Sharp time decay estimates for the discrete Klein–Gordon equation

We establish sharp time decay estimates for the Klein–Gordon equation on the cubic lattice in dimensions d = 2, 3, 4. The ℓ 1 → ℓ ∞ dispersive decay rate is |t|−3/4 for d = 2, |t|−7/6 for d = 3 and |t|−3/2 log|t| for d = 4. These decay rates are faster than conjectured by Kevrekidis and Stefanov (2005). The proof relies on oscillatory integral estimates and proceeds by a detailed analysis of the singularities of the associated phase function. We also prove new Strichartz estimates and discuss applications to nonlinear PDEs and spectral theory.


Introduction and main results
Dispersive estimates play a crucial role in the study of evolution equations. Proving such estimates often boils down to establishing decay estimates for the ∞ norm of the solution at time t in terms of the 1 norm of its initial data. It is by now well-established that the 1 → ∞ decay estimates give rise to a whole family of mixed space-time norm estimates, called Strichartz estimates [12,17,28]. For the continuous Klein-Gordon equation such estimates have been established e.g. by Brenner [6], Pecher [25] and Ginibre and Velo [10,11]; let us also mention the textbook exposition by Nakanishi-Schlag [23]. The dispersive estimates are halfway between those for the Schrödinger equation (for low frequencies) and the wave equation (for high frequencies), see [23, 2.5]. In the discrete case the frequencies are bounded, and one might expect the same decay rate for the discrete Klein-Gordon equation (DKG) as for the discrete Schrödinger equation (DS). In fact, this was conjectured by Kevrekidis and Stefanov [18], who proved the sharp |t| −d/3 decay rate for the DS in any dimension d 1 and for the DKG in one dimension. The obstruction to proving similar estimates for DKG in higher dimensions was that, contrary to the DS, the fundamental solution does not separate variables. Apparently unbeknownst to the authors, earlier, Schultz [26] had already made the striking observation that the decay rate for the discrete wave equation (to which the DKG reduces in the zero mass limit) in dimensions d = 2, 3 is |t| −3/4 and |t| −7/6 , respectively, better than the conjectured estimates. The same decay rate for the d = 2 DKG was established by Borovyk and Goldberg [4], who also proved that the fundamental solution decays exponentially outside the light cone and that the decay rate is independent of additional parameters (mass and wave speeds in the coordinate directions).

Decay estimates
The solution u(t, x) is thus a sum of oscillatory integrals of the form where a : [0, 2π] d → C is a smooth function; in fact, a(ξ) = 1 or a(ξ) = ω(ξ) −1 . We will be interested in obtaining time decay estimates on I(t, x), uniformly in x ∈ Z d . In other words, we want to find (the largest possible) σ > 0 such that for some constant C independent of t. The following theorem is our main result.

Theorem 1.
For the oscillatory integrals (3) the following estimates hold for all t ∈ R: where C is a constant independent of t.

Remark 1.
(a) The exponent in the estimates (5), (6) and (7)  log t |I(t, tv)| = c 4 for d = 4, see [16] for the case d = 4 and [15] for similar observations. (b) We conjecture that for d 5, the following estimate is true, We remark here that the original conjecture of Kevrekidis and Stefanov [18] can be proved by relatively simple stationary phase arguments (see remark 5). However, we expect that the better estimates (8) hold. (c) The estimates (5)- (d) It will be necessary to prove theorem 1 in each case d = 2, 3, 4 separately (see propositions 9, 10 and 12). This is in contrast to the Schrödinger case in [18], where the same proof works in all dimensions.
where v ∈ R d is the velocity, then the oscillatory integral (3) takes the form In fact, the proof of theorem 1 gives more precise information on the set of velocities for which the indicated decay occurs. These velocities are the images, under the map ξ → ∇ω(ξ), of the critical points of the phase function Φ(·, v). The time-decay of J(t, v) is governed by the degeneracy of the phase function at the critical points lying within the support of a. For a fixed value of the parameter v = v 0 a critical point of Φ(·) = Φ(v 0 , ·) is a point ξ = ξ 0 where the gradient ∇Φ(ξ) vanishes; in the case of the phase function in (9) this happens if ∇ω(ξ 0 ) = v 0 . If there are no critical points, then J(t, v 0 ) decays faster than any polynomial in t. Since the Klein-Gordon equation has finite speed of propagation, |∇ω(ξ)| 1, there are no critical points if |v 0 | > 1; this follows from the alternative formula The region |v| 1 in velocity space is called the light cone. Hence, the solution u(t, x) decays rapidly outside the light cone. In fact, since ω(·) is analytic, the decay is exponential; this can be established by the method of steepest descent, like in [4,26], but we will not pursue the issue here. Inside the light cone, there are critical points. Generically (i.e. for most values of v 0 ) these critical points are non-degenerate, that is det Hess φ(ξ 0 ) = 0. We note in passing that the Hessian is invariantly defined (i.e. invariant under changes of coordinates) at a critical point. The stationary phase method (see e.g. [27]) yields a |t| −d/2 decay at such points. However, unlike in the continuous case, there are caustics, i.e. regions inside the light cone |v| < 1 where the solution decays slower, at a rate |t| −σ (possibly with an additional logarithmic loss log k |t|), where d/2 − σ > 0 is called the order of the caustic [9] or the singular index [2]. In the simplest case, Then Σ := {ξ ∈ T d : det Hess φ(ξ) = 0} is a smooth d − 1 dimensional manifold, and there exists a (non-unique) kernel vector field, i.e. a smooth non-zero vector field V along Σ such that Hess φ(ξ)(V(ξ)) = 0 for all ξ ∈ Σ. A phase function Φ satisfying the condition (10) is said to have a corank one singularity. The simplest corank one singularity occurs when, for every ξ ∈ Σ, ker Hess φ(ξ) intersects T ξ Σ transversally, or equivalently, (∇ ξ det Hess φ) · V = 0 on Σ. This singularity is called the (Whitney) 'fold'. In the classification of Arnol'd [3] the fold is called an A 2 singularity [1], and the oscillatory integral (9) reduces to an Airy integral in one variable [14,22] (after integrating out the other d − 1 variables by stationary phase). The next more complicated singularity, called the 'cusp' or A 3 singularity, occurs at points in Σ where (∇ ξ det Hess φ) · V vanishes to first order. A systematization of these ideas gives rise to the Thom-Boardman classes. We refer the interested reader to [1,7,13] for an introduction to singularity theory. The situation becomes more complicated if we allow the parameter v to vary. Then we have to consider a family of functions as opposed to a single function. This introduces (topological) notions of 'typicality' (or transversality) and 'stability' in the space of functions depending on a given number of parameters. In our case, the number of parameters equals the dimension d of the underlying space Z d . Since we are considering d 4 here, the only stable singularities are Thom's seven elementary catastrophes [7, 15.1]; in the terminology of Arnol'd these are the A 2 , A 3 , A 4 , A 5 , D − 4 , D + 4 , D 5 singularities. For the specific phase in (9), we will show in section 2 that all these, except D 5 , appear in d 4 dimensions. However, an additional (unstable) singularity appears in d = 4. More precisely, in d = 2, there are only A k (k 3) singularities. In d = 3, the phase function has a more degenerate D − 4 type singularity. In d = 4, the phase function has only critical points with finite multiplicity, and in the most degenerate case is similar to a hyperbolic singularity (T 4,4,4 in the classification of Arnol'd [1, 15.1]). The critical point of a hyperbolic singularity is (complex) isolated; however, our uniform estimates hold true for more general phase functions having non-isolated critical points. From the knowledge of the type of singularity, we can determine the singular index. For the A k , D k singularities (and hence in d = 2, 3) this follows from the work of Duistermaat [9]. For the unstable singularity in d = 4 we use a result of Karpushkin [16]. For ease of reference, where r is the corank of φ (or the number of active variables) at the critical point. Table 1 lists the normal forms of f , i.e. after a change of coordinates, f reduces to one of the tabulated normal forms in the cases encountered here (at least in d 3; the T 4,4,4 singularity is tabulated for comparison only).

Strichartz estimates
As a consequence of theorem 1 we obtain Strichartz estimates. The proof follows from the (by now) standard argument of Keel-Tao [17]. More precisely, we apply [17, theorem 1.2] to the operator U(t) = e −it √ 1−Δ x . For fixed t, U(t) is a unitary operator on the Hilbert space H = 2 x . Here and henceforth, r x = r (Z d ) denote the spatial Lebesgue spaces. The mixed space-time Lebesgue spaces L q t r x are endowed with the norms If equality holds in the last condition, then (q, r) is said to be sharp σ-admissible. Taking σ as the decay rate in (4)- (7), the combination of Duhamel's formula (2) and similarly for q, r. Then u satisfies the estimate Remark 2.
(a) We call (q, r) a Strichartz pair if it satisfies (11). For any such pair there exists r 0 ∈ [2, r] such that (q, r 0 ) satisfies (11) with equality (in d = 4 we subtract a fixed, arbitrarily small > 0 from the right-hand side). Thus there is a τ ∈ [0, 1] such that can be made arbitrarily small by choosing sufficiently small). Note that the Lebesgue spaces This fact, together with the Gagliardo-Nirenberg and Young's inequality (compare remark 5 in [19]) yields Hence, all the Strichartz estimates in theorem 2 can be subsumed in the inequality where the infimum is taken over all Strichartz pairs (q, r). (b) The Strichartz estimates are also commonly expressed in terms of mapping properties for the operator Note that (1 − Δ x ) −1/2 is a bounded operator on r x for every r ∈ [1, ∞], see [18, lemma 1]; hence (12) follows from (14) and (2).
(c) By [24, lemma 3.9] the following sharp σ-admissible estimates are best possible, in the sense that Strichartz estimates cannot hold for a pair (q, ∞) withq < q.

Discrete nonlinear Klein-Gordon equation
Strichartz estimates can be used in conjunction with a contraction mapping argument to prove global well-posedness for certain nonlinear equations with small initial data. Here we consider the discrete nonlinear Klein-Gordon equation Given the Strichartz estimates, the proof of the following theorem is standard (see e.g. [18, theorem 6]), but we will give proofs of the PDE applications in section 3.

Theorem 3 (Global well-posedness for small data).
Assume that s satisfies (16). There exists > 0 and a constant C so that, whenever u(0) 2 x + u t (0) 2 x , then (15) has a unique global solution. Moreover, the solution satisfies u L q t r x C for any Strichartz pair (q, r).
Interpolating between the bounds of theorem 1 and energy conservation for the linear DKG (1) (without forcing term, i.e. F = 0) we get the following decay estimates for the p -norm of the solution. For 2 p ∞, where σ = σ d is the decay rate for p = ∞, i.e.
where 3/2-means 3/2 − for arbitrary fixed > 0. Before stating the next theorem we define The next theorem is related to a conjecture by Weinstein [30].

Remark 3.
Theorem 4 implies that no standing wave solutions u(t, x) = e iλt φ(x) are possible under the stated smallness assumption. Weinstein [30] proved the existence of an excitation threshold for the nonlinear Schrödinger equation in the continuum and conjectured that, for s 2/d, solutions with sufficiently small initial conditions satisfy lim t→∞ u(t) p On the lattice, Kevrekidis and Stefanov [18] proved that, for s > d/2, suitably small solutions of the nonlinear Schrödinger equation actually decay like the free solution in p x . They also obtained analogues for the nonlinear Klein-Gordon equation in one space dimension. Theorem 4 establishes analogues of this result in dimensions d = 2, 3, 4.

Resolvent estimates and spectral consequences
Here we consider a stationary version of the DKG equation, namely We start with a resolvent estimate for the unperturbed operator √ 1 − Δ. The idea of using Strichartz estimates to prove resolvent estimates is not new. It has appeared e.g. in [5,8,18,20,21]; the authors of [5] attribute the argument to Duyckaerts. The following results are analogues to [29, proposition 3.3] and [18, theorem 4] for the stationary Schrödinger equation. Again, in the Klein-Gordon case, better estimates are possible; in particular, our resolvent estimates hold in d = 3, whereas the estimates in [18,29] require d 4.
Theorem 5. Let d = 3, 4. There exists a constant C such that for all λ ∈ C we have the estimate where σ = σ d is given by (18).
There exists > 0 such that, whenever V σ , then the eigenvalue problem (21) has no nontrivial solution.

Organization of the paper
Section 2 contains the main body of the paper. After some preliminary reductions, we perform a case-by-case study of the singularity structure of the phase function in dimensions d = 2, 3, 4. This yields the proof of theorem 1. Section 3 contains proofs of the PDE applications. Numerical solutions to a system of equations that appears in the proof of the decay estimates are listed in an appendix.

Singularities of the phase function
In this section we consider the oscillatory integrals (9). To conform with standard notation we use (in this section only) the parameters (λ, s) instead of (t, v) and the integration variable x instead of ξ. That is, we consider Here and in the following sx means s · x. Suppose s = s 0 is a fixed vector and Φ(x, s 0 ) has a critical point x 0 (perhaps non-unique, but we consider one of them).
Note that ω(x) 1 for all x ∈ [0, 2π]. From now on we use the abbreviations c j := cos(x 0 j ), s j := sin(x 0 j ) for j = 1, . . . , d. Consider the functions Obviously, φ 1 (x 0 ) = 0 and ∇φ 1 (x 0 ) = 0. Moreover, φ 2 (x 0 ) = 0; otherwise, we would have 0 = φ 1 (x 0 ) + φ 2 (x 0 ) = 2ω(x 0 ) 2. Instead of Φ we will consider the new function and investigate the type of singularities of the critical point x 0 . By the properties of φ 1 and φ 2 just mentioned, the singularity type (for the so-called weighted homogeneous cases) of the functions Φ and φ at x 0 is the same (and the critical value is zero for the latter). But, as we will see, singularities of φ are easier to study since we can avoid radicals. The critical point equation yields s 0 = ∇ω(x 0 ). In order to avoid possible confusion between s j and the components of s 0 we will not use the latter notation any more. We introduce the new variables by η = x − x 0 and, by using (22) and (23), we have For the Hessian matrix we have where δ kj is the Kronecker 'delta', i.e. δ kj = 1 whenever k = j and otherwise δ kj = 0. Moreover, for the determinant of Hessian matrix the following relation holds true, whenever c j = 0, j = 1, . . . , d. Note that det Hess φ(0) is a rational function of for any c j , not only for c j = 0. The following observation about (24) will be useful later: Hess φ(0) is a rank one perturbation of a diagonal matrix. By rank subadditivity we then have the following result. In the following, we will consider each of the cases d = 2, 3, 4 separately. In view of (7) we distinguish each case into d + 1 sub-cases according to how many of the c j are zero.
We will need the following notion of homogeneity.
We say that a function φ : R d → R is homogeneous of degree r 0 with respect to the weight κ if for any λ > 0, For a given weight κ and degree r we will indicate terms of degree > r by . . .. If κ and r are clear from the context, we will not comment this further. For example, in (26) the meaning of '. . .' is the standard one (κ j = 1, r = 5). We will usually normalize κ such that r = 1 (e.g. κ j = 1/5, r = 1 in the previous example).
If φ is analytic (as will be the case here), then it has an expansion in κ-homogeneous polynomials of increasing degrees. The polynomial with the lowest degree will be called the 'principal part', φ pr .

Two dimensions
If d = 2, then (24) can be written as We will show that the function φ has an A 3 type singularity at the point η = 0. Indeed, by using the change of variables y 1 = s 1 η 1 + s 2 η 2 , y 2 = s 2 η 2 we can see that where '. . .' is a sum of homogeneous polynomials of degree strictly bigger than r = 1 with respect to the weight κ = (1/2, 1/4). Hence, the principal part of the function has the form φ pr (y) = − 1 5 Changing variables again, u 1 = (5/2) 1/2 y 2 , u 2 = 1 5 y 1 + 5 2 y 2 2 , we can write this as From table 1 we infer (dropping the quadratic part, as we may) that this is an A 3 type singularity. Case 2: assume that exactly one of the c j is zero. Without loss of generality we assume that c 1 = 0 and c 2 = 0. Then since s 1 = ±1, it is easy to see that det Hess φ(0) = 0. Hence the function φ has a non-degenerate critical point at the origin, i.e. an A 1 type singularity.
Case 3b: the case c 2 − s 2 2 5−2(c 1 +c 2 ) = 0 (and det Hess φ(0) = 0) is analogous. Case 3c: lastly, assume that c j δ jk − s j s k 5−2(c 1 +c 2 ) = 0 for j, k = 1, 2 (this covers all remaining cases). Then under the condition det Hess φ(0) = 0 (or equivalently (27)) the function φ can be written as We claim that φ has only an A k type singularity with k 2. Indeed, a straightforward calculation shows that φ has A k type singularities with k 3 if and only if and Since s 2 = 0, and s 1 = 0 the (28) under the condition (29) can be written as We claim that the system of equations (29) and (30) has no solution satisfying |c j | < 1, j = 1, 2. Indeed, if the pair (c 1 , c 2 ) is a solution to that system, then c 1 and c 2 have opposite signs. Without loss of generality, assume 1 c 1 > 0 and −1 c 2 < 0.
Proof. Assume that (29) holds. The left-hand side of (30) can be written as Since the signatures of c 1 and c 2 are different and since 0 < |c j | < 1, we have Moreover, since c 1 + c 2 = 0 by (29), it follows that (30) cannot hold.
In summary, we have proved the following.

Remark 4.
The following comment appears in [18] (page no. 1853): numerical simulations seem to confirm the validity of (17) since in a two-dimensional numerical experiment for a DKG lattice, the best fit to the decay was found to be O(t −0.675 ). Note that (17) in [18] is their conjectured bound |J(λ, s)| C(1 + |λ|) − d 3 (for d = 2). proposition 9 actually shows that the decay rate is better than the numerical bound. We speculate that this is related to the 'constant problem'. As mentioned in the introduction, the bound of proposition 9 was already proved by Borovyk-Goldberg [4], without reference to the conjecture of Kevrekidis and Stefanov [18].

Three dimensions
.
Case 4: finally, assume c 1 = 0, c 2 = 0, c 3 = 0 and det Hess φ(0) = 0. This will be the longest and most delicate case. We claim that the function has an A k type singularity with k 3. Indeed, due to lemma 7, the rank of the matrix Hess φ(0) is at least 2. Since we are assuming det Hess φ(0) = 0, the rank of that Hessian matrix is exactly 2.
Case 4a: suppose that one of the s j vanishes. Without loss of generality, we will assume that s 1 = 0. Then c 1 = ±1. Since c 2 = 0, c 3 = 0, arguing as in case 3a in d = 2, we see that φ has an A 2 type singularity (we exclude the A 1 case since we assume det Hess φ(0) = 0). i.e. φ would have an A 1 type (or non-degenerate) critical point at η = 0. Without loss of generality, we will assume that c 1 − s 2 1 7−2(c 1 +c 2 +c 3 ) = 0. The quadratic part of φ is given by For the kernel ∇p 2 (η) = 0 we have the relation Moreover, using the identity we find that Since c 1 − s 2 1 7−2(c 1 +c 2 +c 3 ) = 0 the function φ can be written as where r 1 , r 2 , b 2 , b 3 are nonzero real numbers satisfying the conditions: Case 4b(i): assume that We claim that φ has an A 2 type singularity at η = 0. Indeed, the first condition in (31) is equivalent to det Hess φ(0) = 0. Under the second condition we can use change of variables where ' . . . ' means sum of homogeneous polynomials of degree > 1 with respect to the weight (1/2, 1/2, 1/3). Hence, φ has an A 2 type singularity at η = 0. Case 4b(ii): assume and recall that the first condition simply means det Hess φ(0) = 0. We then have the following relation, Changing variables we arrive at Under the conditions (32) we can write or equivalently, A straightforward but tedious computation yields where ' . . . ' comprises a sum of terms with degree > 1 with respect to the weight (1/2, 1/2, 1/4). Under the condition there is no term cy 3 3 with a non-zero coefficient c. Hence all other terms which are not indicated have degree > 1 with respect to the weight (1/2, 1/2, 1/4). We claim that if the following condition (see (34)) is satisfied, then the phase function has A 3 type singularity at η = 0. Indeed, under the conditions (32), we have In summary, we have the following result for d = 3.
Case 3: suppose that two of the c j are zero; without loss of generality, c 1 = 0 = c 2 and c 3 = 0, c 4 = 0. Since |s 1 | = |s 2 | = 1, the matrix (S k j ) 2 k j=1 (see lemma 7 for the notation) has rank 1. Hence the rank of the matrix Hess φ(0) equals 3. Therefore, the corresponding integral decays as O(|λ| −3/2 ). This is sufficient for our result. A solution to the discrete nonlinear Klein-Gordon equation (15) is a fix point of Λ. To apply the contraction mapping argument, we first check that ΛX ⊂ X. Indeed, applying the Strichartz estimate (13) with (q, r) = (1, 2), we have, for u ∈ X, In the second inequality we used that the pair (2s + 1, 2(2s + 1)) is a Strichartz pair and is thus controlled by the X-norm; in the last inequality we used the assumption on the initial data. For sufficiently small, the last expression is bounded by 2C 1,2 ( f , g) 2 × 2 , and hence Λu ∈ X. Similarly, using one verifies that Λ : X → X is a contraction.
Proof of theorem 4. Here we are use the contraction mapping argument in the metric space where t := (1 + |t|), 2 p p d , σ = σ d , and C p denotes the constant in the p decay estimate for the linear equation (17). By the latter, we have, for u ∈ X, It suffices to show that the second term is bounded by the first. The assumptions on p, s and (19) imply that (2s + 1)p p, σ(p − 2)(2s + 1)/p > 1, from which it follows that the second term is bounded by (using that u ∈ X) C p (2C p (u(0), u t (0)) p × p ) 2s+1 ∞ 0 t − τ −σ(p−2)/p τ −σ(p−2)(2s+1)/p dτ.
By the second inequality in (36), the integral is convergent and bounded by a constant times t −σ(p−2)/p . Since (u(0), u t (0)) 2s p × p 2s , we may choose so small that Λu ∈ X. The contractvity of Λ again follows in a similar manner. Banach's fix point theorem then yields the existence of a unique solution, together with the a priori bound (20).
Proof of theorem 5. We first remark that, by standard arguments, the Strichartz estimates (14) can be localized in time to an interval [0, T]. We then apply the localized estimates with (q, r) = (q, r) = (2, 2σ σ−1 ) to the function u(t) := e itλ ψ, which satisfies the equation where H 0 := √ 1 − Δ. Here we assume without loss of generality that λ is in the lower half plane; otherwise we consider e −itλ ψ. The result is that where c(z, T) := e itz L 2 t (0,T) T 1/2 by the assumption that z is in the lower half plane. Dividing by c(z, T) and letting T → ∞ yields the claimed bound.