On the stability of standing waves of Klein-Gordon equations in a semiclassical regime

We investigate the orbital stability and instability of standing waves for two classes of Klein-Gordon equations in the semi-classical regime.


Introduction and results. The nonlinear Klein-Gordon equation
where ε, m > 0, p > 1 for N = 1, 2 and 1 < p < (N + 2)/(N − 2) for N ≥ 3, arises in many physical contexts, e.g. in particle physics. It is also a model case for the mathematical study of nonlinear partial differential equations. We are interested in the study of the nonlinear Klein Gordon equation in presence of a potential depending on the space variable. Two different choices are viable. We can simply add a potential term W (x)u to equation (1). This case has been studied, for the linear wave equation, for example, by Beals and Strauss in [5]. This approach leads us to consider the equation Otherwise, typically when dealing with quantum electrodynamics, the interaction between u and an external electromagnetic field is described substituting in (1) the usual time and space derivatives with the so called Weyl derivative, that is D t = ∂ t + iV (x), D xj = ∂ xj − iA j (x). Here V and (A j ) j are the potentials of the electric and the magnetic external fields. This approach is classical in the linear theory of electromagnetic waves, and can be extended to the nonlinear setting, as in [14, section 7.5.1]. The nonlinear problem is called in literature Klein-Gordon-Maxwell nonlinear problem and in the last ten years it gained the attention of the mathematical community. See for example [6], [13] and the references therein and [4,12,25,36,39]. We will consider the case of zero magnetic potential, that leads us to consider the equation In this paper, we shall state all the results simultaneously for equations (2) and (3) by studying the problem    ε 2 u tt + 2iεV u t − ε 2 ∆u + mu − W u − |u| p−1 u = 0, in R N , where u : R × R N → C, and V, W are real valued potential functions. Equation (4) yields to (3) for the choice W = V 2 as well as to (2) when V = 0. We are interested in standing wave solutions of problem (4). Standing waves are solutions of the form u(t, x) = e iωt/ε ϕ ω (x/ε), which solve (4) with initial data We shall study the stability of standing waves of this equation in the semiclassical regime ε → 0. To ensure existence of solutions to (5) for ε close to 0, we assume the following. The potentials V and W satisfy For the function Furthermore, we assume that Under these hypotheses, it is well-known (see e.g. [2] or [3,Section 8.2]) that when ε is close to 0 the equation (5) admits a family of positive, exponentially decaying, solutions ϕ ω ⊂ H 1 (R N ) (hiding the dependence upon ε). More precisely, there exist ξ ε ∈ R N and and ψ ω is the unique positive and radial solution of The rate of exponential decay is uniform in ε for sufficiently small ε. Indeed, let λ 0 := inf x∈R N Z(x). By assumption (8), we have λ 0 > 0. Then there exists C 0 > 0 depending only on λ 0 such that |ϕ ω (x)| ≤ C 0 e − √ λ0|x|/2 (see e.g. [8,Chapter 3]).
In what follows, we will need the following assumption on the dependence in ω of the family (ϕ ω ).
Actually, since the family ϕ ω is build upon (ψ ω ), which is C 1 in ω, the statement (10) could probably be obtained by rewriting the proofs of [2,3] by using parameter depending versions of the various results used. Since it is not our main concern in this paper we leave this issue aside and simply assume (10). A standing wave of (4) is said to be (orbitally) stable if any solution of (4) starting close to the standing wave remains close for all time, up to the invariances of the equation. More precisely, for fixed ε, we say that e iωt ε ϕ ω x ε is stable if for all η > 0 there exists δ > 0 such that for all (u 0 , the solution u(t, x) of (4) with initial data (u 0 , u 1 ) satisfies Since the pioneering works [7,10,15,16,37,38], the study of orbital stability for standing waves of dispersive PDE has attracted a lot of attention. Among many others, one can refer to [18,19,22]; see also the books and surveys [9,21,33,35] and the references therein. Relatively few works [17,23,26] are concerned with stability at the semi-classical limit for Schrödinger type equations. For Klein-Gordon equations, after the ground works [30,31] revisited some years ago in [34], there has been a recent interest for instability by blow-up [24,[27][28][29].
We study stability within the framework of Grillakis-Shatah-Strauss Theory [15,16]. We first rewrite (4) in Hamiltonian form where U = u v , J = 0 1 −1 0 , and the energy E is defined for It is easy to see that if u solves (4) and v is defined by v := εu t + iV u, then U = u v solves (12). Due to the Hamiltonian form, the energy E is (at least formally) a conserved quantity for the flow of (12). The invariance with respect to phase shift (i.e. if U solves (12), then for any fixed θ ∈ R the function e iθ U also solves (12)) generates another conserved quantity, the charge Q, which is defined by In this Hamiltonian formulation, a standing wave The energy and the charge for a standing wave are given by To work in the context of the theory developed by Grillakis, Shatah and Strauss in [15,16], three assumptions have to be satisfied. First, the Cauchy Problem has to be locally well-posed in H 1 (R N )×L 2 (R N ). This follows from standard results when V ≡ 0 and we shall assume it otherwise. Indeed, when W verifies (6), the local wellposedness of the Cauchy Problem for (2) follows from a simple adaptation of classical methods (see e.g. [32]). No well-posedness result is available for the specific case of (3), see nevertheless [1] for results on a Klein-Gordon equation with a damping term and [11,20] for the Maxwell-Klein-Gordon equation. Second, the map ω → ϕ ω has to be C 1 , which is granted by assumption (10). Third, the spectrum of the linearized operator must decompose into a finite number of negative eigenvalues, a nondegenerate kernel (i.e. containing only the eigenvectors due to the invariances of the equation), and positive spectrum away from 0. This will be proved in Proposition 2. Under these three assumptions the stability of the standing waves depends on two informations. The first one is a slope information given by the sign of the quantity ∂ 2 The second information is related to the number of negative eigenvalues of the linearized operator H ε .
According to the theory developed in [15,16], a standing wave e iωt ε ϕ ω x ε is stable if two conditions are satisfied.
(ii) The Spectral Condition: H ε has exactly one negative eigenvalue.
On the other hand, denote by n(H ε ) the number of negative eigenvalues of H ε and In [16], it was proved that when (iii) is satisfied, then the instability of the standing waves follows from a linear mechanism, in the sense that the 0 solution of the linearized equation around the standing wave is unstable. Note that when n(H ε ) − p(ω) is even, the question of stability or instability is still open. Our main result is the following.
From the theory of [15,16] we infer the following corollary on the stability of the standing wave in the particular case where x 0 is a minimum of Z. (10) hold and that x 0 is non-degenerate local minimum of Z. Then there exists ε 0 > 0 such that for any 0 < ε < ε 0 the standing waves e iωt ϕ ω are stable if p < 1 + 4/N and Note that, conversely to what was happening in the case of Schrödinger equations studied in [23], the values of the potentials V and W at x 0 come into play for the Slope Condition even in the noncritical case. Note also that only the local behavior of Z around x 0 influences the stability or instability.
Notations. Most of the objects we consider will depend both on ε and ω. We will emphasize the most important parameter by indicating it as a subscript, the dependence in the other parameter being understood.

2.
Proofs. In this section, we prove Theorem 1.1 and Corollary 1. We start by focusing on the Slope Condition and then we study the Spectral Condition. We finish by the proof of Corollary 1. For the sake of simplicity in notations and without loss of generality, in the rest of this section we assume that x 0 = 0.
2.1. The Slope Condition. We start with the noncritical case.

Critical case.
We assume now that In the critical case, the term of order ε N in the expansion of ∂ ∂ω Q(ϕ ω ) vanishes and we need to calculate the expansion at a higher order. We first refine (20)- (21).
Then (19) gives From (17), we obtain Expanding the potential V we get where the last cancellation comes from the fact that ψ ω is radial. Coming back to (26) and as in (23), we have It remains to compute the integrals involving the Hessians in (25) and (29). Since our problem is invariant by an orthonormal transformation, we can assume without loss of generality that ∇ 2 V (0) = diag(b 1 , . . . , b N ). Hence ∇ 2 V (0)(y, y) = N j=1 b j y 2 j . Recall also that T λ can be computed explicitly to have Therefore, We have after integration by parts Recall the following expansion in ε for R ω and ϕ ω .
Therefore, since ψ ω is radial, and so Similarly, we have Summarizing, using successively (25), (29), (30), (31) and (24) we have obtained Now, we compute ∂Q(ϕω) ∂ω . First, recall that, coming back to the definition (14) of Q, we have As in (27), (28), and (30) we can expand in ε and get Therefore, Using (32), we finally get 2.2. The spectral condition. We first give some preliminary considerations on the scalar operator The analysis of the spectrum of the operator L ε was essential in the case of Schrödinger equations [17,23] to determine the spectral stability condition. It turns out that it will play also an important role in the analysis for the Klein-Gordon equation.
We define the operator L 0 := −∆ + Z(0) − pψ p−1 ω (recall that ψ ω solves (9)). It is well known (see e.g. [3]) that the spectrum of L 0 consists of one negative eigenvalue, a N -dimensional kernel (generated by ∂ψω ∂yj for j = 1, . . . , N ) and the rest of the spectrum is bounded away from 0. When ε is close to 0, the spectrum of L ε will be close to the spectrum of L 0 . In particular, the 0 eigenvalue, of multiplicity N , will transform into N possibly different eigenvalues close to 0 but shifted either to the positive or to the negative side of the real axis, depending on the sign of the eigenvalues of the Hessian of Z at 0. More precisely, the following proposition was proved in [23] (see [17] for a detailed justification). Proposition 1. The spectrum of L ε consists of positive spectrum away from 0 and a set of N + 1 simple eigenvalues {λ 0 , λ 1 , . . . , λ N } such that As ε → 0, we have λ 0 < 0 and the following asymptotic expansion holds for the other eigenvalues: In the following proposition, we establish the spectral decomposition for H ε and we relate the number of negative eigenvalues of H ε with the number of negative eigenvalues of L ε . Proposition 2. The spectrum of H ε consists into positive essential spectrum away from 0, a finite number of eigenvalues and a nondegenerate kernel, i.e.

ker(H
In addition, the number n(H ε ) of negative eigenvalues of H ε is identical to the number of negative eigenvalues n(L ε ) of the operator L ε .
Therefore, (3) in Theorem 1.1 is a direct consequence of Propositions 1 and 2. In particular, the spectral condition for stability will be satisfied if and only if 0 is a non-degenerate local minimum of Z.
Proof of Proposition 2. Explicitly, H ε = E (ϕ ω ) − ωQ (ϕ ω ) is given by For notational convenience we change variables and replace x/ε by y. We denote V (εy) and W (εy) by V ε and W ε . Then H ε becomes H ε = ε NH ε , whereH ε is the Therefore, to find the spectrum of H ε it is enough to find the spectrum ofH ε . Due to exponential localization of ϕ ω , this operator is a compact perturbation of and therefore by Weyl's Theorem they share the same essential spectrum. For which can easily be factorized into where the last inequality follows from the assumption λ 0 = inf x∈R N Z(x) > 0. We claim that there exists δ > 0 such that Indeed, assume by contradiction that there exists (U n ) = u n v n ⊂ H 1 (R N ) × L 2 (R N ) such that U n 2 H 1 ×L 2 = 1 and LU n , U n → 0 as n → +∞. From (34) this implies that u n → 0 in H 1 (R N ) and v n − i(ω + V )u n L 2 → 0. Therefore U n 2 H 1 ×L 2 → 0, which is a contradiction and proves (35). From (35), we infer that the spectrum of L is contained into [δ, +∞). In particular, this implies that the essential spectrum ofH ε is contained into [δ, +∞). Let us now treat the eigenvalues ofH ε . Recall the definition of L ε and also define another operator R ε by Then a similar factorization as in (34) First we remark that due to the boundedness of ϕ ω , there exists C > 0 such that for any U ∈ H 1 (R N ) × L 2 (R N ) we have Therefore, in (−∞, δ/2), the spectrum ofH ε consists of a finite number of eigenvalues. We claim that the eigenvalues ofH ε will be distributed on one side or the other of the real axis in the same fashion as the eigenvalues of (L ε , R ε ). Indeed, the number of negative eigenvalues ofH ε is given by which, in view of (36), is exactly the same number as Since R ε ϕ ω = 0 and ϕ ω > 0, R ε has a first simple eigenvalue at 0 with eigenvector ϕ ω , and the rest of its spectrum is positive. Therefore R ε has no negative eigenvalue, and we can conclude that n(H ε ) = n(L ε ).
Coming back to the original variable x = εy implies the desired result on the kernel of H ε .
Proof of Corollary 1. The definition of stability given by the theory of [15,16] is the following. The standing wave U = e iωt/ε Φ ω (x/ε) is stable if for any η > 0 there exists δ > 0 such that for all U 0 = u 0 v 0 verifying the solution U of (12) with initial data U 0 satisfies sup t∈R inf θ∈R U (t) − e iθ Φ ω · ε H 1 ×L 2 < η.
More explicitly, we have

MARCO GHIMENTI, STEFAN LE COZ AND MARCO SQUASSINA
This definition is not exactly the same as the one we use (stated in (11)), but our stability is implied by this definition. Indeed, we have from the triangle inequality where the last inequality follows from the boundedness in L ∞ (R N ) of V . With similar arguments, it is rather easy to check that instability in the sense of [15,16] also implies instability in our sense (11). Hence, the corollary simply follows from Theorem 1.1 and a direct application of the theory developed by Grillakis, Shatah, and Strauss in [15,16].