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Bifurcation of Periodics and Subharmonics in Abstract Nonlinear Undamped Wave Equations
journal of differential equations 153, 4160 (1999) Article ID jdeq.1998.3542, available online at http:www.idealibrary.com on
Bifurcation of Periodics and Subharmonics in Abstract Nonlinear Undamped Wave Equations Michal Fec kan Department of Mathematical Analysis, Comenius University Mlynska dolina, 842 15 Bratislava, Slovakia E-mail: Michal.Feckanfmph.uniba.sk Received June 6, 1997; revised June 16, 1998
Periodic and subharmonic bifurcation is investigated for certain abstract wave differential equations modelled after a nonlinear beam equation. 1999 Academic Press Key Words: undamped wave equations; periodic and subharmonic solutions; bifurcation
1. INTRODUCTION The purpose of this paper is to study the existence of periodic and subharmonic solutions of certain abstract wave differential equations modeled after the following partial differential equation u tt +u xxxx +1u xx + p
\|
1
u 2 (s, t) ds,
0
==q(x) cos
|
1
0
+
u 2x(s, t) ds D !xx u
2?t T
(1.1)
u(0, } )=u(1, } )=u xx (0, } )=u xx (1, } )=0, where D xx u=&u xx , D !xx is the !-power of D xx in L 2 (0, 1), 0 ! 1, 1 # R, p # C 2 (R_R, R), p(0, 0)=0, q # H 2 (0, 1) & H 10(0, 1), T>0 and = # R is a small parameter. We are motivated by the papers [10, 11, 17], and [18], where similar partial differential equations are studied. Contrary to these papers, we consider an undamped case. For this reason, we can show arbitrarily many, but finite number of subharmonics, while an infinite number of subharmonics is detected in these papers. Since we deal with an undamped case, the method of the above papers cannot be applied directly. We follow the way of [46, 8, 13, 19] in conjunction with a subharmonic bifurcation method of [7] and [9, p. 193]. Related problems are studied also in [3] and [20]. New methods in that 41 0022-039699 30.00 Copyright 1999 by Academic Press All rights of reproduction in any form reserved.
42
MICHAL FEC8 KAN
direction are established in series of papers [1416]. Studied are perturbed nonlinear Schrodinger equations and perturbed sineGordon equations. If we put u(x, t)=z(t) - 2 sin ?x in (1.1) with ==0 [10, 11], we arrive at an ordinary differential equation given by z +? 2 (? 2 &1 ) z+ p(z 2, ? 2z 2 ) ? 2!z=0.
(1.2)
By assuming 1>? 2,
p(z, ? 2z)=}z
\z 0
for a constant }>0, (1.2) becomes in Duffing's equation 9. It is well-known that Duffing's equation possesses an oneparametric family of periodic solutions. The main result in this paper concerning (1.1) for 12>! 0 and ? 2 <1<3? 2, is, roughly speaking, as follows: If 10 q(x) sin ?x dx{0, then the smaller 1&? 2 >0, the larger the number of subharmonic solutions of (1.2) persists for (1.1) when = is sufficiently small. Moreover, we show that (1.1) for 1 # R"[? 2j 2 | j # N] and 12>! 0, has a small T-periodic solution for any = sufficiently small and almost all T>0. On the other hand, for 1 ! 12, we can show only the existence of a small T-periodic solution for any = sufficiently small provided that ?T # Q and 1? 2 Â Q. A similar result is proved in [19, Theorem 4.1]. The same existence problem with ! = 0 is studied as well. The partial differential equation of (1.1) with periodic boundary value conditions is discussed in the end of this paper. Finally, we have to point out that all above-mentioned results are consequences from Section 5 of more general results which are derived in Sections 2, 3, and 4 of this paper for an abstract version of (1.1) defined on a Hilbert space.
2. REDUCTION TO A FINITEDIMENSIONAL CASE Let X be a Hilbert space with an inner product ( } , } ) and norm | } |. Let A: D(A)/X Ä X be a selfadjoint operator with eigenvectors [u i ] i=1 and with the corresponding eigenvalues [* i ] , * * } } } * 1 2 i0 <0< i=1 forms an orthonormal basis * i0 +1 } } } Ä . We suppose that [u i ] i=1 of X. For 12 _ 0, we put X _= D(A _),
( u, v) =(A 12u, A 12v),
A =A&(* 1 &1) I.
UNDAMPED WAVE EQUATIONS
43
We consider on X _ the norm given by i0
&u& 2_= : a 2i + : i=1
a 2i * 2_ i ,
u= : a i u i .
i=i0 +1
i=1
In this paper, we consider the wave equation u +Au+F(u) Bu==h(t),
(2.1)
where B # L(X 12, X { ),
F # C 2 (X 12, R),
h # C 1 (S T, X { ),
12 { 0 is given and S T =R[0, T ] is a circle. We also suppose that [u i ] i=1 are eigenvectors of B. We are interested in T-periodic (weak) solutions of (2.1) in the sense [1, 2] that u # L (S T, X 12 ), u* # L (S T, X) satisfies (u (t), u i )+ +(* 1 &1)(u(t), u i )+(F(u(t)) Bu(t), u i )==(h(t), u i ) almost all
\i 1,
t # S T.
(2.2)
The norms on L (S T, X 12 ) and L (S T, X { ) are defined by _u_ 12 =ess sup &u( } )& 12 ,
_u_ { =ess sup &u( } )& { ,
respectively. First, we consider for i>i 0 the equation (v (t), u j )+(A 12v(t), A 12u j )=(h(t), u j ) \j i,
almost all
t # S T,
(2.3)
where h # L (S T, V i ) and V i =span[u j ] j i /X {. Lemma 2.1.
Assume that there is a c>0 such that
}
* {j sin
- *j T c 2
}
\j>i 0 ,
then (2.3) has a unique solution v=L i h # L (S T, W i ) satisfying _v_ 12 T
2*
&2{ i
+
1 _h_ { , 2c 2
(2.4)
44
MICHAL FEC8 KAN
where W i =span[u j ] j i /X 12. If in addition, it holds
}
* {&+ sin j
- *j T c~ 2
}
\j>i 0 ,
(2.5)
for some c~ >0 and { + 0, then v* # L (S T, X + ) satisfies _v* _ + T
2*
&2({&+) i
+
1 _h_ { . 2c~ 2
Since h # L (S T, V i ), we have
Proof.
h j # L (S T, R),
h(t)= : h j (t)u j ,
2 2 ess sup : * 2{ j h j (})=_h_ { .
j=i
j=i
The equation (2.3) splits into the system v j (t)+* j v j (t)=h j (t),
j i (2.6)
v(t)= : v j (t)u i . j=i
Since (2.4) holds, (2.6) has the solution v j (t)=
1 - *j +
|
t
sin(- * j (t&s)) h j (s) ds 0
1
1 2 - *j
sin
-* T | j
T 0
T cos - * j s& &t 2
\ \
++ h (s) ds.
(2.7)
j
2
2_ 2 We note that _v_ 2_ =ess sup j=i * j v j ( } ). The formula (2.7) gives
: * j v 2j (t) 2 : j=i
j=i
\|
t
sin(- * j (t&s)) h j (s) ds 0
1
+2 : j=i
4 sin 2
2 : * &2{ j j=i
+
1 : 2c 2 j=i
| |
T
| -* T \ j
T 0
0
T cos - * j s& &t 2
\ \
++
h j (s) ds
2 sin 2 - * j (t&s) ds
0
T
+
2
|
T
0
2 * 2{ j h j (s) ds
T cos 2 - * j s& &t ds 2
\
+
|
T
0
2 * 2{ j h j (s) ds
+
2
45
UNDAMPED WAVE EQUATIONS
2T* &2{ i
|
T
2 : * 2{ j h j (s) ds+
0
j=i
T 2 _h_ 2{ + 2* &2{ i
1 T 2c 2
|
T
2 : * 2{ j h j (s) ds
0
j=i
1 1 T 2 _h_ 2{ =T 2 2* &2{ + 2 _h_ 2{ . i 2c 2 2c
\
+
Similarly, we get 2 &2({&+) T : * 2+ j v* j (t) 2* i j=i
+
1 T 2c~ 2
|
|
0
2 : * 2{ j h j (s) ds j=i
T 0
T
\
2 2 : * 2{ 2* &2({&+) + j h j (s) ds T i j=i
1 _h_ 2{ . 2c~ 2
+
Clearly the proof is finished. K Lemma 2.2.
Let
: k>i0
1 < * \k
for some \>0 and let D 0. Let S(c) be the set of all T>0 satisfying
}
* \j sin
- *j T c 2
}
\j>i 0 ,
(2.8)
where c>0 is a constant. Then the Lebesque measure of the complement (R"S(c)) & [D, D+1] satisfies 2c?
\- * *
m((R"S(c)) & [D, D+1]) : k>i0
k
\ k
+
c 2c 2? + . \ * k - * k * 2\ k
+
Proof. If T # (R"S(c)) & [D, D+1] then * \k |sin(- * k T2)|i 0 . Since |sin x| 2? |x&n?| for any |x&n?| ?2, n # Z + =N _ [0], we obtain 2 - *k T &n?
}
}
46
MICHAL FEC8 KAN
for some n # Z + . Hence T
c
n
} 2?&- * } < 2 - * * , 2?n ?c } T& - * } < - * * . k
k
k
k
\ k
\ k
Consequently, we obtain T n D c c & < & \ \ 2? 2 - * k * k 2? 2 - * k * k - * k <
T D+1 c c , + + 2? 2 - * k * \k 2? 2 - * k * \k
c D+1 c D - * k & \i0
= : k>i0
\
c - *k 2?c + +1 \ *k 2? - * k * \k
+
2c?
+ : \
- *k * k
k>i0
c 2c 2? . \+ : * k k>i - * k * 2\ k 0
The proof is finished. K We put for i>i 0 { Y i =V = i =span[u j ] 1 j i&1 /X .
Let Q i : X { Ä V i , P i =I&Q i be the orthogonal projections. Now we decompose (2.1) in the following way v +Av+F(v+w) Bv==Q i h(t+:),
v # Wi ,
(2.9)
w +Aw+F(v+w) Bw==P i h(t+:),
w # Yi ,
(2.10)
where : # R is a shift parameter. In view of (2.3) and Lemma 2.1, (2.9) has the form v&L i (=Q i h(t+:)&F(v+w) Bv)=0.
(2.11)
47
UNDAMPED WAVE EQUATIONS
We suppose that the following condition holds: There is a w 0 # L (S T, Yi ) satisfying w 0 +Aw 0 +F (w 0 ) Bw 0 =0.
(C1)
The linearization L of (2.11) by v at ==0, w=w 0 , v=0 has the form Lv=v+L i (F(w 0 ) Bv). We compute _Lv_ 12 _v_ 12 &T
2*
&2{ i
+
1 sup |F(w 0 ( } ))| &B i & _v_ 12 , (2.12) 2c 2 ST
where B i =BW i . We note that B i # L(W i , V i ). Theorem 2.3.
Let (2.4) and (C1) be satisfied. Let us take i such that T
2*
&2{ i
+
1 sup |F(w 0 ( } ))| &B i &<1. 2c 2 ST
(2.13)
Then for any = sufficiently small, : # R and w # L (S T, Yi ) near w 0 , there is a unique solution of (2.11) v=v(=, w, :) # L (S T, W i ) such that the mapping v( } , } , } ) is C 1-smooth and v(0, w, :)=0. Proof. By applying the implicit function theorem to (2.11), the proof is finished according to (2.12) and (2.13). K We put the solution of (2.9) from Theorem 2.3 into (2.10) and we arrive at the functional equation w +Aw+F(v(=, w, :)+w) Bw==P i h(t+:).
(2.14)
Since v(0, w, :)=0, we take v(=, w, :)==z(=, w, :). Then (2.11) gives z(=, w, :)=L i (Q i h(t+:)&F(=z(=, w, :)+w) B i z(=, w, :)).
(2.15)
Hence z(0, w 0 , :)=L i (Q i h(t+:)&F(w 0 ) B i z(0, w 0 , :)), z(0, w 0 , :)+L i F(w 0 )B i z(0, w 0 , :)=L i Q i h(t+:), Lz(0, w 0 , :)=L i Q i h(t+:), z(0, w 0 , :)=L &1 (L i Q i h(t+:)).
48
MICHAL FEC8 KAN
We put in (2.14) w=w 0 +=y, and we arrive at the equation y +Ay+F(=z(=, w, :)+w 0 +=y) By 1 + (F(=z(=, w, :)+w 0 +=y)&F(w 0 )) Bw 0 =P i h(t+:). = This equation has the form y +Ay+F(w 0 ) By+(DF(w 0 )( y+z(0, w 0 , :))) Bw 0 +O(=)=P i h(t+:).
(2.16)
Remark 2.4. To get (2.9) and (2.10), it is enough to assume that B(W i )/V i and B(Yi )/Yi . Consequently, [u j ] j 1 need not be eigenvectors of B.
3. SMALL PERIODIC SOLUTIONS \ Theorem 3.1. Suppose F(0)=0 and k=i0 +1 1* k < for some { \>0. Then for almost all T>0 (in the sense of the Lebesque measure) and any h # L (S T, X { ), the equation (2.1) has a small T-periodic solution for = sufficiently small.
Proof. Lemma 2.2 implies that for almost all T>0 the conditions (2.4) and (2.5) with +=\&{ are satisfied for some c>0, respectively c~ >0. The condition (C1) holds with w 0 =0. Now Theorem 2.3 is applied with i=i 0 +1 and :=0. The equation (2.16) has the form y +Ay+O(=)=P i h(t),
y # Yi0 .
(3.1)
The eigenvalues of A on Yi0 are all negative, hence y +Ay=0 has the only zero T-periodic solution. For this reason, (3.1) has a T-periodic solution for any = sufficiently small. The proof is finished. K
4. SUBHARMONIC SOLUTIONS We assume in this section that i 0 =1, i.e., &$=* 1 <0<* 2 } } } and that $ is small and * 2 is bounded off 0. Hence Y2 =span[u 1 ]
UNDAMPED WAVE EQUATIONS
49
and by putting w 0 (t)=z(t) u 1 , the equation of the condition (C1) becomes z &$z+ f (z) z=0,
(4.1)
where f (z)=(F(zu 1 ) Bu 1 , u 1 ). We suppose F(zu 1 )=%z 2,
%{0.
(C2)
Then the system given by z* 1 =z 2 z* 2 =$z 1 &%;z 31 ,
;=(Bu 1 , u 1 )
(4.2)
has a hyperbolic equilibrium (0, 0). We suppose that %;>0. It is wellknown [9] that (4.1) has the one-parametric family of periodic solutions g k (t)=
- 2$ 2
- (2&k ) %;
dn
- $t
\- 2&k , k+ , 2
where dn is the Jacobi elliptic function and k is the elliptic modulus. The period of g k (t) is Tk - $, Tk =2K(k) - 2&k 2, where K(k) is the complete elliptic integral of the first kind. We take k such that Tk - $= pT,
p # N,
and we put w 0 (t)= g k (t) u 1 . We note that Tk is increasing and lim k Ä 0 Tk =- 2?, lim k Ä 1 Tk = . We have
}
|F(w 0 (t))| %
2$ <2$ | ;|. (2&k 2 ) %;
}
We are interested in pT-periodic solutions of (2.1) near w 0 (t) given above. Then the condition (2.13) has the form E= pT
2*
&2{ 2
+
1 2$ &B 2 &<1. 2c 2 | ;|
50
MICHAL FEC8 KAN
The above inequality holds with E<12 when pTc<
| ;| 2 &B 2 & $ - 8* &2{ c 2 +2 2
.
(4.3)
(2.15) for this case implies _z(=, w, :)_ 12 pT
2*
&2{ 2
+
1 3 _Q 2 h(t+:)_ { + _z(=, w, :)_ 12 , 2 2c 4
whenever &=z(=, w, :)( } )+w( } )&w 0 ( } )& 12 ' and '>0 is sufficiently small, fixed. Consequently, by assuming (4.3) we have
2*
_=z(=, w, :)_ 12 |=| 4pT
&2{ 2
+
1 _h(t+:)_ { 2c 2
|=| | ;| _h_ { '2. $ &B 2 &
(4.4)
Hence we have to take w such that &w( } )&w 0 ( } )& 12 '2. We note that w=w 0 +=y. From (2.15) we also obtain z(=, w, :)&z(0, w 0 , :) =L 2 (F(w 0 )&F(=z(=, w, :)+w 0 +=y)) B 2 z(0, w 0 , :) +L 2 F(=z(=, w, :)+w) B 2 (z(0, w 0 , :)&z(=, w, :)). When 1 c>0 then (4.3) implies that at most pTct1$. Hence we obtain _z(=, w, :)&z(0, w 0 , :)_ 12 1
2* +2c c _=z(=, w, :)+=y_ &B & _z(0, w , :)_ 1 4pT 2* +2c 1 _c |=| 4pT 2* + \ 2c _h( } +:)_ + |=| _ y_ + 1 _&B & 4pT 2* + 2c _h( } +:)_ 1 1 K (_h_ , _ y_ ) |=| (4.5) \$ +$ + , &2{ 2
4pT
&2{ 2
2
12
&2{ 2
&2{ 2
2
{
2
0
12
2
1
1
1
12
{
2
{
2
3
12
2
where 1 c>0, K 1 ( } , } ) is a continuous function and c 1 >0 is a constant.
51
UNDAMPED WAVE EQUATIONS
The O(=) term of (2.16) has the form
} } (F(=z(=, w, :)+w +=y)&F(w )) By + | DF(s(=z(=, w, :)+=y)+w ) ds(z(=, w, :)+ y) \ &DF(w )( y+z(0, w , :)) Bw + }} 0
0
1
0
0
0
0
0
12
c 2 |=| (_z(=, w, :)_ 12 +_ y_ 12 ) &B& _y_ 12 +c 3 |=| (_z(=, w, :)_ 12 +_ y_ 12 ) 2 &B& _w 0 _ 12 +c 4 _z(=, w, :)&z(0, w 0 , :)_ 12 &B& _w 0 _ 12
\ ) |=| _ y_ \
K 2 (_h_ { , _ y_ 12 ) |=| _ y_ 212 + K 3 (_h_ { , _ y_ 12
2 12
+
p 3T 3 p 2T 2 + 2 c3 c 1 1 + 2 ), 3 $ $
+ (4.6)
where 1 c>0, c 2 , c 3 , c 4 are positive constants and K 2 ( } , } ), K 3 ( } , } ) are continuous functions. The estimates (4.44.6) imply that the O(=) term of (2.16) is small whenever |=|$ 3 is small and _ y_ 12 is bounded. We note that Tk - $= pT, Tk - 2? and Tk is increasing in k. Hence (4.3) is equivalent to - 2?- $ pT<
| ;|c 2 &B 2 & $ - 8* &2{ c 2 +2 2
.
(4.7)
By taking c=$ 14, (4.7) becomes - 2?- $ pT<
| ;| 2 &B 2 & $
34
- 8* &2{ $ 12 +2 2
.
(4.8)
For simplicity, we suppose DF(w) v=0 \v # W 2 ,
\w # Y2 .
(C3)
Under the above assumptions, (2.16) has the form z &$z+3%;g 2k z+O(=$ 3 )=(h(t+:), u 1 ),
(4.9)
52
MICHAL FEC8 KAN
where y=zu 1 . The unperturbed part of (4.9) given by z &$z+3%;g 2k z=0 is just the linearization of (4.1) at g k . It has, up to a scalar multiple, the only pT-periodic solution g$k . It is well known [7], [9, Theorem 4.6.2] that the bifurcation of pT-periodic solution of (4.9) is determined by a subharmonic Melnikov function given as follows M(:)=
|
pT
(h(s+:), u 1 ) g$k (s) ds. 0
Summarizing, we arrive at the main result of this paper. Theorem 4.1. Consider (2.1) under the assumptions of this section. Let $>0, 1 c>0 be fixed. If there are a, b such that M(a) M(b)<0 then (2.1) has a pT-periodic solution for any = sufficiently small, p # N and any T such that pT # S(c) and (4.7) holds as well. Remark 4.2. We note that pT=Tk - $ in the formula of M(:). Hence M(:) depends also on $.
5. APPLICATIONS TO THE BEAM EQUATION To apply the result of the previous section to (1.1) with ? 2 <1<3? 2, we put X=L 2 (0, 1), F(u)= p
\|
1
Au=u xxxx +1u xx ,
u 2 (s, t) ds,
0
|
1
0
Bu=D !xx u
+
u 2x(s, t) ds ,
X 12 =H 2 (0, 1) & H 10(0, 1),
h(t)=cos
{=(1&!)2 2?t q(x). T
We have u j =- 2 sin ?jx,
* j =? 2j 2 (? 2j 2 &1 ).
Since ? 2 <1<3? 2, we obtain * 2 >4? 4,
* 1 =? 2 (? 2 &1 )=&$,
$>0.
UNDAMPED WAVE EQUATIONS
53
The condition (C2) is satisfied when there is a % # R such that p(z, z? 2 )=%z
\z 0.
(5.1)
In this case ;=? 2!, so we suppose that %>0. The series
: j=2
1 1 = : { 1&! 2 2 * j j=2 (?j) (? j &1 ) (1&!)2
converges if 12>! 0. Hence Lemma 2.2 can be applied only for 12>! 0. Let us put q k (t)=
-2 - 2&k
2
dn
t
\- 2&k , k+ . 2
Hence g k (t)=
-$ - %;
q k (- $t).
The Melnikov mapping now has the form [9, p. 199] M(:)= =
|
pT
cos 0
-$
| - %; |
Tk
cos
0
2? T -$
|
1
- 2q(x) sin ?x dx
0
(s+- $:) q$k (s) ds
1
- 2q(x) sin ?x dx
_
=&
2? (s+:) g$k (s) ds T
0
2? 2 - 2 !
? T -%
_sech
_sin
?pK$(k) K(k)
2? : T
|
1
- 2q(x) sin ?x dx.
0
We see that M( } ) does change the sign on R provided that 10 q(x) sin ?x dx{0. Clearly the condition (C3) holds. We note D !xx
\ : a u += : ? i
i 2
i
i 2
2! 2!
i ai ui .
54
MICHAL FEC8 KAN
We compute &1 * i a 2i =? 4! : i 4!* &! * i a 2i ? 4! : i 4!* 2{ i *i i i 2
i 2
=? 4! : i 4! i 2
1 * a2 (? i (? i &1 )) ! i i 2 2
2 2
* i a 2i
=? 4! : i 2
\? \ 2
1 ? & 2 i 2
++
!
.
Hence
\
&B 2 &=? ! ? 2 &
1 4
+
&!2
2 !.
Furthermore, for c 2 1, we have c 2* 2{ 2 =
c2 1 . 2 2 1&! 4 1&! (4? (4? &1 )) (4? )
Consequently, (4.3) and (4.7) are satisfied when it holds -2 - 1 &?
2
pT<
1 !
4 1&!
4 - 8+2(4? )
c . 1 &? 2
(5.2)
We note that $=? 2 (1 &? 2 ). By applying Theorem 4.1, we obtain the following result. Theorem 5.1. Consider (1.1) with 0 ! 1, ? 2 <1 <3? 2. Let 1 c>0 and let (5.1) hold for %>0. If 10 q(x) sin ?x dx{0 then for any p # N, T satisfying (5.2) and pT # S(c), the equation (1.1) has a pT-periodic solution for any = sufficiently small. Moreover, if 0 !<12, then m(S(c) & [D, D+1]) Ä 1 as c Ä 0 + uniformly with respect to D 0. Roughly speaking, Theorem 5.1 asserts that (1.1) with 0 !<12 and ? 2 <1 <3? 2, has many subharmonics for = sufficiently small when c and ? 2 &1 are sufficiently small. On the other hand, since we do not know the structure of the set S(c), we do not know a concrete case for pT. This result can be related to the KAM theory [12]. Furthermore, Lemma 2.2 gives that the Lebesque measure of S(c) tends to 1 as c Ä 0 + . Consequently, we
55
UNDAMPED WAVE EQUATIONS
put c=(1 &? 2 ) 14 in (5.2), like for (4.8), and the assumptions of Theorem 5.1 become in the following form -2 - 1 &?
2
pT<
1 !
4 1&!
4 - 8+2(4? )
1 , (1 &? 2 ) 34
(5.3)
pT # S((1 &? 2 )). Summarizing we obtain the following result. Theorem 5.2. Let 12>! 0, ? 2 <1 <3? 2 and let (5.1) hold for %>0. If 10 q(x) sin ?x dx{0, then the smaller 1 &? 2 >0, the larger the number of subharmonic solutions of (1.2) persists for (1.1) when = is sufficiently small. Now we apply Theorem 3.1. Theorem 5.3. Let 12>! 0 and 1 # R"[? 2j 2 | j # N] be arbitrary. For almost all T>0, there is a small T-periodic solution of (1.1) for any = sufficiently small. Theorem 5.1 should be applicable also for the case 1 ! 12 and ? 2 <1 <3? 2. The problem is to show the non-emptiness of the set S(c) in conjunction with (5.2). We note that now the condition (2.8) with \={=(1&!)2 is equivalent to j 2(1&!)
T
} 2 j - ? j &1 &n } c 2 2
\j # N"[1],
\n # Z + ,
(5.4)
for some constant c >0. Furthermore, let us consider the case !=1. Then to get a condition similar to (5.3), the condition (5.4) has to be replaced by
}
pT j - ? 2j 2 &1 &n c(1 &? 2 ) 14 2
}
\j # N"[1],
\n # Z + ,
(5.5)
for some constant c>0, p # N and 1 >? 2 could be taken arbitrarily near ? 2. The problem here is to control the constant c in (5.5) when 1 Ä ? 2+ , and also to show the inequality of (5.3) for pT. We do not know to solve this problem. Consequently, we are not able to apply Theorem 5.1 for 1 ! 12, because we do not know to show neither (5.2) nor (5.3). On the other hand, we have the following result.
56
MICHAL FEC8 KAN
Lemma 5.4. that
If T? # Q and 1? 2 Â Q then there is a constant c>0 such
T
} 2 j - ? j &1 &n } c 2 2
\j # N : j 2 1? 2,
\n # Z + .
Proof. We follow the proof of [19, Theorem 4.1] by computing for j 2 1? 2
T
} 2 j - ? j &1 &n } 2 2
=
T
1
T
T
} 2 j \- ? j &1 & \?j&2j?++ + 2 ?j & 4? 1 &n } }
2 2
1 2T T 2 T ?j & 1 &n & 2 2 2 4? 8? j
}
2
1
1
2
2
<\ ? & j +?& 2j ?+ . 2
The assumptions of this lemma imply that T? # Q, T 21 Â Q and T1? Â Q. Hence we have
inf j # N, n # Z+
T j - ? 2j 2 &1 {n 2
}
T 2 T ?j & 1 &n >0, 2 4?
}
\j # N : j 2 1? 2,
\n # Z + .
Since the second term in the end of the above inequality tends to 0 as j Ä , the proof is finished. K The proof of Theorem 3.1 and Lemma 5.4 give the next result. A similar result is proved in [19, Theorem 4.1]. Theorem 5.5. Consider (1.1) with 1 ! 0. If T? # Q and 1? 2 Â Q, then there is a small T-periodic solution of (1.1) for any = sufficiently small. Now we give another criterion for (5.4) following directly from the proof of Lemma 5.4.
57
UNDAMPED WAVE EQUATIONS
Theorem 5.6. T
Let 1 !>0. If there is a constant c>0 such that
T
} 2 ?j & 4? 1 &n } cj 2
2(1&!)
\j # N,
\n # Z + , (5.6)
T j - ? 2j 2 &1 {n 2
2
2
\j # N : j 1? ,
\n # Z + ,
then (5.4) holds with j # N: j 2 1? 2. Now, let us take for (1.1) 1 =2m? 20,
T=20?,
!=0,
where m # Z & =&Z + and 0>0 has the infinite continuous fraction decomposition [13] 0=[a 0 , a 1 , a 2 , } } } ] such that a i M\i # Z + for a constant M # N. Like in the proof of Lemma 5.4 and according to [13, Propositions 1 and 2] as well, we have j2
}
T T 2 T 1 2T j - ? 2j 2 &1 &n j 2 ?j & 1 &n & 2 2 4? 16? 3
}
}
}
= j 2 |0j 2 &m&n| &
1 m2 m2 & 20 M+2 20
for any n # Z + and j # N. On the other hand, since 0=[a 0 , a 1 , a 2 , } } } ] [M, 1, M, 1, } } } ]=
M+- M 2 +4M , 2
from the inequality m 2 (M+2)<20, we obtain m2 <
M+- M 2 +4M . M+2
M 1 implies that 1<(M+- M 2 +4M)(M+2)<2. Consequently, we arrive at the following result. Theorem 5.7. Consider (1.1) with !=0. If T=20?, 1 =2m? 20, where m # [&1, 0] and 0>0 has the infinite continuous fraction decomposition [13] 0=[a 0 , a 1 , a 2 , } } } ] such that a i M \i # Z + for a constant M # N satisfying m 2 (M+2)<20, then there is a small-periodic solution of (1.1) for any = sufficiently small.
58
MICHAL FEC8 KAN
We remark that Theorem 5.3 provides a more general result than Theorems 5.5 and 5.7, while Theorems 5.5 and 5.7 give concrete forms of T and 1 for the existence of periodic solutions of (1.1). For instance, Theorem 5.7 is valid for 0=(M+- M 2 +4M)2, M # N and m # [&1, 0]. A similar result can be derived also for 1 >0 small, but formulas are more tedious. So we leave those computations. Finally, we note that (1.1) can be considered with periodic boundary value conditions of the form u tt +u xxxx +1u xx + p
\
== q 1 (x) cos
\|
2
u 2 (s, t) ds,
0
|
2
0
2?t 2?t +q 2 (x) sin T T
u(x, } )=u(x+2, } )
+
u 2x(s, t) ds D !xx u
+
(5.7)
\x # R,
where !, 1, p are like in (1.1), D !xx is the !-power of D xx in L 2 (S 2 ), q 1, 2 # H 2 (0, 2) and q 1, 2 are 2periodic satisfying
|
2
q 1 (x) cos ?x dx=0,
0
|
2
q 2 (x) sin ?x dx=0.
0
Now we put in (5.7) with ==0 u(x, t)=x(t) sin ?x+ y(t) cos ?x, like for (1.2), and we obtain a system of ordinary differential equations x +? 2 (? 2 &1 ) x+%? 2! (x 2 + y 2 ) x=0,
(5.8)
y +? 2 (? 2 &1 ) y+%? 2! (x 2 + y 2 ) y=0, where (5.1) holds for %>0. By supposing 1 >? 2, (5.8) has a 1-parametric family of periodic solutions given by x k, . =cos ._g k (t),
y k, . =sin ._g k (t),
. # R.
The Melnikov function [7] now is a 2-dimensional mapping M(., :)=(M1 (., :), M2 (., :))
59
UNDAMPED WAVE EQUATIONS
of the form M1 (., :)=&
2? 2 - 2 !
? T -%
?pK$(k) K(k)
_sech M2 (., :)=
2? 2 - 2 !
? T -%
_sech
_sin
|
2
q 1 (x) sin ?x dx,
0
2? :_sin . T
_cos
?pK$(k) K(k)
2? :_cos . T
|
2
q 2 (x) cos ?x dx. 0
Clearly (., :)=(0, 0) is a simple zero point of M provided that
|
2
q 1 (x) sin ?x dx{0,
0
|
2
q 2 (x) cos ?x dx{0.
0
Consequently [7], like for Theorem 5.2, we arrive at the following result. Theorem 5.8. Let 12>! 0 and ? 2 <1 <3? 2. If 20 q 1 (x) sin ?x dx{0 and 20 q 2 (x) cos ?x dx{0, then the smaller 1 &? 2 >0, the larger the number of subharmonic solutions of (5.8) persists for (5.7) when = is sufficiently small. Remark 5.9. Theorems 5.2, 5.3, 5.5 and 5.7 can be also straightforwardly modified to (5.7). The term p( 10 u 2 (s, t) ds, 10 u 2x(s, t) ds) in (1.1), similarly in (5.7), can be also replaced by p(
|
1 0
u 2 (s, t) ds,
|
1 0
u 2x(s, t) ds,
|
1
0
u 2xx(s, t) ds
+
in the above considerations. Of course, the results concerning (2.1) of the previous sections can be applied to other types of partial differential equations than (1.1) and (5.7).
REFERENCES 1. J. M. Ball, Stability theory for an extensible beam, J. Differential Equations 14 (1973), 399418. 2. J. M. Ball, Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl. 42 (1973), 6190. 3. A. K. Benn-Naoum, On the Dirichlet problem for the nonlinear wave equation in bounded domains with corner points, Bull. Belg. Math. Soc. 3 (1996), 345361.
60
MICHAL FEC8 KAN
4. A. K. Benn-Naoum and J. Mawhin, The periodic-Dirichlet problem for some semilinear wave equations, J. Differential Equations 96 (1992), 340354. 5. M. Fec kan, Periodic solutions of certain abstract wave equations, Proc. Amer. Math. Soc. 123 (1995), 465470. 6. M. Fec kan, Periodic oscillations of abstract wave, J. Dynamics Differential Equation, to appear. 7. M. Fec kan, Bifurcation of periodic solutions in differential inclusions, Appl. Math. 42 (1997), 369393. 8. M. Fec kan, On the existence of perioric solutions for a certain type of nonlinear equations, J. Differential Equations 121 (1995), 2841. 5. J. Guckenheimer and P. Holmes, ``Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,'' Springer-Verlag, New York, 1983. 10. P. Holmes and J. Marsden, A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam, Arch. Rational Mech. Anal 76 (1981), 135165. 11. P. Holmes and J. Marsden, Bifurcation to divergence and flutter in flow-induced oscillations: an infinite dimensional analysis, Automatica 14 (1978), 367384. 12. S. B. Kuksin, ``Nearly Integrable Infinite-Dimensional Hamiltonian Systems,'' Lecture Notes in Math., Vol. 1556, Springer-Verlag, Berlin, 1993. 13. P. J. McKenna, On solutions of a nonlinear wave equation when the ratio of the period to the length of the interval is irrational, Proc. Amer. Math. Soc. 93 (1985), 5964. 14. D. W. McLaughin and E. A. Overman, II, Whiskered tori for integrable pde's: chaotic behavior in near integrable pde's, Surveys Appl. Math. 1 (1995), 82203. 15. D. W. McLaughin, E. A. Overman, II, and C. Xiong, Homoclinic orbits in a four-dimensional model of a perturbed NLS equation: a geometric singular perturbed study, Dynam. Rep. (1995), 190287. 16. D. W. McLaughin and J. Shatah, Homoclinic orbits for pde's, preprint. 17. H. M. Rodrigues and M. Silveira, Properties of bounded solutions of linear and nonlinear evolution equations: homoclinics of a beam equation, J. Differential Equations 70 (1987), 403440. 18. S. W. Shaw, Chaotic dynamics of a slender beam rotating about its longitudinal axis, J. Sound Vibration 124 (1988), 329343. 19. M. Yamaguchi, Existence of periodic solutions of second order nonlinear evolution equations and applications, Funkcial Ekvac. 38 (1995), 519538. 20. O. Vejvoda et al., ``Partial Differential Equations: Time-Periodic Solutions,'' Noordhoff, Groningen, 1981.
Bifurcation of Periodics and Subharmonics in Abstract Nonlinear Undamped Wave Equations Michal Fec kan Department of Mathematical Analysis, Comenius University Mlynska dolina, 842 15 Bratislava, Slovakia E-mail: Michal.Feckanfmph.uniba.sk Received June 6, 1997; revised June 16, 1998
Periodic and subharmonic bifurcation is investigated for certain abstract wave differential equations modelled after a nonlinear beam equation. 1999 Academic Press Key Words: undamped wave equations; periodic and subharmonic solutions; bifurcation
1. INTRODUCTION The purpose of this paper is to study the existence of periodic and subharmonic solutions of certain abstract wave differential equations modeled after the following partial differential equation u tt +u xxxx +1u xx + p
\|
1
u 2 (s, t) ds,
0
==q(x) cos
|
1
0
+
u 2x(s, t) ds D !xx u
2?t T
(1.1)
u(0, } )=u(1, } )=u xx (0, } )=u xx (1, } )=0, where D xx u=&u xx , D !xx is the !-power of D xx in L 2 (0, 1), 0 ! 1, 1 # R, p # C 2 (R_R, R), p(0, 0)=0, q # H 2 (0, 1) & H 10(0, 1), T>0 and = # R is a small parameter. We are motivated by the papers [10, 11, 17], and [18], where similar partial differential equations are studied. Contrary to these papers, we consider an undamped case. For this reason, we can show arbitrarily many, but finite number of subharmonics, while an infinite number of subharmonics is detected in these papers. Since we deal with an undamped case, the method of the above papers cannot be applied directly. We follow the way of [46, 8, 13, 19] in conjunction with a subharmonic bifurcation method of [7] and [9, p. 193]. Related problems are studied also in [3] and [20]. New methods in that 41 0022-039699 30.00 Copyright 1999 by Academic Press All rights of reproduction in any form reserved.
42
MICHAL FEC8 KAN
direction are established in series of papers [1416]. Studied are perturbed nonlinear Schrodinger equations and perturbed sineGordon equations. If we put u(x, t)=z(t) - 2 sin ?x in (1.1) with ==0 [10, 11], we arrive at an ordinary differential equation given by z +? 2 (? 2 &1 ) z+ p(z 2, ? 2z 2 ) ? 2!z=0.
(1.2)
By assuming 1>? 2,
p(z, ? 2z)=}z
\z 0
for a constant }>0, (1.2) becomes in Duffing's equation 9. It is well-known that Duffing's equation possesses an oneparametric family of periodic solutions. The main result in this paper concerning (1.1) for 12>! 0 and ? 2 <1<3? 2, is, roughly speaking, as follows: If 10 q(x) sin ?x dx{0, then the smaller 1&? 2 >0, the larger the number of subharmonic solutions of (1.2) persists for (1.1) when = is sufficiently small. Moreover, we show that (1.1) for 1 # R"[? 2j 2 | j # N] and 12>! 0, has a small T-periodic solution for any = sufficiently small and almost all T>0. On the other hand, for 1 ! 12, we can show only the existence of a small T-periodic solution for any = sufficiently small provided that ?T # Q and 1? 2 Â Q. A similar result is proved in [19, Theorem 4.1]. The same existence problem with ! = 0 is studied as well. The partial differential equation of (1.1) with periodic boundary value conditions is discussed in the end of this paper. Finally, we have to point out that all above-mentioned results are consequences from Section 5 of more general results which are derived in Sections 2, 3, and 4 of this paper for an abstract version of (1.1) defined on a Hilbert space.
2. REDUCTION TO A FINITEDIMENSIONAL CASE Let X be a Hilbert space with an inner product ( } , } ) and norm | } |. Let A: D(A)/X Ä X be a selfadjoint operator with eigenvectors [u i ] i=1 and with the corresponding eigenvalues [* i ] , * * } } } * 1 2 i0 <0< i=1 forms an orthonormal basis * i0 +1 } } } Ä . We suppose that [u i ] i=1 of X. For 12 _ 0, we put X _= D(A _),
( u, v) =(A 12u, A 12v),
A =A&(* 1 &1) I.
UNDAMPED WAVE EQUATIONS
43
We consider on X _ the norm given by i0
&u& 2_= : a 2i + : i=1
a 2i * 2_ i ,
u= : a i u i .
i=i0 +1
i=1
In this paper, we consider the wave equation u +Au+F(u) Bu==h(t),
(2.1)
where B # L(X 12, X { ),
F # C 2 (X 12, R),
h # C 1 (S T, X { ),
12 { 0 is given and S T =R[0, T ] is a circle. We also suppose that [u i ] i=1 are eigenvectors of B. We are interested in T-periodic (weak) solutions of (2.1) in the sense [1, 2] that u # L (S T, X 12 ), u* # L (S T, X) satisfies (u (t), u i )+
\i 1,
t # S T.
(2.2)
The norms on L (S T, X 12 ) and L (S T, X { ) are defined by _u_ 12 =ess sup &u( } )& 12 ,
_u_ { =ess sup &u( } )& { ,
respectively. First, we consider for i>i 0 the equation (v (t), u j )+(A 12v(t), A 12u j )=(h(t), u j ) \j i,
almost all
t # S T,
(2.3)
where h # L (S T, V i ) and V i =span[u j ] j i /X {. Lemma 2.1.
Assume that there is a c>0 such that
}
* {j sin
- *j T c 2
}
\j>i 0 ,
then (2.3) has a unique solution v=L i h # L (S T, W i ) satisfying _v_ 12 T
2*
&2{ i
+
1 _h_ { , 2c 2
(2.4)
44
MICHAL FEC8 KAN
where W i =span[u j ] j i /X 12. If in addition, it holds
}
* {&+ sin j
- *j T c~ 2
}
\j>i 0 ,
(2.5)
for some c~ >0 and { + 0, then v* # L (S T, X + ) satisfies _v* _ + T
2*
&2({&+) i
+
1 _h_ { . 2c~ 2
Since h # L (S T, V i ), we have
Proof.
h j # L (S T, R),
h(t)= : h j (t)u j ,
2 2 ess sup : * 2{ j h j (})=_h_ { .
j=i
j=i
The equation (2.3) splits into the system v j (t)+* j v j (t)=h j (t),
j i (2.6)
v(t)= : v j (t)u i . j=i
Since (2.4) holds, (2.6) has the solution v j (t)=
1 - *j +
|
t
sin(- * j (t&s)) h j (s) ds 0
1
1 2 - *j
sin
-* T | j
T 0
T cos - * j s& &t 2
\ \
++ h (s) ds.
(2.7)
j
2
2_ 2 We note that _v_ 2_ =ess sup j=i * j v j ( } ). The formula (2.7) gives
: * j v 2j (t) 2 : j=i
j=i
\|
t
sin(- * j (t&s)) h j (s) ds 0
1
+2 : j=i
4 sin 2
2 : * &2{ j j=i
+
1 : 2c 2 j=i
| |
T
| -* T \ j
T 0
0
T cos - * j s& &t 2
\ \
++
h j (s) ds
2 sin 2 - * j (t&s) ds
0
T
+
2
|
T
0
2 * 2{ j h j (s) ds
T cos 2 - * j s& &t ds 2
\
+
|
T
0
2 * 2{ j h j (s) ds
+
2
45
UNDAMPED WAVE EQUATIONS
2T* &2{ i
|
T
2 : * 2{ j h j (s) ds+
0
j=i
T 2 _h_ 2{ + 2* &2{ i
1 T 2c 2
|
T
2 : * 2{ j h j (s) ds
0
j=i
1 1 T 2 _h_ 2{ =T 2 2* &2{ + 2 _h_ 2{ . i 2c 2 2c
\
+
Similarly, we get 2 &2({&+) T : * 2+ j v* j (t) 2* i j=i
+
1 T 2c~ 2
|
|
0
2 : * 2{ j h j (s) ds j=i
T 0
T
\
2 2 : * 2{ 2* &2({&+) + j h j (s) ds T i j=i
1 _h_ 2{ . 2c~ 2
+
Clearly the proof is finished. K Lemma 2.2.
Let
: k>i0
1 < * \k
for some \>0 and let D 0. Let S(c) be the set of all T>0 satisfying
}
* \j sin
- *j T c 2
}
\j>i 0 ,
(2.8)
where c>0 is a constant. Then the Lebesque measure of the complement (R"S(c)) & [D, D+1] satisfies 2c?
\- * *
m((R"S(c)) & [D, D+1]) : k>i0
k
\ k
+
c 2c 2? + . \ * k - * k * 2\ k
+
Proof. If T # (R"S(c)) & [D, D+1] then * \k |sin(- * k T2)|
}
}
46
MICHAL FEC8 KAN
for some n # Z + . Hence T
c
n
} 2?&- * } < 2 - * * , 2?n ?c } T& - * } < - * * . k
k
k
k
\ k
\ k
Consequently, we obtain T n D c c & < & \ \ 2? 2 - * k * k 2? 2 - * k * k - * k <
T D+1 c c , + + 2? 2 - * k * \k 2? 2 - * k * \k
c D+1 c D - * k & \
= : k>i0
\
c - *k 2?c + +1 \ *k 2? - * k * \k
+
2c?
+ : \
- *k * k
k>i0
c 2c 2? . \+ : * k k>i - * k * 2\ k 0
The proof is finished. K We put for i>i 0 { Y i =V = i =span[u j ] 1 j i&1 /X .
Let Q i : X { Ä V i , P i =I&Q i be the orthogonal projections. Now we decompose (2.1) in the following way v +Av+F(v+w) Bv==Q i h(t+:),
v # Wi ,
(2.9)
w +Aw+F(v+w) Bw==P i h(t+:),
w # Yi ,
(2.10)
where : # R is a shift parameter. In view of (2.3) and Lemma 2.1, (2.9) has the form v&L i (=Q i h(t+:)&F(v+w) Bv)=0.
(2.11)
47
UNDAMPED WAVE EQUATIONS
We suppose that the following condition holds: There is a w 0 # L (S T, Yi ) satisfying w 0 +Aw 0 +F (w 0 ) Bw 0 =0.
(C1)
The linearization L of (2.11) by v at ==0, w=w 0 , v=0 has the form Lv=v+L i (F(w 0 ) Bv). We compute _Lv_ 12 _v_ 12 &T
2*
&2{ i
+
1 sup |F(w 0 ( } ))| &B i & _v_ 12 , (2.12) 2c 2 ST
where B i =BW i . We note that B i # L(W i , V i ). Theorem 2.3.
Let (2.4) and (C1) be satisfied. Let us take i such that T
2*
&2{ i
+
1 sup |F(w 0 ( } ))| &B i &<1. 2c 2 ST
(2.13)
Then for any = sufficiently small, : # R and w # L (S T, Yi ) near w 0 , there is a unique solution of (2.11) v=v(=, w, :) # L (S T, W i ) such that the mapping v( } , } , } ) is C 1-smooth and v(0, w, :)=0. Proof. By applying the implicit function theorem to (2.11), the proof is finished according to (2.12) and (2.13). K We put the solution of (2.9) from Theorem 2.3 into (2.10) and we arrive at the functional equation w +Aw+F(v(=, w, :)+w) Bw==P i h(t+:).
(2.14)
Since v(0, w, :)=0, we take v(=, w, :)==z(=, w, :). Then (2.11) gives z(=, w, :)=L i (Q i h(t+:)&F(=z(=, w, :)+w) B i z(=, w, :)).
(2.15)
Hence z(0, w 0 , :)=L i (Q i h(t+:)&F(w 0 ) B i z(0, w 0 , :)), z(0, w 0 , :)+L i F(w 0 )B i z(0, w 0 , :)=L i Q i h(t+:), Lz(0, w 0 , :)=L i Q i h(t+:), z(0, w 0 , :)=L &1 (L i Q i h(t+:)).
48
MICHAL FEC8 KAN
We put in (2.14) w=w 0 +=y, and we arrive at the equation y +Ay+F(=z(=, w, :)+w 0 +=y) By 1 + (F(=z(=, w, :)+w 0 +=y)&F(w 0 )) Bw 0 =P i h(t+:). = This equation has the form y +Ay+F(w 0 ) By+(DF(w 0 )( y+z(0, w 0 , :))) Bw 0 +O(=)=P i h(t+:).
(2.16)
Remark 2.4. To get (2.9) and (2.10), it is enough to assume that B(W i )/V i and B(Yi )/Yi . Consequently, [u j ] j 1 need not be eigenvectors of B.
3. SMALL PERIODIC SOLUTIONS \ Theorem 3.1. Suppose F(0)=0 and k=i0 +1 1* k < for some { \>0. Then for almost all T>0 (in the sense of the Lebesque measure) and any h # L (S T, X { ), the equation (2.1) has a small T-periodic solution for = sufficiently small.
Proof. Lemma 2.2 implies that for almost all T>0 the conditions (2.4) and (2.5) with +=\&{ are satisfied for some c>0, respectively c~ >0. The condition (C1) holds with w 0 =0. Now Theorem 2.3 is applied with i=i 0 +1 and :=0. The equation (2.16) has the form y +Ay+O(=)=P i h(t),
y # Yi0 .
(3.1)
The eigenvalues of A on Yi0 are all negative, hence y +Ay=0 has the only zero T-periodic solution. For this reason, (3.1) has a T-periodic solution for any = sufficiently small. The proof is finished. K
4. SUBHARMONIC SOLUTIONS We assume in this section that i 0 =1, i.e., &$=* 1 <0<* 2 } } } and that $ is small and * 2 is bounded off 0. Hence Y2 =span[u 1 ]
UNDAMPED WAVE EQUATIONS
49
and by putting w 0 (t)=z(t) u 1 , the equation of the condition (C1) becomes z &$z+ f (z) z=0,
(4.1)
where f (z)=(F(zu 1 ) Bu 1 , u 1 ). We suppose F(zu 1 )=%z 2,
%{0.
(C2)
Then the system given by z* 1 =z 2 z* 2 =$z 1 &%;z 31 ,
;=(Bu 1 , u 1 )
(4.2)
has a hyperbolic equilibrium (0, 0). We suppose that %;>0. It is wellknown [9] that (4.1) has the one-parametric family of periodic solutions g k (t)=
- 2$ 2
- (2&k ) %;
dn
- $t
\- 2&k , k+ , 2
where dn is the Jacobi elliptic function and k is the elliptic modulus. The period of g k (t) is Tk - $, Tk =2K(k) - 2&k 2, where K(k) is the complete elliptic integral of the first kind. We take k such that Tk - $= pT,
p # N,
and we put w 0 (t)= g k (t) u 1 . We note that Tk is increasing and lim k Ä 0 Tk =- 2?, lim k Ä 1 Tk = . We have
}
|F(w 0 (t))| %
2$ <2$ | ;|. (2&k 2 ) %;
}
We are interested in pT-periodic solutions of (2.1) near w 0 (t) given above. Then the condition (2.13) has the form E= pT
2*
&2{ 2
+
1 2$ &B 2 &<1. 2c 2 | ;|
50
MICHAL FEC8 KAN
The above inequality holds with E<12 when pTc<
| ;| 2 &B 2 & $ - 8* &2{ c 2 +2 2
.
(4.3)
(2.15) for this case implies _z(=, w, :)_ 12 pT
2*
&2{ 2
+
1 3 _Q 2 h(t+:)_ { + _z(=, w, :)_ 12 , 2 2c 4
whenever &=z(=, w, :)( } )+w( } )&w 0 ( } )& 12 ' and '>0 is sufficiently small, fixed. Consequently, by assuming (4.3) we have
2*
_=z(=, w, :)_ 12 |=| 4pT
&2{ 2
+
1 _h(t+:)_ { 2c 2
|=| | ;| _h_ { '2. $ &B 2 &
(4.4)
Hence we have to take w such that &w( } )&w 0 ( } )& 12 '2. We note that w=w 0 +=y. From (2.15) we also obtain z(=, w, :)&z(0, w 0 , :) =L 2 (F(w 0 )&F(=z(=, w, :)+w 0 +=y)) B 2 z(0, w 0 , :) +L 2 F(=z(=, w, :)+w) B 2 (z(0, w 0 , :)&z(=, w, :)). When 1 c>0 then (4.3) implies that at most pTct1$. Hence we obtain _z(=, w, :)&z(0, w 0 , :)_ 12 1
2* +2c c _=z(=, w, :)+=y_ &B & _z(0, w , :)_ 1 4pT 2* +2c 1 _c |=| 4pT 2* + \ 2c _h( } +:)_ + |=| _ y_ + 1 _&B & 4pT 2* + 2c _h( } +:)_ 1 1 K (_h_ , _ y_ ) |=| (4.5) \$ +$ + , &2{ 2
4pT
&2{ 2
2
12
&2{ 2
&2{ 2
2
{
2
0
12
2
1
1
1
12
{
2
{
2
3
12
2
where 1 c>0, K 1 ( } , } ) is a continuous function and c 1 >0 is a constant.
51
UNDAMPED WAVE EQUATIONS
The O(=) term of (2.16) has the form
} } (F(=z(=, w, :)+w +=y)&F(w )) By + | DF(s(=z(=, w, :)+=y)+w ) ds(z(=, w, :)+ y) \ &DF(w )( y+z(0, w , :)) Bw + }} 0
0
1
0
0
0
0
0
12
c 2 |=| (_z(=, w, :)_ 12 +_ y_ 12 ) &B& _y_ 12 +c 3 |=| (_z(=, w, :)_ 12 +_ y_ 12 ) 2 &B& _w 0 _ 12 +c 4 _z(=, w, :)&z(0, w 0 , :)_ 12 &B& _w 0 _ 12
\ ) |=| _ y_ \
K 2 (_h_ { , _ y_ 12 ) |=| _ y_ 212 + K 3 (_h_ { , _ y_ 12
2 12
+
p 3T 3 p 2T 2 + 2 c3 c 1 1 + 2 ), 3 $ $
+ (4.6)
where 1 c>0, c 2 , c 3 , c 4 are positive constants and K 2 ( } , } ), K 3 ( } , } ) are continuous functions. The estimates (4.44.6) imply that the O(=) term of (2.16) is small whenever |=|$ 3 is small and _ y_ 12 is bounded. We note that Tk - $= pT, Tk - 2? and Tk is increasing in k. Hence (4.3) is equivalent to - 2?- $ pT<
| ;|c 2 &B 2 & $ - 8* &2{ c 2 +2 2
.
(4.7)
By taking c=$ 14, (4.7) becomes - 2?- $ pT<
| ;| 2 &B 2 & $
34
- 8* &2{ $ 12 +2 2
.
(4.8)
For simplicity, we suppose DF(w) v=0 \v # W 2 ,
\w # Y2 .
(C3)
Under the above assumptions, (2.16) has the form z &$z+3%;g 2k z+O(=$ 3 )=(h(t+:), u 1 ),
(4.9)
52
MICHAL FEC8 KAN
where y=zu 1 . The unperturbed part of (4.9) given by z &$z+3%;g 2k z=0 is just the linearization of (4.1) at g k . It has, up to a scalar multiple, the only pT-periodic solution g$k . It is well known [7], [9, Theorem 4.6.2] that the bifurcation of pT-periodic solution of (4.9) is determined by a subharmonic Melnikov function given as follows M(:)=
|
pT
(h(s+:), u 1 ) g$k (s) ds. 0
Summarizing, we arrive at the main result of this paper. Theorem 4.1. Consider (2.1) under the assumptions of this section. Let $>0, 1 c>0 be fixed. If there are a, b such that M(a) M(b)<0 then (2.1) has a pT-periodic solution for any = sufficiently small, p # N and any T such that pT # S(c) and (4.7) holds as well. Remark 4.2. We note that pT=Tk - $ in the formula of M(:). Hence M(:) depends also on $.
5. APPLICATIONS TO THE BEAM EQUATION To apply the result of the previous section to (1.1) with ? 2 <1<3? 2, we put X=L 2 (0, 1), F(u)= p
\|
1
Au=u xxxx +1u xx ,
u 2 (s, t) ds,
0
|
1
0
Bu=D !xx u
+
u 2x(s, t) ds ,
X 12 =H 2 (0, 1) & H 10(0, 1),
h(t)=cos
{=(1&!)2 2?t q(x). T
We have u j =- 2 sin ?jx,
* j =? 2j 2 (? 2j 2 &1 ).
Since ? 2 <1<3? 2, we obtain * 2 >4? 4,
* 1 =? 2 (? 2 &1 )=&$,
$>0.
UNDAMPED WAVE EQUATIONS
53
The condition (C2) is satisfied when there is a % # R such that p(z, z? 2 )=%z
\z 0.
(5.1)
In this case ;=? 2!, so we suppose that %>0. The series
: j=2
1 1 = : { 1&! 2 2 * j j=2 (?j) (? j &1 ) (1&!)2
converges if 12>! 0. Hence Lemma 2.2 can be applied only for 12>! 0. Let us put q k (t)=
-2 - 2&k
2
dn
t
\- 2&k , k+ . 2
Hence g k (t)=
-$ - %;
q k (- $t).
The Melnikov mapping now has the form [9, p. 199] M(:)= =
|
pT
cos 0
-$
| - %; |
Tk
cos
0
2? T -$
|
1
- 2q(x) sin ?x dx
0
(s+- $:) q$k (s) ds
1
- 2q(x) sin ?x dx
_
=&
2? (s+:) g$k (s) ds T
0
2? 2 - 2 !
? T -%
_sech
_sin
?pK$(k) K(k)
2? : T
|
1
- 2q(x) sin ?x dx.
0
We see that M( } ) does change the sign on R provided that 10 q(x) sin ?x dx{0. Clearly the condition (C3) holds. We note D !xx
\ : a u += : ? i
i 2
i
i 2
2! 2!
i ai ui .
54
MICHAL FEC8 KAN
We compute &1 * i a 2i =? 4! : i 4!* &! * i a 2i ? 4! : i 4!* 2{ i *i i i 2
i 2
=? 4! : i 4! i 2
1 * a2 (? i (? i &1 )) ! i i 2 2
2 2
* i a 2i
=? 4! : i 2
\? \ 2
1 ? & 2 i 2
++
!
.
Hence
\
&B 2 &=? ! ? 2 &
1 4
+
&!2
2 !.
Furthermore, for c 2 1, we have c 2* 2{ 2 =
c2 1 . 2 2 1&! 4 1&! (4? (4? &1 )) (4? )
Consequently, (4.3) and (4.7) are satisfied when it holds -2 - 1 &?
2
pT<
1 !
4 1&!
4 - 8+2(4? )
c . 1 &? 2
(5.2)
We note that $=? 2 (1 &? 2 ). By applying Theorem 4.1, we obtain the following result. Theorem 5.1. Consider (1.1) with 0 ! 1, ? 2 <1 <3? 2. Let 1 c>0 and let (5.1) hold for %>0. If 10 q(x) sin ?x dx{0 then for any p # N, T satisfying (5.2) and pT # S(c), the equation (1.1) has a pT-periodic solution for any = sufficiently small. Moreover, if 0 !<12, then m(S(c) & [D, D+1]) Ä 1 as c Ä 0 + uniformly with respect to D 0. Roughly speaking, Theorem 5.1 asserts that (1.1) with 0 !<12 and ? 2 <1 <3? 2, has many subharmonics for = sufficiently small when c and ? 2 &1 are sufficiently small. On the other hand, since we do not know the structure of the set S(c), we do not know a concrete case for pT. This result can be related to the KAM theory [12]. Furthermore, Lemma 2.2 gives that the Lebesque measure of S(c) tends to 1 as c Ä 0 + . Consequently, we
55
UNDAMPED WAVE EQUATIONS
put c=(1 &? 2 ) 14 in (5.2), like for (4.8), and the assumptions of Theorem 5.1 become in the following form -2 - 1 &?
2
pT<
1 !
4 1&!
4 - 8+2(4? )
1 , (1 &? 2 ) 34
(5.3)
pT # S((1 &? 2 )). Summarizing we obtain the following result. Theorem 5.2. Let 12>! 0, ? 2 <1 <3? 2 and let (5.1) hold for %>0. If 10 q(x) sin ?x dx{0, then the smaller 1 &? 2 >0, the larger the number of subharmonic solutions of (1.2) persists for (1.1) when = is sufficiently small. Now we apply Theorem 3.1. Theorem 5.3. Let 12>! 0 and 1 # R"[? 2j 2 | j # N] be arbitrary. For almost all T>0, there is a small T-periodic solution of (1.1) for any = sufficiently small. Theorem 5.1 should be applicable also for the case 1 ! 12 and ? 2 <1 <3? 2. The problem is to show the non-emptiness of the set S(c) in conjunction with (5.2). We note that now the condition (2.8) with \={=(1&!)2 is equivalent to j 2(1&!)
T
} 2 j - ? j &1 &n } c 2 2
\j # N"[1],
\n # Z + ,
(5.4)
for some constant c >0. Furthermore, let us consider the case !=1. Then to get a condition similar to (5.3), the condition (5.4) has to be replaced by
}
pT j - ? 2j 2 &1 &n c(1 &? 2 ) 14 2
}
\j # N"[1],
\n # Z + ,
(5.5)
for some constant c>0, p # N and 1 >? 2 could be taken arbitrarily near ? 2. The problem here is to control the constant c in (5.5) when 1 Ä ? 2+ , and also to show the inequality of (5.3) for pT. We do not know to solve this problem. Consequently, we are not able to apply Theorem 5.1 for 1 ! 12, because we do not know to show neither (5.2) nor (5.3). On the other hand, we have the following result.
56
MICHAL FEC8 KAN
Lemma 5.4. that
If T? # Q and 1? 2 Â Q then there is a constant c>0 such
T
} 2 j - ? j &1 &n } c 2 2
\j # N : j 2 1? 2,
\n # Z + .
Proof. We follow the proof of [19, Theorem 4.1] by computing for j 2 1? 2
T
} 2 j - ? j &1 &n } 2 2
=
T
1
T
T
} 2 j \- ? j &1 & \?j&2j?++ + 2 ?j & 4? 1 &n } }
2 2
1 2T T 2 T ?j & 1 &n & 2 2 2 4? 8? j
}
2
1
1
2
2
<\ ? & j +?& 2j ?+ . 2
The assumptions of this lemma imply that T? # Q, T 21 Â Q and T1? Â Q. Hence we have
inf j # N, n # Z+
T j - ? 2j 2 &1 {n 2
}
T 2 T ?j & 1 &n >0, 2 4?
}
\j # N : j 2 1? 2,
\n # Z + .
Since the second term in the end of the above inequality tends to 0 as j Ä , the proof is finished. K The proof of Theorem 3.1 and Lemma 5.4 give the next result. A similar result is proved in [19, Theorem 4.1]. Theorem 5.5. Consider (1.1) with 1 ! 0. If T? # Q and 1? 2 Â Q, then there is a small T-periodic solution of (1.1) for any = sufficiently small. Now we give another criterion for (5.4) following directly from the proof of Lemma 5.4.
57
UNDAMPED WAVE EQUATIONS
Theorem 5.6. T
Let 1 !>0. If there is a constant c>0 such that
T
} 2 ?j & 4? 1 &n } cj 2
2(1&!)
\j # N,
\n # Z + , (5.6)
T j - ? 2j 2 &1 {n 2
2
2
\j # N : j 1? ,
\n # Z + ,
then (5.4) holds with j # N: j 2 1? 2. Now, let us take for (1.1) 1 =2m? 20,
T=20?,
!=0,
where m # Z & =&Z + and 0>0 has the infinite continuous fraction decomposition [13] 0=[a 0 , a 1 , a 2 , } } } ] such that a i M\i # Z + for a constant M # N. Like in the proof of Lemma 5.4 and according to [13, Propositions 1 and 2] as well, we have j2
}
T T 2 T 1 2T j - ? 2j 2 &1 &n j 2 ?j & 1 &n & 2 2 4? 16? 3
}
}
}
= j 2 |0j 2 &m&n| &
1 m2 m2 & 20 M+2 20
for any n # Z + and j # N. On the other hand, since 0=[a 0 , a 1 , a 2 , } } } ] [M, 1, M, 1, } } } ]=
M+- M 2 +4M , 2
from the inequality m 2 (M+2)<20, we obtain m2 <
M+- M 2 +4M . M+2
M 1 implies that 1<(M+- M 2 +4M)(M+2)<2. Consequently, we arrive at the following result. Theorem 5.7. Consider (1.1) with !=0. If T=20?, 1 =2m? 20, where m # [&1, 0] and 0>0 has the infinite continuous fraction decomposition [13] 0=[a 0 , a 1 , a 2 , } } } ] such that a i M \i # Z + for a constant M # N satisfying m 2 (M+2)<20, then there is a small-periodic solution of (1.1) for any = sufficiently small.
58
MICHAL FEC8 KAN
We remark that Theorem 5.3 provides a more general result than Theorems 5.5 and 5.7, while Theorems 5.5 and 5.7 give concrete forms of T and 1 for the existence of periodic solutions of (1.1). For instance, Theorem 5.7 is valid for 0=(M+- M 2 +4M)2, M # N and m # [&1, 0]. A similar result can be derived also for 1 >0 small, but formulas are more tedious. So we leave those computations. Finally, we note that (1.1) can be considered with periodic boundary value conditions of the form u tt +u xxxx +1u xx + p
\
== q 1 (x) cos
\|
2
u 2 (s, t) ds,
0
|
2
0
2?t 2?t +q 2 (x) sin T T
u(x, } )=u(x+2, } )
+
u 2x(s, t) ds D !xx u
+
(5.7)
\x # R,
where !, 1, p are like in (1.1), D !xx is the !-power of D xx in L 2 (S 2 ), q 1, 2 # H 2 (0, 2) and q 1, 2 are 2periodic satisfying
|
2
q 1 (x) cos ?x dx=0,
0
|
2
q 2 (x) sin ?x dx=0.
0
Now we put in (5.7) with ==0 u(x, t)=x(t) sin ?x+ y(t) cos ?x, like for (1.2), and we obtain a system of ordinary differential equations x +? 2 (? 2 &1 ) x+%? 2! (x 2 + y 2 ) x=0,
(5.8)
y +? 2 (? 2 &1 ) y+%? 2! (x 2 + y 2 ) y=0, where (5.1) holds for %>0. By supposing 1 >? 2, (5.8) has a 1-parametric family of periodic solutions given by x k, . =cos ._g k (t),
y k, . =sin ._g k (t),
. # R.
The Melnikov function [7] now is a 2-dimensional mapping M(., :)=(M1 (., :), M2 (., :))
59
UNDAMPED WAVE EQUATIONS
of the form M1 (., :)=&
2? 2 - 2 !
? T -%
?pK$(k) K(k)
_sech M2 (., :)=
2? 2 - 2 !
? T -%
_sech
_sin
|
2
q 1 (x) sin ?x dx,
0
2? :_sin . T
_cos
?pK$(k) K(k)
2? :_cos . T
|
2
q 2 (x) cos ?x dx. 0
Clearly (., :)=(0, 0) is a simple zero point of M provided that
|
2
q 1 (x) sin ?x dx{0,
0
|
2
q 2 (x) cos ?x dx{0.
0
Consequently [7], like for Theorem 5.2, we arrive at the following result. Theorem 5.8. Let 12>! 0 and ? 2 <1 <3? 2. If 20 q 1 (x) sin ?x dx{0 and 20 q 2 (x) cos ?x dx{0, then the smaller 1 &? 2 >0, the larger the number of subharmonic solutions of (5.8) persists for (5.7) when = is sufficiently small. Remark 5.9. Theorems 5.2, 5.3, 5.5 and 5.7 can be also straightforwardly modified to (5.7). The term p( 10 u 2 (s, t) ds, 10 u 2x(s, t) ds) in (1.1), similarly in (5.7), can be also replaced by p(
|
1 0
u 2 (s, t) ds,
|
1 0
u 2x(s, t) ds,
|
1
0
u 2xx(s, t) ds
+
in the above considerations. Of course, the results concerning (2.1) of the previous sections can be applied to other types of partial differential equations than (1.1) and (5.7).
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60
MICHAL FEC8 KAN
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