll(u - v)(~-)ll~ , p d r

- ¢"m z FI~, + Hvllz.,,~:z.f~.llul I_"~LF(~)) lu -- Vllz°(~:tf(~)- (4.9)

Ga ther ing (4.6), (4.7) and (4.9) up, we find that

L P c L p

s u p t t ( ~ ( u ) ~ ( v ) ) ( t ) l l l . ~ <c(211u ~- i ~-I t

a 2 +IIvlIZ"I~ ~ ~I lu l I z" (~:L~(~) ) ) I lu -- vlIL"(~:LF(~>, (4.10)

Next ~(V)[[L°(~:L~(~. This can be done using a similar a rgument as in (4.4), thus we obtain

IVl'(u) - a)(v)llL,,,~:c,,l~)~ _< c([lullT,,~:,C~,, + IlvtlT"~:L(~))) suplI(u -- v) ( t ) lh ,2 . t

we est imate ] l ~ ( u ) - qb(LJ)/IL,,(~:c¢,(~), so we begin by est imat ing I I ~ ( u ) -

(4.11)

Asympto t ic behavior of solut ions of a genera l i zed Bouss inesq type equa t ion 1157

It remains to estimate H(~(u) - qb(v)), II ~ , , ~ . , , ~ . Lemma 2.6 leads to

I I (~(u) - ( I ' ( v ) )~( t )b lp <_c ( t - w ) >211(lu1~ : + l v l ~ - e ) ( u - v ) u ~ ( r ) l l q dr

f + c (t r ) "'2111vlvl ~ e ( u ~ - v ~ ) ( r ) l l q d r

2 = I ~ + I : , w h e r c q = 1 + y '

To bound I2 we use a similar a rgument as in (4.4) to obtain

. I t - y/'2 1 <_c ( t - r ) I l t , ( r * " ' , /.p I I ( u - v)(r)l l l .2 d r . (4.12)

Now to estimate l ~ we follow the argument in (4.8) to have

£ 11<c (t r ) ' " : ( l l u ( r )H~ 'pe+ l lv ( r ) l l " : - 1.z, ) l l ( u - v)(r ) l l l pllu~('r)n.2 dr .

Setting - / / 2 = 2 / ~ and using the Hardy Litt lcwood Sobolev theorem it follows that

I t ( ~ ( u ) q ' ( L , ) ) I I z , , , ~ : z , , l ~ , _ < c l l '~ t - , v!l, .,~:~:, ¢,~,)supll(u - v ) ( t )111,2 t

o~ 2 + (llu!lT;:;~:z.c(~,, + [tVllz"~a:ZFC~)))supllu(t)II,.z

t

×llu ~'llz .'~. c('~u~,- (4.13)

Therefore , a combinat ion of (4.11) (4.13)leads to

I I~(u) - q~(v)llt_ 'Ca ~ ~,~,, -< c(!lull~,;z_'~.(a,,- - L II~ .,~ z ct~,, ) supll(u - v)(t)lll.2 t

" ~ ~ ' : )supllu(t)ll l ~llu--vllco~R:L~(~> l

(4.14)

Thus f rom (4.10) and (4.14) it follows that

A ( q b ( u ) - ~ t t , ) ) _ < c { 2 ( A ( u ) ) " ~ - 2 ( A l v ) ) " ' + ( A ( v ) ) ~ - : A ( u ) } A ( u - v )

_< A(u - v) by the choice of a.

This shows that el) is a contraction. Therefore. the contract ion mapping principle gives the existence and uniqueness in X~. It is not difficult to prove the uniqueness in X, this completes the proof. •

Now we are in position to prove theorem 12.

1158 F. I JNARES and M. SCIALOM

Proof of Theorem 1.2. We define

f, u+_(t)=u(t)+ g(t-~r)(lul"-lu)~x('r)dT,

where u(t) is given by theorem 4.1. From lemma 2.7 and HSlder's inequality it follows that

I[u(t)-u_(t)lll.2<_cf'lllul" I u(~)lll.z dr ~+vc

f' f _<c Ilu(r)l[;/. d~'+c lluO-)ll~2~_l))/mllUx(.r)l[ p dr.

L P ( ~ ) c L Z " ( ~ ) and L('(~)cL z~'' ~/~(~) for c ~ > l / ( 1 - y ) and e ~ ( 0 , 4 / 5 ) . However, Therefore,

f+ !

IPu(t)-u+(t)H1.2<_2c Ilu(~-)lll~ p d~-.

By theorem 4.1, the integral on the right-hand side approaches to zero as t --, + w. This gives the desired result. •

R E F E R E N C E S

1. BOUSSINESQ M.J., Th6orie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal , . . . , J. Math. pures appl. 17, 55 108 (1872).

2. ZABUSKY N.J., Nonlinear Partial Differential Equations. Academic Press, New York (1967). 3. BONA J. L. & SACHS R.L., Global existence of smooth solutions and stability theory of solitary waves for a

generalized Boussinesq, Communs math. Phys. I18, 15 29 (1988). 4. TSUTSUMI M & MATAHASH1 T., On the Cauchy problem for the Boussinesq type equation, Math. Japonica

36, 371-379 (1991). 5. LINARES F., Global existence of small solutions for a generalized Boussinesq equation, ,Z diff. Eqns. 106,

257-293 (1993). 6. STRAUSS W.A., Nonlinear scattering theory at low energy,, J. funct. Analysis. 41, 110 133 (1981). 7. PECHER H., Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z. 185, 261-270

(1985). 8. PONCE G. & VEGA k.. Nonlinear small data scattering for the generalized Korteweg-de Vries equation, J.

funct. Analysis. 90, J,45 457 (1990). 9. KENIG C.E., PONCE G. & V E G A L., Oscillatory integrals and regularity of dispersive equations, Indiana Univ.

math. J. 40, 33-69 (1991). 10. STEIN E. M,, Singular huegrals and Differentiability Propertie~ of Functions. Princeton University Press, New York

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Asymptotic behavior of solutions of a generalized Boussinesq type equation