Invariant tori of full dimension for higher-dimensional beam equations with almost-periodic forcing

<jats:p>In this paper, we focus on the class of almost-periodically forced higher-dimensional beam equations <jats:disp-formula><jats:alternatives><jats:tex-math>$$ u_{tt}+(-\Delta +\mu )^{2}u+\psi (\omega t)u=0,\quad \mu >0, t \in \mathbb{R}, x\in \mathbb{R}^{d}, $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mo>−</mml:mo><mml:mi>Δ</mml:mi><mml:mo>+</mml:mo><mml:mi>μ</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:mi>ψ</mml:mi><mml:mo>(</mml:mo><mml:mi>ω</mml:mi><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mspace /><mml:mi>μ</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:mi>R</mml:mi><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mi>R</mml:mi><mml:mi>d</mml:mi></mml:msup><mml:mo>,</mml:mo></mml:math></jats:alternatives></jats:disp-formula> subject to periodic boundary conditions, where <jats:inline-formula><jats:alternatives><jats:tex-math>$\psi (\omega t)$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ψ</mml:mi><mml:mo>(</mml:mo><mml:mi>ω</mml:mi><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:math></jats:alternatives></jats:inline-formula> is real analytic and almost-periodic in <jats:italic>t</jats:italic>. We show the existence of almost-periodic solutions for this equation under some suitable hypotheses. In the proof, we improve the KAM iteration to deal with the infinite-dimensional frequency <jats:inline-formula><jats:alternatives><jats:tex-math>$\omega =(\omega _{1},\omega _{2},\ldots)$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ω</mml:mi><mml:mo>=</mml:mo><mml:mo>(</mml:mo><mml:msub><mml:mi>ω</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>ω</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>)</mml:mo></mml:math></jats:alternatives></jats:inline-formula>.</jats:p>


Introduction
Recently, many researchers focus on some physical models appeared in dynamics of the suspension bridge, nonrelativistic quantum mechanics, supersymmetric field theoriesk and inflation cosmology [1][2][3][4][5][6][7][8][9][10][11]. For those models, there are many remarkable results on the global existence or the blowup of solutions for wave equations [2][3][4], elliptic equations [5,6], and some semilinear evolution equations [7]. As in the fundamental models, the dynamical behavior of the solutions is studied. The decay estimate of the solution at both subcritical and critical initial energy levels was obtained by Xu [2]. Nguyen [8] considered the compactness and stability for the Maxwell equations. Goubet and Manoubi [9] investigated the asymptotic convergence of the solutions.
For one-dimensional Hamiltonian systems, the existence of quasiperiodic solutions or almost-periodic solutions is also very significant in physics. It is well known that the infinite-dimensional KAM theory is powerful to obtain it (see [12][13][14][15][16][17][18][19][20][21]). However, the standard KAM theory fails to study higher-dimensional Hamiltonian PDEs because of the multiplicity of the eigenvalues.
It is worth noting that the first breakthrough in higher-dimensional PDEs is due to Bourgain. Bourgain [22] obtained quasiperiodic solutions for two-dimensional nonlinear Schrödinger equations via the developed Craig-Wayne methods. The Craig-Wayne-Bourgain methods can overcome the difficulty of the asymptotical multiplicity of eigenvalues in higher-dimensional PDEs. However, it should also be pointed out that the KAM theory has some important advantages. We can construct a local normal form in a neighborhood of the solutions using the KAM theory, which turns out the behavior and dynamics of the equation of motion. Thereafter, the infinite KAM theorem was extended to the existence of finite-dimensional tori for higher-dimensional Hamiltonian systems. Geng and You [23,24] constructed KAM theorems for the higher-dimensional beam equation. Yuan [25] obtained a KAM theorem to apply to partial differential equations of higher dimensions.
However, there is a crucial condition in the KAM theorems in [23] and [24] that the nonlinearity f (u) does not explicitly contain the time variable t and the space variable x. Thus their KAM approaches failed in the case of the nonlinearity depending on t or x. Physically, it requires no external force acting when the string is at rest, tending to distort its equilibrium of u = 0. Up to now there are very few results on the reducibility in higher dimensions. Eliasson and Kuksin [26] (also see [27]) showed the reducibility for the linear Schrödinger equations in higher dimensioṅ Eliasson, Grébert, and Kuksin [28] also considered the d-dimensional beam equation, which is a good model for the Klein-Gordon equation. Rui and Liu [29] proved the existence of quasiperiodic solutions for a linear d-dimensional beam equation with a quasiperiodic in time potential.
Comparing with the case of quasiperiodic solutions in higher dimensions, as far as we know, the reducibility results for almost-periodic solutions in higher dimensions have not been previously regarded in the literature. In this paper, we focus on the reducibility of the linear d-dimensional beam equation with almost-periodic forcing with periodic boundary conditions where ψ(ωt) is real analytic and almost-periodic in t. Our aim is to construct almostperiodic solutions of small amplitude for the beam equation (1.1). This equation is an important model of mathematical physics. It is of great interest in applying to many engineering fields.
Our beam equation (1.1) is quite different from the equations mentioned. There is almost-periodic forcing in higher dimensions, because the reducibility is complex and doubtful. Unfortunately, all those KAM theorems fail to handle infinite-dimensional frequency ω = (ω 1 , ω 2 , . . .) in Eq. (1.1). Using the method of Pöschel [19] and Xu and You [30], we succeed in decomposing infinite-dimensional frequency in the reducibility. Our nonresonance condition of an infinite-dimensional frequency benefits a lot from Pöschel [14]. The main difficulty in this problem is estimating measures of small divisors, since the infinite-dimensional frequency will handle at each step of the KAM iteration. The KAM theory in Kuksin [12] and Pöschel [13] cannot be directly applied to the d-dimensional beam equation with almost-periodic forcing, and we will improve the KAM iteration (see Sect. 3). A new strategy to overcome the difficulty are the techniques of decomposing infinitely many frequencies and expanding Hamiltonian into proper series, which are the main achievements of this paper. The author of this paper in [31] and [32] obtained the existence of almost-periodic solutions with almost-periodic forcing using similar techniques. However, Eq. (1.1) is a higher-dimensional equation, and therefore the analysis of Birkhoff normal forms and more precise estimation of new perturbation is very difficult because of the effects of infinite-dimensional frequency, To state the main results of our paper, we need the following assumptions and sets. To dispose the infinite-dimensional frequency, we construct a sequence {b ν } ν≥0 satisfying b 0 = 2b ≥ 2, b ν+1 > b ν , and b ν ∈ Z + . We choose the index set The frequencies can be split up as For fixed ∈ (0, 1), by [14] the frequency ω satisfies the following nonresonance conditions where 0 <α < 1 is arbitrary and fixed, |k| = ∞ i=1 |k i |, O ν is a closed set in R b ν , and O ν is a closed set. The frequencies ω b ν will be chosen properly by the KAM iteration. We also need the following notation: To apply the KAM theory, we introduce the following assumptions: (H1) The function ψ(ωt) is real analytic and almost-periodic with ω ∈ O.
(H2) The function ψ(θ ) has a special series expansion of the form which is absolutely convergent. There exists an absolute constant C such that By assumption (H1) we can expand ψ b j (θ b j 1 ) (j = 0, 1, . . .) into the converging Fourier-Taylor series where λ * = (· · · , λ n∞ , . . .) n∈Z d and Remark 1.2 Assumption (H2) is crucial to have a successful KAM iteration. We will split the Hamiltonian and add some proper parts of perturbations to increase the number of frequencies in the next KAM step. Moreover, for the reducibility, we need to ensure that the new perturbation in the next KAM step is smaller than the previous one. Thus ψ(θ ) needs a special series expansion, and the form of the series is decided by KAM iteration. Remark 1.3 The function ψ(θ ) in (1.1) depends only on t to conserve the partial zeromomentum property in the KAM iteration. Otherwise, in the case of higher dimension the estimate of the new perturbations becomes doubtful, and the terms of new normal form cannot be handled. It is harder than one-dimensional equations in [31] and [32].
The rest of the paper is organized as follows. In Sect. 2, we discuss the Hamiltonian setting corresponding Eq. (1.1). Section 3 is devoted to the reducibility of proving the existence of almost-periodic solutions for the linear d-dimensional beam equation using an improved KAM iteration. The small divisors estimate in reducibility is given in the Appendix.

The Hamiltonian of the higher-dimensional beam equation
In this section, we analyze the Hamiltonian of the higher-dimensional beam equation, which will be transformed into the KAM iteration.
In the following, we reduce the Hamiltonian of the higher-dimensional beam equation (1.1). The Hamilton systems (1.1)-(1.2) are equivalent to the systems By a simple computation, λ n = |n| 2 + μ are the eigenvalues of the operator A = -+ μ subject to periodic boundary conditions with eigenfunctions φ n ( with the corresponding Hamiltonian function Letting θ = ωt, we introduce a pair of action-angle variables Thus the corresponding Hamiltonian function of system (2.2) may be rewritten as with the symplectic structure dJ ∧ dθ + i n∈Z d dq n ∧ dq n . By Assumption (H2) the Hamiltonian (2.4) can be split into the following form where for j = 1, . . . , (2.6) Furthermore, for j = 0, 1, . . . , R j may be rewritten in detail as follows: (2.8) . . , as follows: Remark 2.1 Property of (2.9) is important for higher-dimensional Hamiltonian systems. It ensures the form of perturbations and the obtained normal form in the KAM iteration.
There is a crucial difference from the one-dimensional case. Thus, to conserve this property at each KAM step, we require that ψ does not explicitly depend on space variable x.
Then the normal variables q n ,q m with n = m in the new normal form will not be coupled. Moreover, there are no terms of the forms n R jk(e n +e -n )0 q n q -n and n R jk0(e n +e -n )qnq-n .

Lemma 2.3
For a ≥ 0 and ρ > 0, the gradient Rq is a real analytic map from a neighborhood of the origin of l a,ρ into l a+1,ρ , with For the proof of Lemma 2.3, see [33].
Remark 2. 4 We require thatā > a, which means that the weight of vector fields is a little heavier than that of q,q. The regularity of X R ā,ρ ensures that X R sends a decaying qsequence to a faster decaying sequence.

Reducibility
In this section, we state the important Theorem 3.2 and give a detailed proof of the reducibility to obtain the Theorem 1.1. The main program of proof comes form the KAM iteration, which involves an infinite sequence of change of variables. Thus, at each step of KAM iteration the estimates of the coordinate transformation and the Lebesgue measure of a small devisor are necessary (see Sects. 3.2-3.4). Because of our infinite frequencies in beam equations, we need to improve the program of the KAM iteration (see Sect. 3.5). Since at each KAM step the perturbation must become more smaller than at the previous KAM step, we estimate the new perturbation (see Sect. 3.6). For a high-dimensional beam equation, we need to verify the partial zero-momentum property at each KAM step (see Sect. 3.7). The normal form is obtained by the infinite transforms. Thus the convergence of the infinite transforms needs to be considered (see Sect. 3.8).
We assume that the given analytic function has the following form: with the weighted norm For the Hamiltonian vector field X F = (F J , -F θ , {iF q n }, {-iFq n }) associated with a function F on D(σ , r) × O, its weighted norm is defined as Lemma 3.1 For ε * > 0 sufficiently small and r = ε * , if |J| < r 2 and q a,ρ < r, then for ε = ε(ε * ), we have For the proof, see [24]. Now we state our theorem. and (the smallness condition) X R D(σ ,r),O < ε,ā = a + 1.

Moreover, we have:
(i) For each ω ∈ O * , there exists a real analytic linearly symplectic coordinate transformation The symplectic coordinate transformation Σ ∞ is close to the identity: where C > 0 is an absolute constant. (ii) The symplectic coordinate transformation Σ ∞ transforms the Hamiltonian (2.5) into

5)
where Remark 3.3 The forced term ψ(ωt) is almost periodic with an infinite-dimensional frequency ω = (ω 1 , . . . ). A significantly difficult problem is estimating the small divisors at each KAM step because of treating infinite frequencies at the same time. To overcome this difficulty, we split the infinite frequencies to the sum of some finite frequencies, which means that at each KAM step, we only treat some finite frequencies.
Remark 3.4 By the chosen finitely many frequencies at each KAM step the Hamiltonian H in (2.5) needs to be expanded into the proper series of H = H 1 0 + H 1 + · · · + H j + · · · . We will transform them in a proper order at the KAM iteration. Thus, in the reducibility, we construct the proper Hamiltonian iteration sequences {H 1 l } ∞ l=0 and {H 3 jl } ∞ l=0 , j = l + 1, . . . .
Remark 3.5 Assumption (H2) is crucial. The KAM iteration is successful because of the new perturbation reducing speedily after each KAM step. The new added perturbation ε ν+1 P 3 (ν+1)(ν+1) defined in (3.34) at the next KAM step should be smaller than the previous one. The coefficients ε b j of ψ(θ ) = ∞ j=0 ε b j ψ b j (θ b j 1 ) will be decided by the estimations of the small divisor measure and the new perturbation in reducibility.

Solve the homological equations
At each step of the KAM iteration, we will meet the small divisors in finding the coordinate transforms. Now we first estimate the measure of the small divisor about the frequency ω ∈ O, which will be proved in Appendix.
we have the following inequalities: where λ nl and λ ml are defined in (3.4) l .

Moreover, letting
we get where C is a constant depending on μ and .
We look for a change of variables S ν defined in a domain D ν+1 by the time-one map X 1 F ν of the Hamiltonian vector field X F ν . Let X t F ν be the time-t map of the flow of the Hamiltonian vector field X F ν given by the Hamiltonian (3.9) and for j ≥ ν + 1, Hamiltonian (3.5) ν is transformed into Now the unknown function F ν needs to satisfy the following equation: (3.11) which is equivalent to Thus (3.10) can be rewritten as (3.14)

Estimation on the coordinate transformation
We proceed to estimate X F ν and φ 1 F ν .

Lemma 3.8 Let
Moreover, Note that F ν is a polynomial of degree 2 in q,q. By (3.15), the weighted norm, and the Cauchy inequality we get that for any m ≥ 2, We consider the integral equation , which directly follows from (3.13). Since where J = ( 0 -I I 0 , from the Gronwall inequality we get that Consequently, Lemma 3.8 follows.

Rewrite the new normal form and new perturbation
We expand the Hamiltonian H to deal with infinite frequencies ω. Thus we need to rewrite the new Hamiltonian to increase some new finite frequencies in the next iteration. The Due to the special form of P 1 ν in (3.2) ν , the terms in nm We consider the following form of P 2 ν+1 : Recalling Moreover, we rewrite (3.29) in the form Let λ n,ν+1 (ε) = λ n,ν (ε) + ε νλn,ν (ε).