A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential

We study the one-dimensional nonlinear Klein-Gordon (NLKG) equation with a convolution potential, and we prove that solutions with small analytic norm remain small for exponentially long times. The result is uniform with respect to $c \geq 1$, which however has to belong to a set of large measure.


Introduction
In this paper we study the real one-dimensional nonlinear Klein-Gordon (NLKG) equation with a convolution potential on the segment, with c ∈ [1, +∞), x ∈ I := [0, π], f : R → R an analytic function with a zero of order 3 at the origin, in the case of Dirichlet boundary conditions. In this paper we show that the technique developed in [FG13] allows us to deduce that solutions with small initial data that are analytic in a strip of width ρ > 0 remain analytic in a strip of width ρ/4 for a timescale which is exponentially long with respect to the size of the initial datum; however, we have to assume that both the parameter c and the coefficients of the potential belong to a set of large measure.
In [Pas17] we proved an almost global existence result uniform with respect to c ≥ 1 for the NLKG with a convolution potential. Furthermore, we deduced that for any δ > 0 any solution in H s with initial datum of size O(c −δ ) remains of size O(c −δ ) up to times of order O(c δ(r+1/2) ) for any r ≥ 1. Here we use normal form techniques in order to establish a result valid for exponentially long times, but we have to use analytic norms instead of Sobolev ones.
The issue of long-time existence for small solutions of the NLKG equation on compact manifolds has been quite studied; see for example [DS04], [BDGS07], [Del09], [FZ10], [FHZ17] and [DI17]. However, all results in the aforementioned papers rely on a nonresonance condition which is not uniform with respect to c. We also point out that the almost global existence for small solutions has been established only for the segment [0, π] and in the case of Zoll manifolds, such as the multidimensional spheres S d , d ≥ 1.
The proof combines the argument of [FG13] for the NLS equation on the torus with a diophantine type estimate for the linear frequencies which holds uniformly for c ≥ 1. We mention that a diophantine estimate for the frequencies uniform with respect to c has been already used in [Pas17] in order to prove the almost global existence.
A further aspect that would deserve future work is the study of Nekhoroshev estimates for the NLKG without potential. This is expected to be a quite subtle problem, since for c = 0 the frequencies of NLKG are typically non resonant, while the limiting frequencies are resonant.
The paper is organized as follows. In sect. 2 we state the results of the paper, together with some examples and comments. In sect. 3 we introduce the notations and the spaces which we use for our result. In sect. 4 we define a special class of polynomials. In sect. 5 we show that the nonlinearity appearing in the NLKG equation belongs to that class. In sect. 6 we study the resonances of the system. In sect. 7 we introduce the notion of normal form, and in the last section we prove the main theorem.

Statement of the Main Results
In (1) we assume that the potential has the form V (x) = k≥1 v k cos(kx).
(2) By using the same approach of [FG13], we fix a positive s, and for any M > 0 we consider the probability space and we endow the product probability measure on the space of (c, (v ′ k ) k ).

It is well known that (1) is Hamiltonian with Hamiltonian
where v := u t /c 2 is the momentum conjugated to u, and F (u) is such that ∂ u F = f . Consider now the Sturm Liouville problem with Dirichlet boundary conditions on I: it is well known that all the eigenvalues are distinct, that the solutions (φ k ) k≥1 of (5) given by φ k (x) = π −1/2 sin(kx) form an orthonormal basis of L 2 (I). It is useful to expand both u and v with respect to (φ k ) k≥1 , and to introduce the following change of variables where λ k = k 2 + v k . Indeed, in the coordinates (ξ, η) := ((ξ k ) k≥1 , (η k ) k≥1 ) the Hamiltonian (4) takes the form where where the linear frequencies (ω k ) k≥1 of (9) are given by Equation (1) is a semilinear PDE locally well-posed in the energy space H 1 (I) × L 2 (I) (see [NS11], ch. 2.1). Consider a local solution (u(t), c 2 v(t)) of (1): a standard computations shows that (u, c 2 v) ∈ H 1 (I) × L 2 (I) if and only if (ξ, η) ∈ l 1/2 2 (I) × l 1/2 2 (I) solves the following system Moreover, we will denote by ψ andψ the functions given by the following expansions with respect to the eigenfunctions (φ k ) k≥1 , It is easy to check that (u, c 2 v) ∈ H 1 × L 2 solve equation (1) if and only if (ψ,ψ) ∈ H 1/2 × H 1/2 solve the following equation, where V is the operator that maps ψ to V * ψ.
Now, for ρ > 0 we denote by A ρ := A ρ (I, C × C) the space of functions that are analytic on the complex neighborhood of I given by and continuous on the closure of this strip. The space A ρ , endowed with the following norm, Our main result is the following theorem: Theorem 2.1. Consider the equation (1). For any positive β < 1 and for any ρ > 0 the following holds: there exist γ > 0, τ > 0, and a set R γ : then the solution of (1) with initial datum (ψ 0 ,ψ 0 ) exists for times |t| ≤ e −σρ| log R| β+1 , σ ρ = min(1/8, ρ/4), and satisfies Furthermore, we have that Remark 2.2. In finite dimension n, the standard Nekhoroshev theorem controls the dynamics over timescales of order exp(−αR −1/(τ +1) ) for some α > 0 and for some τ > n − 1 (see [Nek77] and [BGG85]; see also [Loc92] and [Pös93] for a more direct proof in the convex and quasi-convex case respectively).
In the infinite-dimensional context there are only few results, mainly due to Bambusi and Pöschel in the one-dimensional case (see [BN98], [Bam99a], [Bam99b], [Pös99] and [BN02]), and by Faou and Grébert in the multidimensional case (see [FG13]). In particular, in [Bam99b] Bambusi proved a Nekhoroshev result for the one-dimensional NLKG: he was able to control the dynamics of analytic solutions in a strip on a timescale of order O(e −α| log R| β+1 ) for some α > 0 and β < 1 (which is the same timescale we cover in Theorem 2.1), but assuming that the parameter that appears in the equation ranges over a compact interval.
Remark 2.3. Actually, our result is slightly different from the one obtained by Faou ang Grébert in [FG13]: indeed, while they proved that there exists a full measure set of potentials for which each solution of the NLS equation corresponding to an initial datum with small analytic norm remains small for exponentially long times, in our result we prove that for "most" of the values of speeds of light and potentials each solution of (1) corresponding to an initial datum with small analytic norm remains small for exponentially long times.
Such a difference is motivated by the non resonance condition we prove below (see Theorem 6.1), which is an adaptation of the uniform diophantine estimates reported in [Pas17].
By exploiting the same argument used to prove Theorem 2.1 one can immediately deduce the following stability result for solutions with small (with respect to c) initial data.
We say that z ∈ L ρ is real if zj =z j for any j ∈ Z.
Lemma 3.1. Let ψ,ψ be complex valued functions analytic on a neighborhood of I, and let (z j ) j∈Z be the sequence defined by (15) and (16). Then for all µ < ρ we have where K ρ,µ is a positive constant depending only on ρ and µ.
Hence for µ < ρ we have Conversely, assume that z ∈ L ρ : then |ξ k | ≤ e −ρk (ψ,ψ) µ , and thus by (15) we get that for all x ∈ I and y ∈ R with |y| ≤ µ hence ψ andψ are bounded on the strip I µ .
Consider a function G ∈ C 1 (L ρ , C) we define its Hamiltonian vector field by X G := J∇G, where J is the symplectic operator on L ρ induced by the symplectic form (22), and ∇G(z) : Definition 3.2. For a given ρ > 0 we denote by H ρ the space of real Hamiltonians G satisfying Let G 1 , G 2 ∈ H ρ , then we define the Poisson bracket between G 1 and G 2 via the formula We say that the Hamiltonian H is real if H(z) is real for all real z. We associate to a given Hamiltonian H ∈ H ρ the corresponding Hamilton equations, We also denote by Φ t H (z) the time-t flow associated with the previous system. We just remark that if z = (ξ,ξ) and if H is real, then also the flow Φ t H (z) = (ξ(t), η(t)) is real for all t, namely ξ(t) =η(t) for all t. Moreover, the Hamiltonian with N given by (11), leads to the system (14), that is the NLKG equation (1) written in the coordinates (ξ, η).
Remark 3.3. We point out that the Hamiltonian H 0 = k≥1 ω k ξ k η k , which corresponds to the linear part of (9), does not belong to H ρ . However, it generates a flow which maps L ρ to L ρ , and it is explicitly given by On the other hand, we will show in sec. 5 that the nonlinearity N belongs to the space H ρ .

The Space of Polynomials
In this section we define a class of polynomials in C Z . We first introduce some notations about multi-indices: let l ≥ 2 and j = (j 1 , . . . , j l ) ∈ Z l , with j i = (k i , δ i ), we define • the norm of the multi-index j, • the monomial associated with j, • the momentum of j, • the divisor associated to j, where ω k is the k-th linear frequency of the system, and is given by (12).
We then define the set of indices with zero momentum by We also say that an index j = (j 1 , . . . , j l ) ∈ Z l is resonant (we denote it by j ∈ N l ) if l is even and j = i ∪ī for some choice of i ∈ Z l/2 . In particular, if j is resonant then its associated divisor vanishes, namely Ω(j) = 0, and its associated monomial depends only on the actions.
where for all k ≥ 1 I k (z) = ξ k η k denotes the k-th action. Finally, we point out that if z is real then I k (z) = |ξ k | 2 ; we also remark that for odd l the resonant set N l is empty.
Definition 4.1. Let k ≥ 2, we say that a polynomial P (z) = a j z j belongs to P k if P is real, of degree k, which has a zero of order at least 2 in z = 0, and if • P contains only monomials having zero momentum, namely such that M(j) = 0 for a j = 0, thus P is of the form with aj =ā j ; • the coefficients a j are bounded, namely sup j∈I l |a j | < +∞, ∀l = 2, . . . , k.
We endow the space P k with the norm Remark 4.2. In the following sections we will crucially use the fact that polynomials in P k contain only monomial with zero momentum: indeed, this will allow us to control the largest index by the others.
The zero momentum condition is essential to prove the following result.
Proposition 4.3. Let k ≥ 2 and ρ > 0, then P k ⊂ H ρ . Moreover, any homogeneous polynomial P which belongs to P k satisfies Furthermore, if P ∈ P k and Q ∈ P l , then {P, Q} ∈ P k+l−2 , and Proof. Let and hence we get (29).
To prove (30), take l ∈ Z, and exploit the zero momentum condition in order to get We have therefore by summing in l we obtain which gives (30). Now let us assume that P and Q are homogeneous polynomail of degrees k and l respectively, and with coefficients a k , k ∈ I k , and b l , l ∈ I l . It is easy to check that {P, Q} is a monomial of degree k + l − 2 that satisfies the zero momentum condition. Moreover, if we write we have that c j is a sum of coefficients a k b l for which there exists h ≥ 1, and δ ∈ {±1} such that . . , k k , l 2 , . . . , l l ) = j ). Hence, for a given j the zero momentum condition on k and l determines two possible values of (h, δ). This allow us to deduce (31) for monomials; its extension to polynomial follows by the definition of the norm (28).
Remark 4.4. In Proposition 4.3 we used a l 1 -type norm to estimate Fourier coefficients and vector fields, instead of the usual l 2 norms. This choice does not allow to work in Hilbert spaces, and produces a loss of regularity each time the estimates are transported from the Fourier space to the space of analytic functions A ρ . However, by using this approach we can control vector fields in a simple way. This argument has been already used in [FG13].
It may be possible to recover the usual l 2 estimates for Fourier coefficients and vector fields by introducing a more general space of polynomials P k1,k2 (k 1 ≥ 2, 1 < k 2 ≤ ∞), endowed with the following norm In this case one shoud also introduce a more general space of analytic functions L ρ,k := {z ∈ C Z : z ρ,k := j∈Z e ρ|j| |z j | k < +∞}, 1 ≤ k < +∞.

Nonlinearity
First we recall that the nonlinearity N in (11) in the coordinates (ψ,ψ) takes the form where we assume that F is analytic in a neighbourhood of the origin in C × C. Therefore there exist positive constants M and R 0 such that the Taylor expansion of the integrand in the nonlinearity is uniformly convergent on the ball |ψ| + |ψ| ≤ 2R 0 of C × C, and is bounded by M .
Remark 5.1. The constant M in the above estimate can be chosen uniformly with respect to the speed of light c, because the smoothing pseudodifferential operator can be estimated uniformly with respect to c ≥ 1. Indeed it is easy to check that for some constant K ρ > 0 that depends only on ρ.
Thus, formula (11) defines an analytic function on the ball z ρ ≤ R 0 of L ρ , and we can write where for all k ≥ 0 N k is a homogeneous polynomial defined by Hence it is easy to check that N k satisfies the zero momentum condition, thus N k ∈ P k for all k, and N k ≤ M R −k 0 .
Remark 5.2. One could also extend this argument by considering not only zero momentum monomials, but also monomials with exponentially decreasing or power-law decreasing momentum. This would allow to deal with the NLKG equation with a multiplicative potential and with nonlinearities that depend on x, but this problem would require a more technical discussion.

Non resonance conditions
In order to prove Theorem 2.1, we need to show some non resonance properties of the frequencies ω = (ω k ) k≥1 : it will be crucial that these properties hold uniformly (or at least, up to a set of small probability) in (1, +∞)×V. The argument is similar to the one reported in [Pas17].
For r ≥ 3 and j = (j 1 , . . . , j r ) ∈ Z r (j i = (a i , δ i ), 1 ≤ i ≤ r), we define µ(j) as the third largest integer among j 1 ,. . .,j r , and we again mention that j ∈ Z r is called resonant if r is even and j = i ∪ī for some choice of i ∈ Z r/2 . Theorem 6.1. For any sufficiently small γ > 0 there exists a set R γ : and ∃τ > 0 such that ∀(c, (v j ) j ) ∈ ([1, +∞) × V) \ R γ , for any sufficiently large N ≥ 1 and for any r ≥ 1 for any non resonant j ∈ Z r+2 \ N r+2 with µ(j) ≤ N .
Remark 6.2. The non resonance condition in Theorem 6.1 is slightly weaker from the one proved in sec. 2.4 of [FG13]; this is the reason for which Faou and Grébert are able to prove a result which is valid for a set of potentials of full measure, while our result is valid only for "most" of the values of speed of light and potentials. We also mention that a non resonance condition similar to our one has been already used in [Bam03] (see (3.3)) and in [BG06] (see (r-NR)).
Proof. Fix r ≥ 1, j ∈ Z r non resonant and m ∈ Z.
Proof. If σ i = 0 for i = 1, 2, then we can conclude by using Proposition 6.5. Now fix r ≥ 1, a non resonant j ∈ Z r with |j| ≤ N , two positive integers l and m such that m > l ≥ N , and assume that σ 1 = −1, σ 2 = 1. Introduce p j,l,m (c 2 ) := Ω(j) − ω l (c 2 ) + ω m (c 2 ). Now fix δ > 3. If m N δ , then we can conclude by applying Proposition 6.4 and 6.5. So from now on we will assume that m, l > N δ .
We have to distinguish several cases: Case c < λ α l : we point out that, since c c 2 + λ l = cλ that is, the integer multiples of c are accumulation points for the differences between the frequencies as l, m → ∞, provided that α < 1 6 . Case c > λ m : in this case we have (again by denoting m = l + j) that λ m − λ l = j(j + 2l) + (v m − v l ) = 2jl + j 2 + a lm , with |a lm | ≤ C l , so that p j,l,m = Ω(j) ± 2jl ± j 2 ± a lm .
If l > 2 CN τ /γ then the term a lm represents a negligible correction and therefore we can conclude by applying Proposition 6.4. On the other hand, if l ≤ 2 CN τ /γ, we can apply the same Proposition with N ′ := 2CN τ /γ and r ′ := r + 2.
Case λ 1/6 l ≤ c λ 1/2 l : if we rewrite the quantity to estimate where α := r h=1 δ h , we distinguish three cases: • if α = 0, then we just notice that l , which is greater than γ 0 /N τ for τ > −1, since l > N 3 ; • if α < 0, then we just recall that |ω m − ω l | > γ 0 λ 1/2 l , and by choosing γ 0 sufficiently small (actually γ 0 ≤ |α|) we get that also in this case p j,l,m is bounded away from zero. Now it is easy to deduce Theorem 6.1 by exploiting Theorem 6.6.

Normal form
Fix N ≥ 1. For a fixed integer k ≥ 3, we define Definition 7.1. Let N be an integer. We say that a polynomial Z ∈ P k is in N -normal form if it is of the form namely, if Z contains either monomials that depend only of the actions or monomials whose index j satisfies µ(j) > N , i.e. it involves at least three modes with index greater than N .
The reason for introducing such a definition of normal form is given by the observation that the vector field of a monomial of the form z j1 . . . z j k containing at least three modes with index larger than N induces a flow whose dynamics can be controlled for exponentially long (N -dependent) time scales. This will prevent exchanges of energy between low-and high-index modes for such time scales.
In [Bam03] and [BG06] such monomials were neglected, since the contribution of their vector fields was to be small in Sobolev norms, and this will keep all the modes (almost) invariant. The point is that, even though the contribution of the vector fields of such monomials is not necessarily small in analytic norm, it can be controlled for exponentially long times. This key property was already used by Faou, Grébert and Paturel in [FGP10] and by Faou and Grébert in [FG13].
Before we state and prove the aforementioned property we just state an elementary lemma.
Lemma 7.2. Let f : R → R + be a continuous funciton, and let y : R → R + be a differentiable function such that Then For a given N and z ∈ L ρ , we set R N ρ (z) := |j|≥N e ρ|j| |z j |. Observe that if z ∈ L ρ+µ , then Proposition 7.3. Let N ∈ N 0 , and k ≥ 3. Let Z be a homogeneous polynomial of degree k in N -normal form. Denote by z(t) the real solution of the flow associated to the Hamiltonian H 0 + Z. Then Proof. Let a ≥ 1 be fixed, and let I a (t) = ξ a (t)η a (t) be the actions associated to the solution of the Hamiltonian H 0 + Z. Recalling that H 0 = H 0 (I), and writing j = (j 1 , . . . , and by applying Lemma 7.2 we obtain Ordering the multi-indices in such a way that a 1 and a 2 are the largest, and recalling that z(t) is real, we obtain after a summation in a > N Remark 7.4. Estimates (42)-(43) will be fundamental for proving Theorem 2.1. Indeed, take z(t) the solution of the Hamiltonian in N -normal form with corresponding initial datum z 0 , and assume that z 0 ρ = R. Hence, since R N ρ (z 0 ) = O(Re −N ρ ), estimates (42)-(43) ensure that R N ρ (z(t)) remains of order O(Re −N ρ ) and that the norm of z(t) remains of order O(R) up to times t of order O(e N ρ ).
Next we exploit the non resonance condition (33) and the definition of normal forms to estimate the solution of a homological equation.
Proposition 7.5. Assume that the non resonance condition (33) is fulfilled. Let N be fixed, and let Q be a homogeneous polynomial of degree k. Then the homological equation admits a polynomial solution (χ, Z) homogeneous of degree k such that Z is is N -normal form, and such that Proof. Assume that Q = j∈I k Q j z j , we look for Z = j∈I k Z j z j and χ = j∈I k χ j z j such that the homological equation (45) is satisfied. Then Equation (45) can be rewritten in term of polynomial coefficients where Ω(j) is defined as in (25). By setting and by exploiting (33), we can deduce (46).

Proof of the main Theorem
We now prove Theorem 2.1 by exploiting the results from the previous sections. The argument is inspired by the one in [FG13].

Recursive equation
In this subsection we want to construct a canonical transformation T such that the Hamiltonian (H 0 +N )•T is in N -normal form, up to a small remainder term. We use Lie transform in order to generate the transformation T : hence we look for polynomials χ = r k≥3 χ k and Z = r k≥3 Z k in N -normal form and a smooth Hamiltonian R such that d α R(0) = 0 for all α ∈ N Z with |α| ≥ r, and such that The exponential estimate will be obtained by optimizing the choice of r and N .
We recall that if χ and K are two Hamiltonian, we have that for all k ≥ 0 where ad χ K := {χ, K}. On the other hand, if K and L are homogeneous polynomial of degree respectively k and l, than {K, L} is a homogeneous polynomial of degree k + l − 2. Thus, by using Taylor formula where by "+O r " we mean "up to a smooth function R satisfying ∂ α R(0) = 0 for all α ∈ N Z with |α| ≥ r. Now, recall the following relation where B k are the Bernoulli numbers defined by the expansion of the generating function z e z −1 . We just recall that there exists K > 0 such that |B k | ≤ k! K k for all k.
Hence, defining the two differential operators we obtain where C r is a differential operator satisfying C r O 3 = O r . Applying B r to both sides of Eq. (48) we get Plugging the decomposition in homogeneous polynomials of χ, Z and N in the last equation and comparing the terms with the same degree, we have the following recursive equations where We point out that in (50) the condition l i ≤ m − k is a consequence of l i ≥ 3 and l 1 + . . . + l k+1 = m + 2k. Once these recursive equations are solved, we can define the remainder term . By construction we have that R is analytic on a neighbourhood of the origin in L ρ , and that R = O r . Hence, by Taylor formula Lemma 8.1. Assume that the non resonance condition (33) is satisfied. Let r and N be fixed. For m = 3, . . . , r there exists homogeneous polynomials χ m and Z m of degree m that solve (49), with Z m in N -normal form and such that where K is a positive constant that does not depend on r or N .
Estimate (19) is a consequence of Theorem 8.2 and (43): indeed, it just suffices to remark that z(t) is R 2 -close to y(t), which in turn is almost invariant, since k≥1 e ρ|k| ||y k (t)| − |y k (0)|| and arguing as in (58) we can finally deduce (19).