Quasi-periodic oscillations for wave equations under periodic forcing

ATTIACCADEMIANAZIONALELINCEICLASSESCIENZEFISICHEMATEMATICHENATURALI

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ENDICONTI

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INCEI

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ATEMATICA E

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PPLICAZIONI

Massimiliano Berti, Michela Procesi

Quasi-periodic oscillations for wave equations

under periodic forcing

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche,

Matematiche e Naturali. Rendiconti Lincei. Matematica e

Applicazioni, Serie 9, Vol. 16 (2005), n.2, p. 109–116.

Accademia Nazionale dei Lincei

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Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Accademia Nazionale dei Lincei, 2005.

Rend. Mat. Acc. Lincei s. 9, v. 16:109-116 (2005)

Analisi funzionale. Ð Quasi-periodic oscillations for wave equations under periodic forcing. Nota di MASSIMILIANOBERTIe MICHELAPROCESI, presentata (*) dal Socio A. Ambrosetti.

ABSTRACT. Ð Existence of quasi-periodic solutions with two frequencies of completely resonant,

periodically forced, nonlinear wave equations with periodicspatial boundary conditions is established. We consider both the cases the forcing frequency is (Case A) a rational number and (Case B) an irrational number. KEY WORDS: Nonlinear wave equation; Quasi-periodicsolutions; Variational methods; Lyapunov-Schmidt

reduction; Infinite dimensional Hamiltonian systems.

RIASSUNTO. Ð Oscillazioni quasi periodiche per equazioni delle onde con forzante periodica. Si dimostra

l'esistenza di soluzioni quasi periodiche con due frequenze per una classe di equazioni delle onde non lineari completamente risonanti aventi un termine forzante periodico. Consideriamo che la frequenza forzante sia un numero razionale (Caso A), sia irrazionale (Caso B).

1. INTRODUCTION

We present in this Note the results of [7] concerning existence of small amplitude quasi-periodic solutions for completely resonant forced nonlinear wave equations like

vtt vxx‡ f (v1t; v) ˆ 0

v(t; x) ˆ v(t; x ‡ 2p) 

…1:1†

where the nonlinear forcing term

f (v1t; v) ˆ a(v1t)v2d 1‡ O(v2d); d > 1; d 2 N‡

is 2p=v1-periodic in time and the forcing frequency is:

 A) v12 Q

 B) v12 R n Q.

Existence of periodic solutions for completely resonant forced wave equations was first proved in the pioneering paper [10] (with Dirichlet boundary conditions) if the forcing frequency is a rational number (v1 ˆ 1 in [10]). This requires to solve an infinite

dimensional bifurcation equation which lacks compactness property; see [4] and references therein for other results. If the forcing frequency is an irrational number existence of periodic solutions has been proved in [8]: here the bifurcation equation is trivial but a «small divisors problem» appears.

To prove existence of small amplitude quasi-periodic solutions for completely resonant PDE's like (1.1) one has to deal both with a small divisor problem and with an infinite dimensional bifurcation equation. A major point is to understand from which periodicsolutions of the linearized equation

vtt vxxˆ 0 ;

the quasi-periodic solutions branch-off: indeed such linear equation possesses only 2p-periodicsolutions of the form q‡(t ‡ x)‡ q (t x).

For completely resonant autonomous PDE's, existence of quasi-periodic solutions with 2-frequencies have been recently obtained in [9] for the specific nonlinearities f ˆ u3_{‡ O(u}5_{). Here the bifurcation equation is solved by ODE methods.}

In [7] we prove existence of quasi-periodic solutions with two frequencies v1, v2for

the completely resonant forced equation (1.1) in both the two cases A) and B).

The more interesting case is v12 Q (case A) when the forcing frequency v1enters in

resonance with the linear frequency 1. To find out from which solutions of the linearized equation quasi-periodicsolutions of (1.1) branch-off, requires to solve an infinite dimensional bifurcation equation which can not be solved in general by ODE techniques (it is a system of integro-differential equations). However, exploiting the variational nature of equation (1.1) like in [5], the bifurcation problem can be reduced to finding critical points of a suitable action functional which, in this case, possesses the infinite dimensional linking geometry [3].

On the other hand, we avoid the inherent small divisor problem by restricting the parameters to uncountable zero-measure sets as, e.g., in [9, 5].

1.1. Main results. We look for quasi-periodicsolutions v(t; x) of (1.1) of the form v(t; x) ˆ u(v1t; v2t ‡ x)

u(W₁‡ 2k1p; W2‡ 2k2p) ˆ u(W1; W2); 8k1; k2 2 Z

 with frequencies

v ˆ (v1; v2) ˆ (v1; 1 ‡ ")

imposing the frequency v2ˆ 1 ‡ " to be close to the linear frequency 1. Therefore we get

[v2

1@W21‡ …v 2

2 1†@W22‡ 2v1v2@W1@W2]u…W† ‡ f …W1; u† ˆ 0:

…1:2†

We assume that the forcing term f : T  R ! R

f (W₁; u) ˆ a2d 1(W1)u2d 1‡ O(u2d); d 2 N‡; d > 1

is analyticin u but has only finite regularity in W₁. More precisely  (H) f (W₁; u) :ˆ X1

kˆ2d 1

ak(W1)uk; d 2 N‡, d > 1 and the coefficients ak(W1) 2 H1(T)

verify, for some r > 0, X1

kˆ2d 1

jakjH1rk < 1. The function f (W₁; u) is not identically

con-stant in W₁.

We look for solutions u of (1.2) in the multiplicative Banach algebra Hs;s :ˆ u(W) ˆ

X

l2Z2

^uleil
W : ^ul ˆ ^u l and jujs;s:ˆ

X

l2Z2

j^uljejl2js[l1]s< ‡1

( )

where [l1] :ˆ maxfjl1j; 1g and s > 0, s  0.

In [7] we prove the following theorems.

THEOREM A. Let v₁ˆ n=m 2 Q. Assume that fsatisfies assumption (H) and a2d 1(W1) 6ˆ 0, 8W12 T. Let Bgbe the uncountable zero-measure Cantor set

Bg:ˆ " 2 ( "0; "0) : jl1‡ "l2j >_jlg

2j; 8l1; l22 Z n f0g

 

where 0 < g < 1=6.

There exist constants s > 0, s > 2, " > 0, C > 0, such that 8" 2 Bg, j"jg 1 "=m2,

there exists a classical solution u("; W) 2 Hs;s of (1.2) with (v1; v2) ˆ (n=m; 1 ‡ ")

satisfying  u("; W) j"j2(d 1)1 q_"(W) s;s C m2_j"j g v3 1 j"j 1 2(d 1) …1:3†

for an appropriate function q"2 Hs;sn f0g ofthe form q"(W) ˆ q‡(W2)‡ q (2mW1 nW2).

As a consequence, equation (1.1) admits the quasi-periodic solution v("; t; x) :ˆ u("; v1t; x ‡ v2t) with two frequencies (v1; v2) ˆ (n=m; 1 ‡ ") and the map

t ! v("; t;
) 2 Hs_{(T) has the form (}1₎  v("; t; x) j"j2(d 1)1 [q_‡(x ‡ (1 ‡ ")t) ‡ q ((1 ") nt nx)] Hs_(T)ˆ O m2 g v3 1j"j 2d 1 2(d 1)   : Remark that the bifurcation of quasi-periodic solutions u occurs both for v2> 1 and

v2< 1; actually both Bg\ (0; "0) and Bg\ ( "0; 0) are uncountable 8"0> 0.

At the first order the quasi-periodicsolution v("; t; x) of equation (1.1) is the superposition of two waves traveling in opposite directions (in general, both components q‡, q are non trivial).

The bifurcation of quasi-periodic solutions looks quite different if v1 is irrational.

THEOREMB. Let v₁2 R n Q. Assume that fsatisfies assumption (H), Z2p 0

a2d 1(W1)dW1

6ˆ 0 and f (W₁; u) 2 Hs_{(T), s  1, for all u.}

Let Cg  D  ( "0; "0)  (1; 2) be the uncountable zero-measure Cantor set

Cg:ˆ ("; v1) 2 D : v162 Q ; v_v1 262 Q ; jv1l1‡ "l2j > g jl1j ‡ jl2j; jv1l1‡ (2 ‡ ")l2j >_jl g 1j ‡ jl2j; 8 l12 Z n f0g 8 > < > : 9 > = > ;: …1:4†

Fix any 0 < s < s 1=2. There exist positive constants ", C, s > 0, such that, 8("; v1) 2 Cg

with j"jg 1_{< " and "}Z 2p 0

a2d 1(W1) dW1> 0, there exists a nontrivial solution

(1_{) We denote H}s_{(T) :ˆ fu(W) ˆ}P

l2Z^uleilW : jujHs_(T):ˆP_l2Zj^uljesjlj< ‡1g.

u("; W) 2 Hs;sofequation (1.2) with (v1; v2) ˆ (v1; 1 ‡ ") satisfying  u("; W) j"j2(d 1)1 q "(W2)_s;s Cj"j_g j"j 1 2(d 1) …1:5†

for some function q_"(W₂) 2 Hs_{(T) n f0g.}

As a consequence, equation (1.1) admits the non-trivial quasi-periodic solution v("; t; x) :ˆ u("; v1t; x ‡ v2t) with two frequencies (v1; v2) ˆ (v1; 1 ‡ ") and the map

t ! v("; t;
) 2 Hs_{(T) has the form}

 v("; t; x) j"j2(d 1)1 q "(x ‡ (1 ‡ ")t)_Hs_(T)ˆ O  g 1_j"j2d 1 2(d 1)  :

REMARK1. Imposing in the definition of C_g the condition v₁=v₂ˆ v₁=(1 ‡ ") 2 Q we obtain, by Theorem B, the existence of periodic solutions of equation (1.1). They are reminiscent, in this completely resonant context, of the Birkhoff-Lewis periodic orbits with large minimal period accumulating at the origin, see [2].

In Theorem B, existence of quasi-periodic solutions could follow by other hypotheses on f . However the hypothesis that the leading term in the nonlinearity f is an odd power of u is not of technical nature. Actually, the following non-existence result holds:

THEOREM C (Non-existence). Let f (W₁; u) ˆ a(W₁)uD with D even and Z2p

0

a(W₁) dW₁6ˆ 0. 8R > 0 there exists "0> 0 such that 8s  0, s > s 1₂; 8("; v1) 2 Cg with

j"j < "0, equation (1.2) does not possess solutions u 2 Hs;sin the ball jujs;s Rj"j1=(D 1).

Theorem C also highlights that the existence of periodic solutions of nonlinear wave equations in the case of even powers, see [5], is due to the boundary effects imposed by the Dirichlet conditions.

2. SKETCH OF THE PROOF OFTHEOREMA

We sketch now the proof of (the more difficult) Theorem A. To simplify notations we assume v1ˆ 1, " > 0 and a2d 1(W1) > 0.

Instead of looking for solutions of equation (1.2) in a shrinking neighborhood of 0, it is a convenient devise to perform the rescaling

u ! du with d :ˆ j"j1=2(d 1)

enhancing the relation between the amplitude d and the frequency v2ˆ 1 ‡ ". We

obtain the equation

L"u ‡ "f (W1; u; d) ˆ 0

…2:1†

where L":ˆ h @2 W1‡ 2@W1@W2 i ‡ "h(2 ‡ ")@2 W2‡ 2@W1@W2 i  L0‡ "L1 and f (W₁; u; d) :ˆf (W1; du) d2(d 1) ˆ  a2d 1(W1)u2d 1‡ da2d(W1)u2d‡ . . .  :

Equation (2.1) is the Euler-Lagrange equation of the Lagrangian action functional C"2 C1(Hs;s; R) defined by C"(u) :ˆ Z T2 1 2(@W1u) 2_{‡ (@} W1u)(@W2u) ‡ "(2 ‡ ") 2 (@W2u) 2_{‡ "(@} W1u)(@W2u) "F(W1; u; d)  C0(u) ‡ "G(u; d) where F(W₁; u; d) :ˆRu 0 f (W1; j; d)dj and C0(u) :ˆ Z T2 1 2(@W1u) 2_{‡ (@} W1u)(@W2u) G(u; d) :ˆ Z T2 (2 ‡ ") 2 (@W2u) 2_{‡ (@} W1u)(@W2u) F(W1; u; d) :

To find critical points of C" we perform a variational Lyapunov-Schmidt reduction

inspired to [5], see also [1].

The unperturbed functional C0 : Hs;s! R possesses an infinite dimensional linear

space Q of critical points which are the solutions q of the equation L0q ˆ @W1



@W1‡ 2@W2

 q ˆ 0 : The space Q can be written as

Q ˆnq ˆX

l2Z2

^qleil
W2 Hs;s j ^ql ˆ 0 for l1(l1‡ 2l2) 6ˆ 0

o

and, in view of the variational linking argument used for the bifurcation equation, we split Q as Q ˆ Q‡ Q0 Q where (2₎ Q‡:ˆ n q 2 Q : ^qlˆ 0 for l1ˆ 0 o ˆnq‡:ˆ q‡(W2) 2 Hs0(T) o Q0:ˆ n q0 2 R o Q :ˆnq 2 Q : ^qlˆ 0 for l1‡ 2l2ˆ 0 o ˆnq :ˆ q (2W₁ W₂); q (
) 2 H₀s;s(T)o: (2_{) H}s

0(T) denotes the functions of Hs(T) with zero average. Hs;s(T) :ˆ fu(W) ˆPl2Z^uleilW : ul ˆ u l,

jujHs;s_(T):ˆP_l2Zj^uljesjlj[l]s< ‡1g and Hs;s0 (T) its functions with zero average.

We shall also use in Q the H1_{-norm j
j}

H1 which is the natural norm for applying

variational methods to the bifurcation equation.

In order to prove analyticity (in W₂) of the solutions and to highlight the compactness of the problem we perform a finite dimensional Lyapunov-Schmidt reduction [6], decomposing

Hs;sˆ Q1 Q2 P where Q1:ˆ Q1(N) :ˆ n q ˆ X jljN ^qleil
W2 Q o ; Q2 :ˆ Q2(N) :ˆ n q ˆ X jlj>N ^qleil
W2 Q o and P :ˆnp ˆX l2Z2 ^p_leil
W_{2 H} s;sj ^plˆ 0 for l1(2l2‡ l1) ˆ 0 o :

Projecting equation (2.1) onto the closed subspaces Qi (i ˆ 1; 2) and P, setting

u ˆ q1‡ q2‡ p with qi2 Qiand p 2 P, we obtain

L1[q1] ‡ PQ1[f (W1; q1‡ q2‡ p; d)] ˆ 0 (Q1) L1[q2] ‡ PQ2[f (W1; q1‡ q2‡ p; d)] ˆ 0 (Q2) L"[p] ‡ "PP[f (W1; q1‡ q2‡ p; d)] ˆ 0 (P) 8 > < > :

where PQi : Hs;s! Qi are the projectors onto Qi and PP : Hs;s ! P is the projector

onto P.

Step 1 (Solution of the (Q2)-(P)-equations). For " 2 Bg, L" restricted to P has a

bounded inverse satisfying jL_"1[h]j_s;s jhj_s;sg 1 _{. Moreover, also the operator}

L1: Q2! Q2is invertible and jL11[h]js;s jhjs;sN 2:

Fixed points of the nonlinear operator G : Q2 P ! Q2 P defined by

G(q2; p; q1) :ˆ

 L 1

1 PQ2f (W1; q1‡ q2‡ p; d); "L"1PPf (W1; q1‡ q2‡ p; d)

 are solutions of the (Q2)-(P)-equations.

Using the Contraction Mapping Theorem, we can prove that 8R > 0 there exist an integer N0(R) 2 N‡and positive constants "0(R) > 0, C0(R) > 0 such that:

8jq1jH1 2R ; 8" 2 Bg; j"jg 1  "0(R) ; 8N  N0(R) : 0  sN  1 ;

there exists a unique solution

(q2(q1); p(q1)) :ˆ (q2("; N; q1); p("; N; q1)) 2 Q2 P

of the (Q2)-(P)-equations which satisfies jq2("; N; q1)js;s C0(R)N 2 and

jp("; N; q1)js;s C0(R)j"jg 1.

Step 2 (Solution of the (Q1)-equation). There remains to solve the finite dimensional

(Q1)-equation

L1[q1] ‡ PQ1f (W1; q1‡ q2(q1) ‡ p(q1); d) ˆ 0 :

…2:2†

Actually (see [5, 1]) the bifurcation equation (2.2) is the Euler-Lagrange equation of the 114 M. BERTI - M. PROCESI

reduced Lagrangian action functional

F";N : B2R Q1! R; F";N(q1) :ˆ C"(q1‡ q2(q1) ‡ p(q1))

which can be written as

F";N(q1) ˆ const ‡ "(G(q1) ‡ R";N(q1)) where G(q1) :ˆ Z T2 (2 ‡ ") 2 (@W2q1)2‡ (@W1q1)(@W2q1) a2d 1(W1) q2d 1 2d is the «PoincareÂ-Melnikov» function [1] and the remainder

R";N(q1) :ˆ Z T2 F(W₁; q1; d ˆ 0) F(W1; q1‡ q2(q1) ‡ p(q1); d) ‡1 2f (W1; q1‡ q2(q1) ‡ p(q1); d)(q2(q1) ‡ p(q1)) satisfies jR";N(q1)j  C1(R)(d ‡ j"jg 1‡ N 2) for some C1(R)  C0(R).

The problem of finding non-trivial solutions of the (Q1)-equation reduces to find

non-trivial critical points in B2R of the rescaled functional (still denoted F";N)

F";N(q1) ˆ  A(q1) Z T2 a2d 1(W1)q 2d 1 2d  ‡ R";N(q1)

where the quadraticform A(q) :ˆ Z T2 1 2(2 ‡ ")(@W2q) 2_‡(@ W1q)(@W2q) is positive definite on

Q‡, negative definite on Q and zero-definite on Q0. For q1ˆ q‡‡ q0‡ q 2 Q1,

A(q1) ˆa₂‡jq‡j2H1 a

2 jq j2H1

for suitable positive constants a‡, a , bounded away from 0 by constants independent of ".

The geometry of F";N suggests to look for critical points of «linking type». However

we can not directly apply the linking theorem because F";N is defined only in B2R. To

overcome this difficulty we extend F";N to the whole space in such a way that the

extension eF";N coincides with G(q1) outside B2R and so verifies the global hypothesis of

the linking theorem.

The final step is to prove (exploiting the homogeneity of the termRa2d 1q2d1 ) that

9 R> 0 independent on R; "; N; g

and functions 0 < "1(R)  "0(R), N1(R)  N0(R) such that 8j"jg 1  "1(R), N  N1(R)

the functional

eF";N possesses a non trivial critical point q12 Q1 with jq1jH1 R:

Fix R :ˆ R‡ 1, take j"jg 1 " :ˆ "( R) and 0 < s < 1=N2( R). The function

u("; W) :ˆ j"j1=2(d 1)[q₁‡ q2("; N2( R); q1) ‡ p("; N2( R); q1)]

is the solution of equation (1.2) in Theorem A.

REMARK2. In the proof of Theorem B the bifurcation equation could be solved through variational methods as for Theorem A. However there is a simpler technique available. The bifurcation equation reduces, in the limit " ! 0, to a superquadratic Hamiltonian system with one degree of freedom. We prove existence of a non-degenerate, analytic solution by direct phase-space analysis. Therefore it can be continued by the Implicit function Theorem to an analytic solution of the complete bifurcation equation for " small.

Supported by M.I.U.R. «Variational Methods and Nonlinear Differential Equations».

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______________

Pervenuta il 24 febbraio 2005, in forma definitiva il 21 marzo 2005.

M. Berti: SISSA Via Beirut, 2-4 - 34014 TRIESTE

berti@sissa.it M. Procesi: Dipartimento di Matematica UniversitaÁ degli Studi di Roma Tre Largo S. Leonardo Murialdo, 1 - 00146 ROMA

procesi@mat.uniroma3.it