We investigate the orbital stability and instability of standing waves for
two classes of Klein-Gordon equations in the semi-classical regime.Comment: 9 page

______

A. Ambrosetti

A. Azzollini

C. A. Stuart

C. D. Sogge

D. M. A. Stuart

E. Deumens

E. Long

F. A. Berezin

G. Vaira

H. Berestycki

I. Ianni

J. Shatah

L. Aloui

L. Jeanjean

M. Beals

M. Grillakis

M. I. Weinstein

M. Keel

M. Ohta

Marco Ghimenti

Marco Squassina

R. K. Dodd

S. Cuccagna

S. Le Coz

Stefan Le Coz

T. Cazenave

T. D'Aprile

T. Tao

T.-C. Lin

V. Benci

Y. Liu

Y. Yu

Y.-G. Oh

English

arXiv

null null

Crossref

Discrete and Continuous Dynamical Systems

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 regim

Squassina, Marco

PubliCatt

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

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

Ghimenti, Marco

Le Coz, Stefan

Scientific Publications of the University of Toulouse II Le Mirail

HAL: Hyper Article en Ligne

HAL-INSA Toulouse

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

GHIMENTI, MARCO GIPO

LeCoz S

Squassina M.

Archivio della Ricerca - Università di Pisa

arXiv:1201.4073v1  [math.AP]  19 Jan 2012ON THE STABILITY OF STANDING WAVES OFKLEIN-GORDON EQUATIONS IN A SEMICLASSICAL REGIMEMARCO GHIMENTI, STEFAN LE COZ, AND MARCO SQUASSINAAbstract. We investigate the orbital stability and instability of standingwaves for two classes of Klein-Gordon equations in the semi-classical regime.1. Introduction and resultsThe nonlinear Klein-Gordon equation(1.1) ε2utt − ε2∆u+mu− |u|p−1u = 0 (t, x) ∈ R+ × RN ,where ε,m > 0, p > 1 for N = 1, 2 and 1 < p < (N + 2)/(N − 2) for N ≥ 3, is oneof the simplest nonlinear partial differential equations invariant for the Poincarégroup. We are interested in the study of the nonlinear Klein Gordon equation inpresence of a potential depending on the space variable. Two different choices areviable. We can simply add a potential termW (x)u to equation (1.1). This case hasbeen studied, for the linear wave equation, for example, by Beals and Strauss in [3].Otherwise, if we want to fully preserve the invariance for the Poincaré group of(1.1), we have to change the temporal derivative ε2∂tt with a covariant derivative,depending on the potential D2tt, where Dt = ε∂t + iV (x). This second approachis classical, in quantum electrodynamics, when considering electromagnetic waves.The first approach leads us to consider the equation(1.2) ε2utt − ε2∆u+mu−Wu− |u|p−1u = 0, in RN ,while the second one to investigate the equation(1.3) ε2utt + 2iεV ut − ε2∆u+mu− V 2u− |u|p−1u = 0, in RN .In this paper, we treat simultaneously the two previous Klein-Gordon equations bystudying(1.4) ε2utt + 2iεV ut − ε2∆u+mu−Wu− |u|p−1u = 0, in RN ,where u : RN × R → C, ε > 0 and V,W are real valued potential functions.Equation (1.4) formally yields to (1.3) for the choice W = V 2 as well as to (1.2)when V = 0. We shall study the stability of standing waves of this equation in thesemiclassical regime ε→ 0. It admits standing waves, namely solutions of the formu(x, t) = eiωt/εϕω(x/ε), where ω ∈ R and ϕω satisfies(1.5) −∆ϕω +(m− ω2 − 2ωV (εy)−W (εy))ϕω − |ϕω|p−1ϕω = 0, in RN .Date: November 8, 2018.2000 Mathematics Subject Classification. 35D99, 35J62, 58E05, 35J70.Key words and phrases. Klein-Gordon equations, stability, semi-classical limit.Research supported in part by 2009 MIUR project: “Variational and Topological Methods inthe Study of Nonlinear Phenomena”, ANR project ESONSE and GNAMPA project “Dinamicae proprietà di soluzioni concentrate in teorie di campo nonlineari” .12 M. GHIMENTI, S. LE COZ, AND M. SQUASSINATo ensure existence of solutions to (1.5) for ε close to 0, we assume the following.Let V and W be C2. For the functionZ(y) = m− ω2 − 2ωV (y)−W (y), y ∈ RNthere exists x0 ∈ RN such that(1.6) ∇Z(x0) = 0, ∇2Z(x0) is non-degenerate.Furthermore, we assume that(1.7) Z(x0) = m− ω2 − 2ωV (x0)−W (x0) > 0.Under these hypotheses, it is well-know (see e.g. [1] or [2, Section 8.2]) thatwhen ε is close to 0 the equation (1.5) admits a family of positive, exponentiallydecaying, solutions ϕω ⊂ H1(RN ) (hiding the dependance upon ε). More precisely,there exists ξε ∈ RN and ψω ∈ H1(RN ) such that ϕω(·) = ψω(· − ξε) + O(ε2) inH1(RN ) as ε→ 0, where ξε = o(ε) and−∆ψω + Z(x0)ψω = |ψω|p−1ψω, in RN .We are interested in the (orbital) stability or instability of standing waves whenε goes to 0.A standing wave of (1.4) is said to be (orbitally) stable if any solution of (1.4)starting close to the standing wave remains close for all time, up to the invariancesof the equation. More precisely, we say that eiωtε ϕω(xε)is stable if for all η > 0there exists δ > 0 such that for all u0 ∈ H1(RN ) verifying ‖u0 − ϕω‖H1(RN ) < δthe solution u(x, t) of (1.4) with initial data u0 satisfiessupt∈Rinfθ∈R‖u− eiθϕω‖H1(RN ) < η.Since the pioneering works [4,6–8,24,25], the study of orbital stability for standingwaves of dispersive PDE has attracted a lot of attention. Among many others, onecan refer to [10,11,13]; see also the books and surveys [5,12,22,23] and the referencestherein. Relatively few works [9, 14, 16] are concerned with stability at the semi-classical limit for Schrödinger type equations. For Klein-Gordon equations, afterthe ground works [20, 21], there has been a recent interest for stability by blow-up [15, 17–19].To study stability, we first rewrite (1.4) in Hamiltonian form(1.8) ε∂U∂t= JE′(U),where U =(uv), J =(0 1−1 0), andE(U) =12‖v − iV u‖2L2(RN ) +12‖∇u‖2L2(RN ) +m12‖u‖2L2(RN )− 12∫RNW |u|2dx− 1p+ 1‖u‖p+1Lp+1(RN ).It is easy to see that if u solves (1.4) and v is defined by v := εut + iV u, thenU =(uv)solves (1.8). The charge Q(U) is defined byQ(U) = ℑ∫RNūvdx.ON THE STABILITY OF KLEIN-GORDON EQUATIONS 3In particular, for a standing wave u = eiωt/εϕω(x/ε), the charge is given by(1.9) Q(ϕω) := Q(U) = εN(ω‖ϕω‖2L2(RN ) +∫RNV (εy)|ϕω|2).According to the theory developed in [7,8], a standing wave eiωtε ϕω(xε)is stableif two conditions are satisfied.(i) The Slope Condition:∂∂ωQ(ϕω) < 0.(ii) The Spectral Condition: Lε := −∆ + Z(εy) − p|ϕω|p−1 has exactly onenegative eigenvalue and is non degenerate.On the other hand, denote by n(Lε) the number of negative eigenvalues of Lε andset p(ω) = 0 if∂∂ωQ(ϕω) > 0, p(ω) = 1 if∂∂ωQ(ϕω) < 0. Then the standing waveis unstable if n(Lε)− p(ω) is odd.Then we have the followingTheorem 1.1. Assume that conditions (1.6)-(1.7) hold. Then, we have the follow-ing facts.(1) If p < 1 + 4/N , then the Slope Condition ∂∂ωQ(ϕω) < 0 is fulfilled ifZ(x0) < (ω + V (x0))2( 4p− 1 −N)(non-critical case)or ifZ(x0) = (ω + V (x0))2(4p−1 −N),(∆Z(0)−∆V (0)(1 + 2(ω+V (0))Z(0)))< 0,(critical case).(2) If p ≥ 1 + 4/N , then we always have ∂∂ωQ(ϕω) > 0.(3) We have the equality n(Lε) = n(∇2Z(x0)) + 1, where n(∇2Z(x0)) is thenumber of negative eigenvalues of ∇2Z(x0).In particular, the standing waves eiωtϕω are stable if x0 is non-degenerate localminimum of Z, p < 1 + 4/N andZ(x0) < (ω + V (x0))2( 4p− 1 −N).Note that, conversely to what was happening in the case of Schrödinger equationsstudied in [14], the values of the potentials V and W at x0 comes into play for theSlope Condition even in the noncritical case. Note also that only the local behaviorof Z around x0 influences the stability or instability.Notations : Most of the objects we consider will depend both on ε and ω. Wewill emphasize the most important parameter by indicating it as a subscript, thedependence in the other parameter being understood.2. Proof of Theorem 1.1In this section, we prove Theorem 1.1. We start be focusing on the Slope Con-dition and then we study the Spectral Condition. For the sake of simplicity innotations and without loss of generality, in the rest of this section we assume thatx0 = 0.2.1. The Slope Condition. We start with the noncritical case.4 M. GHIMENTI, S. LE COZ, AND M. SQUASSINA2.1.1. Noncritical case. We assume thatZ(0) 6= (ω + V (0))2( 4p− 1 −N).We first rewrite Q(ϕω) by expanding V (εy) and using the exponential decay ofϕω:Q(ϕω) = εN (ω + V (0)) ‖ϕω‖2L2(RN ) +O(εN+1).Therefore, since(2.1)∂∂ωQ(ϕω) = εN‖ϕω‖2L2(RN ) + εN (ω + V (0))∂∂ω‖ϕω‖2L2(RN ) +O(εN+1),to evaluate the sign of the map ω 7→ ∂∂ωQ(ϕω) one should compute the quantity(2.2)∂∂ω‖ϕω‖2L2(RN ) = 2∫RNRωϕω,where Rω(x) :=∂ϕω∂ω (x).We remark that differentiation of (1.5) with respect to ω easily yields(2.3) LεRω = 2(ω + V (εy))ϕω .If we now introduce the rescaling ϕω(x) = λ1p−1ϕλ(√λx), it follows that ϕλ satisfies(2.4) −∆ϕλ + λ−1Z(εy√λ)ϕλ − |ϕλ|p−1ϕλ = 0, in RN .Now, differentiating equation (2.4) with respect to λ and denoting Tλ =∂ϕλ∂λ |λ=1yields(2.5) LεTλ − Z(εy)ϕω −12εy · ∇Z(εy)ϕω = 0.Since 0 is a critical point of Z, a Taylor expansion givesZ(εy) = Z(0) +O(ε2|y|2),(2.6)12εy · ∇Z(εy) = O(ε2|y|2).(2.7)Then, from (2.5), as ε→ 0 we have(2.8) LεTλ = Z(0)ϕω +O(ε2|y2|ϕω), in RN .Then, in turn, taking into account identity (2.3) we getZ(0)∫RNRωϕω =∫RNRωLεTλ +O(ε2)=∫RNLεRωTλ +O(ε2)=∫RN2(ω + V (εy))ϕωTλ +O(ε2)(2.9)= 2(ω + V (0))∫RNϕωTλ +O(ε)= (ω + V (0))∂∂λ‖ϕλ‖2L2(RN )|λ=1 +O(ε)= (ω + V (0))(N2− 2p− 1)‖ϕω‖2L2(RN ) +O(ε).ON THE STABILITY OF KLEIN-GORDON EQUATIONS 5In conclusion, by combining (2.1), (2.2) and (2.8), we have∂∂ωQ(ϕω) = εN(1 +(ω + V (0))2Z(0)(N − 4p− 1))‖ϕω‖2L2(RN ) +O(εN+1).Then, taking into account the fact that Z(0) > 0 and that ϕω converges to ψω inL2(RN ) as ε→ 0, the sign of ∂∂ωQ(ϕω) is the sign ofZ(0)− (ω + V (0))2( 4p− 1 −N).2.1.2. Critical case. We assume now that(2.10) Z(0) = (ω + V (0))2( 4p− 1 −N).In the critical case, the term of order εN in the expansion of ∂∂ωQ(ϕω) vanishes andwe need to calculate the expansion at a higher order. We first refine (2.6)-(2.7).Z(εy) = Z(0) +ε22∇2Z(0)(y, y) +O(ε3|y|3)12εy · ∇Z(εy) = ε22∇2Z(0)(y, y) +O(ε3|y|3).Then (2.5) givesLεTλ = Z(0)ϕω + ε2∇2Z(0)(y, y)ϕω +O(ε3|y3|)ϕω .Now, we have(2.11) Z(0)∫RNRωϕω =∫RNRωLεTλ − ε2∫RN∇2Z(0)(y, y)Rωϕω +O(ε3).From (2.3), we obtain(2.12)∫RNRωLεTλ =∫RNLεRωTλ =∫RN2(ω + V (εy))ϕωTλ.Expanding the potential V we get(2.13)∫RN2V (εy)ϕωTλ =∫RN2V (0)ϕωTλ + 2ε∫RNy · ∇V (0)ϕωTλ+ ε2∫RN∇2V (0)(y, y)ϕωTλ +O(ε3).Note that since ϕω = ψω(· − ξε) +O(ε2) and ξε = o(ε), we have(2.14) 2ε∫RNy · ∇V (0)ϕωTλ = 2ε∫RNy · ∇V (0)ψωTλ + o(ε2) = o(ε2)where the last cancellation comes from the fact that ψω is radial. Coming back to(2.12) and as in (2.9), we have(2.15)∫RNRωLεTλ = (ω + V (0))(N2− 2p− 1)‖ϕω‖2L2(RN )+ ε2∫RN∇2V (0)(y, y)ϕωTλ + o(ε2).It remains to compute the integrals involving the Hessians in (2.11) and (2.15).Since our problem is invariant by an orthonormal transformation, we can assume6 M. GHIMENTI, S. LE COZ, AND M. SQUASSINAwithout loss of generality that ∇2V (0) = diag(b1, . . . , bN). Hence ∇2V (0)(y, y) =∑Nj=1 bjy2j . Recall also that Tλ can be computed explicitly to haveTλ = −1p− 1ϕω −12y · ∇ϕω .Therefore,∫RNbjy2jϕωTλ = −bjp− 1∫RNy2jϕ2ω −bj2N∑k=1∫RNy2j ykϕω∂ϕω∂yk.We have after integration by parts2N∑k=1∫RNy2j ykϕω∂ϕω∂yk= −N∑k=1∫RN(y2j + 2δjky2j )ϕ2ω = −(N + 2)∫RNy2jϕ2ω.Thus∫RN∇2V (0)(y, y)ϕωTλ =N∑j=1∫RNbjy2jϕωTλ = −(1p− 1 −N + 24) N∑j=1bj∫RNy2jϕ2ω.Recall the following expansion in ε for Rω and ϕω .ϕω = ψω + o(ε), Rω =∂ψω∂ω+ o(ε).Therefore, since ψω is radial,∫RNy2jϕ2ω =∫RNy2jψ2ω + o(ε) =1N‖|y|ψω‖2L2(RN ) + o(ε),and so(2.16)∫RN∇2V (0)(y, y)ϕωTλ =−(1p− 1 −N + 24)1N‖|y|ψω‖2L2(RN )∆V (0) + o(ε).Similarly, we have(2.17)∫RN∇2Z(0)(y, y)Rωϕω =−(1p− 1 −N + 24)1N‖|y|ψω‖2L2(RN )∆Z(0) + o(ε).Summarizing, using successively (2.11), (2.15), (2.16), (2.17) and (2.10) we haveobtained(2.18)∫RNRωϕω = −12(ω + V (0))‖ϕω‖2L2(RN )+ ε2(∆Z(0)−∆V (0))NZ(0)(1p− 1 −N + 24)‖|y|ψω‖2L2(RN ) + o(ε2).Now, we compute ∂Q(ϕω)∂ω . First, recall that, coming back to the definition (1.9)of Q, we haveε−N∂Q(ϕω)∂ω= ‖ϕω‖2L2(RN ) + 2ω∫RNRωϕω + 2∫RNV (εy)Rωϕω.ON THE STABILITY OF KLEIN-GORDON EQUATIONS 7As in (2.13), (2.14), and (2.16) we can expand in ε and get2∫RNV (εy)Rωϕω = 2V (0)∫RNRωϕω− ε2(1p− 1 −N + 24)1N‖|y|ψω‖2L2(RN )∆V (0) + o(ε2).Therefore,ε−N∂Q(ϕω)∂ω= ‖ϕω‖2L2(RN ) + 2(ω + V (0))∫RNRωϕω− ε2(1p− 1 −N + 24)1N‖|y|ψω‖2L2(RN )∆V (0) + o(ε2).Using (2.18), we finally getε−N∂Q(ϕω)∂ω= ε2(1p− 1 −N + 24)1N‖|y|ψω‖2L2(RN )××(∆Z(0)−∆V (0)(1 +2(ω + V (0))Z(0)))+ o(ε2).2.2. The Spectral Condition. We define the operator L0 := −∆+Z(0)−pψp−1ω .It is well known (see e.g. [2]) that the spectrum of L0 consists of one negativeeigenvalue, a N -dimensional kernel (generated by ∂ψω∂yj for j = 1, . . . , N) and therest of the spectrum is bounded away from 0. When ε is close to 0, the spectrum ofLε will be close to the spectrum of L0. In particular, the 0 eigenvalue, of multiplicityN , will transform into N possibly different eigenvalues close to 0 but shifted eitherto the positive or to the negative side of the real axis, depending on the sign of theeigenvalues of the Hessian of Z at 0. More precisely, the following proposition wasproved in [14] (see [9] for a detailed justification).Proposition 2.1. The spectrum of Lε consists of positive spectrum away from 0and a set of N + 1 simple eigenvalues {λ0, λ1, . . . , λN} such thatλ0 < λ1 ≤ · · · ≤ λN .As ε → 0, we have λ0 < 0 and the following asymptotic expansion holds for theother eigenvalues:λj = cjε2 + o(ε2), j = 1, ..., N,where cj =12‖ψω‖2L2(RN )‖ ∂ψω∂xj‖2L2(RN )aj and {a1, . . . , aN} are the eigenvalues of the Hessianmatrix ∇2Z(0).Therefore, (3) in Theorem 1.1 is a direct consequence of Proposition 2.1. Inparticular, the spectral condition for stability will be satisfied if and only if 0 is anon-degenerate local minimum of Z.References[1] A. Ambrosetti, M. Badiale, and S. Cingolani, Semiclassical states of nonlinearSchrödinger equations, Arch. Rational Mech. Anal., 140 (1997), pp. 285–300.[2] A. Ambrosetti and A. Malchiodi, Perturbation methods and semilinear elliptic problemson Rn, vol. 240 of Progress in Mathematics, Birkhäuser Verlag, Basel, 2006.8 M. GHIMENTI, S. LE COZ, AND M. SQUASSINA[3] M. Beals and W. Strauss, Lp estimates for the wave equation with a potential, Comm.Partial Differential Equations, 18 (1993), pp. 1365–1397.[4] H. Berestycki and T. Cazenave, Instabilité des états stationnaires dans les équations deSchrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris Sér. I Math., 293 (1981),pp. 489–492.[5] T. Cazenave, Semilinear Schrödinger equations, New York University – Courant Institute,New York, 2003.[6] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinearSchrödinger equations, Comm. Math. Phys., 85 (1982), pp. 549–561.[7] M. Grillakis, J. Shatah, and W. A. Strauss, Stability theory of solitary waves in thepresence of symmetry I, J. Funct. Anal., 74 (1987), pp. 160–197.[8] , Stability theory of solitary waves in the presence of symmetry II, J. Funct. Anal., 94(1990), pp. 308–348.[9] I. Ianni and S. Le Coz, Orbital stability of standing waves of a semiclassical nonlinearSchrödinger-Poisson equation, Adv. Differential Equations, 14 (2009), pp. 717–748.[10] L. Jeanjean and S. Le Coz, An existence and stability result for standing waves of nonlinearSchrödinger equations, Adv. Differential Equations, 11 (2006), pp. 813–840.[11] , Instability for standing waves of nonlinear Klein-Gordon equations via mountain-passarguments, Trans. Amer. Math. Soc., 361 (2009), pp. 5401–5416.[12] S. Le Coz, Standing waves in nonlinear Schrödinger equations, in Analytical and numericalaspects of partial differential equations, Walter de Gruyter, Berlin, 2009, pp. 151–192.[13] S. Le Coz, R. Fukuizumi, G. Fibich, B. Ksherim, and Y. Sivan, Instability of bound statesof a nonlinear Schrödinger equation with a Dirac potential, Phys. D, 237 (2008), pp. 1103–1128.[14] T.-C. Lin and J. Wei, Orbital stability of bound states of semiclassical nonlinear Schrödingerequations with critical nonlinearity, SIAM J. Math. Anal., 40 (2008), pp. 365–381.[15] Y. Liu, M. Ohta, and G. Todorova, Instabilité forte d’ondes solitaires pour des équationsde Klein-Gordon non linéaires et des équations généralisées de Boussinesq, Ann. Inst. H.Poincaré Anal. Non Linéaire, 24 (2007), pp. 539–548.[16] Y.-G. Oh, Stability of semiclassical bound states of nonlinear Schrödinger equations withpotentials, Comm. Math. Phys., 121 (1989), pp. 11–33.[17] M. Ohta and G. Todorova, Strong instability of standing waves for nonlinear Klein-Gordonequations, Discrete Contin. Dyn. Syst., 12 (2005), pp. 315–322.[18] , Instability of standing waves for nonlinear Klein-Gordon equation and related sys-tem, in Nonlinear dispersive equations, vol. 26 of GAKUTO Internat. Ser. Math. Sci. Appl.,Gakkōtosho, Tokyo, 2006, pp. 189–200.[19] , Strong instability of standing waves for the nonlinear Klein-Gordon equation and theKlein-Gordon-Zakharov system, SIAM J. Math. Anal., 38 (2007), pp. 1912–1931 (electronic).[20] J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math.Soc., 290 (1985), pp. 701–710.[21] J. Shatah and W. A. Strauss, Instability of nonlinear bound states, Comm. Math. Phys.,100 (1985), pp. 173–190.[22] C. A. Stuart, Lectures on the orbital stability of standing waves and application to thenonlinear Schrödinger equation, Milan J. Math., 76 (2008), pp. 329–399.[23] T. Tao, Why are solitons stable?, Bull. Amer. Math. Soc., 46 (2009), pp. 1–33.[24] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm.Math. Phys., 87 (1982/83), pp. 567–576.[25] M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations,SIAM J. Math. Anal., 16 (1985), pp. 472–491.Dipartimento di Matematica ApplicataUniversità di PisaVia F. Buonarroti 1/c, 56127 PisaItaliaE-mail address: ghimenti@mail.dm.unipi.itON THE STABILITY OF KLEIN-GORDON EQUATIONS 9Institut de Mathématiques de Toulouse,Université Paul Sabatier118 route de Narbonne, 31062 Toulouse Cedex 9FranceE-mail address: slecoz@math.univ-toulouse.frDipartimento di InformaticaUniversità di VeronaStrada Le Grazie 15, 37134 VeronaItaliaE-mail address: marco.squassina@univr.it

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

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