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Periodic solution and decay for some nonlinear wave equations with sublinear dissipative terms
Nonlinear Analysis, Theory, Methods & Applications, Vol. 10, No. 6, pp. 587-602, 1986.
0362-546X/86 $3.00 + .00 Pergamon Journals Ltd.
Printed in Great Britain.
PERIODIC
SOLUTION
EQUATIONS
AND WITH
DECAY
FOR SOME
SUBLINEAR
NONLINEAR
DISSIPATIVE
WAVE
TERMS
MITSUHIRO NAKAO Department of Mathematics, College of General Education, Kyushu University, Fukuoka 810, Japan
(Received 1 December 1984; received for publication 20 January 1986) Key words and phrases: Periodic solution, nonlinear wave equation, dissipative term. INTRODUCTION IN THISpaper we shall first be concerned with the existence of periodic solution for the nonlinear wave equations in one space dimension of the form:
un - u,:x + p(x, ut) + fl(x,u) = f ( x , t )
on
I×R,
u[at = 0, t E R,
(0.1) (0.2)
where I is an interval on R, say, [0, :r], and f(x, t) is an o~-periodic (in t) function. When p(x, u,) = vu,, v > 0, or p(x, v)v >1 Co[vl r+2 - Ca for some r/> 0, Co, C1 > 0, in other words, p(x, v) is superlinear as Ivl ~ ~ this problem has been considered by many authors even for higher dimensional equations or more strongly nonlinear equations (cf. Ficken and Fleishman [4], Prodi [15], Rabinowitz [17], Amerio and Prouse [1], Clements [3], Biroli [2], Masudu [9], Haraux [5], Kannan and Lakshmikantham [6], Von Wahl [18], Kato and Nakao [7] and Nakao [11, 12, 14] etc.). But, for the sublinear case; p(x, v)v >i Co[v[ 2-r - Cl for some 0 < r < 1, Co, Ca > 0, there are very little result. We are interested here in such sublinear case. If p(x, v) is superlinear the easily obtained estimate (under certain assumptions on/3(x, u)):
f o ' ° f l p ( x , u 3 u , d x d t < ~ C o n s t . < o~ implies immediately the desired estimate for
which is not the case for sublinear dissipative terms. This is the point essentially difficult in sublinear case and a new device is needed to overcome the difficulty. Let us state precise assumptions on the functions p and ft. (A1) p(x, v) is defined and measurable on I x R, continuously differentiable in v E R - {0} for a.e. x ~ I and satisfies the conditions:
kolo[ 2-r - k 1 <~ p(x, o)o <~-k2(] + Ivl) Iol, 0 ko(1 + [v[) -r ~< -~vP(X, v) =- p'(x, v), 587
a.e. x,
a.e. x (v 4: O)
(0.3) (0.4)
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M. NAKAO
and
Ip'(x, v)vl <~ k2(1 + Ivl), a.e. x (u ¢ 0),
(0.5)
where 0 < r < 1 and ki(i = 0, 1, 2) > 0 are some constants. (A2)/3(x, u) is defined and measurable on I x R, continuously differentiable in u for a.e. x and satisfies the conditions: /3(x, u)u t> k0
/3(x, r/) dr/0,
I/3(x, u)l k2(1 +
(0.6)
a.e. x,
(0.7)
lul 1+'~)
and
;-u/3(x,u)
k2(1+ lul"),
(0.8)
where o~~> 0 and k0, k 2 > 0 are constants. A typical example of our equations is
(0.9)
Utt -- Uxx + [Utl-rut "~ lUl°lU = f ( x , t)
with0/0. For a Banach space X we denote by if(co; X), 0 < p < oo, the set of X-valued measurable functions f: R ~ X with IIf(t) ]1%dt
< o~.
Similar notation will be employed freely. Our result concerning the existence of periodic solution reads as follows. THEOREM 1. Let f(x, t) be og-periodic in t andf, ft E are fulfilled with 0 < r < 1, 0 <~ o~ such that
L2(o9;L2(/)).
Assume that (A1) and
(or + 4)r < 4.
(A2)
(0.10)
Then, the problem (0.1), (0.2) admits an co-periodic (in t) solution u(x, t) belonging to W2,~(oo; L2(/)) N wl'oc((D; Hi(/)) N L~(og; H2(/) n /~1(/)), i.e.
uu ~ L~(~o; L2), ut ~ L~(~o;/-)a)
and
u ~ L~(~o; H2 A ~1).
COROLLARY 1. Under the additional assumption f~/3(x, ~]) d~] ~ k6 lul ~+ 2, k6 > 0, the condition (0.10) in theorem 1 is replaced by a weaker one 2(c~ + 2)r < ~ + 4.
(0.10)'
The device for the proof of theorem 1 is useful also to derive decay estimates of solutions of similar nonlinear wave equations. In the second part of this paper we shall consider the energy decay problem for the equations (0.1), (0.2) with initial data u(0) = Uo, ut(O) = Ul. At this time we require a little stronger assumptions on p and/3.
Nonlinear wave equations
589
(A~) p(x, v) is defined and measurable on I × R, continuously differentiable in v (for a.e. x) on R - {0} and satisfies the conditions (for a.e. x): kolo[ 2-r <~ p ( x , v ) o <- kx([vl 2-~ + Ivl2),
(0.11)
and kolvl -r <~ p ' ( x , v) <~ kl(IO[ -r + Iv[)
(v :/: 0)
(0.12)
with some 0 < r < 1 and k0, k l > 0. (A~) fl(x, u) is defined and measurable on I x R, continuously differentiable in u (for a.e. x) on R and satisfies the conditions:
0 <~ ko and
i0
fl(x, rl) dr 1 <<-fl(x, u)u <~ kllUl 2+°~
(0.13)
0 0 <- ~ufi(x, u) <~ k2[ul ~.
(0:14~
When p(x, v)v >- k01vl ~+2 for some r ~> 0 and fl(x, u) =- 0 the decay estimate of solutions for (0.1), (0.2) is included in [10] (see [13] for more general equations). Indeed, if p(x, v) = Ivlrv r > 0 and fl(x, u) =- 0 we know Ilu,(t)ll2 + Ilu~(t)ll2 ~< c(1 + 0 -1#
where [['llp denotes the norm in LP(I) and C is a constant depending on Ilu0/llz + Iluirl2. While our result concerning the energy decay for the problem (0.1), (0.2) (on I × R +) with u(0) = Uo, ut(O) = ul reads as follows. THEOREM 2. Let (Uo, ut) E H 2 ('1 /~1 × /~1 and f, ft E L~o~(R + ; L2). Suppose that (A~) and (A~) are fulfilled with some c~~> 0 and 0 < r < 1. Moreover, suppose that 60( 0 =
]lf(s)ll 2 ds
and 61(0 =
(it+l
=
o(t -(1-r)/r)
IIf,(s)ll~ cu
as
t--> o¢
(0.15)
)1/2
~ M1 < ~.
Then, the solution u(t) of (0.1) (0.2) on 1 × R + with u(x, O) = Uo(X) and ut(x, O) = ul(x), which exists globally in W2~,~(R + ; L 2) N uzl.=ro+.,, loct,, ,/~1) (3 L~(R+; H2 N ~ ) , Ilu,(/)ll2 + IlUx(t)ll2 ~< C0(1 + t) l-lIt
satisfies (0.16)
and also, under the additional assumption (1 - r)(1 + c~) > 1 and 61(0 = o(t -1#) as t---~ % we have
[lut,(t)ll2 + ]lUtx(t)][2 ~< C0(1 + t) -'/r where Co denotes positive constants depending on r, c~, Ilu0llH= +
(0.17)
Ilu,ll.1 and 60, 61, etc.
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M. NAKAO
1. PERIODIC SOLUTION In this section we shall give the proofs of theorem 1 and corollary 1. Let 0 < e < 1 and define
p'(x,v)
if[v[>/e
o'
p ; ( x , v) = p ' i x , e) - p ' ( x , - e)
p'(x, e) + p'(x, -e)
2
2
if [vl < e
and
pE(x, v) =--
fop
p'e (x, rl) dr1.
e--+0p~(x, v) = p(x, v) a.e.
Then, it is easy to see that lim
remain valid with p(x, v) replaced by p~(x, differentiable in all v E R (for a.e. x ~ / ) . We consider the modified problem:
v).
Moreover, note that
uu - Uxx + pe(x, ut) + I (P~) LD.181 = O,
on I x R, and the conditions (0.3)-(0.5)
fl(X, u)
= f(x,
p~(x, v)
t), e > 0,
U(X, t) = U(X, t + 09).
is continuously
(1.1)
With the standard use of Galerkin method and Leray-Schauder degree theory the existence of solution u~(t) of (P~) is guaranteed if we can establish a priori estimate +
~ const.
<
~¢
(1.2)
for assumed smooth solution u~ of (P~) (cf. Prodi [15]). In fact, we shall derive such estimate independent of e. Then, using a standard compactness argument (cf. Lions [8], Amerio and Prouse [1] etc.) we shall be able to get the desired solution of the original problem as a limit of {u~} as e--~ 0 (along a subsequence). Now, our task is to derive the estimate (1.2) independent of e. Hereafter we write u for u~ and denote by C generous positive constants independent of e. Sometimes we use C(M) to denote constants which depend on M in a significant way. Multiplying the equation (1.1) by ut and integrating over I x [0, col we have, by the periodicity and the boundary condition,
f ftP~(X, Ut)utdxdt~Mo(f with
mo
=--
IlfllL(2-')/O-r)(w;L(2-O/O-O),and
[,u,(,~-~r)1/(2-r)
(1.3)
by (0.3) (with p = p~)
f~ fz lutlZ-r dx dt <~ C(M(O2-r)/(l-r) + 1)
(1.4)
Nonlinearwave equations
591
where fo~ denotes the integral over any interval with length to. From (1.4) we have
f~ fl UZdx dt ~ Csupllu,(t)]l~(M~2-r)/tl-r) + l).
(1.5)
Next, multiplying the equation by u we find fo @u~(t)[122+
f ~(x,u)udx) dt~ f {,,ut(t)l,~ + f
f(t)u(t)dx
+ fi lot(x, u,)t lu[ dx}dt.
(1.6)
The third term of the right-hand side of (1.6) is treated as follows.
f fi'Pe(x, ut)[[uldxdt~C[fi(l+ut)'u'dxdt C f~o (1 + c
[[utll2) [lulldt
( f Ilu,[(~dt + 1) + ~i f IluxllNdt.
(1.7)
From (1.5)-(1.7) and the assumption (0.6) we can obtain easily
fo~ {~ llut(t)"2 + F(u(t)) } dt ~ C(Mo) {sup ,,u,(t),,: +1}
(1.8)
where we set
F(u) - ~ Ilux[l~+ Setting E(u(t))
fl(x, rl) do dx.
~ ½11ut(O]l~+ F(u(t)) we see from (1.8) that there exists t* E (0, ~o) such that E(u(t*)) <~C(Mo) \(sup[lut(t)l[ + 1),
and hence, using the inequality sup E(u(t)) ¢
<~E(u(t*)) +
]fu] dx dt .
and (0.6), we obtain ]
sup (llu,(t)l[~, +
%
[lux(t)l[g)~ C(Mo)\(sup[lut(t)ll% + 1)./
(1.9)
For further estimation we differentiate the equation (1.1):
U, - U~x + O's(x, U)U, + fl'(x, u)U = f~(x, t) where
U =- ut.
(1.10)
M. NAKAO
592
Since u is assumed to be smooth U satisfies integration over I x (0, co) yield
Uloz= 0.
Thus, multiplication by Ut and
~ f O'~(x,CO'V,t~ dxdt<- f f {ll3'(x,u)llSS, l + l:,l lU,I}dxdt
(1.11)
Here,
L f lH'(x,u)l lUU,l dx dt <~c L ft (l + luL~)[UU,Idx dt ~< C(1 + Llull~)
f Ilu& IIu& dt
6(1 + Ilull~)(1+ Ilu,ll~/=)f IIc% dt (by (1.9)) where we set
(1.12)
Ilull~= sup ([[u(t)[l~). From (1.11), (1.12) and the assumption (0.4) we see easily t
that
f IIu,(/)llNdt ~< C(Mo,M1)(1 + IIull~)(a + Ilu,II3')
(1.13)
where M1 --- (fo, lLf,(/)[l~dr) 1/2. On the other hand multiplication of the equation (1.10) by U yields
IlUxll2 dt<~f f {IGlZ+lp'(x, ~l Iuu, l+lY(x, u)lu2 +lLuI} dx dt. The second and third terms of the right-hand side are handled as follows.
~o~~ 'P'~(x,U)"UU,' dx dt <~C(6) f fi o'~(x,U) lU,[2dxdt+ 6 f fl P'e(x,U) lUl2dxdt, 6> O, <<-C(Mo,M1, 6)(1
+ I[ull2~)(1 + Ilu,lle) + c ~
j f, (1 + [UI)IU[ dx dt
where we have used (1.11)-'(1.13) and (0.5).
f~fJ [j6'(x'u)lUedxdt<~Cfo) fl (i+lul")Uedxdt <~ C(1 + I[ull:)(1 + I[u,llr)
(by (1.9)).
Thus, choosing 6(>0) sufficiently small we obtain from (1.13) and (1.14)
~
(IIU,(t)II~
+ Ilg~ (t)ll~) dt ~< C(Mo,Mx)(1 +
IluliP)(1 + Ilu,II3~)
(1.14)
Nonlinear wave equations
593
and hence, by a standard technique (cf. (1.9)), sup (IJu.(t)ll~ +
IlU,x(t)ll~) <~C(Mo, M1)(I
Now, using the well known inequality Ilu[K ~ from (1.9) that
3 r )" + IJttlj2cr)(1 + IIU tlJ~
Cllull~/2 I[Uxl[~/~~
(1.15)
~l,xll~ for u E ~1(I) we see
Ilull~ ~ Csup Ilu~(t)lJz ~ C(M0) (1 + Ilu,ll~~) t
and ))u,ll~ -= sup I)ut(t)lJ~ ~ C sup JJur(t)ll~/2)lu~(t)ll~/2 <~C(Mo) (1 + ))u,l15/') sup Ilu~(t)ljy ~. t
t
t
Therefore,
IJu,ll~ -< C(Mo) 1 + sup [lUtx(t)H2/(4-r) t
and
Combining these estimates with (1.15) we obtain
sup(llutt(t)lj22+[]ut~(t)IIz)<~C(Mo,M,)(1 +supJlutx(t)lj~°~+6rV(4-~)).
(1.16)
By our assumption (0.10) on cr and r we see (2o~r + 6r)/(4 - r) < 2 and hence we obtain from (1.16) the desired estimate: sup (]tu,(t)ll2 + ]]u=(t)ll2 ) ~< C(Mo, M1) < oo t
and (see (1.9))
sup (IM(t)ll2 + Ilu~(t)l/2) ~ C(Mo, M~) < ~. t Now the proof of theorem 1 has been finished. Finally we shall prove corollary 1. We know already (see (1.9))
sup
(1 + s.p
This time we can utilize the inequality
to get, since f~fl(x, r/) dr/~> ko[ul ~+2,
Ilulf~ = sup II.(OL -< sup E(u(t)) 1/~÷4~ Ilu~(t)ll~/~+4~ f
t
594
M. NAKAO
Consequently, we can derive, instead of (1.16), sup ([[u.(t)l]~ + IlUtx(t)l[~) ~ C(Mo, M1) (1 + sup I]Utxl[~) with
4roe
~/ = \ 4 + a ~ + 3 r
)
2
. -4--
(by(0.10)'),
which gives the desired a priori estimate for u,(t). 2. D E C A Y
OF SOLUTION
In this section we shall prove theorem 2. We need the following lemma. LEMMA 1. Let q~(t) be a bounded nonnegative function on R + = [0, oo) such that, for arbitrarily fixed T > 0, sup
(~(S) l + r
~<
Co( dp(t) - ~p(t + 1)) + g(t) + e, 0 <~ t <~ T, e >t O,
t~s<~t + l
with r > 0 and g(t) = o(t -1 -l/r) as t---~ ~. Then q~(t) ~< C1(1 + t) -x/r + O(e)
forO < t ~< T
where C~ is a constant depending on ¢(0) and r, and O(e) denotes positive quantities such that lim O(e) = 0. The proof of lemma 1 with e = 0 is found in [10], The case e > 0 is also e---~0
proved similarly (cf. [15]). We consider again the modified problem
I
u , - U ~ x + p~(x, ut)+ f l ( x , u ) = f ( x , t )
(P~)'[u(x,O)=uo,
u,(x,O)=ul
and
on/xR
uloi=O
(2.1)
where p~ is the one defined in the previous section. By the assumption (A~) it is easily seen that p~(x, O) = O,
kolv[ z-r - O(e) ~ p~(x, v)v <~ kz(lvl a-r + Ivl + O(e))Ivl
(2.2)
and ko(Iv[ + O ( e ) ) -r ~< o's(x,
Sincef, ft E L~oc(R + ; L 2) and (u0, u~ such that U e ~-" ,M'2.~I"D+; " loc k"~"
Ul) ~
L2
H2
(q
~1
X
v) <~ k2(1 + [vl-').
(2.3)
~1 the problem (P~)' has a unique solution
o :¢ )f~lW~6c(R + . H,)fqL,oc(R
+.
,H2 , ~ x ) .
In fact this assertion will be easily verified by using the estimates derived below. Moreover, the solution of the original problem will be given by taking the limit of u~ as e ~ 0 along a subsequence. The passage procedure as e ~ 0 is again standard and omitted. Thus our task is
Nonlinear wave equations
595
again to derive the estimates (0.16) and (0.17) (independent of e) for u~. In the sequel we write u for u~. Multiplying the equation (2.1) by u, and integrating over I x (t, t + 1) we have, by (2.3),
ko It+l
f/ (lut] + o(~))-' I,,I ~d~ ds ~
f,+,f,
= E(u(O) - E(u(t + 1)) +
(2.4)
f u , dx ds,
where
E(u(t)) = Ilut(t)[i2 + ~ llux(t)l[2 +
fo ~(x, rl) drI dx.
,
From the above we have easily
f'+' f, (p.,l
+ o ( E ) ) -r I.tl 2 dx ds ~ C ( E ( u ( [ ) )
- E(l~(l -~- 1)) --[- C(~o(t) (2-r)/(1-r) -{- O(E)
(2.5)
D(t) 2-r and hence ft+l
Ilu,lrNcU ~ (\ t ~ ssup ~ t + l Ilut(s)]l~ + o(e)) D(t) 2-r.
(2.6)
t
From (2.6) we see that there exist t a ~ [t, t + 1/4] and
t2 ~ [t + 3/4,
IJut(ti)l}2 <~2 (\ t ~
t + 1] such that i = 1, 2.
(2.7)
Multiplying the equation by u, integrating over I x [q,/2] and using (2.2) we have
;/ {,,ux(s)[,~ + f (ut(t,), u(t,))
-
(ut(t2) , u(t2) ) +
llu,(s)fiNds + tl
+
Ill [ul dx ds ~ 4 sup
t~s~t+ 1
tI
+ ( sup
\t~s<~t+ 1
[lu,(s)l[: + o(e)
+ o(e) + k2
)
Ip~(x, u,)p lul d~ ds ll
I]u(s)][2 (
sup Ilu,(s)ll~/2 +
\t~s~t+ 1
D(t) 2-' + k2
o(~) D(t) ~>'v2
Ilu,(s)[12_-:ds tl
flu(s)ll~:~ d~
(f,2
t
Ilut[I2 ~
tl
Ilull~ ~ tl
)12
+ 60(0
Ilu(s)ll~ ds tl
)1/2
(2.8)
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M. NAKAO
From (2.8), (2.5), (2.6) and the inequality [lull: ~< C][uxll2 we can show
+ f (x,u)udx} ds ~< C ( sup ~ ) ( {t<~s<_t+l
sup [[ut(s)l]~2 + O(E)tU(t) `2-r,/2/
\t~s<_t+1
+ ( sup Ilut(s)l]: + O(e)] D(t) 2-r + D(t) z-2r /
\t<_s~t+l
+ 60(t) 2 + O(e)} = A2(t).
(2.9)
It follows from (2.6), (2.9) and the assumption on [3(x, u) that
ft12t E(u(s))ch'~CA2(t). Thus, there exists t* E (tl, te) such that E(u(t*)) <~CA2o(t), and hence
sup E(u(s)) <- g(u(t*)) + I
t~s
Ip~(x, ut)ut(s)l dX ds
Jt
+ ftt+lfl
If.,
,tr ds < CA~(t).
(2.10)
By the definition of AZ(t) we obtain from the above that sup E(u(s)) <~C{( sup IIU,(s)ll r + o(E)) O(t) 2-r + O(/) 2-2r + 620(0 + O(e)}.
t~s <~t+1
t<~s<~t+1
(2.11)
To derive the decay of E(u(t)) from (2.11) we must show the boundedness of [lu,(/)L. For this we utilize the differentiated equation: Uu - Uxx + p'e(x, U)Ut + ~ ' ( x , u ) U = f t ( x , t)
(2.12)
with U = ut. Multiplying (2.12) by Ut and integrating we have, by (2.3),
ko ft'+l f/ ([Ul + O(e))-r]utl 2 dxds<~ f t + l f, Pe( ' X , U) u2 dxds = e(u(t)) - E(u(t + 1)) +
+1;1
(IL [ + k2 [ul ~[UI)] Ut [ dx ds
where we set this time 1 2 g(g(t)) = ½lls~(t)ll~ + ~ll gx(t)ll2.
(2.13)
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597
Here,
f jf/([f,I + +' k2lul~IUl)IU,[dxds k0 ft+l
<<-Tj, f(Iul+°(~))-'Iu,[ z~as +C
(ftt+lf/
If,[z]o[r dx ds +
It+If/
lul2=[Ule+rdxds)
+ o(e) and moreover
It+lf/ lulZ~lolz+'oxOs<~
sup ([[u(s)lI~Ilu(s)fl~)
t<~s~t+1
ft+lfl t
OZdxds
~< sup E(u(s)) 1+~ sup NU(s)I[r. t~s<~t+ 1
t~s<<-t+ 1
Thus, we have from (2.13) that
ft+1 fl
p'e(x,
U)U2t dxds <~CG(t) 2
(2.14)
and
ftt+lf/ U~dxds<~C(
sup IPU(s)lJ~+ o(e)
\ t<~s~t+l
)r c(t) 2---Dl(t)2
(2.15)
where we set
G2(t) = +
E(U(t)) - E(U(t + 1))
sup I[U(s)[[• ( 6 ~ ( t ) +
t~s-<-t+ 1
sup E(u(s)) '+~)
t<~s<~t+1
+ o(e).
(2.16)
By (2.15) there exist tl ~ [t, t + 1/4], t 2 ~ [t + 3/4, t + 1] such that
I]U,(ti)l]2 <~2Dl(t), i = 1, 2. Using this we can show easily (cf. (2.8))
f t2 I[Ux(s)[I2 ds ~< (et(tl) ' U(t,)) - (Ut(t2) ' U(t2)) tl
IIu,(~)ll~ as +
+ tl
+
;i f, t
If, ul d~ as t
fp'~(x, u)u, Ulaxas
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M. NAKAO
C( sup IIV(s)llzD,(t ) + D~(t) + 6~(t)) \t~s<-t+ l
I'l 2 IIU,(s)ll~ d~ +
+{
ft2fi tl
Ip'~(x, U)U, Ui d~ d~ (2.17)
+ o(e). Now,
f'~ f, Ip',(x,u ) u , ul Ox ds tl
Ip',(x, u)lu~ dxds
ca(o <~ ,'-
tl
0,
[o',(x, u ) [ u ~ dxc~
(11Nil ~-~ + II uIIN + o(e)) as
IlSxll~ d~ + c(a(t)~/( ~<~ + G=(t)) + o(e).
Therefore we have from (2.17) that
.1
liu,(s)ll~ d~ ~ C tl
+ 62(0 \t<-s<-t+l
+ G(t) 4/('+2) + G(t) z) + O(e).
(2.18)
From (2.15) and (2.18) we can obtain as is usual (cf. (2.11))
E(U(s)) <~C(DI(t) 2 + G(t) 4/(~+2) + G(t) 2 + 61(02) + O(e).
sup
(2.19)
t~s~t + l
Moreover we observe that
Dl(t) 2 <~C
sup
IIU,(s)ll2 +
O(e)
G(t) 2
\t6s~t+l
<~½,<-s~t+lsupE(U(s)) + G(t) 4/(2-') + O(e).
(2.20)
It follows from (2.19) and (2.20) that sup
E(U(s)) ~ C(G(t) 4/(2+0 + G(t) '/(2-0 + G(t) 2 + b1(02) + O(e).
(2.21)
t~s~t+l
First we shall derive the boundedness of E(U(t)) from (2.21). Assume that E(U(t)) <~E(U(t + 1)) for some t >i 0. Then, by (2.21) and (2.16), we find sup t~s<<-t + 1
E(U(s))<~ C ( sup E(U(s)) r/(2+r) (61(t)2 + sup E(u(s))l+c~) 2/(2+r) t. t<<-s<<-t+ 1
t<~s<~t + 1
Nonlinear wave equations
599
+ ,~s~t+asupE(U(s))r/(2-r)((~l(t)2+ t<~s~t+ISUpE(u(s))l+°t) 2/(2-r) t<-s~t+lSUpE(U(s)) r/2 (61(t)2 + t~s<~t+lSUpE(u(s))l+cr) + 6l(t)2} + O(E) and hence sup
t~s<~t+1
E(U(s)) <~C {M 2 + m 2/(l-r) + M 4/(2-0
+
sup
E(u(s)) 1+°~
t<-s<~t+1 + r-~,<-t+lsupE(u(s)) (~+~)/(1-~)+ ,<-s~t+asupE(U(S)) 2(1+°0/(2-r)} + O(E) (2.22)
where we set M1 = sup 61(0. Here we note that by (2.4) t
E(u(t)) <<-E(u(O)) +
flf(s)[lt~z~t~ll:~tds + O(e)T, 0 ~ t ~ T,
for any T > 0, and also by the assumption on
60(0
fo~ [[f(s)[[/Z2ZrI¢ll7_~/ds ~< C E~(fn+l n=0
Ilf(s)ll~
)(2-r)/2(1-r)
n
z¢
<<-C~ (l+n)-(2-r)/r <~
(0
n=0
Hence we have
E(u(t)) <~Co + O(e)T
for 0 ~ t ~< T.
(2.23)
Hereafter O(e, T) will stand for constants depending on e and T like lim O(e, T) = 0. By ~'--* 0
(2.22) and (2.23) it follows sup
t<~s~t+1
E(U(s)) <~Co + O(e, T)
and we obtain, for any T > t t> 0,
E(U(t)) <~max
(Co + O(e, T), 0~<-lsupE(U(s))).
Now, by (2.13), sup 0~
E( U(s)) <~E(U(O)) + + (Co + o(e))
1
f0IIf,(s)llzllU,(s)lj=ds
fo'
IIg (s)ll=
and hence sup 0~s~l
E(U(s))<- C(E(U(O)) + Co + O(e)).
(2.24)
M. NAKAO
600 Observe that
E(U(O)) = ~llu.(O)ll~z + ½11ux,(O)ll~ = ½1luxx(O) - p~(x, u,(O)) - ~(x, u(O))
÷ f ( x , 0)11~ + ½11~x,(O)ll~ C(tluollZ,,~ + Ilu~ll~ -~ + Ilulll~ + Iluo[I,~7 +~!
+ IIf(O)ll~+ 1 + o(~)) < ~. Thus we conclude the boundedness of E(U(t))
E(U(t)) ~ Co + O(e, T)
for 0 ~< t ~ T
(2.25)
with some Co = c0(llu0il~,2, Iluxll~, r, ~, Mo, M1). Now, let us return to the inequality (2.11). With the use of (2.25) and the boundedness of E(u(t)) we find, for 0 ~< t ~< T, sup
t~s<~t+ l
E(u(s)) ~ Co(O(t) 2-r q- D(t) 2-2~ + 60(02) + O(s, T) C o ( O ( t ) 2-2r + ~o(/) 2) + O(6, T)
or sup
E(1A(S)) l+r/2(1-r) ~ C o ( g ( u ( t ) )
t~s<~t+ l
- E ( u ( t + 1))) + CoOo(t) (2-r)/(1-r) -4- O(E, r ) .
(2.26)
Applying lemma 1 to (2.26) we obtain
E(u(t)) ~ C0(1 + t) -2(1-')/r + O(e, T)
(2.27)
for 0 ~< t ~< T. Using this we can show also the decay property of E(U(t)). Indeed, by (2.21), (2.16) and (2.25) we have sup
t<_s<~t+l
E(U(s)) <~Co(G(t) '/(z+r) + 61(t) z) + O(e, T)
and hence sup
t<~s<~t+ l
g(U(s)) 1+r/2 ~ C O t E ( U ( t ) ) - E(U(t + 1)) + C t~s.t+lSUp
E(U(s)) r/2 Ql~l(/)2-1-t.s<~t+lSUp E(u(s))l+a) + C(Sx(t)2+r}
+ O(g,
T),
which implies further, by (2.27), sup
t~s~t + l
E(U(s)) 1+~/2 <~Co{E(U(t)) - E(U(t + 1)) + C61(t) 2+r + C(1
+
t) -(x-r)(l+g)(2+r)/r} -b O(E, T).
(2.28)
Applying lemma 1 to (2.28) we can conclude that
E(U(t)) <~CoO + t) -2/r + O(e, T) provided that (1 - r)(1 + c0 > 1 and 61(0 = o(t-x#).
for 0 ~ t ~< T
(2.29)
Nonlinear wave equations
601
Letting e---~ 0 in (2.27) and (2.29) we complete the proofs of (0.16) and (0.17).
Remark. Even if (1 - r) (1 + a~) ~< 1 we can get certain decay rate for llu,(t)[12 + Ilutx(t)ll2. Indeed, if 1 i> (1 - r)(1 + o0 > r/(2 + r) we have from (2.28) that sup E(U(s)) l+l/2't <~Co{E(U(t)) - E(U(t + 1)) + COl(t) 2+r t<~s<-t + 1
+ C(1 +/)-1-2%}
+ O(8, T)
for any 0 < r / < r/0 with
rl°=½ { ( 1 - r ) ( l + °O(2 +
1}
and consequently we have
fJu.(t)[l: + Jlu=(t)ll2 ~ c0.~(1 + t)-" for
½~(1 - r)(1 + c0(2 + r ) _ 1~, 0 < t/ < r L J provided that 61(0 =
o(t-(t+20/(2+O). Also we note that lim E(U(t)) = 0 if 61(0--+ 0 as
t--+ m without any condition on cr and r (cf. [10]). Finally it may be worth mentioning that our argument of this paper is applicable to the more general equations:
u,t+(-1)~Aku+p(x,u,)+fl(x,u)=f(x,t)
on~xR(R
+)
with Druloa = 0, I)~ ~< k - 1, where ff~ is a bounded domain in R", A is the Laplacian and k is a positive integer with 2k > n. REFERENCES 1. AMERIO L. & PROUSE G., Almost Periodic Functions and Functional Equations, Van Nostrand, New York (1971). 2. BIROLI M., Bounded or almost periodic solution of the nonlinear vibrating membrance equation, Ric. Mat. 22, 190-202 (1973). 3. CLEMENTSJ. C., Existence theorems for some nonlinear equation of evolution, Can. J. Math. 22, 726-745 (1970). 4. FtCKEN F. A. & FLEISHMAN B. A., Initial value and time-periodic solutions for nonlinear wave equations, Communs pure appL Math. 10, 331-356 (1957). 5. HARAUX A., Nonlinear evolution equation; global behavior of solutions, Lecture Notes in Mathematics 841, Springer, Berlin (1981). 6. KANNAN R. & LAKSHMIKANTHAMV., Periodic solutions of nonlinear hyperbolic problems, Comp. Math. Appl. 9, 479-486 (1983). 7. KATO H. & NAKAO M., Existence of strong and smooth periodic solutions, Math. Report College Gen. Ed., Kyushu Univ. 14, 57-88 (1983). 8. LIONS J. L., Quelque MOthodes de Rdsolution des ProblOm aux Limites Nbnlindares, Dunod-Gauthier-Villars, Paris (1969). 9. MASUDA K., On the existence of periodic solution of non-linear differential equations, J. Fac. Sci. Univ. Tokyo 1A, 247-257 (1966). 10. NAKAO M., Convergence of solutions of the wave equation with a nonlinear dissipative term, Mem. Fac. Sci., Kyushu Univ. Ser. A, 30,257-265 (1976). 11. NAKAO M., Bounded periodic or almost periodic solutions of nonlinear hyperbolic partial differential equations, J. diff. Eqns 23, 368-386 (1977). 12. NAKAO M., Bounded, periodic and almost periodic classical solutions of some nonlinear wave equations with a dissipative term, J. Math. Soc. Japan 30, 375-394 (1978).
602
M. NAKAO
13. NAKAO M., A difference inequality and its application to nonlinear evolution equations, J. Math. Soc. Japan 30, 747-762 (1978). 14. NAKAO M., Existence of classical periodic solutions of some nonlinear wave equations in one space dimension, Math. Rep. College Gen. Ed., Kyushu Univ. 12, 77-91 (1980). 15. NAKAO M., On solutions to the initial-boundary value problem for (O/Ot)u - Aft(u) = f, J. Math. Soc. Japan 35, 71-83 (1983). 16. PROD1 G., Soluzioni periodiche dell'equazione delle onde con termine dissipativo nonlineare, Rc. Semin. Mat. Padova 36, 37-49 (1966). 17. RABINOWITZ P. H., Periodic solutions of nonlinear hyperbolic partial differential equations I, Communs pure appl. Math. 20, 145-205 (1967); II, ibid. 22, 15-39 (1969). 18. WAHL W. V., Periodic solutions of nonlinear wave equations with a dissipative term, Math. Annln 190, 313-322 (1971).
0362-546X/86 $3.00 + .00 Pergamon Journals Ltd.
Printed in Great Britain.
PERIODIC
SOLUTION
EQUATIONS
AND WITH
DECAY
FOR SOME
SUBLINEAR
NONLINEAR
DISSIPATIVE
WAVE
TERMS
MITSUHIRO NAKAO Department of Mathematics, College of General Education, Kyushu University, Fukuoka 810, Japan
(Received 1 December 1984; received for publication 20 January 1986) Key words and phrases: Periodic solution, nonlinear wave equation, dissipative term. INTRODUCTION IN THISpaper we shall first be concerned with the existence of periodic solution for the nonlinear wave equations in one space dimension of the form:
un - u,:x + p(x, ut) + fl(x,u) = f ( x , t )
on
I×R,
u[at = 0, t E R,
(0.1) (0.2)
where I is an interval on R, say, [0, :r], and f(x, t) is an o~-periodic (in t) function. When p(x, u,) = vu,, v > 0, or p(x, v)v >1 Co[vl r+2 - Ca for some r/> 0, Co, C1 > 0, in other words, p(x, v) is superlinear as Ivl ~ ~ this problem has been considered by many authors even for higher dimensional equations or more strongly nonlinear equations (cf. Ficken and Fleishman [4], Prodi [15], Rabinowitz [17], Amerio and Prouse [1], Clements [3], Biroli [2], Masudu [9], Haraux [5], Kannan and Lakshmikantham [6], Von Wahl [18], Kato and Nakao [7] and Nakao [11, 12, 14] etc.). But, for the sublinear case; p(x, v)v >i Co[v[ 2-r - Cl for some 0 < r < 1, Co, Ca > 0, there are very little result. We are interested here in such sublinear case. If p(x, v) is superlinear the easily obtained estimate (under certain assumptions on/3(x, u)):
f o ' ° f l p ( x , u 3 u , d x d t < ~ C o n s t . < o~ implies immediately the desired estimate for
which is not the case for sublinear dissipative terms. This is the point essentially difficult in sublinear case and a new device is needed to overcome the difficulty. Let us state precise assumptions on the functions p and ft. (A1) p(x, v) is defined and measurable on I x R, continuously differentiable in v E R - {0} for a.e. x ~ I and satisfies the conditions:
kolo[ 2-r - k 1 <~ p(x, o)o <~-k2(] + Ivl) Iol, 0 ko(1 + [v[) -r ~< -~vP(X, v) =- p'(x, v), 587
a.e. x,
a.e. x (v 4: O)
(0.3) (0.4)
588
M. NAKAO
and
Ip'(x, v)vl <~ k2(1 + Ivl), a.e. x (u ¢ 0),
(0.5)
where 0 < r < 1 and ki(i = 0, 1, 2) > 0 are some constants. (A2)/3(x, u) is defined and measurable on I x R, continuously differentiable in u for a.e. x and satisfies the conditions: /3(x, u)u t> k0
/3(x, r/) dr/0,
I/3(x, u)l k2(1 +
(0.6)
a.e. x,
(0.7)
lul 1+'~)
and
;-u/3(x,u)
k2(1+ lul"),
(0.8)
where o~~> 0 and k0, k 2 > 0 are constants. A typical example of our equations is
(0.9)
Utt -- Uxx + [Utl-rut "~ lUl°lU = f ( x , t)
with0
< o~.
Similar notation will be employed freely. Our result concerning the existence of periodic solution reads as follows. THEOREM 1. Let f(x, t) be og-periodic in t andf, ft E are fulfilled with 0 < r < 1, 0 <~ o~ such that
L2(o9;L2(/)).
Assume that (A1) and
(or + 4)r < 4.
(A2)
(0.10)
Then, the problem (0.1), (0.2) admits an co-periodic (in t) solution u(x, t) belonging to W2,~(oo; L2(/)) N wl'oc((D; Hi(/)) N L~(og; H2(/) n /~1(/)), i.e.
uu ~ L~(~o; L2), ut ~ L~(~o;/-)a)
and
u ~ L~(~o; H2 A ~1).
COROLLARY 1. Under the additional assumption f~/3(x, ~]) d~] ~ k6 lul ~+ 2, k6 > 0, the condition (0.10) in theorem 1 is replaced by a weaker one 2(c~ + 2)r < ~ + 4.
(0.10)'
The device for the proof of theorem 1 is useful also to derive decay estimates of solutions of similar nonlinear wave equations. In the second part of this paper we shall consider the energy decay problem for the equations (0.1), (0.2) with initial data u(0) = Uo, ut(O) = Ul. At this time we require a little stronger assumptions on p and/3.
Nonlinear wave equations
589
(A~) p(x, v) is defined and measurable on I × R, continuously differentiable in v (for a.e. x) on R - {0} and satisfies the conditions (for a.e. x): kolo[ 2-r <~ p ( x , v ) o <- kx([vl 2-~ + Ivl2),
(0.11)
and kolvl -r <~ p ' ( x , v) <~ kl(IO[ -r + Iv[)
(v :/: 0)
(0.12)
with some 0 < r < 1 and k0, k l > 0. (A~) fl(x, u) is defined and measurable on I x R, continuously differentiable in u (for a.e. x) on R and satisfies the conditions:
0 <~ ko and
i0
fl(x, rl) dr 1 <<-fl(x, u)u <~ kllUl 2+°~
(0.13)
0 0 <- ~ufi(x, u) <~ k2[ul ~.
(0:14~
When p(x, v)v >- k01vl ~+2 for some r ~> 0 and fl(x, u) =- 0 the decay estimate of solutions for (0.1), (0.2) is included in [10] (see [13] for more general equations). Indeed, if p(x, v) = Ivlrv r > 0 and fl(x, u) =- 0 we know Ilu,(t)ll2 + Ilu~(t)ll2 ~< c(1 + 0 -1#
where [['llp denotes the norm in LP(I) and C is a constant depending on Ilu0/llz + Iluirl2. While our result concerning the energy decay for the problem (0.1), (0.2) (on I × R +) with u(0) = Uo, ut(O) = ul reads as follows. THEOREM 2. Let (Uo, ut) E H 2 ('1 /~1 × /~1 and f, ft E L~o~(R + ; L2). Suppose that (A~) and (A~) are fulfilled with some c~~> 0 and 0 < r < 1. Moreover, suppose that 60( 0 =
]lf(s)ll 2 ds
and 61(0 =
(it+l
=
o(t -(1-r)/r)
IIf,(s)ll~ cu
as
t--> o¢
(0.15)
)1/2
~ M1 < ~.
Then, the solution u(t) of (0.1) (0.2) on 1 × R + with u(x, O) = Uo(X) and ut(x, O) = ul(x), which exists globally in W2~,~(R + ; L 2) N uzl.=ro+.,, loct,, ,/~1) (3 L~(R+; H2 N ~ ) , Ilu,(/)ll2 + IlUx(t)ll2 ~< C0(1 + t) l-lIt
satisfies (0.16)
and also, under the additional assumption (1 - r)(1 + c~) > 1 and 61(0 = o(t -1#) as t---~ % we have
[lut,(t)ll2 + ]lUtx(t)][2 ~< C0(1 + t) -'/r where Co denotes positive constants depending on r, c~, Ilu0llH= +
(0.17)
Ilu,ll.1 and 60, 61, etc.
590
M. NAKAO
1. PERIODIC SOLUTION In this section we shall give the proofs of theorem 1 and corollary 1. Let 0 < e < 1 and define
p'(x,v)
if[v[>/e
o'
p ; ( x , v) = p ' i x , e) - p ' ( x , - e)
p'(x, e) + p'(x, -e)
2
2
if [vl < e
and
pE(x, v) =--
fop
p'e (x, rl) dr1.
e--+0p~(x, v) = p(x, v) a.e.
Then, it is easy to see that lim
remain valid with p(x, v) replaced by p~(x, differentiable in all v E R (for a.e. x ~ / ) . We consider the modified problem:
v).
Moreover, note that
uu - Uxx + pe(x, ut) + I (P~) LD.181 = O,
on I x R, and the conditions (0.3)-(0.5)
fl(X, u)
= f(x,
p~(x, v)
t), e > 0,
U(X, t) = U(X, t + 09).
is continuously
(1.1)
With the standard use of Galerkin method and Leray-Schauder degree theory the existence of solution u~(t) of (P~) is guaranteed if we can establish a priori estimate +
~ const.
<
~¢
(1.2)
for assumed smooth solution u~ of (P~) (cf. Prodi [15]). In fact, we shall derive such estimate independent of e. Then, using a standard compactness argument (cf. Lions [8], Amerio and Prouse [1] etc.) we shall be able to get the desired solution of the original problem as a limit of {u~} as e--~ 0 (along a subsequence). Now, our task is to derive the estimate (1.2) independent of e. Hereafter we write u for u~ and denote by C generous positive constants independent of e. Sometimes we use C(M) to denote constants which depend on M in a significant way. Multiplying the equation (1.1) by ut and integrating over I x [0, col we have, by the periodicity and the boundary condition,
f ftP~(X, Ut)utdxdt~Mo(f with
mo
=--
IlfllL(2-')/O-r)(w;L(2-O/O-O),and
[,u,(,~-~r)1/(2-r)
(1.3)
by (0.3) (with p = p~)
f~ fz lutlZ-r dx dt <~ C(M(O2-r)/(l-r) + 1)
(1.4)
Nonlinearwave equations
591
where fo~ denotes the integral over any interval with length to. From (1.4) we have
f~ fl UZdx dt ~ Csupllu,(t)]l~(M~2-r)/tl-r) + l).
(1.5)
Next, multiplying the equation by u we find fo @u~(t)[122+
f ~(x,u)udx) dt~ f {,,ut(t)l,~ + f
f(t)u(t)dx
+ fi lot(x, u,)t lu[ dx}dt.
(1.6)
The third term of the right-hand side of (1.6) is treated as follows.
f fi'Pe(x, ut)[[uldxdt~C[fi(l+ut)'u'dxdt C f~o (1 + c
[[utll2) [lulldt
( f Ilu,[(~dt + 1) + ~i f IluxllNdt.
(1.7)
From (1.5)-(1.7) and the assumption (0.6) we can obtain easily
fo~ {~ llut(t)"2 + F(u(t)) } dt ~ C(Mo) {sup ,,u,(t),,: +1}
(1.8)
where we set
F(u) - ~ Ilux[l~+ Setting E(u(t))
fl(x, rl) do dx.
~ ½11ut(O]l~+ F(u(t)) we see from (1.8) that there exists t* E (0, ~o) such that E(u(t*)) <~C(Mo) \(sup[lut(t)l[ + 1),
and hence, using the inequality sup E(u(t)) ¢
<~E(u(t*)) +
]fu] dx dt .
and (0.6), we obtain ]
sup (llu,(t)l[~, +
%
[lux(t)l[g)~ C(Mo)\(sup[lut(t)ll% + 1)./
(1.9)
For further estimation we differentiate the equation (1.1):
U, - U~x + O's(x, U)U, + fl'(x, u)U = f~(x, t) where
U =- ut.
(1.10)
M. NAKAO
592
Since u is assumed to be smooth U satisfies integration over I x (0, co) yield
Uloz= 0.
Thus, multiplication by Ut and
~ f O'~(x,CO'V,t~ dxdt<- f f {ll3'(x,u)llSS, l + l:,l lU,I}dxdt
(1.11)
Here,
L f lH'(x,u)l lUU,l dx dt <~c L ft (l + luL~)[UU,Idx dt ~< C(1 + Llull~)
f Ilu& IIu& dt
6(1 + Ilull~)(1+ Ilu,ll~/=)f IIc% dt (by (1.9)) where we set
(1.12)
Ilull~= sup ([[u(t)[l~). From (1.11), (1.12) and the assumption (0.4) we see easily t
that
f IIu,(/)llNdt ~< C(Mo,M1)(1 + IIull~)(a + Ilu,II3')
(1.13)
where M1 --- (fo, lLf,(/)[l~dr) 1/2. On the other hand multiplication of the equation (1.10) by U yields
IlUxll2 dt<~f f {IGlZ+lp'(x, ~l Iuu, l+lY(x, u)lu2 +lLuI} dx dt. The second and third terms of the right-hand side are handled as follows.
~o~~ 'P'~(x,U)"UU,' dx dt <~C(6) f fi o'~(x,U) lU,[2dxdt+ 6 f fl P'e(x,U) lUl2dxdt, 6> O, <<-C(Mo,M1, 6)(1
+ I[ull2~)(1 + Ilu,lle) + c ~
j f, (1 + [UI)IU[ dx dt
where we have used (1.11)-'(1.13) and (0.5).
f~fJ [j6'(x'u)lUedxdt<~Cfo) fl (i+lul")Uedxdt <~ C(1 + I[ull:)(1 + I[u,llr)
(by (1.9)).
Thus, choosing 6(>0) sufficiently small we obtain from (1.13) and (1.14)
~
(IIU,(t)II~
+ Ilg~ (t)ll~) dt ~< C(Mo,Mx)(1 +
IluliP)(1 + Ilu,II3~)
(1.14)
Nonlinear wave equations
593
and hence, by a standard technique (cf. (1.9)), sup (IJu.(t)ll~ +
IlU,x(t)ll~) <~C(Mo, M1)(I
Now, using the well known inequality Ilu[K ~ from (1.9) that
3 r )" + IJttlj2cr)(1 + IIU tlJ~
Cllull~/2 I[Uxl[~/~~
(1.15)
~l,xll~ for u E ~1(I) we see
Ilull~ ~ Csup Ilu~(t)lJz ~ C(M0) (1 + Ilu,ll~~) t
and ))u,ll~ -= sup I)ut(t)lJ~ ~ C sup JJur(t)ll~/2)lu~(t)ll~/2 <~C(Mo) (1 + ))u,l15/') sup Ilu~(t)ljy ~. t
t
t
Therefore,
IJu,ll~ -< C(Mo) 1 + sup [lUtx(t)H2/(4-r) t
and
Combining these estimates with (1.15) we obtain
sup(llutt(t)lj22+[]ut~(t)IIz)<~C(Mo,M,)(1 +supJlutx(t)lj~°~+6rV(4-~)).
(1.16)
By our assumption (0.10) on cr and r we see (2o~r + 6r)/(4 - r) < 2 and hence we obtain from (1.16) the desired estimate: sup (]tu,(t)ll2 + ]]u=(t)ll2 ) ~< C(Mo, M1) < oo t
and (see (1.9))
sup (IM(t)ll2 + Ilu~(t)l/2) ~ C(Mo, M~) < ~. t Now the proof of theorem 1 has been finished. Finally we shall prove corollary 1. We know already (see (1.9))
sup
(1 + s.p
This time we can utilize the inequality
to get, since f~fl(x, r/) dr/~> ko[ul ~+2,
Ilulf~ = sup II.(OL -< sup E(u(t)) 1/~÷4~ Ilu~(t)ll~/~+4~ f
t
594
M. NAKAO
Consequently, we can derive, instead of (1.16), sup ([[u.(t)l]~ + IlUtx(t)l[~) ~ C(Mo, M1) (1 + sup I]Utxl[~) with
4roe
~/ = \ 4 + a ~ + 3 r
)
2
. -4--
(by(0.10)'),
which gives the desired a priori estimate for u,(t). 2. D E C A Y
OF SOLUTION
In this section we shall prove theorem 2. We need the following lemma. LEMMA 1. Let q~(t) be a bounded nonnegative function on R + = [0, oo) such that, for arbitrarily fixed T > 0, sup
(~(S) l + r
~<
Co( dp(t) - ~p(t + 1)) + g(t) + e, 0 <~ t <~ T, e >t O,
t~s<~t + l
with r > 0 and g(t) = o(t -1 -l/r) as t---~ ~. Then q~(t) ~< C1(1 + t) -x/r + O(e)
forO < t ~< T
where C~ is a constant depending on ¢(0) and r, and O(e) denotes positive quantities such that lim O(e) = 0. The proof of lemma 1 with e = 0 is found in [10], The case e > 0 is also e---~0
proved similarly (cf. [15]). We consider again the modified problem
I
u , - U ~ x + p~(x, ut)+ f l ( x , u ) = f ( x , t )
(P~)'[u(x,O)=uo,
u,(x,O)=ul
and
on/xR
uloi=O
(2.1)
where p~ is the one defined in the previous section. By the assumption (A~) it is easily seen that p~(x, O) = O,
kolv[ z-r - O(e) ~ p~(x, v)v <~ kz(lvl a-r + Ivl + O(e))Ivl
(2.2)
and ko(Iv[ + O ( e ) ) -r ~< o's(x,
Sincef, ft E L~oc(R + ; L 2) and (u0, u~ such that U e ~-" ,M'2.~I"D+; " loc k"~"
Ul) ~
L2
H2
(q
~1
X
v) <~ k2(1 + [vl-').
(2.3)
~1 the problem (P~)' has a unique solution
o :¢ )f~lW~6c(R + . H,)fqL,oc(R
+.
,H2 , ~ x ) .
In fact this assertion will be easily verified by using the estimates derived below. Moreover, the solution of the original problem will be given by taking the limit of u~ as e ~ 0 along a subsequence. The passage procedure as e ~ 0 is again standard and omitted. Thus our task is
Nonlinear wave equations
595
again to derive the estimates (0.16) and (0.17) (independent of e) for u~. In the sequel we write u for u~. Multiplying the equation (2.1) by u, and integrating over I x (t, t + 1) we have, by (2.3),
ko It+l
f/ (lut] + o(~))-' I,,I ~d~ ds ~
f,+,f,
= E(u(O) - E(u(t + 1)) +
(2.4)
f u , dx ds,
where
E(u(t)) = Ilut(t)[i2 + ~ llux(t)l[2 +
fo ~(x, rl) drI dx.
,
From the above we have easily
f'+' f, (p.,l
+ o ( E ) ) -r I.tl 2 dx ds ~ C ( E ( u ( [ ) )
- E(l~(l -~- 1)) --[- C(~o(t) (2-r)/(1-r) -{- O(E)
(2.5)
D(t) 2-r and hence ft+l
Ilu,lrNcU ~ (\ t ~ ssup ~ t + l Ilut(s)]l~ + o(e)) D(t) 2-r.
(2.6)
t
From (2.6) we see that there exist t a ~ [t, t + 1/4] and
t2 ~ [t + 3/4,
IJut(ti)l}2 <~2 (\ t ~
t + 1] such that i = 1, 2.
(2.7)
Multiplying the equation by u, integrating over I x [q,/2] and using (2.2) we have
;/ {,,ux(s)[,~ + f (ut(t,), u(t,))
-
(ut(t2) , u(t2) ) +
llu,(s)fiNds + tl
+
Ill [ul dx ds ~ 4 sup
t~s~t+ 1
tI
+ ( sup
\t~s<~t+ 1
[lu,(s)l[: + o(e)
+ o(e) + k2
)
Ip~(x, u,)p lul d~ ds ll
I]u(s)][2 (
sup Ilu,(s)ll~/2 +
\t~s~t+ 1
D(t) 2-' + k2
o(~) D(t) ~>'v2
Ilu,(s)[12_-:ds tl
flu(s)ll~:~ d~
(f,2
t
Ilut[I2 ~
tl
Ilull~ ~ tl
)12
+ 60(0
Ilu(s)ll~ ds tl
)1/2
(2.8)
596
M. NAKAO
From (2.8), (2.5), (2.6) and the inequality [lull: ~< C][uxll2 we can show
+ f (x,u)udx} ds ~< C ( sup ~ ) ( {t<~s<_t+l
sup [[ut(s)l]~2 + O(E)tU(t) `2-r,/2/
\t~s<_t+1
+ ( sup Ilut(s)l]: + O(e)] D(t) 2-r + D(t) z-2r /
\t<_s~t+l
+ 60(t) 2 + O(e)} = A2(t).
(2.9)
It follows from (2.6), (2.9) and the assumption on [3(x, u) that
ft12t E(u(s))ch'~CA2(t). Thus, there exists t* E (tl, te) such that E(u(t*)) <~CA2o(t), and hence
sup E(u(s)) <- g(u(t*)) + I
t~s
Ip~(x, ut)ut(s)l dX ds
Jt
+ ftt+lfl
If.,
,tr ds < CA~(t).
(2.10)
By the definition of AZ(t) we obtain from the above that sup E(u(s)) <~C{( sup IIU,(s)ll r + o(E)) O(t) 2-r + O(/) 2-2r + 620(0 + O(e)}.
t~s <~t+1
t<~s<~t+1
(2.11)
To derive the decay of E(u(t)) from (2.11) we must show the boundedness of [lu,(/)L. For this we utilize the differentiated equation: Uu - Uxx + p'e(x, U)Ut + ~ ' ( x , u ) U = f t ( x , t)
(2.12)
with U = ut. Multiplying (2.12) by Ut and integrating we have, by (2.3),
ko ft'+l f/ ([Ul + O(e))-r]utl 2 dxds<~ f t + l f, Pe( ' X , U) u2 dxds = e(u(t)) - E(u(t + 1)) +
+1;1
(IL [ + k2 [ul ~[UI)] Ut [ dx ds
where we set this time 1 2 g(g(t)) = ½lls~(t)ll~ + ~ll gx(t)ll2.
(2.13)
Nonlinear wave equations
597
Here,
f jf/([f,I + +' k2lul~IUl)IU,[dxds k0 ft+l
<<-Tj, f(Iul+°(~))-'Iu,[ z~as +C
(ftt+lf/
If,[z]o[r dx ds +
It+If/
lul2=[Ule+rdxds)
+ o(e) and moreover
It+lf/ lulZ~lolz+'oxOs<~
sup ([[u(s)lI~Ilu(s)fl~)
t<~s~t+1
ft+lfl t
OZdxds
~< sup E(u(s)) 1+~ sup NU(s)I[r. t~s<~t+ 1
t~s<<-t+ 1
Thus, we have from (2.13) that
ft+1 fl
p'e(x,
U)U2t dxds <~CG(t) 2
(2.14)
and
ftt+lf/ U~dxds<~C(
sup IPU(s)lJ~+ o(e)
\ t<~s~t+l
)r c(t) 2---Dl(t)2
(2.15)
where we set
G2(t) = +
E(U(t)) - E(U(t + 1))
sup I[U(s)[[• ( 6 ~ ( t ) +
t~s-<-t+ 1
sup E(u(s)) '+~)
t<~s<~t+1
+ o(e).
(2.16)
By (2.15) there exist tl ~ [t, t + 1/4], t 2 ~ [t + 3/4, t + 1] such that
I]U,(ti)l]2 <~2Dl(t), i = 1, 2. Using this we can show easily (cf. (2.8))
f t2 I[Ux(s)[I2 ds ~< (et(tl) ' U(t,)) - (Ut(t2) ' U(t2)) tl
IIu,(~)ll~ as +
+ tl
+
;i f, t
If, ul d~ as t
fp'~(x, u)u, Ulaxas
598
M. NAKAO
C( sup IIV(s)llzD,(t ) + D~(t) + 6~(t)) \t~s<-t+ l
I'l 2 IIU,(s)ll~ d~ +
+{
ft2fi tl
Ip'~(x, U)U, Ui d~ d~ (2.17)
+ o(e). Now,
f'~ f, Ip',(x,u ) u , ul Ox ds tl
Ip',(x, u)lu~ dxds
ca(o <~ ,'-
tl
0,
[o',(x, u ) [ u ~ dxc~
(11Nil ~-~ + II uIIN + o(e)) as
IlSxll~ d~ + c(a(t)~/( ~<~ + G=(t)) + o(e).
Therefore we have from (2.17) that
.1
liu,(s)ll~ d~ ~ C tl
+ 62(0 \t<-s<-t+l
+ G(t) 4/('+2) + G(t) z) + O(e).
(2.18)
From (2.15) and (2.18) we can obtain as is usual (cf. (2.11))
E(U(s)) <~C(DI(t) 2 + G(t) 4/(~+2) + G(t) 2 + 61(02) + O(e).
sup
(2.19)
t~s~t + l
Moreover we observe that
Dl(t) 2 <~C
sup
IIU,(s)ll2 +
O(e)
G(t) 2
\t6s~t+l
<~½,<-s~t+lsupE(U(s)) + G(t) 4/(2-') + O(e).
(2.20)
It follows from (2.19) and (2.20) that sup
E(U(s)) ~ C(G(t) 4/(2+0 + G(t) '/(2-0 + G(t) 2 + b1(02) + O(e).
(2.21)
t~s~t+l
First we shall derive the boundedness of E(U(t)) from (2.21). Assume that E(U(t)) <~E(U(t + 1)) for some t >i 0. Then, by (2.21) and (2.16), we find sup t~s<<-t + 1
E(U(s))<~ C ( sup E(U(s)) r/(2+r) (61(t)2 + sup E(u(s))l+c~) 2/(2+r) t. t<<-s<<-t+ 1
t<~s<~t + 1
Nonlinear wave equations
599
+ ,~s~t+asupE(U(s))r/(2-r)((~l(t)2+ t<~s~t+ISUpE(u(s))l+°t) 2/(2-r) t<-s~t+lSUpE(U(s)) r/2 (61(t)2 + t~s<~t+lSUpE(u(s))l+cr) + 6l(t)2} + O(E) and hence sup
t~s<~t+1
E(U(s)) <~C {M 2 + m 2/(l-r) + M 4/(2-0
+
sup
E(u(s)) 1+°~
t<-s<~t+1 + r-~,<-t+lsupE(u(s)) (~+~)/(1-~)+ ,<-s~t+asupE(U(S)) 2(1+°0/(2-r)} + O(E) (2.22)
where we set M1 = sup 61(0. Here we note that by (2.4) t
E(u(t)) <<-E(u(O)) +
flf(s)[lt~z~t~ll:~tds + O(e)T, 0 ~ t ~ T,
for any T > 0, and also by the assumption on
60(0
fo~ [[f(s)[[/Z2ZrI¢ll7_~/ds ~< C E~(fn+l n=0
Ilf(s)ll~
)(2-r)/2(1-r)
n
z¢
<<-C~ (l+n)-(2-r)/r <~
(0
n=0
Hence we have
E(u(t)) <~Co + O(e)T
for 0 ~ t ~< T.
(2.23)
Hereafter O(e, T) will stand for constants depending on e and T like lim O(e, T) = 0. By ~'--* 0
(2.22) and (2.23) it follows sup
t<~s~t+1
E(U(s)) <~Co + O(e, T)
and we obtain, for any T > t t> 0,
E(U(t)) <~max
(Co + O(e, T), 0~<-lsupE(U(s))).
Now, by (2.13), sup 0~
E( U(s)) <~E(U(O)) + + (Co + o(e))
1
f0IIf,(s)llzllU,(s)lj=ds
fo'
IIg (s)ll=
and hence sup 0~s~l
E(U(s))<- C(E(U(O)) + Co + O(e)).
(2.24)
M. NAKAO
600 Observe that
E(U(O)) = ~llu.(O)ll~z + ½11ux,(O)ll~ = ½1luxx(O) - p~(x, u,(O)) - ~(x, u(O))
÷ f ( x , 0)11~ + ½11~x,(O)ll~ C(tluollZ,,~ + Ilu~ll~ -~ + Ilulll~ + Iluo[I,~7 +~!
+ IIf(O)ll~+ 1 + o(~)) < ~. Thus we conclude the boundedness of E(U(t))
E(U(t)) ~ Co + O(e, T)
for 0 ~< t ~ T
(2.25)
with some Co = c0(llu0il~,2, Iluxll~, r, ~, Mo, M1). Now, let us return to the inequality (2.11). With the use of (2.25) and the boundedness of E(u(t)) we find, for 0 ~< t ~< T, sup
t~s<~t+ l
E(u(s)) ~ Co(O(t) 2-r q- D(t) 2-2~ + 60(02) + O(s, T) C o ( O ( t ) 2-2r + ~o(/) 2) + O(6, T)
or sup
E(1A(S)) l+r/2(1-r) ~ C o ( g ( u ( t ) )
t~s<~t+ l
- E ( u ( t + 1))) + CoOo(t) (2-r)/(1-r) -4- O(E, r ) .
(2.26)
Applying lemma 1 to (2.26) we obtain
E(u(t)) ~ C0(1 + t) -2(1-')/r + O(e, T)
(2.27)
for 0 ~< t ~< T. Using this we can show also the decay property of E(U(t)). Indeed, by (2.21), (2.16) and (2.25) we have sup
t<_s<~t+l
E(U(s)) <~Co(G(t) '/(z+r) + 61(t) z) + O(e, T)
and hence sup
t<~s<~t+ l
g(U(s)) 1+r/2 ~ C O t E ( U ( t ) ) - E(U(t + 1)) + C t~s.t+lSUp
E(U(s)) r/2 Ql~l(/)2-1-t.s<~t+lSUp E(u(s))l+a) + C(Sx(t)2+r}
+ O(g,
T),
which implies further, by (2.27), sup
t~s~t + l
E(U(s)) 1+~/2 <~Co{E(U(t)) - E(U(t + 1)) + C61(t) 2+r + C(1
+
t) -(x-r)(l+g)(2+r)/r} -b O(E, T).
(2.28)
Applying lemma 1 to (2.28) we can conclude that
E(U(t)) <~CoO + t) -2/r + O(e, T) provided that (1 - r)(1 + c0 > 1 and 61(0 = o(t-x#).
for 0 ~ t ~< T
(2.29)
Nonlinear wave equations
601
Letting e---~ 0 in (2.27) and (2.29) we complete the proofs of (0.16) and (0.17).
Remark. Even if (1 - r) (1 + a~) ~< 1 we can get certain decay rate for llu,(t)[12 + Ilutx(t)ll2. Indeed, if 1 i> (1 - r)(1 + o0 > r/(2 + r) we have from (2.28) that sup E(U(s)) l+l/2't <~Co{E(U(t)) - E(U(t + 1)) + COl(t) 2+r t<~s<-t + 1
+ C(1 +/)-1-2%}
+ O(8, T)
for any 0 < r / < r/0 with
rl°=½ { ( 1 - r ) ( l + °O(2 +
1}
and consequently we have
fJu.(t)[l: + Jlu=(t)ll2 ~ c0.~(1 + t)-" for
½~(1 - r)(1 + c0(2 + r ) _ 1~, 0 < t/ < r L J provided that 61(0 =
o(t-(t+20/(2+O). Also we note that lim E(U(t)) = 0 if 61(0--+ 0 as
t--+ m without any condition on cr and r (cf. [10]). Finally it may be worth mentioning that our argument of this paper is applicable to the more general equations:
u,t+(-1)~Aku+p(x,u,)+fl(x,u)=f(x,t)
on~xR(R
+)
with Druloa = 0, I)~ ~< k - 1, where ff~ is a bounded domain in R", A is the Laplacian and k is a positive integer with 2k > n. REFERENCES 1. AMERIO L. & PROUSE G., Almost Periodic Functions and Functional Equations, Van Nostrand, New York (1971). 2. BIROLI M., Bounded or almost periodic solution of the nonlinear vibrating membrance equation, Ric. Mat. 22, 190-202 (1973). 3. CLEMENTSJ. C., Existence theorems for some nonlinear equation of evolution, Can. J. Math. 22, 726-745 (1970). 4. FtCKEN F. A. & FLEISHMAN B. A., Initial value and time-periodic solutions for nonlinear wave equations, Communs pure appL Math. 10, 331-356 (1957). 5. HARAUX A., Nonlinear evolution equation; global behavior of solutions, Lecture Notes in Mathematics 841, Springer, Berlin (1981). 6. KANNAN R. & LAKSHMIKANTHAMV., Periodic solutions of nonlinear hyperbolic problems, Comp. Math. Appl. 9, 479-486 (1983). 7. KATO H. & NAKAO M., Existence of strong and smooth periodic solutions, Math. Report College Gen. Ed., Kyushu Univ. 14, 57-88 (1983). 8. LIONS J. L., Quelque MOthodes de Rdsolution des ProblOm aux Limites Nbnlindares, Dunod-Gauthier-Villars, Paris (1969). 9. MASUDA K., On the existence of periodic solution of non-linear differential equations, J. Fac. Sci. Univ. Tokyo 1A, 247-257 (1966). 10. NAKAO M., Convergence of solutions of the wave equation with a nonlinear dissipative term, Mem. Fac. Sci., Kyushu Univ. Ser. A, 30,257-265 (1976). 11. NAKAO M., Bounded periodic or almost periodic solutions of nonlinear hyperbolic partial differential equations, J. diff. Eqns 23, 368-386 (1977). 12. NAKAO M., Bounded, periodic and almost periodic classical solutions of some nonlinear wave equations with a dissipative term, J. Math. Soc. Japan 30, 375-394 (1978).
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M. NAKAO
13. NAKAO M., A difference inequality and its application to nonlinear evolution equations, J. Math. Soc. Japan 30, 747-762 (1978). 14. NAKAO M., Existence of classical periodic solutions of some nonlinear wave equations in one space dimension, Math. Rep. College Gen. Ed., Kyushu Univ. 12, 77-91 (1980). 15. NAKAO M., On solutions to the initial-boundary value problem for (O/Ot)u - Aft(u) = f, J. Math. Soc. Japan 35, 71-83 (1983). 16. PROD1 G., Soluzioni periodiche dell'equazione delle onde con termine dissipativo nonlineare, Rc. Semin. Mat. Padova 36, 37-49 (1966). 17. RABINOWITZ P. H., Periodic solutions of nonlinear hyperbolic partial differential equations I, Communs pure appl. Math. 20, 145-205 (1967); II, ibid. 22, 15-39 (1969). 18. WAHL W. V., Periodic solutions of nonlinear wave equations with a dissipative term, Math. Annln 190, 313-322 (1971).