Long time dynamics for damped Klein-Gordon equations
LLONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS
N. BURQ, G. RAUGEL, W. SCHLAGA bstract . For general nonlinear Klein-Gordon equations with dissipation we show thatany finite energy radial solution either blows up in finite time or asymptotically approachesa stationary solution in H ˆ L . In particular, any global solution is bounded. The resultapplies to standard energy subcritical focusing nonlinearities | u | p ´ u , 1 ă p ă p d ` q{p d ´ q as well as any energy subcritical nonlinearity obeying a sign condition of the Ambrosetti-Rabinowitz type. The argument involves both techniques from nonlinear dispersive PDEsand dynamical systems (invariant manifold theory in Banach spaces and convergencetheorems).
1. I ntroduction
Nonlinear dispersive evolution equations such as the wave and Schrödinger equationshave been investigated for decades. For defocusing power-type energy subcritical orcritical nonlinearities the theory is developed, while the energy supercritical powers arewide open. For semilinear focusing equations the picture is less complete for long-termdynamics. These equations exhibit finite-time blowup, small data global existence andscattering, as well as time-independent solutions (solitons). For the energy critical waveequation l u “ u , p t , x q P R ` , p u p q , B t u p qq P H p R q ˆ L p R q , in the radial setting, Duyckaerts, Kenig, and Merle [18] achieved a breakthrough byshowing that all global trajectories can be described as a superposition of a finite numberof rescalings of the ground state W p r q “ p ` r { q ´ plus a radiation term which isasymptotic to a free wave. This work introduces the novel exterior energy estimates. Thesubcritical case appears to require di ff erent techniques, however. Nakanishi and the thirdauthor [40] described the asymptotics of solutions provided the energy is only slightlylarger than the ground state energy. The trichotomy in forward time of (i) blowup in finitetime (ii) global existence and scattering to zero (iii) global existence and scattering to theground state, can be naturally formulated in terms of the center-stable manifold associatedwith the ground state. The authors thank F. Merle, E. Hebey and M. Willem for fruitful discussions. In particular, they thankE. Hebey for having indicated them the paper of Cazenave [9]. The first author was partially funded byANR through ANR-13-BS01-0010-03 (ANAÉ). The third author was partially supported by the NSF throughDMS-1160817. a r X i v : . [ m a t h . A P ] M a y N. BURQ, G. RAUGEL, W. SCHLAG
In this paper, we develop a robust approach to the problem of long-term asymptotics ofthe general energy subcritical Klein-Gordon equations with (arbitrarily small) dissipation.The focusing damped subcritical Klein-Gordon equation in R d , 1 ď d ď d ě
7, see [7]), is B t u ` α B t u ´ ∆ u ` u ´ | u | θ ´ u “ , p u p q , B t u p qq “ p ϕ , ϕ q P H , (1.1)where H “ H p R d q ˆ L p R d q , α ě ă θ ă θ ˚ , with θ ˚ “ d ` d ´ . We will limit our study to the case of radial functions H rad “ H rad p R d q ˆ L rad p R d q . The energy functional E θ below, also called Lyapunov functional in the dissipative case α ą
0, plays an important role in the analysis of the behaviour of the solutions of (1.1).This energy functional is given by(1.3) E θ p ϕ , ϕ q “ ż R d ˆ | ∇ ϕ | ` ϕ ` ϕ ´ θ ` | ϕ | θ ` ˙ dx For the Klein-Gordon equation (1.1), it is known (see [46], [3], [14], [39] and [10] forexample) that (1.1) admits a unique positive radial stationary solution p Q g , q (the groundstate solution), which minimizes the energy E θ p ., q in the class of all nonzero stationarysolutions p Q , q in H , that is,0 ă E θ p Q g , q “ min t E p Q , q | Q P H p R d q , Q ‰ , ´ ∆ Q ` Q ´ | Q | θ ´ Q “ u The behaviour of solutions of (1.1) with initial data p ϕ , ϕ q P H with energy E θ p ϕ , ϕ q ă E θ p Q g , q is rather well understood in the case α ě α “ α “ α ą
0, respectively. For adescription of this phenomenon in the case α “
0, we refer for example to the book [40].It is also well-known that this equation has an infinite number of radial equilibriumpoints p e (cid:96) , q with a prescribed number (cid:96) ě nodal solutions , seefor example [4]). Unfortunately, one knows almost nothing about the uniqueness and thehyperbolicity of those nodal solutions. ([15] obtains uniqueness results for nodal solutionsbut for sub-linear nonlinearities). In the Hamiltonian case ( α “ (cid:126) u p t q of (1.1) whose initial data p ϕ , ϕ q have an energy E θ p ϕ , ϕ q much larger than the one of the ground state p Q , q .In 1985 Cazenave [9] established the following dichotomy for the Hamiltonian case α “
0: solutions of (1.1) either blow up in finite time or are global and bounded in H ,provided 1 ă θ ă `8 , if d “ , θ ď d “ ă θ ď dd ´ if d ě ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 3 of Cazenave should extend to the case α ą α ą
0, gave an independent proof of the boundeness of the global solutions of (1.1), when d ě ă θ ă ` min p dd ´ , d q (for the case d “
1, see his earlier paper [21]).Unfortunately, the proofs of Cazenave [9] and of Feireisl [23] do not seem to extend tononlinearities satisfying dd ´ ă θ ă d ` d ´ , when d ě
3, where one needs to use Strichartzestimates in the various a priori estimates rather than Galiardo-Nirenberg-Sobolev in-equalities. In this paper, we restrict our study to the dissipative radial case ( α ą
0) andshow the following dichotomy.
Theorem 1.1.
Let α ą and d ď . Then, (1) either the solutions of (1.1) in H rad blow up in finite positive time, (2) or they are global in positive time and converge to an equilibrium point.In particular, all global in positive times solutions are bounded for positive times. We notice that this theorem is a particular case of Theorem 1.2 below. In [7], we willpartly generalise this dichotomy to non-radial solutions.Actually the above dichotomy holds for some more general nonlinearities and, in thispaper, we consider the damped Klein-Gordon equation in R d , d ď d ě B t u ` α B t u ´ ∆ u ` u ´ f p u q “ , p u p q , B t u p qq “ p ϕ , ϕ q P H rad , p KG q α where f : y P R ÞÑ f p y q P R is an odd C -function, f p q “
0, which satisfies the followingAmbrosetti-Rabinowitz type condition: there exists a constant γ ą ż R d ` p ` γ q F p ϕ q ´ ϕ p x q f p ϕ p x qq ˘ dx ď , @ ϕ P H p R d q , p H . q f where F p y q “ ş y f p s q ds .We also need to impose a growth condition on f , when d ě
2. We assume that, | f p y q| ď C max ` | y | β , | y | θ ´ ˘ , @ y P R , | f p y q ´ f p y q| ď C ` | y ´ y | β ` | y ´ y | θ ´ ˘ , @ y , y P R , p H . q f where 1 ă θ ă θ ˚ , 0 ă β ă θ ´ θ ˚ “ ˚ ´ ˚ “ 8 if d “ , ˚ “ dd ´ if d ě
3. We notice that, when d ě θ ˚ “ d ` d ´ .In other words, the growth of f is energy subcritical for large y “
0, and we also assumethat f is β -Hölder continuous. For sake of simplicity in the proofs below, we may assume,without loss of generality, that 0 ă β ă min p θ ´ , d ´ q .We remark that our argument does not depend on the existence or uniqueness of aground state solution. Note that Hypothesis p H . q f alone does not imply the existence N. BURQ, G. RAUGEL, W. SCHLAG and uniqueness of a ground state solution. We further note that Hypothesis p H . q f mayactually be replaced by the following weaker one: ż R d ` p ` γ q F p ϕ q ´ ϕ p x q f p ϕ p x qq ˘ dx ď , for } ϕ } H large enough . p H . bis q f But, for sake of simplicity, we assume p H . q f throughout. A classical example of a function f satisfying hypotheses p H . q f and p H . q f is as follows: f p u q “ m ÿ i “ a i | u | p i ´ u ´ m ÿ j “ b j | u | q j ´ u , with 1 ă q j ă p i ď d ` d ´ , @ i , j and a i , b j ě , a m ą . (1.4)In Section 2, we shall prove that the equation p KG q α generates a local dynamical systemon H as well as on H rad , for α ě
0. We denote S α p t q , α ě
0, this local dynamical system.As in the particular case of the Klein-Gordon equation (1.1), we introduce the energyfunctional (also called Lyapunov functional in the case of positive damping α ą
0) on H :(1.5) E p ϕ , ϕ q “ ż R d ˆ | ∇ ϕ | ` ϕ ` ϕ ´ F p ϕ q ˙ dx . The natural first step in the study of the dynamics of the equation p KG q α consists instudying the boundedness or unboundedness of its global (in positive times) solutions.As already mentioned above, under restrictions on the growth rate of the nonlinearity,Cazenave [9] and Feireisl [23] established this boundedness. In this paper, taking advan-tage of the fact that all the functions are radial, we will show the boundedness of the globalsolutions of p KG q α , for α ą
0, by using “dynamical systems” arguments. Indeed, we willshow that each global solution (cid:126) u p t q converges to an equilibrium point as t goes to `8 .If the equation p KG q α admits a ground state solution and is Hamiltonian, the functional K : ϕ P H p R d q ÞÑ K p ϕ q P R defined as(1.6) K p ϕ q “ ż R d ` | ∇ ϕ | ` ϕ ´ ϕ f p ϕ q ˘ dx , has played a decisive role in the description of the dynamics of the solutions with initialenergy smaller or slightly larger than the one of the ground state (see [42], [40] for example).It will also be important in our situation. First we shall prove in Lemma 2.7, that if (cid:126) u p t q “ S α p t qp ϕ , ϕ qp t q ” p u p t q , B t u p t qq satisfies K p u p t qq ď ´ δ (where δ ą K p u p t qq ě η for some finite η on the maximal interval of existence, the solution exists and is bounded for all positivetimes.In order to prove that each global solution (cid:126) u p t q “ S α p t qp ϕ , ϕ qp t q converges to anequilibrium point as t goes to `8 , we argue by contradiction. If this trajectory (cid:126) u p t q is ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 5 unbounded in positive time, then there exists a sequence of times t n , t n Ñ n Ñ`8 `8 , suchthat K p u p t n qq Ñ n Ñ`8 t n , we show in Theorem 3.5, that the ω -limit set ω p ϕ , ϕ q of p ϕ , ϕ q is non-empty and contains at least one equilibrium point p Q ˚ , q of the equation p KG q α . We recall that the ω -limit set ω p ϕ , ϕ q of p ϕ , ϕ q is defined as follows: ω p ϕ , ϕ q “ t (cid:126) w P H rad | D a sequence τ n ě , so that τ n Ñ n Ñ`8 `8 , and S α p τ n qp ϕ , ϕ q Ñ n Ñ`8 (cid:126) w u . (1.7)Then, in Section 3.2, taking advantage of the fact that the linearized Klein-Gordon equationaround p Q ˚ , q in the space H rad has a kernel which is at most one-dimensional, we show,by using classical convergence arguments based on invariant manifold theory, that thetrajectory converges to this equilibrium in positive infinite time, and is therefore bounded. Theorem 1.2.
Let α ą . Assume that ď d ď and that f satisfies the conditions p H . q f and p H . q f . Let p ϕ , ϕ q P H rad , then (1) either S α p t qp ϕ , ϕ q blows up in finite time, (2) or S α p t qp ϕ , ϕ q exists globally and converges to an equilibrium point p Q ˚ , q of p KG q α ,as t Ñ `8 . For the case d ě
7, we refer the reader to [7].To place this result into context, we now briefly recall various related convergencetheorems. Since we are considering the equation p KG q α in the radial setting, the linearizedKlein-Gordon operator around the equilibrium p Q ˚ , q has a kernel of dimension less thanor equal to 1, that is, either 0 does not belong to the spectrum of the elliptic selfadjointoperator L ” ´ ∆ ` I ´ f p Q ˚ q or 0 is a simple eigenvalue of L (see Section 2, Lemma 2.10). If 0 is a simple eigenvalueof L , then the dynamical system S α p t q admits a C local center manifold W c pp Q ˚ , qq ofdimension 1 at p Q ˚ , q . Since the ω -limit set of any element p ϕ , ϕ q P H rad belongs tothe set of equilibria, if the trajectory of S α p t qp ϕ , ϕ q ” (cid:126) u p t q were precompact in H rad ,we could directly conclude by using the convergence results contained in [5] or in [26]for example that the whole trajectory S α p t qp ϕ , ϕ q converges to p Q ˚ , q , when t goes toinfinity. Unfortunately, we do not know that the trajectory S α p t qp ϕ , ϕ q is bounded andthus we do not even know that the ω -limit set of p ϕ , ϕ q is bounded and connected.However, adapting the proof of [5, Lemma 1] and using the asymptotic phase propertyof the local center unstable and local center manifolds around p Q ˚ , q (see Appendix Afor these concepts), we easily obtain that the entire trajectory S α p t qp ϕ , ϕ q convergesto p Q ˚ , q as t goes to infinity. An alternative way for proving the convergence of thetrajectory S α p t qp ϕ , ϕ q towards p Q ˚ , q would be to use (instead of dynamical systemsarguments) a Łojasiewicz-Simon inequality (see Sections 3.2 and 3.3 in the monographof L. Simon [45] and also [28, Theorem 2.1]) together with functional arguments as inJendoubi and Haraux (see [27] or [28]). The proof of the Łojasiewicz-Simon inequality N. BURQ, G. RAUGEL, W. SCHLAG in [45] uses a Lyapunov-Schmidt decomposition. In the special case where the kernel of L is one-dimensional, this proof also shows that the set of equilibria of p KG q α passingthrough p Q ˚ , q is a C -curve. Using this Łojasiewicz-Simon inequality and introducing anappropriate functional like in [28], we could show that the ω -limit set of every precompacttrajectory converges to an equilibrium point. Unfortunately, the trajectory S α p t qp ϕ , ϕ q isnot a priori bounded and it seems di ffi cult to adapt the functional part of the proof of [28,Theorem 3.1]. Moreover, there is an additional di ffi culty in the construction of such anappropriate functional coming from the fact that we need to use Strichartz estimates. Sowe have not been able to follow this route.The plan of this paper is as follows. Section 2 is devoted to basic properties of theKlein-Gordon equation p KG q α . In particular, we recall the local existence and uniquenessof mild solutions of the equation p KG q α . In Section 2.2, we introduce the functional K ,which not only plays an important role in the proof of Theorem 1.2 but also defines thewell-known Nehari manifold N as the locus of the radial zeros of the functional K . InLemma 2.7, we give a su ffi cient condition on K for blow-up in finite time of the solutionsof p KG q α . We end this section by describing the spectral properties of the linearizedKlein-Gordon equation around a (radial) equilibrium point. Section 3 is the core of thispaper. In Section 3.1 (see Theorem 3.5) we show that if a solution (cid:126) u p t q does not blowup in finite positive time, then the ω -limit set ω p (cid:126) u p qq contains at least one equilibriumpoint. In Section 3.2 we show that the whole trajectory (cid:126) u p t q converges to this equilibriumpoint and is therefore bounded. In Section 4, we apply the classical invariant manifoldtheory, recalled in Appendix A, in order to construct the local unstable, center unstable andcenter manifolds about equilibrium points of the Klein-Gordon equation p KG q α and theunstable, center unstable and center manifolds about equilibrium points of the localizedKlein-Gordon equation (4.7). In Appendix A, we recall the existence theorems for localcenter-stable, local center-unstable and local center manifolds together with their foliationsand exponential attraction properties with asymptotic phase in the formulation of Chen,Hale and Tan (see [11]). Finally, in Appendix B, we recall the classical convergence theorem(see [1], [25] or [26]) in the generalised form given by Brunovský and Poláˇcik in [5].Such a convergence theorem is needed in case the dynamics near the equilibriumexhibits a nontrivial center manifold. As a result of dissipation and the radial condition,this center manifold can be at most one-dimensional. For the nonlinearities (1.4), it isknown that the kernel of the linearized operator about the ground state is trivial, see [10].But, due to the lack of precise description of the bound states, we cannot guarantee that thelocal center manifold is absent about a bound state. The local strongly unstable manifoldis finite-dimensional. The local strongly stable manifold is infinite-dimensional in starkcontrast to the Hamiltonian scenario for which the local center manifold is the largestpiece. The convergence theorem in [5] then guarantees that, if the ω -limit set is not asingle equilibrium point p Q ˚ , q , and if p Q ˚ , q is stable for the restriction of S α p t q to thelocal center manifold of p Q ˚ , q (for this definition of stability, see (3.41) and Appendix B),then this ω -limit set must contain a point on the unstable manifold of p Q ˚ , q , distinct from ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 7 p Q ˚ , q . But this contradicts the fact that, due to the properties of the Lyapunov functional(1.5), the ω -limit set is contained in the set of equilibrium points.2. B asic properties Local existence results.
Consider the linear equation, with α ě B t u ` α B t u ´ ∆ u ` u “ G , p u , B t u q ˇˇˇ t “ “ p u , u q P H p R d q ˆ L p R d q . (2.1)Since v p t q “ e α t u p t q satisfies v tt ´ ∆ v ` p ´ α q v “ e α t G , p v , v t q ˇˇˇ t “ “ p u , u ` α u q . (2.2)We deduce that the solution of (2.1) is given by u p t q “ e ´ α t ” cos p t a ´ ∆ ` ´ α q ` α sin p t a ´ ∆ ` ´ α q a ´ ∆ ` ´ α ı u ` e ´ α t sin p t a ´ ∆ ` ´ α q a ´ ∆ ` ´ α u ` ż t sin pp t ´ s q a ´ ∆ ` ´ α q a ´ ∆ ` ´ α e ´p t ´ s q α G p s q ds “ S ,α p t q u ` S ,α p t q u ` ż t S ,α p t ´ s q G p s q ds . (2.3)Clearly, the regimes 0 ď α ă α “
1, and α ą ff erent behaviours.The dispersion relation for α ă α “ X is a Banach space, then we let L p ,β t p X q be the space with norm } f } L p ,β t p X q “ } e β t } f p t q} X } L pt , β P R In this section, the β in these weighted estimates has nothing to do with the regularityin ( p H . q f ). Lemma 2.1.
Let ď α ă and assume d ě for simplicity. Set p “ dd ´ and σ “ ´ d , σ “ ´ σ . The solution u of (2.1) satisfies the following Strichartz-type estimates for any ď β ď α , } u } L ,β t B σ p , X L ,β t H x ď C p α q ” }p u , u q} H ˆ L ` } G } L ,β t B σ p , ` L ,β t L x ı (2.4) where C p α q is uniform on compact intervals of r , q .Proof. This follows from (2.2) and the Keel-Tao endpoint for the Klein-Gordon equation,see for example Lemma 2.46 in [40]. (cid:3)
N. BURQ, G. RAUGEL, W. SCHLAG
Lemma 2.1 does not hold for α ě
1. Indeed, for α “ β p α q “ α if 0 ď α ď β p α q “ α ´ a α ´ α ą
1. Exploiting the exponential decay in (2.3) we can now state the following space-time averaged estimates.
Lemma 2.2.
Let α ą . In all dimensions d ě the solution u of (2.1) satisfies the followingenergy bounds with decay sup t ě e t β p α q }p u , B t u qp t q} H ˆ L ď C p α q ” }p u , u q} H ˆ L ` ż e s β p α q } G p s q} ds ı (2.5) as well as the exponentially weighted Strichartz estimates, in dimensions d ě , and with ď β ă β p α q , } u } L q ,β t L px ď C p α, β q “ }p u , u q} H ˆ L ` } G } L ˜ q ,β t L ˜ p x ‰ (2.6) where q ` dp “ d ´ “ q ` d ˜ p ´ , ď p , ˜ p ă 8 , ď q , ˜ q, and q ` d ´ p ď d ´ , q ` d ´
12 ˜ p ď d ´ .The constant C p α, β q is uniform on compact subsets of tp α, β q | α P p , , ď β ă β p α qu Proof.
Taking the Fourier transform of (2.3) yieldsˆ u p t , ξ q “ m α p t , ξ q p u p ξ q ` ˜ m α p t , ξ q p u p ξ q ` ż t ˜ m α p t ´ s , ξ q e ´p t ´ s q α p G p s , ξ q ds The multipliers satisfy the estimates | m α p t , ξ q| ` | ˜ m α p t , ξ q| ď C p α q e ´ β p α q t which proves (2.5). For (2.6) we introduce the Littlewood-Paley decomposition1 “ P À α ` ÿ j P j “ P À α ` P ą α where the P j are associated to frequencies 2 j ą α and P À α f “ f for all Schwartz functionswith support in t| ξ | ď ` α u . Let K ˘ λ p t q be the propagator defined by, cf. (2.3), r K ˘ λ p t q f sp x q “ e ´ α t ż R d e ˘ it ? ξ ` ´ α e ix ¨ ξ χ p ξ { λ q ˆ f p ξ q d ξ where χ is the usual Littlewood-Paley bump function supported on an annulus, and λ ą α ` } K ˘ λ p t q} ď e ´ α t λ d x t λ y ´ d ´ À e ´ α t t ´ d ´ λ d ` for all t ą
0. Proceeding as for the wave equation (see Keel-Tao), and ignoring theexponential decay for the frequencies Á α , yields the Strichartz estimates (2.6) for P ą α u ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 9 with β “
0. On the other hand, by the same logic we can also derive Strichartz estimatesfor the transformed equation (2.2) which yields (2.6) with β “ α for the piece P ą α u .Interpolating between these two cases we obtain Strichartz inequalities for all 0 ď β ď α for those frequencies. Smaller frequencies require smaller β . Indeed, for the remainingpiece P À α u we use the energy bound (2.5) and Bernstein’s inequality. To be precise, theenergy estimate } P À α u p t q} ď C p α q ” e ´ t β p α q }p u , u q} H ˆ L ` ż t e ´p t ´ s q β p α q } P À α G p s q} ds ı implies via Bernstein’s inequality that e β t } P À α u p t q} q ď C p α q ” e ´ t p β p α q´ β q }p u , u q} H ˆ L ` ż t e ´p t ´ s qp β p α q´ β q e β s } P À α G p s q} ˜ q ds ı Taking L pt norms on both sides, and applying Young’s inequality to the Duhamel integralyields (2.6) for all frequencies. (cid:3) We now turn to the nonlinear equation p KG q α . We write (cid:126) u “ p u , B t u q . Theorem 2.3.
Let d ď . Let f : R Ñ R be a C odd function, satisfying the assumption p H . q f .Then for every data (cid:126) u in H “ H p R d q ˆ L p R d q ( resp. in H rad ) the equation p KG q α has a uniquestrong solution u P X ” X T : “ C pr , T s , H p R d qq X C pr , T s , L p R d qq ( resp. in C pr , T s , H rad p R d qq X C pr , T s , L rad p R d qq ), where T only depends on } (cid:126) u } H .Moreover, if ď d ď , the solution belongs toL θ ˚ pp , T q , L θ ˚ p R d qq where θ ˚ “ d ` d ´ and the estimate (2.23) below holds.Furthermore, the following properties hold. (1) The solution p t , (cid:126) u q P r , T s ˆ H ÞÑ (cid:126) u p t q ” p u p t q , B u p t qq P H is continuous. (2) For any ď τ ď T, the map (cid:126) u P H ÞÑ S α p τ q (cid:126) u ” (cid:126) u p τ q P H is Lipschitz continuous onthe bounded sets of H (see (2.25) ). (3) The map (cid:126) u P H ÞÑ u p t q P X X L θ ˚ pp , T q , L θ ˚ p R d qq is a C -map. (4) Let T ˚ be the maximal time of existence. If T ˚ ă 8 , then lim sup t Ñ T ˚ } (cid:126) u p t q} H “ `8 (5) If (cid:126) u P H p R d q ˆ H p R d q , thenu P C pr , T q , H p R d qq X C pr , T q , H p R d qq (6) The energy (1.5) decreases: (2.7) E p (cid:126) u p t qq ´ E p (cid:126) u p t qq “ ´ α ż t t }B t u p s q} L dsand, in particular, (2.8) E p (cid:126) u p t qq ` α ż t }B t u p s q} L ds ď E p (cid:126) u p qq (7) If } (cid:126) u p q} ! , then the solution exists globally, and } (cid:126) u p t q} H converges exponentially to as t Ñ 8 .Proof.
We have | f p u q| À | u | ` | u | θ . We begin with dimensions d “ ,
2. In that case theSobolev embeddings and (2.5) imply, for any β , e β t } (cid:126) u p t q} H À } (cid:126) u p q} H ` ż t e ´ β p t ´ s q ` } u p s q} ` } u p s q} θ θ ˘ ds À } (cid:126) u p q} H ` β ´ p ´ e ´ β T q max ď s ď T e β s ` } (cid:126) u p s q} H ` } (cid:126) u p s q} θ H ˘ (2.9)where the implicit constant is of the form C p α q as above. For T small we discard theexponential weight whence } (cid:126) u p t q} H À } (cid:126) u p q} H ` T max ď s ď T ` } (cid:126) u p s q} H ` } (cid:126) u p s q} θ H ˘ This immediately shows that we can set up a contraction in the space X and that T onlydepends on } (cid:126) u p q} H . Moreover, global existence for small data follows from (2.9) by themethod of continuity. This also implies the exponential decay.We shall establish the persistence of H ˆ H regularity later in this proof. Taking it forgranted for now, and using the density of H p R d q ˆ H p R d q in H p R d q ˆ L p R d q , one showsthat(2.10) E p (cid:126) u p t qq P C pp´ ˜ T , T qq , and ddt E p (cid:126) u p t qq “ ´ α }B t u p t q} L . Integrating this implies the above identities for the energy.We now continue with the dimensions d ě
3. If θ ď ˚ “ dd ´ , then the same energybounds su ffi ce. As usual, larger θ requires the Strichartz bounds.The local wellposedness for small times does not require the exponential weights in theStrichartz estiimates and are identical to standard proofs for the wave equation.We first recall the main lines of the proof of the local existence and uniqueness ofthe solution. The local existence is proved by using the classical strict contraction fixedpoint theorem with parameters. In the fixed point argument below, we will use theStrichartz inequality (2.6) given in Lemma 2.2. Let θ ˚ “ ˚ ´ “ d ` d ´ , p ˜ p , ˜ q q “ p , q and p p , q q “ p θ ˚ , θ ˚ q . We remark that these pairs satisfy the conditions of Lemma 2.2 and inparticular q ě d ď ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 11
Let K ą B H p , K q the ball of center0 and radius K in H . Using the notation of the previous lemma, we set(2.11) M ” M p α q “ p C p α q ` C p α, qq K ” C p α q K and T ą Y ” Y T ” t (cid:126) u P L pp , τ q , H q with u P L θ ˚ pp , τ q , L θ ˚ p R d qq| } u } L p H qX W , p L qX L θ ˚ p L θ ˚ q ď M u . (2.12)We consider the mapping F : p (cid:126) u , (cid:126) u q P B H p , K q ˆ Y ÞÑ F p (cid:126) u , (cid:126) u q ” p F , F qp (cid:126) u , (cid:126) u q P Y , defined by(2.13) p F p (cid:126) u , (cid:126) u qqp t q “ S ,α p t q u ` S ,α p t q u ` ż t S ,α p t ´ s q f p u p s qq ds , and F p (cid:126) u , (cid:126) u q “ B t F p (cid:126) u , (cid:126) u q , where (cid:126) u “ p u , u q and (cid:126) u “ p u , B t u q . Fix some (cid:126) u P H with } (cid:126) u } H ă K . Consider the map F p (cid:126) u , . q : (cid:126) u P Y ÞÑ F p (cid:126) u , (cid:126) u q P Y and simply write F p (cid:126) u , (cid:126) u q “ F (cid:126) u .An application of Lemma 2.2 implies(2.14) } F p u , q} Y ď C p α q K ď M . Applying again Lemma 2.2 and using the hypothesis p H . q f , we get } F (cid:126) u ´ F (cid:126) v } Y ď C p α q ż T } f p u p s qq ´ f p v p s qq} L ds ď C p α q ż T } ż f p v p s q ` λ p u p s q ´ v p s qqqp u p s q ´ v p s qq d λ } L ds ď C p α q C ż T }p ` | u p s q| θ ´ ` | v p s q| θ ´ q| u p s q ´ v p s q| } L ds ď C p α q C “ T } u ´ v } L p L q ` ż T }| u p s q| θ ´ | u p s q ´ v p s q| } L ds ` ż T }| v p s q| θ ´ | u p s q ´ v p s q| } L ds ‰ (2.15)where C “ C p f q . We next estimate the term B “ ż T }| u p s q| θ ´ | u p s q ´ v p s q| } L ds . Applying the Hölder inequality, we obtain(2.16) B ď ż T } u p s q} θ ´ L θ } u p s q ´ v p s q} L θ ds . We next write 2 θ as 2 θ “ η ` p ´ η q θ ˚ which is equivalent to(2.17) η “ d ` ´ θ p d ´ q ă θ ă θ ˚ implies 0 ă η ă
1. Using the above decomposition of θ in (2.16)together with a Hölder inequality, we get B ď ż T } u p s q} p θ ´ q ηθ L } u p s q} θ ˚p θ ´ qp ´ η q θ L θ ˚ } u p s q ´ v p s q} ηθ L } u p s q ´ v p s q} θ ˚p ´ η q θ L θ ˚ ds ď } u p s q} p θ ´ q ηθ L p L q } u p s q ´ v p s q} ηθ L p L q ż T } u p s q} θ ˚p θ ´ qp ´ η q θ L θ ˚ } u p s q ´ v p s q} θ ˚p ´ η q θ L θ ˚ ds . (2.18)Applying again the Hölder inequality to the integral term, we obtain, ż T } u p s q} θ ˚p θ ´ qp ´ η q θ L θ ˚ } u p s q ´ v p s q} θ ˚p ´ η q θ L θ ˚ ds ď T η ´ ż T } u p s q ´ v p s q} θ ˚ L θ ˚ ds ¯ ´ ηθ ˆ ´ ż T } u p s q} θ ˚ L θ ˚ ds ¯ p θ ´ qp ´ η q θ . (2.19)The estimates (2.18) and (2.19) together with the Young inequality give B ď CT η M θ ´ θ p θ ˚ p ´ η q` η q “ } u ´ v } L p L q ` } u ´ v } L θ ˚ p L θ ˚ q ‰ . (2.20)We next choose T ą C p α q C “ T ` T η M θ ´ θ p θ ˚ p ´ η q` η q ‰ “ . The estimates (2.15) to (2.20) imply that, for 0 ă T ď T , } F (cid:126) u ´ F (cid:126) v } Y ď C p α q C “ T ` T η M θ ´ θ p θ ˚ p ´ η q` η q ‰ } (cid:126) u ´ (cid:126) v } Y ď } (cid:126) u ´ (cid:126) v } Y . (2.22)From the estimates (2.14) and (2.22), we deduce that F is a strict contraction and thus hasa unique fixed point (cid:126) u ” (cid:126) u p (cid:126) u q in Y satisfying(2.23) } (cid:126) u p (cid:126) u q} Y ď C p α q} (cid:126) u } H The fact that (cid:126) u p t q “ p u p t q , B t u p t qq also belongs to C pr , T s , H q is standard and left to thereader. Likewise, we leave it to the reader to verify that the map p t , (cid:126) u q P r , T s ˆ H ÞÑ (cid:126) u p t q P H is jointly continuous.We now turn to property (2). To show that (cid:126) u P H Ñ (cid:126) u p τ q ” S α p τ q (cid:126) u P H is Lipschitzcontinuous on the bounded sets of H , we choose (cid:126) u and (cid:126) v in the ball B H p , K q . Let T ą M be defined in (2.11). Arguing as above (see the inequality (2.22)),we obtain the following inequality for 0 ď T ď T ,(2.24) } F p (cid:126) u , (cid:126) u q ´ F p (cid:126) v , (cid:126) v q} Y T ď C p α q} (cid:126) u ´ (cid:126) v } H ` } (cid:126) u ´ (cid:126) v } Y T , ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 13 and thus, the fixed points (cid:126) u p (cid:126) u q and (cid:126) v p (cid:126) v q satisfy:(2.25) } (cid:126) u p (cid:126) u q ´ (cid:126) v p (cid:126) v q} Y T ď C p α q} (cid:126) u ´ (cid:126) v } H . If the solutions (cid:126) u p (cid:126) u q and (cid:126) v p (cid:126) v q exist on a time interval r , T ˚ q , where T ˚ ą T , we repeatthe above proof by considering now the ball in H of center (cid:126) u p (cid:126) u qp T q and radius K ą v p (cid:126) v qp T q also belongs to this new ball and replacing the non-linearity f p . q by f p . ` u p (cid:126) u qp T qq ´ f p u p (cid:126) u qp T qq . Repeating this process a finite number of timesshows that the map is Lipschitz continuous up to any time ˜ T ă T ˚ and therefore on all of r , T ˚ q . The above inequality also implies the uniqueness of the solution of p KG q α .We next want to show the property (3), namely that the map (cid:126) u P H ÞÑ u p (cid:126) u q P X X L θ ˚ pp , T q , L θ ˚ p R d qq is a C -map. To this end, we will first go back to the mapping F : p (cid:126) u , (cid:126) u q P B H p , K q ˆ Y ÞÑ F p (cid:126) u , (cid:126) u q P Y which has been defined by (2.13). And then, for t ě T , proceed like in the proof of theproperty (2). Clearly the map F p (cid:126) u , (cid:126) u q is di ff erentiable with respect to the variable (cid:126) u sinceit is a linear map in (cid:126) u . The di ff erentiability with respect to the variable (cid:126) u P Y is provedas follows (we only indicate the main arguments and leave the details to the reader). Let (cid:126) h “ p h , k q P Y be small. Applying Lemma 2.2, one sees that the proof of the di ff erentiabilityreduces to proving that(2.26) } f p u ` h q ´ f p u q ´ f p u q h } L pp , T q , L q “ o p} (cid:126) h } Y q . As above, using the hypothesis p H . q f , using the fact that 0 ă β ď d ´ and the classicalSobolev embeddings, we may write } f p u ` h q ´ f p u q ´ f p u q h } L pp , T q , L q “ ››› ż p f p u p s q ` λ h p s qq ´ f p u p s qqq h p s q d λ ››› L pp , T q , L q ď C ż T }p| h p s q| β ` | h p s q| θ ´ q| h p s q| } L ds ď C “ T } h } ` β L p H q ` ż T }| h p s q| θ ´ | h p s q| } L ds ‰ . (2.27)We remark that the last term in the right-hand side of the inequality (2.27) can be estimatedas in the inequalities (2.18) and (2.19). We thus deduce from the inequalities (2.18), (2.19)and (2.27) that(2.28) } f p u ` h q ´ f p u q ´ f p u q h } L pp , T q , L q “ O p} (cid:126) h } ` δ Y q , where δ “ min p β, η q and η ą F p (cid:126) u , (cid:126) u q is di ff erentiable with respect to the variable (cid:126) u P Y . The derivative of F p (cid:126) u , (cid:126) u q with respect to p (cid:126) u , (cid:126) u q is given by D F p (cid:126) u , (cid:126) u q “ p D F , D F qp (cid:126) u , (cid:126) u q ,where D F p (cid:126) u , (cid:126) u q “ B t D F p (cid:126) u , (cid:126) u q and(2.29) p D F p (cid:126) u , (cid:126) u qp (cid:126) v , (cid:126) v qqp t q “ S ,α p t q v ` S ,α p t q v ` ż t S ,α p t ´ s q f p u p s qq v p s q ds . We let to the reader to check that this derivative is continuous with respect to p (cid:126) u , (cid:126) u q .Finally, we remark that, with the choice of the time T made in (2.21), the mapping F p (cid:126) u , . q : (cid:126) u P Y T ÞÑ F p (cid:126) u , (cid:126) u q P Y T is a uniform contraction on B H p , K q . We may thusapply the uniform contraction principle as stated for example in [12, Theorem 2.2 on Page25], which implies that (cid:126) u P B H p , K q ÞÑ (cid:126) u p (cid:126) u q P Y T is of class C .We now return to the H ˆ H -regularity question, that is, prove the regularity property(5). Assuming this regularity for now, taking a derivative of p KG q α yields B t v ` α B t v ´ ∆ v ` v ´ f p u q v “ v stands for any of the derivatives B x j u , 1 ď j ď d . The data for (2.30) belong to H by assumption. We now perform the same estimates as in (2.15)-(2.20) to conclude that } (cid:126) v } Y ď C }p u , u q} H ˆ H ` } (cid:126) v } Y , see especially (2.20), (2.22). As above, these estimates require T to be su ffi ciently small.To be precise, the smallness here is determined by u alone through the constant M , see(2.20). It follows that } (cid:126) v } Y ď C }p u , u q} H ˆ H which is the desired regularity estimate. In order to pass from an a priori bound toa regularity statement we follow a standard procedure involving di ff erence quotients:letting (cid:126) e j be the coordinate vectors in R d we define with h ą v p h q j p x q : “ h ´ p u p x ` h (cid:126) e j q ´ u p x qq . By the argument leading to the a priori estimate we obtain ›› (cid:126) v p h q j ›› Y ď C }p u , u q} H ˆ H uniformly in h ą
0. Passing to suitable weak limits, we obtain the H ˆ L regularity ofthe derivatives of u , as desired.Finally, we turn to the case of small data. We will only provide a sketch of the mainargument. In the hypothesis p H . q f , we can choose β ą ă β ă
1. We recall that, for any y P R ,(2.31) | f p y q| ď C p| y | β ` | y | θ ´ q| y | ď C p| y | ` β ` | y | θ ˚ q . Proceeding as before, applying Lemma 2.2, using the inequality (2.31), one gets, for t ě } u } L θ ˚ ,β pp , t q , L θ ˚ q ` } e β s (cid:126) u } L pp , t q , H q ď C r}p u , u q} H ˆ L ` } u ` β } L ,β pp , t q , L q ` }| u | θ ˚ } L ,β pp , t q , L q s . ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 15
Applying the Hölder inequality, one deduces from the above inequality that, for t ě } u } L θ ˚ ,β pp , t q , L θ ˚ q ` } e β s (cid:126) u } L pp , t q , H q ď C r}p u , u q} H ˆ L ` } e β s (cid:126) u } ` β L pp , t q , H q ` } u } θ ˚ L θ ˚ ,β pp , t q , L θ ˚ q s , (2.32)where we used that β ą
0. For small data the method of continuity implies global existenceand smallness of the norms on the left-hand side. In particular, we have exponentialconvergence to zero in the energy (see also [35]). (cid:3)
In Section 3, we will linearize the equation p KG q α around an equilibrium point. Moregenerally, we can linearize the Klein-Gordon equation p KG q α along any solution of theequation p KG q α . This leads us to consider the following a ffi ne equation(2.33) w tt ` α w t ´ ∆ w ` w ´ f p u ˚ p t , x qq w “ G , p w , w t qp q ” (cid:126) w p q “ (cid:126) w P H , where u ˚ p t , x q P X τ X L θ ˚ pp , τ q , L θ ˚ p R d qq , τ ą
0, and G P L pp , τ q , L p R d qq . Theexistence (and uniqueness) of a solution (cid:126) w ” p w , B t w q P C pr , τ q , H q is classical if thedimension d is equal to 1 ,
2. So we will state this existence result and the correspondingStrichartz estimates only in the case where d ě Proposition 2.4.
Let d ě and α ě . Assume that u ˚ p t , x q P X τ X L θ ˚ pp , τ q , L θ ˚ p R d qq andthat G P L pp , τ q , L p R d qq . Then the equation (2.33) admits a unique solution (cid:126) w ” p w , B t w q P C pr , τ q , H q . Moreover, the solution (cid:126) w of (2.33) satisfies the following bound, for ď τ ă τ , (2.34) } (cid:126) w } L pp ,τ q , H q ` } w } L q pp ,τ q , L px q ď C p α, τ q “ } (cid:126) w } H ` } G } L pp ,τ q , L x q ‰ , where q ` dp “ d ´ , ď p ă 8 , q ě , and q ` d ´ p ď d ´ . The constant C p α, τ q ” C p α, τ, u ˚ q ě depends only on α , τ and the normof u ˚ in the space X τ X L θ ˚ pp , τ q , L θ ˚ p R d qq .If u ˚ , G and the initial data are radial functions, then (cid:126) w is a radial solution.Proof. This proposition can be proved in the same way as Theorem 2.3, by consideringthe term f p u ˚ p t , x qq w ` G as a non-linearity. The changes are minor in the fixed pointargument used in the proof of Theorem 2.3. Here Y and F “ p F , F q “ p F , B t F q simplybecome: Y ” Y T ” t (cid:126) w P L pp , τ q , H q with w P L θ ˚ pp , τ q , L θ ˚ p R d qqu . and p F p (cid:126) w , (cid:126) w qqp t q “ S ,α p t q w ` S ,α p t q w ` ż t S ,α p t ´ s qp f p u ˚ p s qq w p s q ` G p s qq ds . We obtain estimates similar to (2.22), where now M is replaced by the norm of u ˚ in X τ X L θ ˚ pp , τ q , L θ ˚ p R d qq . If the time T defined in (2.21) is larger than τ , then we haveproved the existence (and uniqueness) of the solution (cid:126) w p (cid:126) w q P Y T and the estimates (2.34) follow from Lemma 2.2. If T ă τ , we repeat the above proof by taking as initial data p (cid:126) w p (cid:126) w qqp T q and by replacing f p u ˚ p t , x qq w p t , x q ` G p t , x q by f p u ˚ p t ` T , x qq w p t ` T , x q ` G p t ` T , x q We repeat this argument a finite number of times till we reach the time τ . (cid:3) Definition of the functional K and the Nehari manifold. We introduce the func-tional K : ϕ P H p R d q ÞÑ K p ϕ q P R , defined by K p ϕ q “ ż R d p| ∇ ϕ | ` ϕ ´ ϕ f p ϕ qq dx , and introduce the Nehari manifold(2.35) N “ t ϕ P H rad p R d q | K p ϕ q “ u . The Nehari manifold arises naturally in the study of elliptic equations. The “Ambrosetti-Rabinowitz" hypothesis p H . q f allows to prove the following lemmas, which will be usedalong this paper. The first one is trivial. Lemma 2.5.
Assume that Hypothesis p H . q f holds. Then, for any p ϕ, ψ q P H p R d q ˆ L p R d q , wehave (2.36) γ p} ϕ } H ` } ψ } L q ď p ` γ q E pp ϕ, ψ qq ´ K p ϕ q . Proof.
We simply write γ p} ϕ } H ` } ψ } L q “ p ` γ q E pp ϕ, ψ qq ´ K p ϕ q ´ p ` γ q} ψ } L ` ż R d ` p ` γ q F p ϕ q ´ ϕ p x q f p ϕ p x qq ˘ dx ď p ` γ q E pp ϕ, ψ qq ´ K p ϕ q , (2.37)where the integral is nonpositive by p H . q f . (cid:3) Corollary 2.6.
Suppose (cid:126) u p t q “ p u p t q , B t u p t qq is a strong solution of p KG q α defined on the maximalinterval ď t ă T ˚ . Assume inf ď t ă T ˚ K p u p t qq ą ´8 . Then T ˚ “ 8 , i.e., the solution is global.Proof. By Lemma 2.5, we have for some finite M and all 0 ď t ă T ˚ } (cid:126) u p t q} H ď p ` γ q E p u p t q , B t u p t qq ` M ď p ` γ q E p u p q , B t u p qq ` M where the second line holds by the decrease of the energy. Since finite time blowup meansthat } (cid:126) u p t q} H goes to infinity in finite time along some subsequence, we obtain the result. (cid:3) ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 17
The proof of the next lemma uses a convexity argument and follows the lines of theproof of [42] and [40, Corollary 2.13]. We denote the nonlinear evolution by S α p t q . Lemma 2.7.
Assume that the hypotheses p H . q f and p H . q f hold. Assume that p u p t q , B t u p t qq isa solution of p KG q α defined on r , T ˚ q where T ˚ P p , is maximal. If K p u p t qq ď ´ δ (where δ ą ), for t ď t ă T ˚ , then T ˚ ă 8 , i.e., the solution blows up in finite time. From Lemmas 2.5 and 2.7 we immediately deduce the following result.
Corollary 2.8.
Assume that the initial energy E p (cid:126) u q is negative. Then the solution blows-up infinite time T ˚ ă `8 .Proof of Lemma 2.7. We assume without loss of generality that t “
0. We also assumetowards a contradiction that T ˚ “ 8 . In order to show that S α p t qp u , u q blows up in finitetime, we use a convexity argument as in [42]. Assume that S α p t qp u , u q exists for all t ě y p t q “ } u p t q} L ` α ż t } u p s q} L ds . We have(2.38) y p t q “ p u p t q , u p t qq ` α } u p t q} L “ p u p t q , u p t qq ` α } u p q} L ` α ż t p u p s q , u p s qq ds and(2.39) : y p t q “ } u p t q} L ` p u p t q , : u p t q ` α u p t qq“ } u p t q} L ` p u p t q , p ∆ u ´ u ` f p u qqp t qq“ } u p t q} L ´ K p u p t qq . Thus,(2.40) : y p t q ě } u p t q} L ` δ ě δ. We deduce from (2.40) that lim t Ñ`8 y p t q “ `8 , and therefore lim t Ñ`8 y p t q “ `8 .Next, we note that : y p t q “ } u p t q} L ´ K p u p t qq“ p ` γ q} u p t q} L ` γ } u p t q} H ´ p ` γ q E p t q´ ż R d ` p ` γ q F p u p t qq ´ u p t q f p u p t qq ˘ dx (2.41)where we have set for simplicity E p t q “ E pp u p t q , u p t qqq . But, we have E p t q “ ´ α } u p t q} L and E p t q “ E p q ` ż t E p s q ds “ E p q ´ α ż t } u p s q} L ds . Using p H . q f and the definition of y p t q , we can also write, for t ě : y p t q ě p ` γ q} u p t q} L ` γ } u p t q} H ´ p ` γ q E p q ` α p ` γ q ż t } u p s q} L ds . (2.42)For the sake of illustration, assume first that α “
0. Since y p t q Ñ 8 , we infer from (2.42)that for large t : y p t q ě p ` γ q} u p t q} L (2.43)Then | y p t q| ď } u p t q} L } u p t q} L whence : y p t q ě ` γ y p t q y p t q This implies that d dt p y ´ δ p t qq ă δ “ γ {
2. Since y ´ δ p t q Ñ t Ñ 8 we must have ddt p y ´ δ qp t q ă t “ t ą ddt p y ´ δ qp t q ď ddt p y ´ δ qp t q ă t ě t .But then y ´ δ p t q “ t ą t which is a contradiction.For α ą
0, we claim that there exists c ą : y p t q y p t q ´ c y p t q ą d dt ` y ´p c ´ q ˘ p t q “ ´p c ´ q y ´ c ´ p t qp : y p t q y p t q ´ c y p t qq ă y p t q : y p t q ´ c y p t q ě ´ } u } L ` α ż t } u p s q} L ds ¯ (2.45) ¨ ´ p ` γ q} u p t q} L ` γ } u p t q} H ´ p ` γ q E p q ` α p ` γ q ż t } u p s q} L ds ¯ ´ c ” } u } L } u } L ` α ´ż t } u p s q} L ds ¯ ´ż t } u p s q} L ds ¯ ` α } u p q} L ı . ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 19
But, for any ε ą
0, we estimate the term in brackets as follows: c ” } u } L } u } L ` α ´ż t } u p s q} L ds ¯ ´ż t } u p s q} L ds ¯ ` α } u p q} L ı ď c p ` ε q ´ } u } L } u } L ` α ´ż t } u p s q} L ds ¯ ´ż t } u p s q} L ds ¯ ¯ ` c ´ ` ε ¯ α } u p q} L ď c p ` ε q ´ } u } L ` α ż t } u p s q} L ds ¯´ } u } L ` α ż t } u p s q} L ds ¯ ` c ´ ` ε ¯ α } u p q} L . Setting b “ c p ` ε q , C “ c α p ` ε q} u p q} L , we may replace the right-hand side of thisinequality by ď y p t q ´ b } u } L ` b α ż t } u p s q} L ds ¯ ` C From the last inequality and from (2.45), we deduce that y : y p t q ´ c y p t q ě y p t q ! p ` γ ´ b q} u p t q} L ` α p ` γ ´ b q ż t } u p s q} L ds ` γ } u p t q} H ´ p ` γ q E p q ) ´ C “ y p t q Ψ p t q ´ C (2.46)where Ψ p t q is defined by the term in braces.We now adjust the constants c ą ε ą ` γ ´ b ą
0, 1 ` γ ´ b ą
0. Wenow pick η ą ` γ ´ b ą η, γ ´ η ´ αη ą Ψ p t q from below: Ψ p t q “ „´ ` γ ´ b ´ η ¯ } u p t q} L ` α p ` γ ´ b q ż t } u p t q} L ds ` γ } ∇ u p t q} L ` ´ γ ´ η ´ αη ¯ } u p t q} L ` η y p t q ´ p ` γ q E p q ı ě η y p t q ´ p ` γ q E p q ` q p t q where q p t q ě
0. From (2.46), we infer that, for t ě y p t q : y p t q ´ c y p t q ě y p t qr η y p t q ´ p ` γ q E p q ` q p t qs ´ C . (2.47)Since y p t q , y p t q Ñ 8 as t Ñ 8 , we are done. (cid:3)
Spectral properties.
Suppose we have a stationary solution ϕ P H p R d q to p KG q α ,namely, ´ ∆ ϕ ` ϕ ´ f p ϕ q “ C ,β for some β ą
0. Solving p KG q α for u “ ϕ ` v yields v tt ` α v t ´ ∆ v ` v ´ f p ϕ q v “ N p ϕ , v q (2.48)where N p ϕ , v q “ f p ϕ ` v q ´ f p ϕ q ´ f p ϕ q v . Set L “ ´ ∆ ` I ´ f p ϕ q . Rewrite (2.48) inthe form B t ˆ vv t ˙ “ ˆ ´ L ´ α ˙ ˆ vv t ˙ ` ˆ N p ϕ , v q ˙ (2.49)Denoting the matrix operator on the right-hand side by A α , and setting (cid:126) v : “ ` vv t ˘ , we maywrite (2.49) in the form B t (cid:126) v “ A α (cid:126) v ` (cid:126) N The spectral properties of L stated in the following lemma are standard, see for exam-ple [32] and the references cited here. Lemma 2.9.
The operator L is self-adjoint with domain H p R d q . The spectrum σ p L q consists ofan essential part r , , which is absolutely continuous, and finitely many eigenvalues of finitemultiplicity all of which fall into p´8 , s . The eigenfunctions are C ,β with β ą and the onesassociated with eigenvalues below are exponentially decaying. Over the radial functions, alleigenvalues are simple.Proof. The essential spectrum equals r , by the Weyl criterion. The Agmon-Kurodatheory on asymptotic completeness guarantees that there are no imbedded eigenvaluesand no singular continuous spectrum. Thus, the spectral measure restricted to r , ispurely absolutely continuous. The Birman-Schwinger criterion shows (due to the rapiddecay of the potential f p ϕ q ) that there are only finitely many eigenvalues of L which are ď
1, counted with multiplicity. The C ,β property of the eigenfunctions is standard ellipticregularity (Schauder estimates) since ϕ is smooth, and so f p ϕ q is Hölder regular.For the sake of completeness we remark that the threshold 1 may be an eigenvalue ora resonance. To illustrate what this means, consider R . Then this distinction refers tothe to fact that solutions to L ψ “ ψ either decay like | x | ´ (which means ψ P L is aneigenfunction) or like | x | ´ , the latter implying that ψ R L p R q (this is the resonant case).We remark that over the radial functions only the resonant case can occur. However, noneof this finer analysis at energy 1 is relevant for our purposes.The exponential decay of the eigenfunctions with eigenvalues below 1 is known asAgmon’s estimate. The simplicity of the radial eigenfunctions is immediate from thereduction to an ODE on p , with a Dirichlet condition at r “
0. Let us elaborate on thekernel of L , since it is important in our construction. We set L v “ v ‰ H . Then ´ ∆ v ` v ´ f p ϕ q v “ ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 21
We already notes that v P C ,β p R d q , and that v p r q decays exponentially. Set u p r q “ r d ´ v p r q .Then u p q “ u p r q Ñ r Ñ 8 (exponentially in fact), and it satisfies the equation ´ u p r q ` u p r q ´ p d ´ qp d ´ q u p r q r ´ f p ϕ q u p r q “ , r ą e r and onedecaying like e ´ r as r Ñ 8 . Only the latter can lie in the kernel and it does so if and onlyif it satisfies the boundary condition u p q “
0. In this case the kernel has dimension 1otherwise it consists only of t u . (cid:3) We now analyze the spectral properties of the matrix operator A α . Lemma 2.10. ‚ The operator A α has discrete spectrum if and only if L does. The essentialspectrum of A α lies strictly to the left of the imaginary axis, i.e., in (cid:60) p z q ă ´ δ p α q forsome δ p α q ą . The spectrum of A α on the imaginary axis is either empty or t u . In thelatter case, is an eigenvalue of A α and this occurs if and only if is an eigenvalue of L .Then dim p Ker p L qq “ , in which case is a simple eigenvalue. The eigenvalues of A α areprecisely ´ α ˘ b α ´ µ where µ P σ p L q is an eigenvalue. – If α ě , then the discrete spectrum of A α lies only on the real axis. – If ă α ă , in addition to real eigenvalues, there may also be eigenvalues on the line (cid:60) p z q “ ´ α resulting from eigenvalues of L in the gap p , s . ‚ The essential spectrum of L gives rise to essential spectrum σ ess p A α q of A α as follows: – If ă α ď , σ ess p A α q is contained in the line (cid:60) p z q “ ´ α and consists of ´ α ˘ i β , β ě a ´ α . – If α ą , σ ess p A α q consists of the entire line (cid:60) p z q “ ´ α and of the interval r´ α ´ a α ´ , ´ α ` a α ´ s Proof.
We need to address the solvability of the system A α ˆ u u ˙ “ z ˆ u u ˙ over the domain H rad p R d q ˆ H rad p R d q of A α . This means that u “ zu ´ L u ´ α u “ zu which is the same as u “ zu p L ` α z ` z q u “ A α if and only if2 α z ` z P σ p´ L q Taking λ P σ p L q , this means that z “ ´ α ˘ a α ´ λ, λ P σ p L q . (2.51)This relation establishes all the claims concerning the point spectrum of A α . Let now τ belong to the resolvent set ρ p A α q of A α . Then, for any p , v q P H rad , the system(2.52) p A α ´ τ Id q ˆ u u ˙ “ ˆ v ˙ has a unique solution p u , u q in H rad , which implies that ´ L u ´ p τ ` ατ q u “ v has a unique solution u and thus τ ` ατ ” ´ λ does not belong to the spectrum of ´ L ,that is, τ ‰ ´ α ˘ a α ´ λ, λ P σ p L q and we are done. (cid:3) The discrete spectrum of A α (and therefore of L ) is important to our analysis. In fact,the strongly unstable manifold of the linear evolution e tA α as t Ñ 8 corresponds exactlyto spectrum of A α in the right-half plane which occurs if and only if L exhibits negativeeigenvalues. In the generality we assume here we cannot determine whether this is the caseor not, and so our arguments need to be flexible enough to account for both possibilities.However, consider the following additional condition, where γ is as in p H . q f : for any φ P H , ż R d “ φ p x q f p φ p x qq ´ p ` γ q φ p x q f p φ p x qq ‰ dx ě ϕ ‰ x L ϕ , ϕ y “ ż R d p| ∇ ϕ | ` ϕ ´ f p ϕ q ϕ q dx “ ´ γ ż R d f p ϕ q ϕ dx ` ż R d rp ` γ q f p ϕ q ϕ ´ f p ϕ q ϕ s dx ď ´ γ } ϕ } H ă K p ϕ q “
0. Therefore, L has negative eigenvalues. We leave it tothe reader to check that the class of nonlinearities f given by a sum and di ff erence ofpure powers as in (1.4) satisfy (2.53). Hence, for such nonlinearities all nonzero stationarysolutions are linearly unstable. In other words, under the additional condition (2.53) allnonzero equilibria give rise to a strongly unstable manifold of e tA α . ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 23 F igure
1. The spectrum of A α for 0 ă α ă
13. P roof of T heorem p ϕ , ϕ q P H rad ,we will first show that, if S α p t qp ϕ , ϕ q does not blow up in finite time, then there existsa sequence of times t n going to `8 such that S α p t n qp ϕ , ϕ q converges to an equilibriumpoint p Q ˚ , q .3.1. Convergence to an equilibrium p Q ˚ , q along a subsequence. Denote the evolutionoperator of p KG q α by S α p t q and for p ϕ , ϕ q P H rad , let (cid:126) u p t q : “ S α p t qp ϕ , ϕ q . We have thefollowing trichotomy for the forward evolution of p KG q α :(FTB) (cid:126) u p t q blows up in finite positive time.(GEB) (cid:126) u p t q exists globally and the trajectory t (cid:126) u p t q , t ě u is bounded in H rad ,(GEU) (cid:126) u p t q exists globally and the trajectory t (cid:126) u p t q , t ě u is unbounded in H rad . Remark 3.1.
Several remarks have to be made at this stage. (i):
From Corollary 2.8, we know that if E p ϕ , ϕ q ă
0, then S α p t qp ϕ , ϕ q blows upin finite time. Thus, in the study of the cases (GEB) and (GEU), we only need toconsider solutions (cid:126) u p t q ” S α p t qp ϕ , ϕ q such that, for any t ě E p u p t q , B t u p t qq ě . (ii): Assume now that a solution (cid:126) u p t q ” S α p t qp ϕ , ϕ q of p KG q α satisfies the properties p H . q f , p H . q f and (3.1). Assume moreover, that the exponent θ in p H . q f satisfiesthe bound(3.2) θ ă ` d . Then, arguing exactly as in [23, Lemma 4.2], one can prove that every globalsolution S α p t qp ϕ , ϕ q is bounded in H . In this proof, the upper bound (3.2) of θ plays a crucial role. (iii): Now, let us turn to the case where 1 ` d ď θ ď dd ´ . We consider a globalsolution p u p t q , B t u p t qq “ S α p t qp ϕ , ϕ q . In this case, arguing as in [23, Page 59] byintroducing the auxiliary equation satisfied by B t (cid:126) u p t q : “ pB t u p t q , B t u p t qq , one showsthat B t (cid:126) u p t q converges to p , q in L p R d qˆ H ´ p R d q . From this convergence property,we deduce that K p u p t qq converges to 0 as t goes to infinity.We first make a simple observation concerning the case (GEU). Later in Section 3.2, weshall show that (GEU) cannot occur. Lemma 3.2.
Assume that the hypothesis p H . q f and p H . q f hold. In the case (GEU), we mayassume that there exist a sequence of times t n and a sequence of numbers δ n , δ n ď , such thatt n Ñ `8 as n
Ñ `8 and that (3.3) K p u p t n qq “ δ n , with lim n Ñ`8 δ n “ . Proof. If K p u p t qq ě T ă t , then the trajectory is bounded by Lemma 2.5.So there exists a sequence τ n Ñ 8 with K p u p τ n qq ă
0. If K p u p t qq ă ´ δ for all times T ă t ă 8 , where δ ą (cid:3) For the case (GEB) we shall now also construct such a sequence, albeit with δ n “ K p u qp t n q possibly being positive. Proposition 3.3.
In the case (GEB) there exists a sequence t n Ñ 8 with K p u p t n qq Ñ and }B t u p t n q} L Ñ as n Ñ 8 .Proof.
Taking the inner product in L of the equation p KG q α with u and integrating by partsyields ż ` | ∇ u | ` u ´ f p u q u ˘ dx ` α ż u B t u dx “ ż pB t u q dx ´ ddt ż B t uu dx . (3.4)Notice that for smooth (or, more precisely, H ˆ H ) initial data, this integration by partsis justified. Moreover, for H ˆ L initial data, we conclude that the map p ϕ , ϕ q P H ˆ L ÞÑ ż B t uu dx P R is C with derivative given by (3.4). In the sequel, we shall take (3.4) as a definition for ddt ż B t uu dx . ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 25
We wish to choose a sequence t n Ñ `8 so that each term on the right-hand side of (3.4),when evaluated at t n , tends to 0 as n Ñ `8 . First, we rewrite (3.4) as ż R d ` | ∇ u | ` u ` pB t u q ´ f p u q u ˘ dx ` α ż R d u B t u dx “ ż R d pB t u q dx ´ ddt ż B t uu dx . (3.5)We now make the following Claim:
There exists a sequence t n Ñ `8 such that lim n Ñ`8 }B t u p t n q} L “ n Ñ`8 }B t u p t n q} L ´ ddt ş pB t uu qp t n , x q dx “ . Since the energy is nonnegative, we have that for any 0 ă T ă ˜ T ,2 α ż ˜ TT }B t u p s q} L ds “ E p (cid:126) u p T qq ´ E p (cid:126) u p ˜ T qq ď K “ E p (cid:126) u p qq , and ˇˇˇˇˇż ˜ TT ddt ż pB t uu qp s , x q dxds ˇˇˇˇˇ “ ˇˇ xB t u p ˜ T q , u p ˜ T qy ´ xB t u p T q , u p T qy ˇˇ ď ˜ K , where ˜ K ą T , and ˜ T . We now distinguish two cases: Case 1 : If there exists T ą t ě T ,2 }B t u p t q} L ´ ddt xB t u p t q , u p t qy does not change sign (and, for example, is nonnegative), then for any T ď T ă ˜ T , ż ˜ TT ˆ }B t u p t q} L ` ˇˇˇˇ }B t u p t q} L ´ ddt xB t u p t q , u p t qy ˇˇˇˇ˙ dt “ ż ˜ TT ˆ }B t u p t q} L ` ˆ }B t u p t q} L ´ ddt xB t u p t q , u p t qy ˙˙ dt ď K ` ˜ K . This allows us to show that there exists a sequence t n Ñ `8 , such that }B t u p t n q} L ` ˇˇˇˇ }B t u p t n q} L ´ ddt |xB t u p t n q , u p t n qy ˇˇˇˇ Ñ t n Ñ `8 . (3.6) Case 2 : There exists a sequence of times τ m Ñ `8 such that A p τ m q : “ }B t u p τ m q} L ´ ddt xB t u p τ m q , u p τ m qy “ . To conclude, we need
Lemma 3.4.
There exists a subsequence τ m j and η ą such that the function A p t q is uniformlycontinuous on I “ ď τ mj r τ m j ´ η , τ m j ` η s . Proof.
We write A p t q “ E p (cid:126) u p t qq ` ż R d “ F p u p t , x qq ´ f p u q u p t , x q ‰ dx ` αµ p t q (3.7)where µ p t q “ x u p t q , B t u p t qy . Since E p (cid:126) u p t qq is continuous and has a limit as t Ñ `8 , E p (cid:126) u p t qq is uniformly continuous on r , `8q . Since τ m Ñ `8 , there exist a subsequence (that westill denote τ m for ease of notation) and e P H rad p R d q such that u p τ m q converges weakly to e in H p R d q as m goes to infinity. Thus, using the fact that the injection H rad p R d q Ñ L p p R d q ,2 ă p ă ˚ , is compact, we deduce that u p τ m q converges strongly to e p x q in L p p R d q as τ m goes to infinity. Furthermore, since t ÞÑ ş f p u q u p t , x q dx is continuous and ż “ F p u p τ m , x qq ´ f p u q u p τ m , x q ‰ dx Ñ ż “ F p e p x qq ´ f p e q e p x q ‰ dx as τ m Ñ `8 , we obtain the uniform continuity on I of the middle term in (3.7).Integration by parts shows that µ p t ` δ q “ µ p t q ` ż t ` δ t ż R d ` pB t u q ´ α p u B t u q ´ | ∇ u | ´ u ` f p u q u ˘ dxds Since (cid:126) u p t q “ S α p t qp ϕ , ϕ q , t ě
0, is bounded in H , we deduce that µ p t q is uniformlycontinuous on r , `8q . (cid:3) Now, the construction of a sequence t n Ñ `8 such that (3.6) holds follows by a standardinductive procedure. Indeed, assume that we have constructed a sequence t t ă ¨ ¨ ¨ ă t N u such that @ ď n ď N , }B t u p t n q} L ď ´ n , | A p t n q| ď ´ n . Let (cid:15) “ ´p N ` q . Since A p τ m j q “
0, according to Lemma 3.4 there exists η ą t P r τ m j ´ η, τ m j ` η s one has | A p t q| ď ´p N ` q . Then, since lim j Ñ`8 ż τ mj ` ητ mj ´ η }B t u } L p s q ds “ , we obtain that for j large enough, there exists s j P r τ m j ´ η, τ m j ` η s such that }B t u p s j q} L ď ´p N ` q . Choosing t N ` “ s j for j large enough ensures that t N ` ă t N ` and }B t u p t N ` q} L ď ´p N ` q , | A p t N ` q| ď ´p N ` q . From the
Claim above, and (3.5), we deduce thatlim n Ñ`8 ż R d ` | ∇ u | ` u ´ f p u q u ˘ p t n , x q dx “ n Ñ`8 K p u p t n qq “ (cid:3) ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 27
Next, by means of these vanishing results for K , we deduce the convergence to anequilibrium along a subsequence. Theorem 3.5.
Let α ą and (cid:126) u : “ p ϕ , ϕ q P H rad so that the solution (cid:126) u p t q exists for all timest ą . Let t n be a sequence of times such that K p u p t n qq “ δ n converges to , then there exists anequilibrium point (cid:126) u ˚ “ p Q ˚ , q P H rad such that (after possibly extracting a subsequence), (cid:126) u p t n q converges to p Q ˚ , q in H .Proof. From Lemma 2.5 we conclude thatsup n ě }p u p t n q , B t u p t n qq} H ă 8 We recall that without loss of generality, we may assume that E p u p t q , B t u p t qq ě , @ t ě . Since the left-hand side is non-increasing, there exists (cid:96) ě t Ñ`8 E p u p t q , B t u p t qq “ (cid:96) ě . In fact, from the equality valid for any t ď t , E p u p t q , B t u p t qq ´ E p u p t q , B t u p t qq “ α ż t t }B t u p s q} L ds , we deduce that ş t t }B t u p s q} L ds tends to 0, as t , t Ñ 8 .We consider the equations B tt u n ` α B t u n ´ ∆ u n ` u n ´ f p u n q “ p u n p q , B t u n p qq “ p u p t n q , B t u p t n qqp KG q n α By Theorem 2.3, there exists T ą C ą n , the solution p u n p t q , B t u n p t qq is in C pr´ T , T s , H q and, for ´ T ď t ď T , }p u n p t q , B t u n p t qq} H ď C . (3.9)In the case d “ d “
2, the inequality (3.9) implies that } u n } L pp´ T , T q , L p q ď C , for any2 ď p ă `8 . In the case 3 ď d ď
6, the estimate (2.23) in Theorem 2.3 implies that(3.10) } u n } L θ ˚ pp , T q , L θ ˚ q ď C . where θ ˚ “ d ` d ´ . By uniqueness, u n p t q “ u p t n ` t q . For any s , t P r´ T , T s , ż R d | u n p t q ´ u n p s q| dx “ ż R d ˇˇˇˇż ts B t u n p σ q d σ ˇˇˇˇ dx ď | t ´ s | ż R d ż ts |B t u n p σ q| d σ dx ď | t ´ s | ż t ` t n s ` t n }B t u p σ q} L d σ whence } u n p t q ´ u n p s q} L ď | t ´ s | ż t ` t n s ` t n }B t u p σ q} L d σ (3.11) ď T ż t n ` Tt n ´ T }B t u p σ q} L d σ ÝÑ n Ñ `8 . For s , t P r´ T , T s , and fixed p P p , ˚ q , interpolation gives the existence of a P p , q suchthat } u n p t q ´ u n p s q} L p ď } u n p t q ´ u n p s q} aL ˚ } u n p t q ´ u n p s q} ´ aL (3.12) À | t ´ s | ´ a ˆż t n ` Tt n ´ T }B t u p σ q} L d σ ˙ ´ a with a uniform constant in n . Fix 2 ă p ă p ă ˚ and set X : “ L p p R d q X L p p R d q . Thechoice of p , p depends on the nonlinearity f p u q through the parameters β, θ in p H . q f .We consider the family of functions p u n p t qq n in C pr´ T , T s ; X q . By the property (3.9), ď n P N , t Pr´ T , T s u n p t q Ă bounded set of H rad p R d q . Due to the compact embedding of H rad p R d q into X , we deduce that ď n P N , t Pr´ T , T s u n p t q Ă compact set of X Moreover, by (3.12), the family p u n p t qq n is equicontinuous in C pr´ T , T s ; X q . Thus, by thetheorem of Ascoli, (after possibly extracting a subsequence) the sequence u n p t q convergesin C pr´ T , T s ; X q to a function u ˚ p t q P C pr´ T , T s ; X q .Moreover, by (3.11) and (3.12), u ˚ p t q is constant on the time interval r´ T , T s . We shallsimply write u ˚ p t q ” u ˚ . Remark that we deduce from K p u n p qq Ñ u n p t q towards u ˚ in C pr´ T , T s ; X q thatlim n Ñ`8 } u n p q} H “ ż R d f p u ˚ q u ˚ dx . (3.13)For this implication we need to choose p , p close to 2 , ˚ , respectively, dependingon p H . q f .To summarize, we know that ‚ u n p t q Ñ u ˚ as n Ñ `8 in C pr´ T , T s ; X q and u ˚ : “ u ˚ p t q‚ B t u n p t q Ñ n Ñ `8 in L pp´ T , T q ; L p R d qq‚ p u n p t q , B t u n p t qq n is uniformly bounded in n in L pp´ T , T q ; H q and, in particular in L pp´ T , T q ; H q . ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 29
Taking these properties into account, one shows that p u n , B t u n q converges in the sense ofdistributions (i.e., D pp´ T , T q ˆ R d q ) towards p u ˚ , q as n Ñ `8 and that p u ˚ , q is anequilibrium point of p KG q α . Since p u n p q , B t u n p qq is uniformly bounded in H , with respectto n , there exists a subsequence (that we still label by n ) such that u n p q á u ˚ as n Ñ `8 weakly in H p R d q .Since u ˚ is an equilibrium point of p KG q α , the following equality holds: ż R d f p u ˚ q u ˚ dx “ ż R d p| ∇ u ˚ | ` p u ˚ q q dx . (3.14)The equalities (3.13) and (3.14) imply thatlim n Ñ`8 } u n p q} H “ } u ˚ } H (3.15)and thus, since u n p q á u ˚ as n Ñ `8 weakly in H p R d q , the convergence of u n p q towards u ˚ takes place in the strong sense in H p R d q . Moreover, the strong convergence of u n p q towards u ˚ in L p R d q and the property (3.11) imply the strong convergence of u n p s q towards u ˚ in L p R d q , uniformly in s P r´ T , T s . In summary, u n p . q Ñ u ˚ in C pp´ T , T q , L p R d qq . To finish the proof of Theorem 3.5 it remains to prove(3.16) B t u n p q Ñ L p R d q . As a first step towards the proof of property (3.16), we consider the equation satisfied by˜ u n : “ u n ´ u ˚ , namely $’&’% B tt ˜ u n ´ ∆ ˜ u n ` ˜ u n “ f p u n q ´ f p u ˚ q ´ α B t ˜ u n ˜ u n p q “ u n p q ´ u ˚ Ñ n Ñ `8 in H p R d qB t ˜ u n p q “ B t u n p q (3.17)We write u n ´ u ˚ “ w n ` v n where w n and v n are solutions of the following equations: $’&’% B tt w n ´ ∆ w n ` w n “ f p u n q ´ f p u ˚ q ´ α B t u n w n p q “ u n p q ´ u ˚ B t w n p q “ $’&’% B tt v n ´ ∆ v n ` v n “ v n p q “ B t v n p q “ B t u n p q . (3.19) The classical energy estimates for the Klein-Gordon equation imply that, for ´ T ď t ď T , }p w n , B t w n qp t q} H ď C ” } u n p q ´ u ˚ } H ` α ? T ˆż T ´ T }B t u n p s q} L ds ˙ ` ż T ´ T } f p u n qp s q ´ f p u ˚ q} L ds ı (3.20)Taking into account Hypothesis p H . q f , one has ż T ´ T } f p u n qp s q ´ f p u ˚ q} L ds ď C ż T ´ T }p u n p s q ´ u ˚ qp| u n | β ` | u ˚ | β ` | u n | θ ´ ` | u ˚ | θ ´ q} L ds (3.21)Here 0 ă β ă θ ´ ff ects the constant C .Actually, we can choose 0 ă β ă θ ´ ď β p {p p ´ q ď p . Applying the Hölderinequality, we obtain, ż T ´ T }p u n p s q ´ u ˚ qp| u n | β ` | u ˚ | β q} L ds ď CT } u n ´ u ˚ } L p I , L p q p} u n } β L p I , L p q ` } u ˚ } β L p I , L p q qď CT } u n ´ u ˚ } L p I , L p q p} u n } β L p I , H q ` } u ˚ } β L p I , H q q . (3.22)where 2 ď p ď p ă ˚ is fixed. Since u n Ñ u ˚ in C p I , X q , we conclude that the right-handside of (3.22) vanishes in the limit n Ñ 8 . We next estimate the term(3.23) ż T ´ T }p u n ´ u ˚ q| u ˚ | θ ´ q} L ds ď T } u n ´ u ˚ } L p L q } u ˚ } θ ´ L p L q ď C } u n ´ u ˚ } L p L q , which tends to 0 as n Ñ 8 . To bound the remaining term in (3.21), we argue as in theproof of Theorem 2.3. Indeed, from the estimates (2.15) to (2.19), we deduce that ż T ´ T }p u n p s q ´ u ˚ p s qq| u n p s q| θ ´ } L ds ďp T q η C } u n } p θ ´ q ηθ L p I , L q } u n ´ u ˚ } ηθ L p I , L q ˆ “ } u n } p ´ η q θ ˚ L θ ˚ p I , L θ ˚ q ` p T q ´ η } u ˚ } p ´ η q θ ˚ L p I , L θ ˚ q ‰ ď C p ` T q} u n ´ u ˚ } ηθ L p I , L q , (3.24)where, by (2.17), η “ d ` ´ θ p d ´ q . The right-hand side of the inequality (3.24) tends to 0 as n goes to infinity.Finally, in view of (3.20), (3.21), (3.22), (3.23) and (3.24), we conclude that }p w n p t q , B t w n p t q} H Ñ n Ñ `8 , (3.25) ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 31 uniformly in ´ T ď t ď T .By construction, v n “ p u n ´ u ˚ q ´ w n and, in particular, B t v n “ B t u n ´ B t w n . From (3.25)and the properties of }B t u n } L p I ; L p R d qq , we infer that }B t v n } L pp´ T , T q ; L p R d qq ď }B t u n } L pp´ T , T q ; L p R d qq ` ? T }B t w n } C pr´ T , T s ; L p R d qq Ñ n Ñ 8 .In the final step of the proof we shall turn this L t averaged vanishing of }B t v n p t q} L x as n Ñ 8 into vanishing in the uniform sense in t . The main tool for this is the following“observation inequality” for equation (3.19). Lemma 3.6.
For any T ą , there exists a positive constant c p T q ą , independent of n, suchthat }B t v n p q} L p R d q ď c p T q ż T ´ T ż R d |B t v n | dxds . (3.27) Proof.
For sake of simplicity, we set: B t v n p q ” B t u n p q “ v n . If ˆ v n denotes the Fourier transform of v n , we haveˆ v n p t , ξ q “ sin ´ t a | ξ | ` ¯a | ξ | ` v n p ξ q and therefore }B t ˆ v n p t , ¨q} L “ ż R d ˇˇˇˇ sin ˆ t b | ξ | ` ˙ˇˇˇˇ | ˆ v n p ξ q| d ξ as well as ż T ´ T }B t ˆ v n p t , ¨q} L dt “ ż T ´ T ż R d ˇˇˇˇ sin ˆ t b | ξ | ` ˙ˇˇˇˇ | ˆ v n p ξ q| d ξ dt “ ż R d ˜ż T ´ T ˇˇˇˇ sin ˆ t b | ξ | ` ˙ˇˇˇˇ dt ¸ | ˆ v n p ξ q| d ξ ě ˜ c p T q| ż R d ˆ v n p ξ q| d ξ , (3.28)where ˜ c p T q ą
0, since T ą
0. Indeed ż T ´ T ˇˇˇˇ sin ˆ t b | ξ | ` ˙ˇˇˇˇ dt “ ż T ´ T ¨˚˝ ´ cos ´ t a | ξ | ` ¯ ˛‹‚ dt “ T ´ sin ´ T a | ξ | ` ¯ a | ξ | ` . One easily sees that, for any T ą
0, there exists ˜ c p T q ą | ξ | ,(3.29) T ´ sin ´ T a | ξ | ` ¯ a | ξ | ` ě ˜ c p T q . The estimate (3.27) is then a direct consequence of (3.28), (3.29) and Plancherel’s theorem. (cid:3)
From the property (3.27) and the estimate (3.26), one deduces that }B t u n p q} L ď c p T q ” }B t u n } L pp´ T , T q ; L q ` ? T }B t w n } C pr´ T , T s ; L p R d qq ı Ñ n Ñ `8 and the theorem is proved. (cid:3)
Convergence property.
Let (cid:126) u “ p ϕ , ϕ q P H rad be so that the solution (cid:126) u p t q “ S α p t q (cid:126) u ” p u p t q , B t u p t qq exists globally and may be unbounded. Theorem 3.5 asserts thatthere exists a sequence of times t n Ñ `8 such that (cid:126) u p t n q Ñ p Q ˚ , q strongly in H rad , where Q ˚ is an equilibrium of p KG q α . We shall now show by contradiction that then necessar-ily (cid:126) u p t q Ñ p Q ˚ , q strongly in H rad as t Ñ 8 and hence the trajectory is bounded. Inother words, Theorem 3.5 implies that the ω -limit set ω p (cid:126) u q is not empty and contains anequilibrium point p Q ˚ , q P H rad . We recall that the ω -limit set of (cid:126) u is defined as ω p (cid:126) u q “ t (cid:126) w P H rad | D a sequence s n ě , so that s n Ñ n Ñ`8 `8 , and S α p s n q (cid:126) u Ñ n Ñ`8 (cid:126) w u . Below we will show that the ω -limit set ω p (cid:126) u q reduces to the singleton p Q ˚ , q , and that theentire trajectory converges to this point in the strong sense. And this concludes the proofof Theorem 1.2.Before proving that the entire trajectory (cid:126) u p t q “ S α p t q (cid:126) u converges to p Q ˚ , q , we willemphasize that the ω -limit set ω p (cid:126) u q is contained in the set E rad of radial equilibriumpoints of p KG q α . Lemma 3.7.
The ω -limit set ω p (cid:126) u q satisfies the property (3.31) ω p (cid:126) u q Ă E rad . Proof.
Let (cid:126) v “ p v , v q P ω p (cid:126) u q . Then, there exists a sequence s n Ñ n Ñ`8 `8 such that S α p s n q (cid:126) u ” (cid:126) u p s n q Ñ n Ñ`8 (cid:126) v .On the one hand, we know by (3.8) that the energy satisfies E p (cid:126) u p s n qq Ñ (cid:96) “ E pp Q ˚ , qq as n Ñ `8 , and E p (cid:126) u p s n qq Ñ E p (cid:126) v q . If (cid:126) v is not an equilibrium point, then for some time σ ą E p S α p σ q (cid:126) v q ď E p (cid:126) v q ´ δ “ (cid:96) ´ δ (3.32) ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 33 where δ ą
0. Since E p (cid:126) u p s n ` σ qq Ñ (cid:96) and E p (cid:126) u p s n ` σ qq Ñ E p S α p σ q (cid:126) v q , we arrive at a contradiction and (3.31) holds. (cid:3) Remark 3.8.
Let us fix a positive time τ ą and introduce the ω -limit set ω τ p (cid:126) u q of the discretedynamical system defined by the iterates S α p τ q m , m P N , that is, ω τ p (cid:126) u q “ t (cid:126) w P H rad | D a sequence k n ě , so that k n Ñ n Ñ`8 `8 , and S α p τ q k n (cid:126) u Ñ n Ñ`8 (cid:126) w u . Obviously, ω τ p (cid:126) u q Ă ω p (cid:126) u q . Using the fact that ω p (cid:126) u q is contained in E rad and that the Lipschitzproperty of S α p t q : (cid:126) v P H Ñ S α p t q (cid:126) v P H , which is uniform with respect to t P r , τ s (see thearguments in Step 1 of Section 4 and especially the estimates (4.11) , (4.12) , and (4.13) ), one canshow that (3.33) ω τ p (cid:126) u q “ ω p (cid:126) u q . To prove that the ω -limit ω p (cid:126) u q is a singleton and that the entire trajectory converges tothis point, we will apply a generalization of the classical convergence theorem of Aulbach[1], Hale-Massat [25] and Hale-Raugel [26], due to Brunovský and P. Poláˇcik [5], whichuses local invariant manifold theory. For more details on these convergence theorems,we refer the reader to Appendix B and especially to Lemma B.3 that we shall applybelow. The behaviour of S α p t q (cid:126) u “ (cid:126) u p t q heavily depends on the spectral properties of thelinearized operator L about Q ˚ and the linearized operator ˜ Σ α p t q “ e A α t about p Q ˚ , q (seethe definitions (2.48), (2.49) or (4.3) with ϕ “ Q ˚ ). Lemma 2.10 describes the spectrum ofthe operator A α .Before proving this convergence result, we need to recall some notation given in Section 4.There we introduce the modified (localized) Klein-Gordon equation (4.7) and show thatthis localized equation defines a globally defined flow ¯ S α p t q on H rad , such that,(3.34) (cid:126) u p t q “ S α p t qpp Q ˚ , q ` (cid:126) v q “ p Q ˚ , q ` ¯ S α p t q (cid:126) v , as long as (cid:126) u p t q P B r , where B r ” B pp Q ˚ , q , r q is the open ball of center p Q ˚ , q and radius r ą
0, with r ď p C p α, τ qq ´ r (see Remark 4.2). In other terms, if we set S ˚ α p t q (cid:126) u “ p Q ˚ , q ` ¯ S α p t qp (cid:126) u ´ p Q ˚ , qq , then S α p t q (cid:126) u and S ˚ α p t q (cid:126) u coincide as long as S α p t q (cid:126) u P B r .In Section 4, we define the (global) stable, unstable, center stable, center unstable, andcenter manifolds W i ˚ pp Q ˚ , qq of S ˚ α p t q about p Q ˚ , q , where i “ s , u , cs , cu , c respectively.Since S α p t q (cid:126) u and S ˚ α p t q (cid:126) u coincide as long as S α p t q (cid:126) u P B r , we may define the local stable,unstable, center stable, center unstable, and center manifolds W iloc pp Q ˚ , qq of S α p t q about pp Q ˚ , qq as follows:(3.35) W iloc pp Q ˚ , qq “ W i ˚ pp Q ˚ , qq X B r , i “ s , u , cs , cu , c . We begin our proof with the particular case where p Q ˚ , q is the (hyperbolic) trivialequilibrium p , q of p KG q α . We remark that in that case L “ ´ ∆ ` I and the entirespectrum of A α lies in a half-plane of the form (cid:60) z ă ´ δ ă
0. In the terminology ofSection 4 and of Appendix A, this means that the local stable manifold W uloc pp , qq is awhole neighborhood of p , q and that then necessarily p , q is an isolated equilibrium,and the perturbative equation (2.48) around p , q exhibits exponential decay of solutionsin H rad for small data. Actually, this exponential decay to zero had already been provedin Theorem 2.3. In particular, (cid:126) u p t q Ñ p , q in that case as t Ñ 8 .Let us come back to the general case. If Q ˚ ‰
0, then Lemma 2.10 states that A α has either a trivial kernel, or a one-dimensional kernel. The former case means that thedynamics near p Q ˚ , q is hyperbolic , whereas in the latter case it is not. In the hyperbolicscenario, we have no central part, which means that the invariant manifolds constructed inSection 4 and in Appendix A only involve stable and unstable manifolds W sloc pp Q ˚ , qq and W uloc pp Q ˚ , qq . In both cases, the (local) unstable manifold W uloc p Q ˚ , q is finite-dimensionalsince L has only finitely many eigenvalues (and thus only finitely many eigenvalues withpositive real part).In the non-hyperbolic case, the kernel of A α is one-dimensional, the local center manifold W cloc pp Q ˚ , qq is a C -curve containing p Q ˚ , q . We notice that we can also choose r ą W cloc pp Q ˚ , qq “ W c ˚ pp Q ˚ , qq X B r is a connected curve. Moreover, asremarked above, the (local) unstable manifold W uloc p Q ˚ , q is finite-dimensional. In order toprove the convergence to p Q ˚ , q , we would like to directly apply the classical convergencetheorem of [5] or [26], which is the case (1) of Theorem B.4. However, we do not know thatthe trajectory (cid:126) u p t q is bounded and thus we also cannot ascertain that the ω -limit set ω p (cid:126) u q is connected. So we will apply the more general convergence Theorem B.2 of Brunovskýand Poláˇcik, and more precisely their local Lemma B.3, which are recalled in Appendix B.To this end, we need to show that p Q ˚ , q is stable for S α p t q restricted to the local centermanifold (see the definition (3.41) below). In order to prove this stability, we shall usethe same arguments as Brunovský and Poláˇcik in the proof of Lemma B.3. Like them, wewill make use of the attraction of the center unstable manifold with asymptotic phase ofSection 4 (see also Appendix A). Notice that the hyperbolic case can be considered as aspecial case, where the local center unstable (respectively, center) manifold reduces to thelocal unstable manifold (respectively, to p Q ˚ , q ). In the non-hyperbolic case, the centermanifold is present and the dynamics is more delicate to analyze.We proceed by contradiction and assume that (cid:126) u p t q Ñ p Q ˚ , q . Since (cid:126) u p t q does notconverge to p Q ˚ , q , there exists β ą β ă r with the following property: for any0 ă β ď β , if (cid:126) u p t q P B H pp Q ˚ , q , β q , there exists a first time τ ą (cid:126) u p t ` τ q P B β ,for 0 ď τ ă τ , and (cid:126) u p t ` τ q R B β . In other words, (cid:126) u p t ` τ q belongs to the sphere S pp Q ˚ , q , β q . ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 35
We first fix β ą β ď β . By Theorem 3.5, there exists n p β q such that, for n ě n p β q , (cid:126) u p t n q P B β . Moreover, there exists a first time τ n p β q ą (cid:126) u p t n ` τ q P B β for 0 ď τ ă τ n p β q (cid:126) u p t n ` τ q R B β for τ “ τ n p β q . (3.36)Since (cid:126) u p t n q Ñ p Q ˚ , q as n Ñ `8 , we remark that τ n p β q Ñ `8 as n Ñ `8 . We nowinvoke the asymptotic phase property of the center-unstable manifold, see (A.9) (or also(4.29) in Theorem 4.1). Thus, there exists ξ n : “ ξ p (cid:126) u p t n qq P W culoc p Q ˚ , q such that, for t ě } S ˚ α p t q (cid:126) u p t n q ´ S ˚ α p t q ξ n } H ď c ρ t } (cid:126) u p t n q ´ ξ n } H , (3.37)where 0 ă ρ ă
1. And, by continuity of the map ξ p¨q , ξ n Ñ p Q ˚ , q as n Ñ `8 . In particular, (3.37) implies that } S α p τ n p β qq (cid:126) u p t n q ´ S ˚ α p τ n p β qq ξ n } H Ñ n Ñ `8 . (3.38)Since W cu ˚ pp Q ˚ , qq is finite-dimensional and by (3.38), S ˚ α p τ n p β qq ξ n is bounded, the se-quence S ˚ α p τ n p β qq ξ n , n P N , contains a convergent subsequence. We conclude that up topassing to a subsequence one has (cid:126) u p t n ` τ n p β qq “ S α p τ n p β qq (cid:126) u p t n q Ñ p ˜ u , ˜ u q P ¯ B β as n Ñ `8 . By the invariance property of W cu ˚ pp Q ˚ , qq and by (3.38),(3.39) p ˜ u , ˜ u q P W culoc pp Q ˚ , qq . We remark that, by (3.31) and (3.36), p ˜ u , ˜ u q is an equilibrium point p ˜ Q , q ” p ˜ Q p β q , q and }p ˜ Q p β q , q ´ p Q ˚ , q} H “ β. (3.40)If p Q ˚ , q is an isolated equilibrium point, then (3.40) with β ď r leads to a contradiction.We remark that, in the hyperbolic case, p Q ˚ , q is necessarily an isolated equilibrium whichends the proof in this case.Let us now focus on the case where p Q ˚ , q is not isolated. Before completing the proofin this case, we recall a definition of Brunovský and Poláˇcik, see Appendix B. We say that p Q ˚ , q is stable for S α p t q| W cloc pp Q ˚ , qq if, @ (cid:15) ą D θ ą (cid:126) v P W cloc pp Q ˚ , qq , } (cid:126) v ´ p Q ˚ , q} H ď θ implies that, for t ě } S α p t q (cid:126) v ´ p Q ˚ , q} H ď (cid:15). (3.41)We now complete our proof. By construction and (3.39), the element p ˜ Q p β q , q belongsto W culoc pp Q ˚ , qq . Since p ˜ Q p β q , q is an equilibrium point, it necessarily belongs to thelocal center manifold W cloc pp Q ˚ , qq (see Section 4 and Appendix A for more explanations), which, as we saw earlier, is a C one-dimensional embedded manifold passing through p Q ˚ , q .Since (3.40) holds for any small β ą
0, we see that this curve segment contains equilibriain the omega -limit set ω p (cid:126) u q which are arbitrarily close to, but distinct from, p Q ˚ , q . Infact, we can say even more than that. First, we place an order on the curve ˜ W cr pp Q ˚ , qq if r ą v ´ ă p Q ˚ , q ă v ` if v ´ (respectively v ` ) isto the “left” ( resp. “right”) of p Q ˚ , q on the curve segment ˜ W cr pp Q ˚ , qq . By intersectingthe tangent line to this curve at p Q ˚ , q with the spheres of radius β for all small β , we seethat there are two possibilities:(1) Either there exist two families of equilibria p Q ´ m , q and p Q ` m , q with p Q ´ m , q ăp Q ˚ , q ă p Q ` m , q such that p Q ˘ m , q Ñ p Q ˚ , q as m Ñ `8 . (3.42) A simple dynamical argument based on (3.42) implies that S α p t q| W cloc pp Q ˚ , qq is in factstable. We can now directly apply Lemma B.3 of Brunovský and Poláˇcik to thetime 1 map S α p q , which implies that the ω -limit set ω p (cid:126) u q and thus the ω -limitset ω p (cid:126) u q contain an element of W uloc pp Q ˚ , qqzp Q ˚ , q . This contradicts the fact that ω p (cid:126) u q P E (cid:96) . Instead of directly applying Lemma B.3 to the map S α p q , we can alsoargue for the flow S α p t q as at the end of the proof of [5, Lemma 1] of Brunovskýand Poláˇcik and directly show that p ˜ Q p β q , q P W uloc pp Q ˚ , qqzp Q ˚ , q , where ˜ Q p β q isas in (3.40). But this contradicts the fact that p ˜ Q p β q , q is an equilibrium and so weagain obtain the desired convergence.(2) Or there exists β ą p ˜ Q p β q , q on the “left" (say) of p Q ˚ , q in W cloc pp Q ˚ , qq X B β . But then, the abovearguments (and in particular the properties (3.40)) imply that, for every 0 ď β ď β ,there exists an equilibrium p ˜ Q ` p β q , q in ω p (cid:126) u q satisfying the properties (3.40). Thisimplies that on the right of p Q ˚ , q , W cloc pp Q ˚ , qq consists only of equilibria and thatthe ω -limit set ω p (cid:126) u q contains a curve C of equilibria with end point p Q ˚ , q (as foran interval). We then choose an equilibrium p ˜ Q ` p β q , q in the interior of C and closeto p Q ˚ , q . We repeat the above proof with p Q ˚ , q replaced by p ˜ Q ` p β q , q . And weagain obtain the same contradiction as in Case (1). Remark 3.9.
In the particular case of a wave type or reaction-di ff usion equation, the proofof the Łojasiewicz-Simon inequality (see Sections 3.2 and 3.3 in the monograph of L. Simon[45] and also [28, Theorem 2.1]) shows that, when the kernel of L is one-dimensional, theset of equilibria of p KG q α passing through p Q ˚ , q is a C -curve. We could have used thisproperty in the proof above to avoid the last arguments and apply Theorem B.2. However,in view of possible extensions, we chose not to use this property.4. I nvariant manifold theory for the K lein -G ordon equation In Section 3.2, in order to prove the convergence of any global solution (in positive time)towards an equilibrium point p ϕ , q of p KG q α , we used the properties of the local unstable, ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 37 local center unstable and local center manifolds W iloc pp ϕ , qq , i “ u , cu , c about p ϕ , q forthe flow S α p t q . There, we defined these local manifolds as the intersections of the globalmanifolds W i ˚ pp ϕ , qq , i “ u , cu , c about p ϕ , q for the global flow S ˚ α p t q , with the ball ofcenter p ϕ , q and radius r ą
0, where r ą S ˚ α p t q was defined by S ˚ α p t q (cid:126) u “ p ϕ , q ` ¯ S α p t qp (cid:126) u ´ p ϕ , qq , where ¯ S α p t q is the global flow defined by the localized Klein-Gordon equation (4.7) below.In this section, we construct the global invariant manifolds W i pp , qq , i “ u , cu , c , for theglobal flow ¯ S α p t q and obtain the attraction property of W cu pp , qq by applying the generalinvariant manifold theory recalled in Appendix A.Let p ϕ , q P H rad be an equilibrium point of p KG q α , that is, ϕ is a radial solution of theelliptic equation(4.1) ´ ∆ ϕ ` ϕ ´ f p ϕ q “ . Solving the equation p KG q α in the neighborhood of p ϕ , q leads one to solve the equation(4.2) v tt ` α v t ` L v ´ g p v q “ , p v , v t qp q ” (cid:126) v p q P H rad . where L “ ´ ∆ ` I ´ f p ϕ q , g p v q “ f p ϕ ` v q ´ f p ϕ q ´ f p ϕ q v . (4.3)The equation (4.2) can be written in matrix form as follows B t ˆ vv t ˙ “ ˆ ´ L ´ α ˙ ˆ vv t ˙ ` ˆ g p v q ˙ ” A α (cid:126) v ` ˆ g p v q ˙ (4.4)We denote by ˜ Σ α p t q “ e A α t the linear group generated by A α and ˜ S α p t q the local flowdefined by the equation (4.2). We notice that(4.5) S α p t q (cid:126) u “ S α p t qpp ϕ , q ` (cid:126) v q “ p ϕ , q ` ˜ S α p t q (cid:126) v , where (cid:126) v “ (cid:126) u ´ p ϕ , q . When α ą
0, according to Lemma 2.10, the radius ρ p σ ess p ˜ Σ α p τ qqq of the essential spectrumof ˜ Σ α p τ q satisfies: ρ p σ ess p ˜ Σ α p τ qqq ď δ p α, τ q ă A α can have a finite number of negative eigenvalues µ ´ j p α q ă µ ` j p α q ą λ ´ j p τ, α q ” exp p µ ´ j p α q τ q ă λ ` j p τ, α q ” exp p µ ` j p α q τ q ą Σ α p τ q , τ ą
0, it is a simple eigenvalue (and is asimple eigenvalue of ˜ Σ α p τ q for any τ ą Σ α p τ q , τ ą
0. In this case, wewill construct a local center unstable manifold W culoc pp , qq of the equilibrium p , q of ˜ S α p t q ,a foliation of a neighborhood of p , q in H rad over W culoc pp , qq as well as a local center manifold W cloc pp , qq by applying Theorems A.2 and A.5 to ˜ S α p t q . We choose τ ą τ will be made more precise later). And we set L “ ˜ Σ α p τ q . The spectrum σ p L q can be decomposed as in Hypothesis (HA.5.1) and one can defineconstants C ě C ě η ą ε ą S α p t q is only a local flow and thus ˜ S α p τ q will not satisfy the hypothesis(HA.3). Moreover, we need to show that the Lipschitz-constant Lip p R q can be chosen assmall as needed, which is not true for ˜ S α p t q . Therefore, we need to make a localization inthe following way, for instance. Let r ą ff function χ : R Ñ r , s such that χ p s q “ | s | ď , | s | ě . (4.6)And, we consider the modified Klein-Gordon equation,(4.7) v tt ` α v t ` L v ´ g p v q χ ` } (cid:126) v } H r ˘ “ , (cid:126) v p q “ (cid:126) v P H rad , where 0 ă r ď r is fixed. To simplify the notation, we set h p (cid:126) v q “ g p v q χ ` } (cid:126) v } H r ˘ . We first show that, for any (cid:126) v P H , the equation (4.7) admits a unique solution (cid:126) v p t q ” ¯ S α p t q (cid:126) v P C pr , `8q , H q (we leave to the reader to show that ¯ S α p t q (cid:126) v also belongs to C pp´8 , s , H q ). To this end, it is su ffi cient to show that, for any (cid:126) v P H , the solution (cid:126) v p t q ” ¯ S α p t q (cid:126) v of (4.7) exists on the time interval r , τ s and remains bounded there.We will do that in two steps. We will give the proof only in the case where d ě
3, the case d ď (cid:126) v p t q of (4.7) is given by the Duhamelformula,(4.8) (cid:126) v p t q “ ˜ Σ α p t q (cid:126) v ` ż t ˜ Σ α p t ´ s qp , g p v p s qq χ ` } (cid:126) v p s q} H r ˘ q t ds , and also remark that, as long as (cid:126) v p s q R B H p , ? r q , the term h p (cid:126) v p s qq vanishes. Step 1:
Let (cid:126) v P H so that } (cid:126) v } H ď mr with p C p α, τ qq ´ ď m ď M ” M p mr q “ C p α, τ q mr , where C p α, τ q ě (cid:126) v p t q on the time interval r , τ s , we argue as inthe proof of Theorem 2.3 and introduce the space Y ” t (cid:126) v P L pp , τ q , H q with v P L θ ˚ pp , τ q , L θ ˚ p R d qq| } v } L p H qX W , p L qX L θ ˚ p L θ ˚ q ď M p mr qu . ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 39
Like there we introduce the mapping F : Y Ñ Y defined by p F (cid:126) v qp t q “ ˜ Σ α p t q (cid:126) v ` ż t ˜ Σ α p t ´ s qp , h p (cid:126) v p s qqq t ds . The application of Proposition 2.4 implies(4.9) } F p q} Y ď C p α, τ q mr ď M p mr q . We next show that F is a strict contraction from Y into Y . Due to the hypothesis p H . q f ,we may write, for v , v in H p R d q , |p g p v q ´ g p v qqp x q| “ | f p ϕ p x q ` v p x qq ´ f p ϕ p x q ` v p x qq ´ f p ϕ p x qqp v p x q ´ v p x qq|“ | ż p f p ϕ ` v ` σ p v ´ v qq ´ f p ϕ qqp v ´ v q d σ |ď C |p| v | β ` | v | β ` | v | θ ´ ` | v | θ ´ qp v ´ v q| , (4.10)where 0 ă β ă min p θ ´ , d ´ q and C ” C p f , ϕ q is a constant depending only on f and on ϕ . For (cid:126) v i P Y , i “ ,
2, Proposition 2.4 and the inequality (4.10) imply, } F (cid:126) v ´ F (cid:126) v } Y ď C p α, τ q ż τ } h p (cid:126) v p s qq ´ h p (cid:126) v p s qq} L ds ď C p α, τ q ż τ }p g p v q ´ g p v qq χ ` } (cid:126) v } H r ˘ ` g p v q ` χ ` } (cid:126) v } H r ˘ ´ χ ` } (cid:126) v } H r ˘˘ } L ds ď C p α, τ q C “ ż τ }p| v p s q| β ` | v p s q| β q| v p s q ´ v p s q| } L ds ` ż τ }p| v p s q| θ ´ ` | v p s q| θ ´ q| v p s q ´ v p s q| } L ds ` ż τ }p| v p s q| β ` ` | v p s q| θ q} L | ` χ ` } (cid:126) v } H r ˘ ´ χ ` } (cid:126) v } H r ˘˘ | ds ‰ ” B ` B ` B . (4.11)Arguing as in the proof of Theorem 2.3, by using the Sobolev embeddings, the Hölderinequality and the fact that 0 ă β ă d ´ , we obtain the following inequality for B : B ď C p α, τ q C ż τ p} v } β H ` } v } β H q} v ´ v } H ds ď C p α, τ q τ CM p rm q β } v ´ v } L p H q (4.12)The bound of the term B is obtained as in the proof of Theorem 2.3 (see (2.20)):(4.13) B ď C p α, τ q C τ η M p rm q θ ´ θ p θ ˚ p ´ η q` η q “ } v ´ v } L p L q ` } v ´ v } L θ ˚ p L θ ˚ q ‰ . where η ą B . We firstremark that, since χ ` } (cid:126) w } H r ˘ vanishes if } (cid:126) w } H ě ? r , we may write | ` χ ` } (cid:126) v } H r ˘ ´ χ ` } (cid:126) v } H r ˘˘ | ď ż | χ ` } (cid:126) v ` σ p (cid:126) v ´ (cid:126) v q} H r ˘` (cid:126) v ` σ p (cid:126) v ´ (cid:126) v q r , p (cid:126) v ´ (cid:126) v qq H | d σ ď ? r } (cid:126) v ´ (cid:126) v } H . (4.14)The estimate (4.14), together with the estimates (4.12) and (4.13) with v “
0, imply that(4.15) B ď ? mC C p α, τ q r τ M p rm q β ` C p α, τ q τ η M p rm q θ ´ θ p θ ˚ p ´ η q` η q s} (cid:126) v ´ (cid:126) v } L p H q . Choosing r ą K p r , τ q ” C p α, τ q τ CM p r q β ` C p α, τ q C τ η M p r q θ ´ θ p θ ˚ p ´ η q` η q ` ? C C p α, τ q r τ M p r q β ` C p α, τ q τ η M p r q θ ´ θ p θ ˚ p ´ η q` η q s ď , (4.16)we deduce from the inequalities (4.11) to (4.16) that(4.17) } F (cid:126) v ´ F (cid:126) v } Y ď } (cid:126) v ´ (cid:126) v } Y , which implies with (4.9), that, for any (cid:126) v P Y ,(4.18) } F (cid:126) v } Y ď M p mr q . Therefore, F is a strict contraction and admits a unique fixed point (cid:126) v p (cid:126) v q in Y . Theuniqueness of the solution (cid:126) v of the equation (4.7) on the time interval r , τ s is proved asin the proof of Theorem 2.3.Let next (cid:126) v , i , i “ ,
2, be so that } (cid:126) v , i } H ď mr , and let (cid:126) v i , i “ ,
2, be the correspondingsolutions of the equation (4.7) on the time interval r , τ s ; by the above proof, they belongto Y . Applying Proposition 2.4 and repeating the above proof, we show that(4.19) } (cid:126) v ´ (cid:126) v } Y ď C p α, τ q} (cid:126) v , ´ (cid:126) v , } Y . As in the proof of Theorem 2.3, one also shows that (cid:126) v P B H p , mr q ÞÑ (cid:126) v p (cid:126) v q P Y is a C -function.In the remaining part of the proof, we set m “ Step 2 :
We begin by showing that for every (cid:126) v P H , (cid:126) v p t q “ ¯ S α p t q (cid:126) v exists on r , `8q .Let first (cid:126) v P H satisfying } (cid:126) v } H ď r , then, by Step 1, (cid:126) v p t q stays in the ball B H p , M p r qq for 0 ď t ď τ . Let next (cid:126) v P H be such that } (cid:126) v } H ě r and let (cid:126) v p t q “ ¯ S α p t q (cid:126) v be themild local solution of (4.7). By continuity of this solution, there exists a time t ą (cid:126) v p t q R B H p , ? r q , for 0 ď t ď t . We have, for 0 ď t ď t ,(4.20) (cid:126) v p t q “ ˜ Σ α p t q (cid:126) v , ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 41 and, in particular, for 0 ď t ď inf p t , τ q ,(4.21) } (cid:126) v p t q} H ` } v } L θ ˚ pp , t q , L θ ˚ q ď C p α, τ q} (cid:126) v } H . If at a time t , (cid:126) v p t q enters into the ball B H p , r q , then, according to Step 1, for t ď t ď t ` τ , (cid:126) v p t q still exists, stays in the ball B H p , M p r qq and satisfies the estimates givenin Step 1. We thus have proved that, for every (cid:126) v P H , (cid:126) v p t q exists on the time interval r , τ s . Consequently, for every (cid:126) v P H , ¯ S α p t q (cid:126) v exists on r , `8q . Likewise, one showsthat ¯ S α p t q (cid:126) v exists on p´8 , sq . Arguing as in the proof of Theorem 2.3, one shows thecontinuity properties of ¯ S α p t q (cid:126) v with respect to p t , (cid:126) v q and the fact that, for any t P R , (cid:126) v P H ÞÑ ¯ S α p t q (cid:126) v P H is a C -map.We are now able to prove that ¯ S α p t q satisfies the assumptions p HA . q , p HA . . q , and p HA . . q . We first prove the last part of assumption p HA . q . ¯ S α p t q is Lipschitz continuous,with a Lipschitz constant which is uniform in 0 ď t ď τ . The idea is that it is true if (cid:126) v , and (cid:126) v , belong to B H p , r q by (4.19). If (cid:126) v , P B H p , r q and (cid:126) v , R B H p , r q , we estimatethe di ff erence up to the first time t ď τ when (cid:126) v p t q enters the ball B H p , r q , and thenapply the estimate proved in the first case up to time τ . Finally, if both initial data areoutside B H p , r q , we apply the linear estimates up to the first time when one solutionenters B H p , r q and then the estimate of the second case. As a consequence, to conclude,it remains to show that, if } (cid:126) v , } H ď r and } (cid:126) v , } H ě r so that } (cid:126) v p t q} H ě r for any t ě (cid:126) v ´ (cid:126) v satisfies the estimate (4.19). Using Proposition 2.4, the inequalities (4.10),(4.11), and (4.15), we obtain, for 0 ď t ď τ , } (cid:126) v ´ (cid:126) v } Y ď C p α, τ q “ } (cid:126) v , ´ (cid:126) v , } H ` ż τ } h p (cid:126) v p s qq ds ‰ ď C p α, τ q “ } (cid:126) v , ´ (cid:126) v , } H ` ż τ } g p v q ` χ ` } (cid:126) v } H r ˘ ´ χ ` } (cid:126) v } H r ˘˘ } L ds ‰ ď C p α, τ q} (cid:126) v , ´ (cid:126) v , } H ` B , (4.22)where B had already been defined and used in (4.11). As before, the inequality (4.14)holds. Therefore, we deduce from the estimates (4.22), (4.15) and the condition (4.16) that,for 0 ď t ď τ , } (cid:126) v ´ (cid:126) v } Y ď C p α, τ q} (cid:126) v , ´ (cid:126) v , } H ` } (cid:126) v ´ (cid:126) v } Y . (4.23)And thus the inequality (4.19) holds. From all the above results, one infers that ¯ S α p t q isLipschitz continuous and that(4.24) sup ď t ď τ Lip p ¯ S α p t qq “ D ď C p α, τ q . Likewise, one shows that this estimate also holds for ´ τ , ď t ď
0. Thus, Hypothesis p HA . q is satisfied. We next show that the hypotheses p HA . . q and p HA . . q hold. To this end, we set¯ S α p τ q “ ˜ Σ α p τ q ` R p τ q ” L p τ q ` R p τ q ¯ S α p´ τ q “ ˜ Σ α p´ τ q ` ˜ R p τ q ” L p τ q ´ ` ˜ R p τ q . (4.25)Let (cid:126) v P H and (cid:126) v p t q “ ¯ S α p t q (cid:126) v ; then, R p τ q writes(4.26) R p τ q “ ż τ ˜ Σ α p t ´ s qp , h p v p s qqq t ds . To prove that the conditions (A.23), (A.24), and (A.29) hold, we will show that Lip p R p τ qq and Lip p ˜ R p τ qq go to zero as r goes to zero (we will only show it for R p τ q , since the proofis similar for ˜ R p τ q ). To show this property, we are going back to the three cases consideredabove. If (cid:126) v , and (cid:126) v , belong to B H p , r q , then the estimates (4.11) to (4.19) imply that(4.27) } R p τ q (cid:126) v , ´ R p τ q (cid:126) v , } Y ď K p r , τ q C p α, τ q} (cid:126) v , ´ (cid:126) v , } H . The estimate (4.22) shows that the same property (4.27) holds if (cid:126) v , belongs to B H p , r q and (cid:126) v , is so that } (cid:126) v p t q} H ě r for any 0 ď t ď τ . Finally, we remark that if (cid:126) v i p t q R B H p , r q , i “ ,
2, for 0 ď t ď τ , then R p τ q (cid:126) v , ´ R p τ q (cid:126) v , “
0. Combining all the above cases andusing the estimate (4.24), we finally obtain that, in every case,(4.28) } R p τ q (cid:126) v , ´ R p τ q (cid:126) v , } Y ď K p r , τ q C p α, τ q} (cid:126) v , ´ (cid:126) v , } H . Since K p r , τ q goes to zero as r goes to zero, Lip p R p τ qq goes to zero as r goes to zeroand the condition (A.23) is satisfied provided r is chosen small enough. Likewise theconditions (A.24) and (A.29) hold, provided r is chosen small enough. From now on, wefix r ą r “ r in (4.7).We have seen that, for r ą S α p t q satisfies the hypotheses of TheoremsA.2 and A.5. We can thus state the following result concerning the invariant manifolds of¯ S α p t q . For the notations and definitions of the di ff erent invariant manifolds, we refer thereader to Appendix A below.As in the assumption (HA.5.1), we denote by P i the spectral (continuous) projectionassociated to the spectral set σ i and let H rad , i be the image H rad , i “ P i H rad , where i “ cu , cs , u , s , c . Theorem 4.1.
Let α ą be fixed.1) There exists a C globally Lipschitz continuous map g cu : H rad , cu Ñ H rad , s so that the C centerunstable manifold W cu pp , qq of ¯ S α p t q at p , q W cu pp , qq “ t (cid:126) v cu ` g cu p (cid:126) v cu q | (cid:126) v cu P H rad , cu u satisfies all the properties given in Theorem A.1. ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 43
2) There exists a C globally Lipschitz continuous map g u : H rad , u Ñ H rad , cs so that the C (strongly) unstable manifold W u pp , qq of ¯ S α p t q at p , q W u pp , qq “ t (cid:126) v u ` g u p (cid:126) v u q | (cid:126) v u P H rad , u u satisfies all the properties described in the statement (2) of Theorem A.5.3) Moreover, there exists a continuous mapping (cid:96) : H rad ˆ H rad , s Ñ H rad , cu , such that, for any (cid:126) v P H rad , the manifold M (cid:126) v “ t (cid:126) v ` (cid:96) p (cid:126) v , (cid:126) v s q | (cid:126) v s P H rad , s u satisfies all the properties in TheoremA.2. In particular, t M (cid:126)ξ | (cid:126)ξ P W cu pp , qqu is a foliation of H rad over W cu pp , qq .4) In particular, there exist ˜ c ą , ă ρ ă , and, for any (cid:126) v P H rad , a unique element (cid:126)ξ p (cid:126) v q P W cu pp , qq such that, for t ě , (4.29) } ¯ S α p t q (cid:126) v ´ ¯ S α p t q (cid:126)ξ p (cid:126) v q} H ď ˜ c ρ t } (cid:126) v ´ (cid:126)ξ p (cid:126) v q} H . Moreover, the map (cid:126) v P H rad ÞÑ (cid:126)ξ p (cid:126) v q P W cu pp , qq is continuous.5) There exists a C globally Lipschitz continuous map g c : H rad , c Ñ H rad , s ‘ H rad , u withg c p q “ , so that the center manifold W c p q of ¯ S α p t q at p , q W c pp , qq “ t x c ` g c p x c q | x c P H rad , c u “ W cu pp , qq X W cs pp , qq satisfies all the properties given in statement (4) of Theorem A.5. Let us go back to the “actual” variable (cid:126) u “ (cid:126) v ` p ϕ , q t . We set S ˚ α p t q (cid:126) u “ p ϕ , q t ` ¯ S α p t qp (cid:126) u ´ p ϕ , qq . Then the invariant manifolds of S ˚ α p t q are defined by(4.30) W i ˚ pp ϕ , qq “ p ϕ , q t ` W i pp , qq , i “ cu , c , u , s . Remark 4.2.
We emphasize that the proof given in Step 1 above shows that if, for example, r “ r ,m “ p C p α, τ qq ´ , and } (cid:126) u } H ď mr , then, for ď t ď τ , } ¯ S α p t q (cid:126) u } Y ď r { , which implies that, for ď t ď τ , ¯ S α p t q (cid:126) u “ S α p t q (cid:126) u . In other terms, if (cid:126) u belongs to the ballB H rad pp ϕ , q , r q of center p ϕ , q and radius r ď p C p α, τ qq ´ r , then S ˚ α p t q (cid:126) u “ S α p t q (cid:126) u . Thisallows one to define the local invariant manifolds W iloc pp ϕ , qq of S α p t q about p ϕ , q as (4.31) W iloc pp ϕ , qq “ W i ˚ pp ϕ , qq X B H rad pp ϕ , q , r q , i “ cu , c , u , s . Remark 4.3.
1) In the above theorem, M coincides with the (strongly) stable manifold ˜ W s pp , qq .2) If Ker p L q “ t u , then the center unstable manifold W cu pp , qq coincides with the unstablemanifold W u pp , qq of p , q , while M coincides with the stable manifold W s pp , qq . Remark 4.4.
In the case where α “ , we can also apply Theorems A.1 and A.2 below in orderto prove the existence of the strong unstable manifold and the existence of a center stable manifoldaround any equilibrium point of p KG q α as well as the existence of a foliation of H rad over theunstable manifold. This gives an alternative proof to the construction of a center stable manifold,by the Hadamard method in [40] (for more details, see [7] ). A ppendix A. G lobal invariant manifolds and foliations by the L yapunov -P erronmethod In this appendix, we recall the basic properties of invariant manifold theory that weapplied to the equation p KG q α in Section 4. We reproduce the theorems of Chen, Hale andTan about global invariant manifolds and foliations as given in [11]. For classical resultson invariant manifolds, we also refer the reader to the books [8], [29], [30], and [41] forexample as well as to [2] and to [13].Let X be a Banach space with norm } ¨ } X and S p t q : X Ñ X be a non-linear semigroup,satisfying the following hypotheses: (HA.1) : S p . q . : p t , x q P r , `8q ˆ X ÞÑ S p t q x P X is continuous and there exists aconstant τ ą ď t ď τ Lip p S p t qq “ D ă `8 . (HA.2): There exists τ , 0 ă τ ď τ such that S p τ q can be decomposed as S p τ q “ L ` R , where L : X Ñ X is a bounded linear operator and R : X Ñ X is a global Lipschitzcontinuous map, satisfying the following properties. (HA.2.1): There are subspaces X i , i “ ,
2, of X and continuous projections P i : X Ñ X i such that P ` P “ I , X “ X ‘ X , L leaves X i , i “ ,
2, invariant and L commuteswith P i , i “ ,
2. The restrictions L i of L to X i satisfy the following properties.The map L has a bounded inverse and there exist constants 0 ď β ă β , C i ě i “ ,
2, such that, for k ě } L ´ k P } L p X , X q ď C β ´ k , } L k P } L p X , X q ď C β k . (A.1) (HA.2.2): The maps L and R satisfy the condition(A.2) p a C ` ? C q β ´ β Lip p R q ă . Chen, Hale and Tan considered the following quantity, for γ P p β , β q ,(A.3) λ p γ q “ C β ´ γ ` C γ ´ β . A short computation shows that, under the condition (A.2), there exist γ i , i “ ,
2, with β ă γ ă γ ă β such that,(A.4) λ p γ q Lip p R q “ λ p γ q Lip p R q “ , and λ p γ q Lip p R q ă , @ γ P p γ , γ q . In the trivial case, where Lip p R q “
0, one sets γ “ β and γ “ β .We are now able to state the first theorem, concerning the existence of an invariantmanifold, which is a graph over X . ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 45
Theorem A.1.
Assume that the hypotheses (HA.1), (HA.2) hold and that R p q “ . Then thereexists a globally Lipschitz map g : X Ñ X with g p q “ , and (A.5) Lip p g q ď min γ ď γ ď γ C C Lip p R q γβ p γ ´ β qp ´ λ p γ q Lip p R qq , so that the Lipschitz submanifold G “ t x ` g p x q | x P X u satisfies the following properties: (i): (Invariance) The restriction to G of the semi-flow S p t q , t ě , can be extended to aLipschitz continuous flow on G. In particular, S p t q G “ G, for any t ě , and for any ξ P G, there exists a unique negative semi-orbit u p t q P G of S p . q , t ď , so that u p q “ ξ . (ii): (Lyapunov exponent) If a negative semi-orbit u p t q , t ď , of S p . q is contained in G, then, (A.6) lim sup t Ñ´8 | t | ln | u p t q| ď ´ τ ln γ . Conversely, if a negative semi-orbit u p t q , t ď , of S p . q is contained in X satisfies (A.7) lim sup t Ñ´8 | t | ln | u p t q| ă ´ τ ln γ . then, it is contained in G. (iii): (Smoothness) If the map S p τ q : X Ñ X is of class C , then g : X Ñ X is of class C ,that is, G is a C -submanifold of X. The second theorem states the existence of a foliation of X over the invariant manifold G . Theorem A.2.
Assume that the hypotheses (HA.1), (HA.2) hold and that R p q “ . Then, thereexists an invariant foliation of X over G as follows. (i): (Invariance) There exists a continuous mapping (cid:96) : X ˆ X Ñ X such that, for any ξ P G, (cid:96) p ξ, P ξ q “ P ξ and the manifold M ξ “ t x ` (cid:96) p ξ, x q | x P X u passing through ξ satisfies: (A.8) S p t q M ξ Ă M S p t q ξ , t ě , and (A.9) M ξ “ t y P X | lim sup t Ñ8 t ln | S p t q y ´ S p t q ξ | ď τ ln γ u . Moreover, the map (cid:96) : X ˆ X Ñ X is uniformly Lipschitz continuous in the X direction. (ii): (Completeness) Suppose in addition that “ min γ ď γ ď γ C C Lip p R qp β ´ γ qp ´ λ p γ q Lip p R qq ‰ ¨ “ min γ ď γ ď γ C C Lip p R q γβ p γ ´ β qp ´ λ p γ q Lip p R qq ‰ ă . (A.10) Then, for any x P X, M x X G consists of a single point. In particular, (A.11) M ξ X M η “ H , @ ξ, η P G , X “ ď ξ P G M ξ . In other terms, t M ξ | ξ P G u is a foliation of X over G.Moreover, the mapping x P X ÞÑ ξ p x q “ M x X G is a continuous map from X into G Ă X. (iii): (Smoothness) If the map S p τ q : X Ñ X is of class C , then (cid:96) : X ˆ X Ñ X is of classC in the X direction. Hence, M ξ is a C -submanifold of X, for any ξ P G. Comments on the proof of Theorems A.1 and A.2:
Theorems A.1 and A.2 are proved in [11] by first showing the corresponding results for themap S p τ q and at the end coming back to the continuous dynamical system. This meansthat Theorems A.1 and A.2 still hold for iterates of maps S p τ q . It su ffi ces to replace t P R by n τ , n P N . Theorems A.1 and A.2 are proved in [11] by using the Lyapunov-Perronmethod.The property that the mapping x P X ÞÑ ξ p x q “ M x X G is a continuous map from X into G Ă X is not stated in the main Theorem 1.1 of [11]. It is merely a consequence of theproof of [11, Lemma 3.4]. Indeed, given x P X , the intersection points ξ p x q of M x with G are the solutions of(A.12) ξ p x q ” y ` (cid:96) p x , y q “ (cid:96) p x , y q ` g p (cid:96) p x , y qq , where y P X . This leads to study the fixed points of the map F x p y q ” F p x , y q “ g p (cid:96) p x , y qq , depending on the parameter x P X . One can check that the condition (A.10)implies that F x : X Ñ X is a strict contraction and therefore has a unique fixed point y p x q . The continuity property of y p x q with respect to x P X is a direct consequence of thecontinuity of F with respect to the variable x P X and of the uniform contraction principle (see [12, Theorem 2.2 on Page 25]). It follows that ξ p x q “ y p x q ` (cid:96) p x , y p x qq P G is alsocontinuous with respect to x P X . Remark A.3.
If the equilibrium point of S p . q is hyperbolic, then we may choose β ă ă β . Inthis case, G is the classical unstable manifold W u p q and M ξ , ξ P G, defines an invariant foliationof X over W u p q , with M being the classical stable manifold W s p q . And the solutions on M decay exponentially to , as t goes to `8 .If is a non-hyperbolic equilibrium point and β ă β ă with β close to , then TheoremsA.1 and A.2 allow for the construction of the center-unstable manifold G “ W cu p q of and afoliation over it. If is a non-hyperbolic equilibrium point and ă β ă β with β close to , thenTheorems A.1 and A.2 give the strongly unstable manifold G “ W su p q of and a foliation overit. If γ ă , the existence of the foliation implies that each positive semi-orbit of S p t q convergesexponentially to an orbit of G and is synchronized with this orbit in time. This property is oftencalled “attraction" of G with asymptotic phase".We emphasize that the construction in Theorems A.1 and A.2 is also interesting in the case whereS α p . q depends on a parameter α and β p α q ă ă β p α q with β p α q arbitrarily close to as α converges say to α “ . ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 47
Mutatis mutandis, repeating the arguments of the proofs of Theorems A.1 and A.2,one can also show the existence of a Lipschitz manifold ˜ G “ t x ` ˜ g p x q | x P X u where˜ g : X Ñ X is a globally Lipschitz map with ˜ g p q “
0, such that ˜ G is invariant and suchthat, if a semi-orbit u p t q , t ě
0, of S p . q is contained in ˜ G , then,(A.13) lim sup t Ñ8 t ln | u p t q| ď τ ln ˜ γ ´ , where β ă ˜ γ ´ ă ˜ γ ´ ă β is made more precise below, and also the existence of afoliation ˜ M ξ (in reverse time) of X over ˜ G .If S p t q is a non-linear group, these properties can be proved by reversing the time inTheorems A.1 and A.2. In Section 3, the existence of a center manifold played an importantrole. We can derive this existence by defining the center manifold as the intersection ofthe center stable and center unstable manifolds. The center stable manifold is constructedlike the Lipschitz manifold ˜ G “ t x ` ˜ g p x q | x P X u described above. Since throughoutthe paper we are only dealing with groups, we will quickly show the existence of ˜ G byreversing the time in Theorem A.1. The constants appearing in the proof below are maybenot optimal, but we are not looking here for optimality.In addition to the hypothesis (HA.2), we assume now that (HA.3) : S p . q . : p t , x q P p´8 , `8q ˆ X ÞÑ S p t q x P X is continuous and there exists aconstant τ ą ´ τ ď t ď τ Lip p S p t qq “ D ă `8 . (HA.4): S p´ τ q can be decomposed as S p´ τ q “ L ´ ` ˜ R , where τ and L : X Ñ X have been introduced in the hypothesis (HA.2) and where˜ R : X Ñ X is a global Lipschitz continuous map, satisfying the following property:(A.14) p a C ` ? C q β ´ β β β Lip p ˜ R q ă . We remark that the linear map L ´ satisfies the hypothesis (HA.2.1) with P (resp. P )replaced by P (resp. P ), C (resp. C ) replaced by C (resp. C ), and β (resp. β ) replacedby β ´ (resp. β ´ ). Indeed, we have }p L ´ q ´ k P } L p X , X q ď C p β ´ q ´ k , }p L ´ q k P } L p X , X q ď C p β ´ q k . (A.15)We next set(A.16) ˜ λ p ˜ γ q “ C β ´ ´ ˜ γ ` C ˜ γ ´ β ´ . As above, a short computation shows that, under the condition (A.14), there exist ˜ γ i , i “ ,
2, with β ´ ă ˜ γ ă ˜ γ ă β ´ such that,(A.17) ˜ λ p ˜ γ q Lip p ˜ R q “ ˜ λ p ˜ γ q Lip p ˜ R q “ , and ˜ λ p ˜ γ q Lip p ˜ R q ă , @ ˜ γ P p ˜ γ , ˜ γ q . We may now apply Theorem A.1 to the nonlinear semigroup ˜ S p t q “ S p´ t q and we obtainthe following result. Theorem A.4.
Assume that the hypotheses (HA.2), (HA.3), and (HA.4) hold and that R p q “ ˜ R p q “ . Then there exists a globally Lipschitz map ˜ g : X Ñ X with ˜ g p q “ and (A.18) Lip p ˜ g q ď min ˜ γ ď ˜ γ ď ˜ γ C C Lip p ˜ R q β β p β ´ { ˜ γ qp ´ ˜ λ p ˜ γ q Lip p ˜ R qq , so that the Lipschitz submanifold ˜ G “ t x ` ˜ g p x q | x P X u satisfies the following properties: (i): (Invariance) ˜ G is invariant under S p t q , i.e., S p t q ˜ G “ ˜ G, for any t ě . (ii): (Lyapunov exponent) If a positive semi-orbit u p t q , t ě , of S p . q is contained in ˜ G, then, lim sup t Ñ8 t ln | u p t q| ď τ ln 1˜ γ . Conversely, if a positive semi-orbit u p t q , t ě , of S p . q in X, satisfies (A.19) lim sup t Ñ8 t ln | u p t q| ă τ ln 1˜ γ . then, it is contained in ˜ G. (iii): (Smoothness) If the map S p τ q : X Ñ X is of class C , then ˜ g : X Ñ X is of class C ,that is, ˜ G is a C -submanifold of X. We next consider the classical case where S p . q is a non-linear group satisfying theassumption (HA.3) as well as (HA.5): The point 0 is an equilibrium point of S p . q . And there exists τ , 0 ă τ ď τ such that S p τ q and S p´ τ q can be decomposed as follows S p τ q “ L ` R , S p´ τ q “ L ´ ` ˜ R , where L : X Ñ X is a bounded linear operator, R : X Ñ X and ˜ R : X Ñ X are globalLipschitz continuous maps, satisfying the following properties. (HA.5.1): The spectrum σ p L q of L can be written as σ p L q “ σ s Y σ c Y σ u , where σ s , σ c and σ u are closed subsets of t λ P C | | λ | ă u , t λ P C | | λ | “ u , and t λ P C | | λ | ą u . ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 49
There exists η ą σ s Ă t λ P C | | λ | ă ´ η u , σ u Ă t λ P C | | λ | ą ` η u We set: σ cu “ σ c Y σ u and σ cs “ σ c Y σ s . Let P i be the spectral (continuous) projectorassociated to the spectral set σ i and let X i be the image X i “ P i X , where i “ cu , cs , u , s , c .We have that P cu ` P s “ I “ P cs ` P u . The linear map L leaves X i invariant and commuteswith P i , i “ cu , cs , u , s , c .Now we choose 0 ă ε ă η {
2. The restrictions L i of L to X i satisfy the following properties.There exist constants C ě C ě k ě } L ´ kcu P cu } L p X , X q ď C p ´ ε q ´ k , } L ks P s } L p X , X q ď C p ´ η q k , (A.21)and }p L ´ cs q ´ k P cs } L p X , X q ď C pp ` ε q ´ q ´ k , }p L ´ u q k P u } L p X , X q ď C pp ` η q ´ q k . (A.22)We further assume that the maps R and ˜ R satisfy the conditions. (HA.5.2): The following inequalities hold(A.23) p a C ` ? C q η ´ ε Lip p R q ă , and(A.24) p a C ` ? C q η ´ ε p ` ε qp ` η q Lip p ˜ R q ă . (HA.5.3): We define the function λ p γ q as in (A.3), that is,(A.25) λ p γ q “ C ´ ε ´ γ ` C γ ´ ` η , and the quantities γ i , i “ ,
2, with 1 ´ η ă γ ă γ ă ´ ε , satisfying (A.4).Likewise, we define the function ˜ λ p ˜ γ q as in (A.16), that is,(A.26) ˜ λ p ˜ γ q “ C p ` ε q ´ ´ ˜ γ ` C ˜ γ ´ p ` η q ´ . and the quantities ˜ γ i , i “ ,
2, with p ` η q ´ ă ˜ γ ă ˜ γ ă p ` ε q ´ , satisfying(A.17).We next introduce the function λ ˚ p γ ˚ q :(A.27) λ ˚ p γ ˚ q “ C ` η ´ γ ˚ ` C γ ˚ ´ ´ ε , and the quantities γ ˚ i , i “ ,
2, with 1 ` ε ă γ ˚ ă γ ˚ ă ` η , satisfying(A.28) λ ˚ p γ ˚ q Lip p R q “ λ p γ ˚ q Lip p R q “ , and λ ˚ p γ q Lip p R q ă , @ γ ˚ P p γ ˚ , γ ˚ q . We finally require that the following inequality holds:min γ ď γ ď γ C C Lip p R q γ p ´ ε qp γ ´ ` η qp ´ λ p γ q Lip p R qq ˆ min ˜ γ ď ˜ γ ď ˜ γ C C Lip p ˜ R qp ` ε qp ` η qp ` η ´ { ˜ γ qp ´ ˜ λ p ˜ γ q Lip p ˜ R qq ă . (A.29)Applying Theorems A.1 and A.4 to the above flow S p . q , we obtain the following prop-erties, which are used in Sections 3 and 4. Theorem A.5.
Assume that the hypotheses (HA.3) and (HA.5) are satisfied. Then, the followingproperties hold. (1)
There exists a globally Lipschitz map g cu : X cu Ñ X s with g cu p q “ , so that the Lipschitzcenter unstable manifold W cu p q W cu p q “ t x c ` x u ` g cu p x c ` x u q | x c P X c , x u P X u u satisfies all the properties described in Theorem A.1. In particular, if S p τ q is of class C ,then g cu : X cu Ñ X s is of class C . (2) There exists a globally Lipschitz map g u : X u Ñ X cs with g u p q “ , so that the Lipschitzunstable (also called strongly unstable) manifold W u p q W u p q “ t x u ` g u p x u q | x u P X u u satisfies all the properties described in Theorem A.1 with γ replaced by γ ˚ and γ i replacedby γ ˚ i , i “ , . In particular, if S p τ q is of class C , then g u : X u Ñ X cs is of class C .And, if a negative semi-orbit u p t q , t ď , of S p . q is contained in W u p q , then, (A.30) lim sup t Ñ´8 | t | ln | u p t q| ď ´ τ ln γ ˚ . (3) There exists a globally Lipschitz map g cs : X cs Ñ X u with g cs p q “ so that the Lipschitzcenter stable manifold W cs p q W cs p q “ t x c ` x s ` g cs p x c ` x s q | x c P X c , x s P X s u satisfies all the properties described in Theorem A.4. In particular, if S p τ q is of class C ,then g cs : X cs Ñ X u is of class C . (4) There exists a globally Lipschitz map g c : X c Ñ X s ‘ X u with g c p q “ , so that theLipschitz center manifold W c p q W c p q “ t x c ` g c p x c q | x c P X c u “ W cu p q X W cs p q satisfies the following properties:(i) W c p q is invariant under S p t q , i.e., S p t q W c p q “ W c p q , for any t ě .(ii) The properties (ii) of Theorem A.1 and the properties (ii) of Theorem A.4 hold. Inparticular, if a trajectory u p t q , t P p´8 , of S p . q is contained in W c p q , then (A.31) lim sup t Ñ´8 | t | ln | u p t q| ď ´ τ ln γ , lim sup t Ñ8 t ln | u p t q| ď τ ln 1˜ γ . ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 51
Moreover, W c p q contains all the equilibria of S p t q .(iii) If the map S p τ q : X Ñ X is of class C , then g c : X c Ñ X s ‘ X u is of class C , that is,W c p q is a C -submanifold of X. (5) If moreover the condition (A.10) holds with β “ ´ ε and β “ ´ η , then one has afoliation of X over W cu p q as defined in Theorem A.2.Proof. (1) Statements (1) and (5) are direct consequences of Theorem A.1 and Theorem A.2respectively, applied to the case where β “ ´ ε and β “ ´ η .(2) Statement (2) is a direct consequence of Theorem A.1, applied to the case where β “ ` η and β “ ` ε .(3) Statement (3) is a direct consequence of Theorem A.4, applied to the case where β ´ “ p ` ε q ´ and β ´ “ p ` η q ´ .Let us next prove the statement (4). We are looking for the trajectories u p t q , which satisfyboth properties of (A.31). These two properties together are satisfied only by the elementsin W cu p q X W cs p q .Thus, we are looking for the elements x “ x c ` x s ` x u so that(A.32) x c ` x u ` g cu p x c ` x u q “ x c ` x s ` g cs p x c ` x s q “ x c ` g cu p x c ` x u q` g cs p x c ` g cu p x ` x u qq , or also for the elements x u P X u satisfying(A.33) x u “ g cs p x c ` g cu p x c ` x u qq . In other terms, given x c P X c , we are looking for the fixed point of the map x u P X u ÞÑ F p x c , x u q “ g cs p x c ` g cu p x c ` x u qq P X u . We notice that the Lipschitz constant of F p x c , . q satisfies Lip p F p x c , . qq ď Lip p g cs q ˆ Lip p g cu q . By Theorems A.1 and A.4 and the assumption (A.29), we have, for any x P X Lip p F p x c , . q ď min γ ď γ ď γ C C Lip p R q γ p ´ ε qp γ ´ ` η qp ´ λ p γ q Lip p R qq ˆ min ˜ γ ď ˜ γ ď ˜ γ C C Lip p ˜ R qp ` ε qp ` η qp ` η ´ { ˜ γ qp ´ ˜ λ p ˜ γ q Lip p ˜ R qq ă . (A.34)Therefore, x u P X u Ñ F p x c , x u q P X u is a strict contraction, uniformly in x c . Thus, for any x c P X c , there exists a unique fixed point h p x c q P X u of F p x c , . q . And g c p x c q is given by g c p x c q “ x c ` h p x c q ` g cu p x c ` h p x c qq . The regularity of the map g c is proved by using the regularity of the mappings g cu and g cs and by applying the uniform contraction principle of [12, Theorem 2.2 on Page 25]. (cid:3) Remark A.6.
1. If the equilibrium point is hyperbolic (that is, σ c “ H ), then one can choose ε “ η in the hypotheses (HA.5.1) and (HA.5.2). The center unstable manifold W cu p q and the(strongly) unstable manifold W u p q coincide (that is, g cu “ g u ). And the center manifold W c p q reduces to .2. In the above theorem, we have only stated those properties which are used in this paper. We leaveit to the reader to state the existence of the (strongly) stable manifold. A ppendix B. C lassical convergence results
In the study of asymptotic behaviour of dynamical systems, one often encounters thefollowing question: knowing that the ω -limit set of a relatively compact trajectory containsan equilibrium point x , does this ω -limit set reduce to the point x , i.e., does the trajectoryconverge to x ? This question is especially interesting in the case of gradient systems (thatis, systems which admit a strict Lyapunov functional). In fact, consider a gradient systemwith a hyperbolic equilibrium x . Then x is isolated and the whole trajectory convergesto this point x . If the equilibrium x is not hyperbolic and the spectrum of the linearizeddynamical system around x intersects the unit circle, then x could lie in a continuumof equilibria, which could be contained in the ω -limit set. If x belongs to a normallyhyperbolic manifold of equilibria, we can still have convergence to x , under additionalhypotheses.In the proof of Theorem 1.2, we use the convergence property to an equilibrium point inorder to prove the boundedness of the orbits, which are global in forward time. We recallhere the general convergence property in the form proved by Brunovský and Poláˇcik in[5], who extended earlier convergence results, proved for example by Aulbach [1] in thefinite-dimensional frame, or by Hale and Raugel [26], who generalised the convergenceproperty of Aulbach to the infinite-dimensional setting (see also the paper [25] of 1982,and [43] for applications). In the case of the one-dimensional parabolic equation withseparate boundary conditions, convergence proofs had been given before in [38] and [47].Let X be a Banach space and Φ : X Ñ X be a continuous map admitting a fixed point y . Without loss of generality, we may choose y “
0. Brunovský and Poláˇcik assumedthe following hypotheses: ‚ (HB.1) There exists a neighborhood U of 0 in X so that the restriction Φ ˇˇ U : U Ñ X is of class C . ‚ (HB.2) The spectrum σ p DF p qq can be written as σ p DF p qq “ σ s Y σ c Y σ u , where σ s , σ c and σ u are closed subsets of t λ P C | | λ | ă u , t λ P C | | λ | “ u , and t λ P C | | λ | ą u .As in Appendix A, we introduce the spectral projectors P i of B “ DF p q associated withthe spectral sets σ i , i “ s , c , u and the images X i “ P i X . We recall that these spaces are all B -invariant and X “ X s ‘ X c ‘ X u . We also denote X cu “ X c ‘ X u .As we have seen in Appendix A, the hypotheses (HB.1) and (HB.2) allow one to constructLipschitz continuous local center unstable and local center manifolds W culoc p q , W cloc p q of Φ at 0 as graphs over X cu and X c , respectively, and also the local unstable manifold W uloc p q as a graph over X u , by extending the map Φ into a global Lipschitz continuous and C mapping ˜ Φ , which coincides with Φ on the ball B X p , δ q of center 0 and radius δ ą δ being small enough), and by applying Theorems A.1 and A.5. These local invariantmanifolds are defined in the following way(B.1) W iloc p q “ ˜ W i δ p q , i “ cu , c , u , where ˜ W cu δ p q , ˜ W c δ p q and ˜ W u δ p q are the global center stable, center and unstable manifoldsof ˜ Φ around 0. ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 53
On the other hand, Theorem A.2 in Appendix A on the invariant foliations implies that W culoc p q is exponentially attractive in X with asymptotic phase (see Appendix A for moredetails). Likewise, one can show that W cloc p q is exponentially attractive in backward timein W culoc p q with asymptotic phase. These asymptotic phase properties are among the keyarguments in the proof of the main convergence theorem B.2 below. Remark B.1.
Actually, the hypothesis (HB.1) can be replaced by the weaker hypothesis: (HB.1bis)
There exists a neighborhood U of in X so that the restriction Φ ˇˇ U : U Ñ X is Lipschitzcontinuous and di ff erentiable at every fixed point contained in U. Before stating the main convergence result of [5], we introduce the concept of stabilityrestricted to W cloc p q . We say that 0 is stable for the map Φ ˇˇ W cloc p q , if, for any ε ą
0, thereexists η ą y P W cloc p q with } y } X ď η , we have(B.2) } Φ n p y q} X ď ε , @ n “ , , , . . . . As pointed out in [5], this stability is independent of the choice of the local centermanifold W cloc p q . The independence of this stability on the choice of the local centermanifold can be proved by using foliations as in the paper of [6], who actually showed thatthe flows on di ff erent local center manifolds are conjugated (under some more restrictivehypotheses, which can be easily removed). As also remarked in [5], the fact that thestability is independent of the choice of the local centre manifold, is not needed in theproof of Theorem B.2 below. Theorem B.2.
Assume that the hypotheses (HB.1) (or (HB.1bis)) and (HB.2) hold. Let x P Xbe such that the fixed point belongs to the ω -limit set ω p x q of x . Assume that either X cu isfinite-dimensional or that the trajectory Φ n p x q , n “ , , ¨ ¨ ¨ , of x is relatively compact. Assume,moreover, that is stable for the map Φ ˇˇ W cloc p q , where W cloc p q is a local center manifold of .Then either Φ n p x q converges to as n Ñ 8 , or ω p x q contains a point of the local unstablemanifold W uloc p q of , distinct from . Theorem B.2 generalises the above mentioned convergence result of [26] in two ways.Firstly, the hypotheses do not require that ω p x q consists only of fixed points. Secondly, itdoes not require that the trajectory Φ n p x q , n “ , , ¨ ¨ ¨ , of x be relatively compact. But,of course, it requires the additional stability property defined above.In [5], Brunovský and Poláˇcik have proved the following lemma (see [5, Lemma 1]) andhave obtained Theorem B.2 as a direct consequence of it. We emphasize that Lemma B.3 isreally a local result anf that Lemma B.3 will hold for any mapping Φ ˚ : y P U ÞÑ Φ ˚ y P X coinciding with Φ in U . In particular, Φ ˚ need not be well defined outside U , which isthe case in our application in Section 3. Lemma B.3.
Assume that the hypotheses (HB.1) (or (HB.1bis)) and (HB.2) hold, that δ ą issmall enough so that B X p , δ q Ă U and that is stable for the map Φ ˇˇ W cloc p q . Let x k P X andp k P N be sequences satisfying the following properties: (1) x k Ñ as k Ñ `8 . (2) Φ j p x k q P B X p , β q for j “ , , , . . . , p k and Φ p k ` p x k q R B X p , β q , where ă β ă δ . (3) In the case, where dim X cu “ 8 , the set t Φ j ˚ p x k q | k P N , j “ , . . . , p k u is relativelycompact.Then Φ p k p x k q contains a subsequence converging to an element of W uloc p qzt u . As an easy consequence of Theorem B.2, Brunovský and Poláˇcik have obtained thefollowing more classical theorem.
Theorem B.4.
Assume that the hypotheses (HB.1) (or (HB.1bis)) and (HB.2) hold. Let x be apoint in X such that the fixed point belongs to the ω -limit set ω p x q of x and such that ω p x q is contained in the set Fix p Φ q of fixed points of Φ . Assume that either X cu is finite-dimensional orthat the trajectory Φ n p x q , n “ , , ¨ ¨ ¨ , of x is relatively compact. Assume moreover that one ofthe following two properties holds: (1) dim X c “ and the trajectory Φ n p x q , n “ , , ¨ ¨ ¨ , of x is relatively compact. (2) dim X c “ m ă 8 and there is a submanifold M Ă X with dim M “ m such that P M Ă Fix p Φ q .Then ω p x q “ t u .Proof. We give the proof, because it is short.First assume that (2) holds. Then, if δ ą M and W uloc p q coincide since M Ă W uloc p q , and they both have the same dimension m . The assumption M Ă Fix p Φ q thus implies that 0 is stable for the map Φ ˇˇ W cloc p q . Since W uloc p qzt u containsno fixed point if δ ą ω p x q P Fix p Φ q , Theorem B.2 implies that ω p x q “ t u .In the case (1), we first remark that, since the trajectory Φ n p x q , n “ , , ¨ ¨ ¨ , of x isrelatively compact and since ω p x q consists only of fixed points, the omega-limit set ω p x q is connected (see for example [26, Lemma 2.7]). If ω p x q contains more than one fixedpoint, then all fixed points near 0 are contained in W cloc p q and thus 0 belongs to a curveof fixed points. If 0 belongs to the relative interior of this curve, one applies the case (2),which leads to a contradiction. If 0 does not belong to the relative interior of this curve,we consider a fixed point y ˚ near 0, contained in the relative interior of this curve of fixedpoints and in ω p x q . Replacing Φ by Φ p y ˚ ` x q , we are now back to the case (2). Applyingthe case (2), we obtain that ω p x q “ y ˚ , which also leads to a contradiction. (cid:3) Suppose we consider an element x P X such that we do not a priori know that thetrajectory t Φ n p x q | n P N u is bounded. Then, even if dim X c “
1, we cannot directly applycase (1) of Theorem B.4. Indeed, the proof of case (1) uses the connectedness propertyof ω p x q . One can then try to apply the more general Theorem B.2 in order to obtain aconvergence result.In Section 3.2 we encountered such a case. We did not know there that the forwardtrajectory t S α p t q (cid:126) u | t ě u is bounded. Thus, as in the proof of Lemma B.3, we used theproperty that W culoc p q is exponentially attractive in X with asymptotic phase together withthe fact that dim X c “
1, to obtain that S α p t q has the stability property (3.41) (or (B.2)). ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 55
Then, we applied Theorem B.2 to the time τ -map Φ “ S α p τ q , where τ ą S α p t q , it is alsovalid in the case of a more general semi-flow and allows us to state the following generalresult. Corollary B.5.
Assume that the map Φ “ S p τ q where S p t q : R ˆ X Ñ X is a continuous dynamicalsystem and that τ ą is a small enough positive time, so that Φ “ S p τ q satisfies the hypotheses(HB.1) (or (HB.1bis)) and (HB.2). Let x be a point in X such that the equilibrium point belongsto the ω -limit set ω p x q of x and such that ω p x q is contained in the set of equilibrium points ofS p t q . Assume that either X cu is finite-dimensional or that the trajectory Φ n p x q , n “ , , ¨ ¨ ¨ , ofx is relatively compact. Assume moreover that dim X c “ . Then ω p x q “ t u . Let us finally notice that, in the case of gradient systems generated by some evolutionaryequations with an analytic non-linearity or satisfying the hypotheses (1) or (2) of TheoremB.4, one also obtains convergence results based on the Łojasiewicz inequality (see [45, 27,28]) and on the construction of appropriate energy functionals, when the trajectories arerelatively compact. The convergence proofs given [45, 27, 28] require in an essential waythat the trajectory S p t q x , t ě
0, be relatively compact and thus do not seem to be applicablein Section 3.2 above. R eferences [1] B. Aulbach,
Continuous and Discrete Time Dynamics Near Manifolds of Equilibria , Springer Verlag, Berlin-Heidelberg, 1984.[2] P. W. Bates and C.K.R.T. Jones,
Invariant manifolds for semilinear partial di ff erential equations , Dynamicsreported, , pp. 1–38, Dynam. Report. Ser. Dynam. Systems Appl., 2, Wiley, Chichester, 1989[3] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state , Arch. RationalMech. Anal. (1983), pp. 313–345.[4] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions , Arch.Rational Mech. Anal. (1983), pp. 347–375.[5] P. Brunovský and P. Poláˇcik, On the local structure of ω -limit sets of maps , Z. Angew. Math. Phys. (1997),pp. 976–986.[6] A. Burchard, B. Deng, and K. Lu, Smooth conjugacy of centre manifolds , Proc. Roy. Soc. Edinburgh Sect. A , (1992), pp. 61–77.[7] N. Burq, G. Raugel and W. Schlag,
Long time dynamics for damped Klein-Gordon equations II [8] J. Carr,
Application of Centre Manifold Theory , Applied Mathematical Sciences, , Springer-Verlag, NewYork, 1981.[9] T. Cazenave, Uniform Estimates for solutions of non-linear Klein-Gordon equations , J. Functional Analysis, (1985), pp. 36–55.[10] C.- C. Chen and C.- S. Lin, Uniqueness of the ground state solutions of ∆ u ` f p u q “ in R n , n ě
3, Comm.PDE, (1991), pp. 1549–1572.[11] X.-Y. Chen, J. K. Hale and B. Tan, Invariant Foliations for C Semigroups in Banach Spaces , J. of Di ff erentialEquations, (1997) , pp. 283–318.[12] S.-N. Chow and J. K. Hale, Methods of bifurcation theory , Grundlehren Math. Wiss.,
Springer-Verlag,New York-Berlin, 1982.[13] S.-N. Chow, X.-B. Lin, and K. Lu,
Smooth invariant foliations in infinite dimensional spaces , J. Di ff erentialEquations (1991), pp. 266–291. [14] C. V. Co ff man, Uniqueness of the ground state solution for ∆ u ´ u ` u “ and a variational characterizationof other solutions , Arch. Rational Mech. Anal. (1972), pp. 81–95.[15] C. Cortázar, M. García-Huidobro and C. S. Yarur, On the uniqueness of sign changing bound state solutionsof a semilinear equation , Ann. Inst. H. Poincaré Anal. Non Linéaire (2011), pp. 599–621.[16] O. Costin, M. Huang and W. Schlag, On the spectral properties of L ˘ in three dimensions , Nonlinearity (2012), pp. 125–164.[17] T. Duyckaerts and F. Merle Dynamics of threshold solutions for energy-critical wave equation , Int. Math. Res.Notices, , (2008).[18] T. Duyckaerts, C. Kenig and F. Merle, Classification of radial solutions of the focusing, energy-critical waveequation , Camb. J. Math. 1 (2013), no. 1, pp. 75–144.[19] J. A. Esquivel-Avila,
The dynamics of a nonlinear wave equation , J. Math. Anal. Appl. (2003), pp. 135–150.[20] J. A. Esquivel-Avila,
Qualitative analysis of a nonlinear wave equation , Discrete And Continuous DynamicalSystems (2004), pp. 787–804.[21] E. Feireisl, Convergence to an equilibrium for semilinear wave equations on unbounded intervals , Dynam. Syst.Appl. (1994), pp. 423–434.[22] E. Feireisl, Long-time behavior and convergence for semilinear wave equations on R N . J. Dynam. Di ff erentialEquations (1997), pp. 133–155.[23] E. Feireisl, Finite energy travelling waves for nonlinear damped wave equations , Quarterly Journal of Appliedmathematics
LVI (1998), pp. 55–70.[24] F. Gazzola and M. Squassina,
Global solutions and finite time blow up for damped semilinear wave equations ,Ann. I. H. Poincaré – Analyse non-linéaire (2006), pp. 185–207.[25] J. K. Hale and P. Massatt, Asymptotic behavior of gradient-like systems , Dynamical Systems II (A. R. Bednarekand L. Cesari, eds.), Academic Press 1982, pp. 85–101.[26] J. K. Hale and G. Raugel,
Convergence in Gradient-Like Systems with Applications to P.D.E. , Z. Angew. Math.Phys. (1992), pp. 63–124.[27] A. Haraux and M. A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipationand analytic nonlinearity , Calc. Var. Partial Di ff erential Equations (1999), pp. 95–124.[28] A. Haraux and M. A. Jendoubi, On the convergence of global and bounded solutions of some evolution equations ,J. Evol. Equ. (2007), pp. 449–470.[29] D. Henry, Geometric theory of semilinear parabolic equations , Lecture Notes In Mathematics , Springer-Verlag, New York, 1981.[30] M. W. Hirsch, C. C. Pugh and M. Shub,
Invariant manifolds , Lecture Notes in Mathematics , Springer-Verlag, Berlin-New York, 1977.[31] S. Ibrahim, N. Masmoudi and K. Nakanishi,
Scattering threshold for the focusing nonlinear Klein-Gordonequation , Analysis and PDE, (2011), pp. 405–460.[32] A. Ionescu and W. Schlag Agmon-Kato-Kuroda theorems for a large class of perturbations.
Duke Math. J. 131(2006), no. 3, 397–440.[33] M. Keel and T. Tao,
Endpoint Strichartz Estimates , American Journal of Mathematics (1998), pp. 955–980[34] C. Keller, Stable and unstable manifolds for the nonlinear wave equation with dissipation, J. Di ff erentialEquations (1983), pp. 330–347.[35] C. Keller, Large-time asymptotic behavior of solutions of nonlinear wave equations perturbed from a stationaryground state . Comm. Partial Di ff erential Equations (1983), pp. 1073–1099.[36] C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linearwave equation , Acta Math., , (2008), pp. 147-212.[37] H. Lindblad and C. Sogge,
On existence and scattering with minimal regularity for semilinear wave equations ,J. Funct. Anal. (1995), pp. 357–426.[38] H. Matano,
Convergence of solutions of one-dimensional semilinear parabolic equations , Journal of Mathematicsof Kyoto University (1978), pp. 221–227.[39] K. McLeod, Uniqueness of Positive Radial Solutions of ∆ u ` f p u q “ in R n , II , Trans. AMS. (1993), pp.495–505. ONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS 57 [40] K. Nakanishi and W. Schlag,
Invariant manifolds and dispersive Hamiltonian Evolution Equations , ZürichLectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2011.[41] J. Palis and W. de Melo,
Geometric theory of dynamical systems. An introduction . Translated from thePortuguese by A. K. Manning. Springer-Verlag, New York-Berlin, 1982.[42] I. E. Payne and D. H. Sattinger,
Saddle points and instability of nonlinear hyperbolic equations , Israel J. Math. (1975), pp. 273–303.[43] G. Raugel, Dynamics of Partial Di ff erential Equations on Thin Domains, CIME Course, Montecatini Terme ,Lecture Notes in Mathematics , Springer Verlag, (1995), pp. 208–315.[44] W. Schlag, Spectral theory and nonlinear partial di ff erential equations: a survey , Discrete Contin. Dyn. Syst. (2006), pp. 703–723.[45] L. Simon, Theorems on regularity and singularity of energy minimizing maps , Lect. in Math., ETH Zürich,Birkhäuser, 1996.[46] W. A. Strauss,
Existence of solitary waves in higher dimensions , Commun. Math. Phys. (1977), pp. 149–162.[47] T. J. Zelenyak, Stabilization of solutions of boundary value problems for a second order parabolic equation withone space variable , Di ff . Equations (1968), pp. 17–22.,N icolas B urq : U niv P aris -S ud , L aboratoire de M ath ´ ematiques d ’O rsay , O rsay C edex , F-91405; CNRS,O rsay cedex , F-91405, F rance G enevi ` eve R augel : CNRS, L aboratoire de M ath ´ ematiques d ’O rsay , O rsay C edex , F-91405; U niv P aris -S ud , O rsay cedex , F-91405, F rance W ilhelm S chlag : U niversity of C hicago , D epartment of M athematics , 5734 S outh U niversity A venue ,C hicagohicago