Asymptotics of solutions with a compactness property for the nonlinear damped Klein-Gordon equation
aa r X i v : . [ m a t h . A P ] F e b ASYMPTOTICS OF SOLUTIONS WITH A COMPACTNESSPROPERTY FOR THE NONLINEAR DAMPED KLEIN-GORDONEQUATION
RAPHA¨EL C ˆOTE AND XU YUAN
Abstract.
We consider the nonlinear damped Klein-Gordon equation ∂ tt u + 2 α∂ t u − ∆ u + u − | u | p − u = 0 on [0 , ∞ ) × R N with α >
0, 2 N p >
2. We study thebehavior of solutions for which it is supposed that only one nonlinear objectappears asymptotically for large times, at least for a sequence of times.We first prove that the nonlinear object is necessarily a bound state. Next,we show that when the nonlinear object is a non-degenerate state or a degen-erate excited state satisfying a simplicity condition, the convergence holds forall positive times, with an exponential or algebraic rate respectively. Last, weprovide an example where the solution converges exactly at the rate t − tothe excited state. Introduction
Setting of the problem.
We consider the nonlinear focusing damped Klein-Gordon equation ∂ tt u + 2 α∂ t u − ∆ u + u − f ( u ) = 0 ( t, x ) ∈ [0 , ∞ ) × R N , (1.1)where f ( u ) = | u | p − u , α >
0, 2 N p satisfies2 < p < p ∗ ( N ) with p ∗ ( N ) = ∞ if N = 2 ,N + 2 N − N = 3 , , . It follows from [4, Theorem 2.3] that the Cauchy problem for (1.1) is locally well-posed in the energy space: for any initial data ( u , v ) ∈ H ( R N ) × L ( R N ),there exists a unique (in some class) maximal solution u ∈ C ([0 , T max ) , H ( R N )) ∩ C ([0 , T max ) , L ( R N )) of (1.1). Moreover, if the maximal time of existence T max isfinite, then lim t ↑ T max k ~u ( t ) k H × L = ∞ .Setting F ( u ) = p +1 | u | p +1 and E ( ~u ) = 12 Z R N (cid:8) |∇ u | + u + ( ∂ t u ) − F ( u ) (cid:9) d x, for any H × L solution ~u = ( u, ∂ t u ) of (1.1), it holds E ( ~u ( t )) − E ( ~u ( t )) = − α Z t t k ∂ t u ( t ) k L d t. (1.2)One can easily construct finite time blow-up solutions by adequately truncating aconstant in space solution, whose initial data lead to finite time blow-up for theinferred ODE y ′′ + 2 αy ′ + y − f ( y ) = 0 (and using finite speed of propagation). Onthe other hand, solutions to (1.1) which are globally defined for positive time, thatis for which T max = + ∞ , are believed to possess much more structure, in the spiritof a soliton resolution: it roughly asserts that any global solution (maybe under a Mathematics Subject Classification. genericity condition) splits for large times into a sum of decoupled rigid nonlinearobjects, which should be here stationary solutions, especially in view of decay ofenergy (1.2).Let us first recall from [6] (see also references therein) some features on stationarysolution, namely a solution to the elliptic equation − ∆ q + q − f ( q ) = 0 , q ∈ H ( R N ) . (1.3)We call the solutions of (1.3) bound states , and denote B the set of bound states: B = { q : q is a nontrivial solution of (1.3) } . Standard elliptic arguments (see e.g . [19] or [6, Theorem 8.1.1]) show that if q ∈ B ,then q is of class C ( R N ) and has exponential decay as | x | → + ∞ , as well as itsfirst and second-order derivatives.Let W ( v ) = 12 Z R N (cid:8) |∇ v | + v − F ( v ) (cid:9) d x, for v ∈ H . We call the solutions of (1.3) which minimize the functional W by ground states ;the set of ground states is denoted by GG = { q ∈ B : ∀ q ∈ B , W ( q ) W ( q ) } . Ground states are well studied objects. They are unique up to space translation(for rather general nonlinearities): there exists a radial positive function q of class C , exponentially decreasing, along with its first and second-order derivatives, suchthat G = { q ( x − x ) : x ∈ R N } . We refer to Berestycki-Lions [2], Gidas-Ni-Nirenberg [18], Kwong [22], Serrin-Tang[26] (however, a positive bound state may not be a ground state, see [12]). Itis well-known (see e.g.
Grillakis-Shatah-Strauss [20]) that the ground state q isunstable in the energy space. This result was also known in the physics literatureas the Derrick’s Theorem [15].In dimension 1, B = G (due to ODE arguments). In contrast, for any N > G ( B : see [6, Remark 8.1.16]. Functions q ∈ B \ G are referred to as excited states .As a matter of fact, much less is known about excited states.Here are some references on the construction of excited states. Berestycki-Lions[3] showed the existence of infinitely many radial nodal ( i.e . sign changing) so-lutions (see also [21, 24] and the references therein). For the massless version ofequation (1.3), the existence of excited states that are nonradial sign-changing andwith arbitrary large energy was first proved in Ding [16] by variational argument.Later, del Pino-Musso-Pacard-Pistoia [13] constructed more explicit solutions tothe massless equation (1.3) with a centered soliton crowned with negative spikes(rescaled solitons) at the vertices of a regular polygon of radius 1. Then, followingsimilar general strategy in [13], they constructed sign changing, non radial solu-tions to (1.3) on the sphere S N ( N >
4) whose energy is concentrated along specialsubmanifolds of S N in [14].We can now go back to (1.1), and recall some previous results related to the longtime dynamics of global solutions.Under some conditions on N and p , results in [17, 23] state that for any sequence oftime, any global bounded solution of (1.1) converges to a sum of decoupled boundstates after extraction of a subsequence of times. Also in [17], Feireisl constructedglobal solutions that behave as sum of an even number of ground states ( i.e. multi-solitons). SYMPTOTICS OF DAMPED NLKG EQUATION 3
In [4], for dimension N ≥
2, Burq, Raugel and Schlag proved the convergence ofany global radial solution to one (radial) bound state, for the whole sequence oftime.In [10], it is given a complete description of 2-soliton solutions (that is, solutionswhich, on at least a sequence of time, behave as the sum of two decoupled groundstates), in dimension N
5. Building on the tools developed there, [9] gave acomplete description of global solutions in dimension N = 1, that is, the solitonresolution in that case.We aim at considering the behavior of solution without conditions on symmetry(like radiality). A complete description seems out of reach, because of the lackof understanding of the dynamics around general excited states, and because thesystem of centers of mass of the involved bound states may have itself a veryintricate dynamics.1.2. Main results.
In this paper, we are instead interested in understanding thebehavior of solutions to (1.1) for which only one nonlinear object appears for largetimes, at least for a sequence of time. More precisely, we define packed solutions asfollows.
Definition 1.1.
A maximal solution ~u = ( u, ∂ t u ) ∈ C ([0 , T max ) , H × L ) of (1.1)is called a packed solution if there exist ( W , W ) ∈ H × L , and a time sequence t n → T max and a position sequence y n ∈ R N such thatlim n →∞ {k u ( t n ) − W ( · − y n ) k H + k ∂ t u ( t n ) − W ( · − y n ) k L } = 0 . (1.4)We say that ~W = ( W , W ) is a cluster point for ~u at ( t n , y n ) n .Observe that any cluster point ( W , W ) is actually a bound state ( q, Proposition 1.2.
Let ~u = ( u, ∂ t u ) be a packed solution of (1.1) . Then ~u ∈ C ([0 , + ∞ ) , H × L ) is globally defined for positive times, and if ( W , W ) ∈ H × L is a cluster point for ~u at ( t n , y n ) n , then W = q is a bound state of (1.3) and W = 0 . Furthermore, the energy is bounded below, ∂ t u ∈ L ([0 , + ∞ ) , L ) and forall t > , E ( ~u ( t )) − E ( q,
0) = 2 α Z + ∞ t k ∂ t u ( s ) k L d s. (1.5)Notice that it is unclear whether a packed solution is globally bounded in H × L (recall that from arguments of [5] – see also [4] and [9] – if p NN − , then anyglobal solution to (1.1) is globally bounded in H × L , but this is not known forhigher powers of p ).It turns out that the description of the convergence depends deeply on the boundstate. More specifically, consider the linearized operator L q of the energy around abound state q : L q = − ∆ + 1 − f ′ ( q ) , hL q v, v i = Z R N (cid:8) |∇ v | + v − f ′ ( q ) v (cid:9) d x. (1.6)Due to the invariances of equations, L q always has a important kernel: denote theΩ ij are the angular derivatives that areΩ ij = x i ∂ x j − x j ∂ x i for 1 i < j N, (1.7)and consider the vector space Z q spanned by the infinitesimal generator of theinvariance of the equation on q : Z q = Span { ∂ x n q, n = 1 , · · · , N ; Ω ij q, i < j N } . (1.8)One always has Z q ⊂ ker L q . Then we define non-degenerate and degenerate state. R. C ˆOTE AND X. YUAN
Definition 1.3.
Let q ∈ B .(i) q is called a non-degenerate state if Z q = ker L q .(ii) q is called a degenerate state if Z q ( ker L q .The most relevant example is of course the ground state q which is non-degenerate.We will comment further on degenerate excited state in the comment paragraphbelow. For now, let us simply mention one way to understant degeneracy (wedenote ′ for Gateau differentials). The condition that q is a bound state writes E ′ ( q ) = 0. Then hL q v, w i = E ′′ ( q ) · ( v, w ), so that the condition that q is degenerate,is equivalent to the fact that for some φ / ∈ Z q , the linear form E ′′ ( q ) · ( φ, · ) = 0.Our first result is that if one cluster point of a packed solution is a non-degeneratestate, then the convergence holds of all positive time, and occurs with an exponentialrate. More precisely, we have the following. Theorem 1.4.
Let ~u = ( u, ∂ t u ) be a packed solution of (1.1) , with cluster point q ∈ H at ( t n , y n ) n . If q is a non-degenerate state, we have convergence holdingfor all time and exponential decay, i.e. there exist µ > and z ∞ ∈ R N such that ∀ t > , k u ( t ) − q ( · − z ∞ ) k H + k ∂ t u ( t ) k L . e − µt . Next, we consider degree-1 excited states where ker L q has one extra dimension notrelated to the geometric invariances of the equation (1.1) and which also involvesa condition on the third-order Gateau differentials of E , according to the nextdefinition. Definition 1.5.
Let q be a degenerate excited state. q is called a degree-1 excitedstate if there exists φ ∈ H such thatker L q = Z q ⊕ Span { φ } and E ′′′ ( q ) · ( φ, φ, φ ) = 0 . (1.9)Again, we will comment on this definition in the paragraph below, the main pointbeing that degree-1 excited states are somehow the simplest degenerate boundsstates.Our second result is concerned with cluster points which are degree-1 excited states. Theorem 1.6.
Let ~u = ( u, ∂ t u ) be a packed solution of (1.1) , with cluster point q ∈ H at ( t n , y n ) n . If q is a degree-1 excited state, then the convergence ~u ( t ) → ( q, holds for all time as t → + ∞ , and the rate of convergence has algebraic decay,i.e. there exists z ∞ ∈ R N such that ∀ t > , k u ( t ) − q ( · − z ∞ ) k H + k ∂ t u ( t ) k L . t − . Last, we show that the convergence rate in Theorem 1.6 can be sharp: we providean example where the solution converges exactly at the rate t − to the degree-1excited state. Theorem 1.7.
Let q be a degree-1 excited state. Then, there exists a global solution ~u = ( u, ∂ t u ) ∈ C ([0 , + ∞ ) , H × L ) of (1.1) such that k u ( t ) − q k H + k ∂ t u ( t ) k L ∼ t − as t → + ∞ . Comments.
Let us first observe that Theorem 1.4 (and its proof) holds also indimension 1, but of course, they are in that case a direct consequence of the completedescription [9] of global solutions in 1D (as mentioned above, excited states onlyexist for N > N >
2. Therestrictions to N < p < N +2 N − are to ensure a nice local well posednesstheory, and sufficient smoothness on the non-linearity so that Taylor expansionmake sense up to order 2. In this perspective let us remind that our analysisencompasses the most physically relevant nonlinearity, the cubic one f ( u ) = u . SYMPTOTICS OF DAMPED NLKG EQUATION 5
Regarding Theorem 1.4: the ground state is of course non-degenerate, but oneshould keep in mind that is not so easy to construct degenerate excited states. Asa matter of fact, the constructions in [1, 24] (see also [13, 25] for the massless case)yield non-degenerate excited states as well. This means that the scope of Theorem1.4 is rather large and does certainly not restrict to the ground state.We now discuss degree-1 excited states: as we mentioned, they should be under-stood as the simplest degenerate case. Already here, very little is known, and to ourknowledge, our results are the first describing precisely the dynamics in a degener-ate setting. From this point of view, the condition that dim ker L q = dim( Z q ) + 1 isvery natural. Regarding the extra condition E ′′′ ( q ) · ( φ, φ, φ ) = 0, let us note thatit is generic; as we will see in Lemma 2.1, it is equivalent with E ′′′ ( q ) being nonidentically 0 on (ker L q ) .It is remarquable that one already observes a drastic change in the dynamic indegree-1 degeneracy, when compared to non-degeneracy. The convergence here isindeed merely polynomial in time, which is a surprise: such slow rate of convergenceis usually observed due to the interaction with another nonlinear object (as in[10, 9]), and this is not the case here. As it is seen in the proofs, the derivation ofthe main bootstrap regime is noticeably more involved in degree-1 degeneracy, andrelies on the very specific algebra of the main ODE system at leading order (seeSection 3).One setting where excited states are better understood is the case of radial func-tions. Among these, radial bound states q are either non-degenerate or satisfy thefirst condition in the degree-1 degeneracy definition (1.9): indeed, among radialfunctions, the geometric kernel Z q is trivial and dim ker rad L q
1, see for example[4, Section 2.3].All the arguments in the proofs below can taken word for word to the radial setting,and so our results hold for any packed radial solution converging to a radial boundstate q which is either non-degenerate (Theorem 1.4) or such that E ′′′ ( q ) | (ker L q ) =0 (Theorems 1.6 and 1.7). 2. Preliminaries
Proof of Proposition 1.2.
Proof of Proposition 1.2.
Denote ~W ( t ) = ( W ( t ) , ∂ t W ( t )) the solution to (1.1) withinitial data ~W (0) = ( W , W ) and ~u n ( t, x ) = ~u ( t n + t, x + y n ) the solution to (1.1)with initial data ~u n (0 , x ) = ~u ( t n , x + y n ). We can assume that ~W ∈ C ([0 , T ] , H × L ) for some T > H × L and ~u n (0) → ~W (0), we infer that ~u n isdefined on [0 , T ] for n large enough and that ~u n → ~W in C ([0 , T ] , H × L ) . (2.1)This immediately prove that ~u is globally defined for positive times. Indeed, if T max < + ∞ , then for large enough n , T max > t n + T → T max + T , a contradiction:hence T max = + ∞ .As ~E ( ~u ( t n )) → ~E ( W , W ) we infer from the energy dissipation identity that ∂ t u ∈ L ([0 , + ∞ ) , L ). Assume that ~W is not a stationnary solution. Then we canfurthermore assume that ∂ t W = 0 on [0 , T ] × R N , so that k ∂ t W k L ([0 ,T ]) ,L ) > k ∂ t u k L ([ t n ,t n + T ] ,L ) = k ∂ t u n k L ([0 ,T ]) ,L ) → k ∂ t W k L ([0 ,T ]) ,L ) as n → + ∞ . R. C ˆOTE AND X. YUAN
Let t ′ n be a subsequence of t n such that for all n ∈ N , t ′ n +1 > t ′ n + T and k ∂ t u k L ([ t ′ n ,t ′ n + T ] ,L ) > k ∂ t W k L ([0 ,T ]) ,L ) . There holds k ∂ t u k L ([0 , + ∞ ) ,L ) > X n k ∂ t u k L ([ t ′ n ,t ′ n + T ] ,L ) > X n k ∂ t W k L ([0 ,T ]) ,L ) = + ∞ , which is a contradiction. As a consequence, ~W is a stationary solution, which meansthat for all t > ∂ t W ( t ) = 0 and W ( t ) = q for some bound state q . In particular, W = q and W = 0.Furthermore, the energy dissipation identity writes for all 0 t t n : E ( ~u ( t )) − E ( ~u ( t n )) = 2 α Z t n t k ∂ t u ( s ) k L d s. Letting n → + ∞ , we see that the left-hand side has a limit E ( ~u ( t )) − E ( q, ∂ t u ∈ L ([ t, + ∞ ) , L ). This completes the proof of Proposition 1.2. (cid:3) Notation.
Let q be a bound state. Let I q be a subset of { ( i, j ) : 1 i < j N } such that { ∂ x n q, n N ; Ω ij q, ( i, j ) ∈ I q } is a basis of Z q . For any ( i, j ) ∈ I q and ϑ ∈ R , we recall the Givens rotation: G ij ( ϑ ) = · · · · · · · · · · · · cos ϑ · · · − sin ϑ · · · · · · sin ϑ · · · cos ϑ · · · · · · · · · · · · , (2.2)where cos ϑ and sin ϑ appear at the intersections i th and j th rows and columns.That is, the non-zero elements of the Givens matrix G i,j ( ϑ ) = ( g nm ) nm are givenby: g nn = 1 for n = i, j, g ii = g jj = cos ϑ, and g ij = − g ji = − sin ϑ. For K ∈ N ∗ and r >
0, we denote by B R K ( r ) (respectively, S R K ( r )) be the ball(respectively, the sphere) of R K of center 0 and of radius r .We denote h· , ·i the L scalar product for real-valued functions u, v ∈ L , h u, v i := Z R N u ( x ) v ( x )d x. For vector-valued functions ~u = (cid:18) u u (cid:19) , ~v = (cid:18) v v (cid:19) , the notation h· , ·i is also the L scalar product, h ~u, ~v i := X k =1 , h u k , v k i , k ~u k H := k u k H + k u k L . We also define ¯ p = min { , p } > p > SYMPTOTICS OF DAMPED NLKG EQUATION 7
Spectral theory of linearized operator.
In this section, we introduce somespectral properties of the linearized operator for any bound state q ∈ B .For θ = ( θ ij ) ( i,j ) ∈ I q ∈ R I q , denote the rotation R θ = G i j ( θ i j ) · · · G i Iq j Iq ( θ i Iq j Iq ) . For ( z, θ ) ∈ R N + I q , we introduce the following transformation T ( z,θ ) linked to thesymmetries of (1.1): for f ∈ L , T ( z,θ ) f := f ( R θ ( · − z )) . Observe that for all q ∈ B , Z q is generated by taking partial derivatives of T ( z,θ ) q with respect to ( z, θ ) at ( z, θ ) = ( , ): ∂ x n q = − ∂∂z n T ( z,θ ) q | ( z,θ )=( , ) , Ω ij q = ∂∂θ ij T ( z,θ ) q | ( z,θ )=( , ) . (2.3)First, we recall standard properties of the linearized operator L q . Lemma 2.1. (i) Spectral properties.
The self-adjoint operator L q has essentialspectrum [1 , + ∞ ) , a finite number K > of negative eigenvalues and its kernel isof finite dimension M with M > N . Let ( Y k ) k =1 , ··· ,K be an L orthogonal familyof eigenfunctions of L q with negative eigenvalues ( − λ k ) k =1 , ··· ,K , i.e. h Y k , Y k ′ i = δ kk ′ and L q Y k = − λ k Y k , λ k > . (2.4)(ii) Coercivity. Denote Π q the L -orthogonal projection on ker L q . There exists c > such that for all η ∈ H , hL q η, η i > c k η k H − c − (cid:18) k Π q η k L + K X k =1 h η, Y k i (cid:19) . (2.5)(iii) Cancellation. We have, for all ψ , ψ ∈ ker L q and ψ ∈ Z q h f ′′ ( q ) ψ ψ , ψ i = 0 . (2.6) Proof.
Proof of (i) and (ii). See the proof of [7, Lemma 1].Proof of (iii). Without loss of generality, we first consider ψ = Ω ij q for ( i, j ) ∈ I q . For any ψ ∈ ker L q , we have − ∆ ψ + ψ − f ′ ( q ) ψ = 0 . Consider the transformation T ( z,θ ) with ( z, θ ) = ( , θ ) for the above identity, − ∆( T ( z,θ ) ψ ) + T ( z,θ ) ψ − f ′ ( T ( z,θ ) q )( T ( z,θ ) ψ ) = 0 . Note that, from p >
2, we can take the derivative of above identity with respect to θ ij , and then let θ = . It follows that f ′′ ( q ) ψ ψ = − ∆ ˜ ψ + ˜ ψ − f ′ ( q ) ˜ ψ = L q ˜ ψ where ˜ ψ = Ω ij ψ . Thus, by integration by parts and ψ ∈ ker L q , h f ′′ ( q ) ψ ψ , ψ i = h f ′′ ( q ) ψ ψ , ψ i = hL q ˜ ψ , ψ i = h ˜ ψ , L q ψ i = 0 . Proceeding similarly for all the parameters in the transformation T ( z,θ ) , we completethe proof of (iii). (cid:3) As a consequence of the above Lemma, in the case when q is furthermore assumedto be a degree-1 excited state, we now choose φ (introduced in Definition 1.5) withmore rigid properties: namely we claim that there exists (a unique) φ ∈ H suchthat for all n = 1 , · · · , N , ( i, j ) ∈ I q , h φ, ∂ x n q i = h φ, Ω ij q i = 0 and ker L q = Z q ⊕ Span { φ } , (2.7) R. C ˆOTE AND X. YUAN and such that − (4 α k φ k L ) − E ′′′ ( φ, φ, φ ) = 1 . (2.8)(2.7) essentially means that φ is the L -orthogonal supplement of Z q . Moreover,due to (2.6) and the fact that E ′′′ ( q ) is not identically 0 on L q (in view of (1.9)),for such a φ one has E ′′′ ( q )( φ, φ, φ ) = 0: and so, by considering the transformations φ → − φ and φ → λφ , the condition (2.8) can be met.2.4. Modulation around a bound state.
Let q ∈ B be a degree-1 degeneratebound state. Given time dependent C functions z, θ, a , with values in R N , R I q and R , we denote Q = T ( z,θ ) q = q ( R θ ( · − z )) , Φ( t, x ) = T ( z,θ ) φ = φ ( R θ ( · − z )) , (2.9) V ( t, x ) = a ( t )Φ( t, x ) , G = f ( Q + V ) − f ( Q ) − f ′ ( Q ) V. (2.10)It will convenient to encompass both non-degenerate and degree-1 degenerate casesat once by setting a ≡ q is non-degenerate.For all ( i, j ) ∈ I q and ( i ′ , j ′ ) ∈ I q , we denote as follows the derivatives:Ψ ij = ∂Q∂θ ij , Φ ij = ∂ Φ ∂θ ij , Ψ i ′ j ′ ij = ∂ Ψ ij ∂θ i ′ j ′ . (2.11)Finally, we introduce the exponential directions. For k = 1 , · · · , K , we denoteΥ k = T ( z,θ ) Y k , Υ ijk = ∂ Υ k ∂θ ij , and ν ± k = − α ± q α + λ k , ζ ± k = α ± q α + λ k and ~Z ± k = (cid:18) ζ ± k Υ k Υ k (cid:19) . (2.12)The importance of ~Z ± k come from the following observation: if ~v = ( v , v ) is asolution to the linearized (1.1) equation ∂ t (cid:18) v v (cid:19) = (cid:18) v − αv − L q v (cid:19) , then with a ± k := h ~v, ~Z ± k i , there hold dd t a ± k = ν ± k a ± k .Observe that all the function introduced are at least of class C and have pointwiseexponential decay.By direct computation and the definition of Givens rotations in (2.2), we have, forall n = 1 , · · · , N and ( i, j ) ∈ I q ,Ψ ij ∈ Span { (Ω ij q ) ( R θ ( · − z )) : ( i, j ) ∈ I q } ,∂ x n Q ∈ Span { ( ∂ x q ) ( R θ ( · − z )) , · · · , ( ∂ x N q ) ( R θ ( · − z )) } . (2.13)Moreover, by the chain rule, we have, for all ( i, j ) ∈ I q and k = 1 , · · · , K , ∂ t Q = − ˙ z · ∇ Q + X ( i,j ) ∈ I q ˙ θ ij Ψ ij ,∂ t Φ = − ˙ z · ∇ Φ + X ( i,j ) ∈ I q ˙ θ ij Φ ij ,∂ t Υ k = − ˙ z · ∇ Υ k + X ( i,j ) ∈ I q ˙ θ ij Υ ijk ,∂ t Ψ ij = − ˙ z · ∇ Ψ ij + X ( i ′ ,j ′ ) ∈ I q ˙ θ i ′ j ′ Ψ i ′ j ′ ij . (2.14)As a consequence of (2.3), we have the following expansions for small θ ∈ R I q . SYMPTOTICS OF DAMPED NLKG EQUATION 9
Lemma 2.2.
For | θ | ≪ small, we have, for all ( i, j ) ∈ I q , ( i ′ , j ′ ) ∈ I q , k =1 , . . . , K and n = 1 , . . . , N , Ψ ij = (Ω ij q ) ( R θ ( · − z )) + O H ( | θ | ) , Φ ij = (Ω ij φ ) ( R θ ( · − z )) + O H ( | θ | ) , Υ ijk = (Ω ij Y k ) ( R θ ( · − z )) + O H ( | θ | ) ,∂ x n Q = ( ∂ x n q ) ( R θ ( · − z )) + O H ( | θ | ) , Ψ i ′ j ′ ij = (Ω i ′ j ′ Ω ij q ) ( R θ ( · − z )) + O H ( | θ | ) . (2.15)If q is a non-degenerate bound state, we use same notations as above for degree-1degenerate bound states, but with a = 0 and φ = 0.For future reference, we state the following Taylor formulas involving the functions F and f , and omit its proof. Lemma 2.3.
For all s ∈ R , and x ∈ R N , we have | f ′ ( Q + s ) − f ′ ( Q ) | . | s | + | s | p − , (2.16) | f ( Q + s ) − f ( Q ) − f ′ ( Q ) s | . s + | s | p , (2.17) (cid:12)(cid:12)(cid:12)(cid:12) f ( Q + s ) − f ( Q ) − f ′ ( Q ) s − f ′′ ( Q ) s (cid:12)(cid:12)(cid:12)(cid:12) . | s | + | s | p , (2.18) | f ( Q + V + s ) − f ( Q + V ) − f ′ ( Q + V ) s | . s + | s | p , (2.19) | F ( Q + V + s ) − F ( Q + V ) − f ( Q + V ) s | . | Q + V | p − s + | s | p +1 , (2.20) (cid:12)(cid:12)(cid:12)(cid:12) F ( Q + V + s ) − F ( Q + V ) − f ( Q + V ) s − f ′ ( Q + V ) s (cid:12)(cid:12)(cid:12)(cid:12) . | s | + | s | p +1 , (2.21) where all the implied constants in the . are uniform in the space variable of Q or V . First, we introduce the standard modulation result around the non-degenerate stateor degree-1 excited state q . Proposition 2.4 (Properties of the modulation) . There exists < γ ≪ suchthat for any < γ < γ , T T , and any solution ~u = ( u, ∂ t u ) of (1.1) on [ T , T ] satisfying sup t ∈ [ T ,T ] (cid:26) inf ξ ∈ R N k u ( t ) − q ( · − ξ ) k H + k ∂ t u ( t ) k L (cid:27) < γ, (2.22) there exist unique C functions [ T , T ] → R N × R N × R I q × R I q × R × R t ( z ( t ) , ℓ ( t ) , θ ( t ) , β ( t ) , a ( t ) , b ( t )) such that, if we define ~ϕ = ( ϕ , ϕ ) by ~u = (cid:18) u∂ t u (cid:19) = (cid:18) Q − ℓ · ∇ Q (cid:19) + X ( i,j ) ∈ I q β ij (cid:18) ij (cid:19) + (cid:18) a Φ b Φ (cid:19) + (cid:18) ϕ ϕ (cid:19) , (2.23) where β = ( β ij ) ( i,j ) ∈ I q , it satisfies, for all t ∈ [ T , T ] , k ~ϕ ( t ) k H + | θ ( t ) | . k u ( t ) − q k H + k ∂ t u ( t ) k L . γ, | ℓ ( t ) | + | β ( t ) | + | a ( t ) | + | b ( t ) | . k u ( t ) − q k H + k ∂ t u ( t ) k L . γ, (2.24) and for all n = 1 , · · · , N and ( i, j ) ∈ I q , h ϕ , ∂ x n Q i = h ϕ , Ψ ij i = h ϕ , Φ i = 0 , (2.25) h ϕ , ∂ x n Q i = h ϕ , Ψ ij i = h ϕ , Φ i = 0 . (2.26) Proof.
The proof of the decomposition result relies on a standard argument basedon the Implicit function Theorem (See e.g. [11, Appendix B]) and we omit it. (cid:3)
Second, we derive the equation of ~ϕ from (1.1) and (2.23). Lemma 2.5 (Equation of ~ϕ ) . In the contex of Proposition 2.4, we have ( ∂ t ϕ = ϕ + Mod + G ,∂ t ϕ = ∆ ϕ − ϕ − αϕ + f ( Q + V + ϕ ) − f ( Q + V ) + Mod + G + G, (2.27) where Mod := (cid:0) ˙ z − ℓ (cid:1) · ∇ Q − X ( i,j ) ∈ I q (cid:0) ˙ θ ij − β ij (cid:1) Ψ ij − ( ˙ a − b )Φ , Mod := (cid:0) ˙ ℓ + 2 αℓ (cid:1) · ∇ Q − X ( i,j ) ∈ I q (cid:0) ˙ β ij + 2 αβ ij (cid:1) Ψ ij − (cid:16) ˙ b + 2 αb (cid:17) Φ , and G = f ( Q + V ) − f ( Q ) − f ′ ( Q ) V is defined in (2.10) , G := a ˙ z · ∇ Φ − a X ( i,j ) ∈ I q ˙ θ ij Φ ij ,G := X ( i,j ) ∈ I q ˙ θ ij ( ℓ · ∇ Ψ ij ) + X ( i,j ) ∈ I q β ij ( ˙ z · ∇ Ψ ij ) − ( ℓ · ∇ ) ( ˙ z · ∇ ) Q − X ( i ′ ,j ′ ) ∈ I q X ( i,j ) ∈ I q ˙ θ i ′ j ′ β ij Ψ i ′ j ′ ij + b ˙ z · ∇ Φ − X ( i,j ) ∈ I q b ˙ θ ij Φ ij . Proof.
First, from (2.14) and (2.23), ∂ t ϕ = ∂ t u − ∂ t Q − a∂ t Φ − ˙ a Φ= ϕ + (cid:0) ˙ z − ℓ (cid:1) · ∇ Q − X ( i,j ) ∈ I q (cid:0) ˙ θ ij − β ij (cid:1) Ψ ij − ( ˙ a − b )Φ+ a ˙ z · ∇ Φ − a X ( i,j ) ∈ I q ˙ θ ij Φ ij = ϕ + Mod + G . Using (2.14) and (2.23) again, ∂ t ϕ = ∂ tt u + ˙ ℓ · ∇ Q − X ( i,j ) ∈ I q ˙ β ij Ψ ij − ˙ b Φ+ ℓ · ∇ ∂ t Q − X ( i,j ) ∈ I q β ij ∂ t Ψ ij − b∂ t Φ= ∂ tt u + ˙ ℓ · ∇ Q − X ( i,j ) ∈ I q ˙ β ij Ψ ij − ˙ b Φ + G . From (1.1), (2.23), − ∆ Q + Q − f ( Q ) = 0 and − ∆Φ + Φ − f ′ ( Q )Φ = 0, ∂ tt u =∆ u − u − α∂ t u + f ( u )=∆ ϕ − ϕ − αϕ + f ( Q + V + ϕ ) − f ( Q + V )+ 2 αℓ · ∇ Q − α X ( i,j ) ∈ I q β ij Ψ ij − αb Φ + G. Therefore, ∂ t ϕ = ∆ ϕ − ϕ − αϕ + f ( Q + V + ϕ ) − f ( Q + V ) + G + G + (cid:16) ˙ ℓ + 2 αℓ (cid:17) · ∇ Q − X ( i,j ) ∈ I q (cid:16) ˙ β ij + 2 αβ ij (cid:17) Ψ ij − (cid:16) ˙ b + 2 αb (cid:17) Φ . (cid:3) SYMPTOTICS OF DAMPED NLKG EQUATION 11
Third, we derive the control of geometric parameters from orthogonality condi-tions (2.25) and (2.26). Our goal here is to get an ODE system on the modulationsparameters, at leading order, with bounds on the remainder terms as squares orhigher powers of | a | and N := k ~ϕ k H + | ℓ | + | β | + | b | . (2.28)Recall that we defined ¯ p = min { , p } > Lemma 2.6.
In the context of Proposition 2.4, the following holds. (i) Control of non-degenerate directions.
We have | ˙ z − ℓ | + (cid:12)(cid:12) ˙ θ − β (cid:12)(cid:12) . N + | a |N , (2.29) | ˙ ℓ + 2 αℓ | + | ˙ β + 2 αβ | . N + | a |N + | a | ¯ p . (2.30)(ii) Control of extra direction. For q be a degree-1 excited state, we have | ˙ a − b | . N + | a |N , (2.31) | ˙ b + 2 αb + 2 αa | . N + | a |N + | a | ¯ p . (2.32) Proof.
Proof of (i). First, we differentiate the orthogonality h ϕ , ∂ x n Q i = 0 in (2.25),0 = dd t h ϕ , ∂ x n Q i = h ∂ t ϕ , ∂ x n Q i + h ϕ , ∂ t ∂ x n Q i . Using (2.25) and (2.27), h ∂ t ϕ , ∂ x n Q i = h Mod , ∂ x n Q i + h G , ∂ x n Q i . From (2.7), (2.13), (2.15), (2.24), the expression of Mod and change of variables, h Mod , ∂ x n Q i = h ( ˙ z − ℓ ) · ∇ Q, ∂ x n Q i − X ( i,j ) ∈ I q (cid:16) ˙ θ ij − β ij (cid:17) h Ψ ij , ∂ x n Q i − ( ˙ a − b ) h Φ , ∂ x n Q i = h ( ˙ z − ℓ ) · ∇ q, ∂ x n q i − X ( i,j ) ∈ I q (cid:16) ˙ θ ij − β ij (cid:17) h Ω ij q, ∂ x n q i + O ( γ ( | ˙ z − ℓ | + | ˙ θ − β | )) . From the expression of G and (2.24), |h G , ∂ x n Q i| . | a | (cid:16) | ˙ z − ℓ | + | ℓ | + | ˙ θ − β | + | β | (cid:17) . γ (cid:16) | ˙ z − ℓ | + | ˙ θ − β | (cid:17) + | a |N . Next, using again (2.14), ∂ t ∂ x n Q = ∂ x n ∂ t Q = − ˙ z · ∇ ∂ x n Q + X ( i,j ) ∈ I q ˙ θ ij ∂ x n Ψ ij . (2.33)Thus, from (2.24) and the Sobolev embedding Theorem, |h ϕ , ∂ t ∂ x n Q i| . k ~ϕ k H (cid:16) | ˙ z − ℓ | + | ℓ | + | ˙ θ − β | + | β | (cid:17) . γ (cid:16) | ˙ z − ℓ | + | ˙ θ − β | (cid:17) + N . Combining above estimates, we have h ( ˙ z − ℓ ) · ∇ q, ∂ x n q i − X ( i,j ) ∈ I q (cid:16) ˙ θ ij − β ij (cid:17) h Ω ij q, ∂ x n q i = O (cid:16) γ (cid:16) | ˙ z − ℓ | + | ˙ θ − β | (cid:17) + N + | a |N (cid:17) . This gives N inequalities. Proceeding similarly for the I q orthogonality conditions h ϕ , Ω ij Q i = 0 in (2.25), and using the fact that the family { ∂ x n q, n N ; Ω ij q, ( i, j ) ∈ I q } is linearly independent, so that its Gram matrix is invertible, we obtain | ˙ z − ℓ | + (cid:12)(cid:12) ˙ θ − β (cid:12)(cid:12) . γ (cid:0) | ˙ z − ℓ | + (cid:12)(cid:12) ˙ θ − β (cid:12)(cid:12)(cid:1) + N + | a |N , which implies (2.29), upon taking γ small enough.Second, we differentiate the orthogonality h ϕ , ∂ x n Q i = 0 in (2.26),0 = dd t h ϕ , ∂ x n Q i = h ∂ t ϕ , ∂ x n Q i + h ϕ , ∂ t ∂ x n Q i . From (2.27), we have h ∂ t ϕ , ∂ x n Q i = h ∆ ϕ − ϕ + f ′ ( Q ) ϕ , ∂ x n Q i − α h ϕ , ∂ x n Q i + h G , ∂ x n Q i + h R, ∂ x n Q i + h G, ∂ x n Q i + h Mod , ∂ x n Q i , where R = f ( Q + V + ϕ ) − f ( Q + V ) − f ′ ( Q ) ϕ . Based on − ∆ ∂ x n q + ∂ x n q − f ′ ( q ) ∂ x n q = 0, integration by parts and (2.26), h ∆ ϕ − ϕ + f ′ ( Q ) ϕ , ∂ x n Q i − α h ϕ , ∂ x n Q i = 0 . Then, by the expression of G , (2.24) and (2.29), |h G , ∂ x n Q i| . (cid:16) | ˙ z − ℓ | + | ˙ θ − β | + | ℓ | + | β | (cid:17) ( | ℓ | + | β | + | b | ) . N ( N + | a | + 1) . N . Next, using (2.16), (2.19) and (2.24), | R | . | f ( Q + V + ϕ ) − f ( Q + V ) − f ′ ( Q + V ) ϕ | + | ( f ′ ( Q + V ) − f ′ ( Q )) ϕ | . | ϕ | + | ϕ | p + ( | V | + | V | p − ) | ϕ | . | ϕ | + | ϕ | p + | a | (cid:0) | Φ | + | Φ | p − (cid:1) | ϕ | . It follows that |h R, ∂ x n Q i| . Z R N (cid:0) | ϕ | + | ϕ | p + | ϕ || V | (cid:1) d x . N + | a |N . For q be a non-degenerate state, we have a = 0 which implies G = 0. For q be adegree-1 excited state, from (2.18), we have (cid:12)(cid:12)(cid:12)(cid:12) G − f ′′ ( Q ) V (cid:12)(cid:12)(cid:12)(cid:12) . | V | + | V | p . (cid:0) | a | + | a | p (cid:1) (cid:0) | Φ | + | Φ | p (cid:1) . It follows that (cid:12)(cid:12)(cid:12)(cid:12) h G, ∂ x n Q i − h f ′′ ( Q ) V , ∂ x n Q i (cid:12)(cid:12)(cid:12)(cid:12) . (cid:0) | a | + | a | p (cid:1) Z R N (cid:0) | Φ | + | Φ | p (cid:1) | ∂ x n Q | d x . | a | + | a | p . Note that, by (2.6), (2.13) and change of variables, h f ′′ ( Q ) V , ∂ x n Q i = 0 . SYMPTOTICS OF DAMPED NLKG EQUATION 13
Then, from (2.7), (2.13), (2.15), (2.24), the expression of Mod and change ofvariables, h Mod , ∂ x n Q i = h ( ˙ ℓ + 2 αℓ ) · ∇ Q, ∂ x n Q i − X ( i,j ) ∈ I q (cid:16) ˙ β ij + 2 αβ ij (cid:17) h Ψ ij , ∂ x n Q i − (˙ b + 2 αb ) h Φ , ∂ x n Q i = h ( ˙ ℓ + 2 αℓ ) · ∇ q, ∂ x n q i − X ( i,j ) ∈ I q (cid:16) ˙ β ij + 2 αβ ij (cid:17) h Ω ij q, ∂ x n q i + O (cid:16) γ (cid:16) | ˙ ℓ + 2 αℓ | + | ˙ β + 2 αβ | (cid:17)(cid:17) . Last, from (2.24), (2.29) and (2.33), |h ϕ , ∂ t ∂ x n Q i| . k ~ϕ k H (cid:16) | ˙ z − ℓ | + | ℓ | + | ˙ θ − β | + | β | (cid:17) . N ( N + | a | + 1) . N . In conclusion of the previous estimates, the orthogonality condition h ϕ , ∂ x n Q i = 0gives the following, h ( ˙ ℓ + 2 αℓ ) · ∇ q, ∂ x n q i − X ( i,j ) ∈ I q (cid:16) ˙ β ij + 2 αβ ij (cid:17) h Ω ij q, ∂ x n q i + O (cid:16) γ (cid:16) | ˙ ℓ + 2 αℓ | + | ˙ β + 2 αβ | (cid:17)(cid:17) = O (cid:0) N + | a |N + | a | + | a | p (cid:1) . Proceeding similarly for the orthogonality conditions h φ , Ω ij Q i = 0 in (2.26) andusing again the fact that family { ∂ x n q, n N ; Ω ij q, ( i, j ) ∈ I q } is linearly inde-pendent, we find (2.30) for γ small enough.Proof of (ii). We prove (2.32); the proof of (2.31) is same as (2.29). We differentiatethe orthogonality h ϕ , Φ i = 0 in (2.26),0 = dd t h ϕ , Φ i = h ∂ t ϕ , Φ i + h ϕ , ∂ t Φ i . Using again (2.27), h ∂ t ϕ , Φ i = h ∆ ϕ − ϕ + f ′ ( Q ) ϕ , Φ i − α h ϕ , Φ i + h G , Φ i + h R, Φ i + h G + Mod , Φ i , where R = f ( Q + V + ϕ ) − f ( Q + V ) − f ′ ( Q ) ϕ . From − ∆Φ + Φ − f ′ ( Q )Φ = 0, integration by parts and (2.26), h ∆ ϕ − ϕ + f ′ ( Q ) ϕ , Φ i − α h ϕ , Φ i = 0 . By the expression of G , (2.24) and (2.29), |h G , Φ i| . (cid:16) | ˙ z − ℓ | + | ˙ θ − β | + | ℓ | + | β | (cid:17) ( | ℓ | + | β | + | b | ) . N . Recall that, from (2.16), (2.19) and (2.24), | R | . | ϕ | + | ϕ | p + | a | (cid:0) | Φ | + | Φ | p − (cid:1) | ϕ | . Based on above inequality and the Sobolev embedding Theorem, |h R, Φ i| . Z R N (cid:0) | ϕ | + | ϕ | p + | a | (cid:0) | Φ | + | Φ | p − (cid:1) | ϕ | (cid:1) | Φ | d x . k ~ϕ k H + k ~ϕ k p H + | a |k ~ϕ k H . N + | a |N . Then, from (2.18) and p > (cid:12)(cid:12)(cid:12)(cid:12) G − f ′′ ( Q ) V (cid:12)(cid:12)(cid:12)(cid:12) . | V | + | V | p . (cid:0) | a | + | a | p (cid:1) (cid:0) | Φ | + | Φ | p (cid:1) . It follows that (cid:12)(cid:12)(cid:12)(cid:12) h G − f ′′ ( Q ) V , Φ i (cid:12)(cid:12)(cid:12)(cid:12) . (cid:0) | a | + | a | p (cid:1) Z R N (cid:0) | Φ | + | Φ | p +1 (cid:1) d x . | a | + | a | p . Note that, by our normalization choice for φ (2.8), and a change of variables, h f ′′ ( Q ) V , Φ i = a E ′′′ ( q )( φ, φ, φ ) = − αa k φ k . Thus, from (2.7), (2.13), the expression of Mod , the above estimates and changeof variables, h Mod + G, Φ i = − (˙ b + 2 αb ) h Φ , Φ i + h f ′′ ( Q ) V , Φ i + h G − f ′′ ( Q ) V , Φ i = − (˙ b + 2 αb + 2 αa ) h φ, φ i + O (cid:0) | a | + | a | p (cid:1) . Last, from (2.14), (2.24), (2.29) and the Sobolev embedding Theorem, |h ϕ , ∂ t Φ i| . k ~ϕ k H (cid:16) | ˙ z − ℓ | + | ˙ θ − β | + | ℓ | + | β | (cid:17) . N ( N + | a | + 1) . N . Combining the above estimates, we have(˙ b + 2 αb + 2 αa ) h φ, φ i = O (cid:0) N + | a |N + | a | + | a | p (cid:1) , which means (2.32). (cid:3) Last, we derive the control of exponential directions.
Lemma 2.7.
Let a ± k = h ~ϕ, ~Z ± k i for k = 1 , · · · , K . Then (cid:12)(cid:12)(cid:12)(cid:12) dd t a ± k − ν ± k a ± k (cid:12)(cid:12)(cid:12)(cid:12) . N + a . (2.34) Proof.
By (2.27),dd t a ± k = h ∂ t ~ϕ, ~Z ± k i + h ~ϕ, ∂ t ~Z ± k i = h ∆ ϕ − ϕ + f ′ ( Q ) ϕ , Υ k i + (cid:0) ζ ± k − α (cid:1) h ϕ , Υ k i + h R + G, Υ k i + h ζ ± k Mod + Mod , Υ k i + h ζ ± k G + G , Υ k i + h ~ϕ, ∂ t ~Z ± k i . From λ k = ν ± k ζ ± k , L q Y k = − λ k Y k and integration by parts, h ∆ ϕ − ϕ + f ′ ( Q ) ϕ , Υ k i = h ϕ , (∆ − f ′ ( Q )) Υ k i = ν ± k ζ ± k h ϕ , Υ k i . Combining above identity with ν ± k = ζ ± k − α and the definition of a ± k , h ∆ ϕ − ϕ + f ′ ( Q ) ϕ , Υ k i + (cid:0) ζ ± k − α (cid:1) h ϕ , Υ k i = ν ± k (cid:0) ζ ± k h ϕ , Υ k i + h ϕ , Υ k i (cid:1) = ν ± k a ± k . Recall that, from (2.16), (2.17) and (2.19), | R | + | G | . | ϕ | + | ϕ | p + | V | + | V | p . Therefore, by the Sobolev embedding Theorem, and as Υ k is exponentially local-ized, we have |h R + G, Υ k i| . Z R N (cid:0) ϕ + | ϕ | p + | V | + | V | p (cid:1) | Υ k | d x . N + a . Note that, from h ψ, Y k i = 0 for any ψ ∈ ker L q and k = 1 , · · · , K , h ζ ± k Mod + Mod , Υ k i = 0 . SYMPTOTICS OF DAMPED NLKG EQUATION 15
Next, by the expression of G , G , (2.24), (2.29) and (2.30), (cid:12)(cid:12) h ζ ± k G + G , Υ k i (cid:12)(cid:12) . (cid:0) | ˙ z − ℓ | + | ˙ θ − β | + | ℓ | + | β | (cid:1) ( | ℓ | + | β | + | a | + | b | ) . N ( N + | a | ) ( N + | a | + 1) . N + a . Last, from (2.14) (2.24), (2.29) and the Sobolev embedding Theorem, (cid:12)(cid:12) h ~ϕ, ∂ t ~Z ± k i (cid:12)(cid:12) . Z R N ( | ϕ | + | ϕ | ) | ∂ t Υ k | d x . k ~ϕ k H (cid:16) | ˙ z − ℓ | + | ˙ θ − β | + | ℓ | + | β | (cid:17) . N ( N + | a | + 1) . N . Gathering these estimates, and proceeding similarly for all k = 1 , · · · , K , we ob-tain (2.34). (cid:3) Energy estimates.
Let q be a non-degenerate or degree-1 excited state. For µ > ρ = 2 α − µ , and consider the nonlinearenergy functional E = Z R N (cid:8) |∇ ϕ | + (1 − ρµ ) ϕ + ( ϕ + µϕ ) (cid:9) d x − Z R N { F ( Q + V + ϕ ) − F ( Q + V ) − f ( Q + V ) ϕ } d x. (2.35)Analogous energy functionals were introduced in [10, 9], in order to take full ad-vantage of the damping of (1.1), as is shown in the Lemma below. Recall that inthe previous paragraph, in (2.28), we set N = k ~ϕ k H + | ℓ | + | β | + | b | . Lemma 2.8.
In the context of Proposition 2.4, there exist < γ ≪ and <µ < min k, ± (1 , α, ν ± k ) such that the following hold, for all t ∈ [ T , T ] . (i) Coercivity and bound. µ k ~ϕ k H − µ − K X k =1 (cid:0) ( a + k ) + ( a − k ) (cid:1) E µ − k ~ϕ k H . (2.36)(ii) Time variation. dd t E + 2 µ E µ − (cid:0) N + a N (cid:1) . (2.37) Proof.
Proof of (i). First, from (2.20) and the Sobolev embedding Theorem, Z R N | F ( Q + V + ϕ ) − F ( Q + V ) − f ( Q + V ) ϕ | d x . Z R N ( | Q + V | p − | ϕ | + | ϕ | p +1 )d x . k ~ϕ k H + k ~ϕ k p +1 H . It follows that the right-hand side of (2.36).Second, from the definition of E in (2.35), we decompose E = h− ∆ ϕ + ϕ − f ′ ( Q ) ϕ , ϕ i − ρµ h ϕ , ϕ i + E + E + E , where E = Z R N ( ϕ + µϕ ) d x, E = − Z R N ( f ′ ( Q + V ) − f ′ ( Q )) ϕ d x, E = − Z R N (cid:8) F ( Q + V + ϕ ) − F ( Q + V ) − f ( Q + V ) ϕ − f ′ ( Q + V ) ϕ (cid:9) d x. By (2.5) and (2.25), h− ∆ ϕ + ϕ − f ′ ( Q ) ϕ , ϕ i > c k ϕ k H − c − K X k =1 (cid:0) ( a + k ) + ( a − k ) (cid:1) . From the AM-GM inequality, we have E = Z R N ( ϕ + µϕ ) d x > Z R N ϕ d x − µ Z R N ϕ d x. Then, using (2.16), (2.21), (2.24) and the Sobolev embedding Theorem, we have |E | . Z R N (cid:0) | V | + | V | p − (cid:1) | ϕ | d x . (cid:16) γ + γ p − (cid:17) k ~ϕ k H , |E | . Z R N (cid:0) | ϕ | + | ϕ | p +1 (cid:1) d x . k ~ϕ k H + k ~ϕ k p +1 H . Combining the above estimates, we have E > (cid:0) c − ρµ − µ (cid:1) k ϕ k H + 12 k ϕ k L − c − K X k =1 (cid:0) ( a + k ) + ( a − k ) (cid:1) + O (cid:16)(cid:16) γ + γ p − (cid:17) k ~ϕ k H + k ~ϕ k H (cid:17) , which implies the left-hand side of (2.36) for γ > µ > t E =2 Z R N ∂ t ϕ (cid:8) − ∆ ϕ + (1 − ρµ ) ϕ − (cid:2) f ( Q + V + ϕ ) − f ( Q + V ) (cid:3)(cid:9) d x − Z R N { ( f ( Q + V + ϕ ) − f ( Q + V ) − f ′ ( Q + V ) ϕ ) ( ∂ t Q + ∂ t V ) } d x + 2 Z R N { ( ϕ + µϕ )( ∂ t ϕ + µ∂ t ϕ ) } d x = I + I + I . Estimate on I . We claim I = − Z R N { ∆ ϕ − ϕ + [ f ( Q + V + ϕ ) − f ( Q + V )] } ϕ d x − ρµ Z R N ϕ ϕ d x + O (cid:0) N + | a |N (cid:1) . (2.38)By (2.25), (2.27), − ∆ Mod + Mod − f ′ ( Q ) Mod = 0 and integration by parts, I =2 Z R N {− ∆ ϕ + (1 − ρµ ) ϕ − [ f ( Q + V + ϕ ) − f ( Q + V )] } ϕ d x + I , + I , + I , + I , , where I , = − Z R N [∆ ϕ − (1 − ρµ ) ϕ ] G d x, I , = − Z R N [ f ( Q + V + ϕ ) − f ( Q + V )] G d x, I , = − Z R N [( f ′ ( Q + V ) − f ′ ( Q )) ϕ ] Mod d x, I , = − Z R N [ f ( Q + V + ϕ ) − f ( Q + V ) − f ′ ( Q + V ) ϕ ] Mod d x. SYMPTOTICS OF DAMPED NLKG EQUATION 17
Using (2.19), (2.24), (2.29) and integration by parts, we have |I , | . Z R N ( | ∆ G | + | G | ) | ϕ | d x . | a |N (cid:16) | ˙ z − ℓ | + | ˙ θ − β | + N (cid:17) . | a |N ( N + | a | + 1) . | a |N , |I , | . Z R N | G | ( | ϕ | + | ϕ | p )d x . | a |N (cid:16) | ˙ z − ℓ | + | ˙ θ − β | + N (cid:17) . | a |N ( N + | a | + 1) . | a |N . Then, using (2.16), (2.19), (2.24) and (2.29), we have |I , | . Z R N (cid:0) | V | + | V | p − (cid:1) | ϕ || Mod | d x . k ~ϕ k H (cid:0) | a | + | a | p − (cid:1) (cid:16) | ˙ z − ℓ | + | ˙ θ − β | + | ˙ a − b | (cid:17) . N (cid:0) | a | + | a | p − (cid:1) ( N + | a | ) . N + | a |N , and |I , | . Z R N (cid:0) ϕ + | ϕ | p (cid:1) | Mod | d x . (cid:0) N + N p (cid:1) (cid:16) | ˙ z − ℓ | + | ˙ θ − β | + | ˙ a − b | (cid:17) . N (cid:0) N + N p (cid:1) ( N + | a | ) . N . Gathering the above estimates, we obtain (2.38).
Estimate on I . We claim |I | . N . (2.39)By (2.14), ∂ t Q + ∂ t V = − ℓ · ∇ Q + X ( i,j ) ∈ I q β ij Ψ ij + b Φ − Mod − G . Based on above identity, we decompose I = I , + I , + I , + I , + I , , where I , = 2 Z R N ( f ( Q + V + ϕ ) − f ( Q + V ) − f ′ ( Q + V ) ϕ ) ( − b Φ) d x, I , = 2 Z R N ( f ( Q + V + ϕ ) − f ( Q + V ) − f ′ ( Q + V ) ϕ ) G d x, and I , = 2 Z R N ( f ( Q + V + ϕ ) − f ( Q + V ) − f ′ ( Q + V ) ϕ ) Mod d x, I , = 2 Z R N ( f ( Q + V + ϕ ) − f ( Q + V ) − f ′ ( Q + V ) ϕ ) (cid:0) ℓ · ∇ Q (cid:1) d x, I , = − X ( i,j ) ∈ I q β ij Z R N ( f ( Q + V + ϕ ) − f ( Q + V ) − f ′ ( Q + V ) ϕ ) Ψ ij d x. From (2.19) and the Sobolev embedding Theorem, |I , | . | b | Z R N ϕ | Φ | d x . N . Using (2.19), (2.24), (2.29), (2.31) and the Sobolev embedding Theorem, we have |I , | . Z R N (cid:0) ϕ + | ϕ | p (cid:1) | G | d x . (cid:0) N + N p (cid:1) (cid:16) | a |N + | a || ˙ z − ℓ | + | a || ˙ θ − β | (cid:17) . (cid:0) N + N p (cid:1) ( | a |N + | a |N ( | a | + N )) . N , |I , | . Z R N (cid:0) ϕ + | ϕ | p (cid:1) | Mod | d x . (cid:0) N + N p (cid:1) (cid:16) ˙ z − ℓ | + | ˙ θ − β | + | ˙ a − b | (cid:17) . (cid:0) N + N p (cid:1) ( | a | + N ) N . N , |I , | . Z R N (cid:0) ϕ + | ϕ | p (cid:1) | ℓ · ∇ Q | d x . N , and |I , | . X ( i,j ) ∈ I q Z R N (cid:0) ϕ + | ϕ | p (cid:1) | β ij Ψ ij | d x . N . Combining above estimates with (2.24), we obtain (2.39).
Estimate on I . We claim I = − µ E + 2 Z R N { ∆ ϕ − ϕ + [ f ( Q + V + ϕ ) − f ( Q + V )] } ϕ d x + 2 ρµ Z R N ϕ ϕ d x − ρ − µ ) Z R N ( ϕ + µϕ ) d x + O (cid:0) N + a N (cid:1) . (2.40)By direct computation and (2.27), we decompose, I = − µ E + 2 Z R N { ∆ ϕ − ϕ + [ f ( Q + V + ϕ ) − f ( Q + V )] } ϕ d x + 2 ρµ Z R N ϕ ϕ d x − ρ − µ ) Z R N ( ϕ + µϕ ) d x + I , + I , + I , + I , + I , , where I , = 2 Z R N ( ϕ + µϕ ) G d x, I , = 2 Z R N ( ϕ + µϕ ) ( G + µG ) d x, I , = 2 Z R N ( ϕ + µϕ ) (Mod + µ Mod ) d x, and I , = 2 µ Z R N [ f ( Q + V + ϕ ) − f ( Q + V ) − f ′ ( Q + V ) ϕ ] ϕ d x, I , = − µ Z R N (cid:2) F ( Q + V + ϕ ) − F ( Q + V ) − f ( Q + V ) ϕ − f ′ ( Q + V ) ϕ (cid:3) d x. Using (2.17), |I , | . Z R N ( | ϕ | + | ϕ | ) V d x . a N . SYMPTOTICS OF DAMPED NLKG EQUATION 19
Next, from (2.24), (2.29) and the AM-GM inequality, |I , | . Z R N ( | ϕ | + | ϕ | ) ( | G | + | G | ) d x . N ( | a | + | b | + | ℓ | + | β | ) (cid:16) | ˙ z − ℓ | + | ˙ θ − β | + | ℓ | + | β | (cid:17) . N ( | a | + N ) ( N + | a | + 1) . a N + N . Note that, from (2.25) and (2.26), we have I , = 2 h ϕ , Mod + µ Mod i + 2 µ h ϕ , Mod + µ Mod i = 0 . Last, from (2.19), (2.21) and the Sobolev embedding Theorem, |I , | . Z R N (cid:0) | ϕ | + | ϕ | p (cid:1) | ϕ | d x . N + N p +1 , |I , | . Z R N (cid:0) | ϕ | + | ϕ | p +1 (cid:1) d x . N + N p +1 . Gathering the above estimates, we obtain (2.40).Last, combining (2.38), (2.39) and (2.40), we conclude (2.37). (cid:3)
We observe from the definition and the energy property (1.2) that a packed solution ~u with degree-1 excited state cluster point q satisfieslim t →∞ E ( ~u ( t )) = 12 Z R N (cid:0) |∇ q | + q − F ( q ) (cid:1) d x = E ( q, . (2.41)More precisely, we expand the energy for a solution close to degree-1 excited state. Lemma 2.9.
In the context of Proposition 2.4, for the case q be a degree-1 excitedstate, we have E ( ~u ) = E ( q,
0) + 23 α k φ k L a + O (cid:0) N + | a | ¯ p +1 (cid:1) . (2.42) Proof.
By (2.23) and an elementary computation, we find | u | = | Q | + | V | + | ϕ | + 2 Q ( V + ϕ ) + 2 V ϕ , |∇ u | = |∇ Q | + |∇ V | + |∇ ϕ | + 2 ∇ Q · ∇ ( V + ϕ ) + 2 ∇ V · ∇ ϕ , and F ( u ) = F ( Q ) + f ( Q )( V + ϕ ) + 12 f ′ ( Q )( V + ϕ + 2 V ϕ ) + 16 f ′′ ( Q ) V + F ( Q + V + ϕ ) − F ( Q ) − f ( Q )( V + ϕ ) − f ′ ( Q )( V + ϕ ) − f ′′ ( Q )( V + ϕ ) + 16 f ′′ ( Q ) (cid:0) ( V + ϕ ) − V (cid:1) . Based on the above identities and change of variables, we have E ( ~u ) = E ( q, − Z R N f ′′ ( Q ) V d x + E + E + E + E + E + E , where E = 12 Z R N (cid:0) |∇ V | + | V | − f ′ ( Q ) V (cid:1) d x,E = Z R N ( ∇ V · ∇ ϕ + V ϕ − f ′ ( Q ) V ϕ ) d x,E = 12 Z R N (cid:0) |∇ ϕ | + | ϕ | − f ′ ( Q ) ϕ + ( ∂ t u ) (cid:1) d x,E = Z R N ( ∇ Q · ∇ ( V + ϕ ) + Q ( V + ϕ ) − f ( Q )( V + ϕ )) d x, and E = − Z R N f ′′ ( Q ) (cid:0) ( V + ϕ ) − V (cid:1) d x,E = − Z R N (cid:18) F ( Q + V + ϕ ) − F ( Q ) − f ( Q )( V + ϕ ) − f ′ ( Q )( V + ϕ ) − f ′′ ( Q )( V + ϕ ) (cid:19) d x. First, from (2.8) we have E ′′′ ( φ, φ, φ ) = − α k φ k L . Therefore, by change of variables, − Z R N f ′′ ( Q ) V d x = − a E ′′′ ( φ, φ, φ ) = 23 α k φ k L a . Then, from − ∆Φ + Φ − f ′ ( Q )Φ = 0, V = a Φ and integration by parts, E = a Z R N ( − ∆Φ + Φ − f ′ ( Q )Φ) Φd x = 0 ,E = a Z R N ( − ∆Φ + Φ − f ′ ( Q )Φ) ϕ d x = 0 . By the Sobolev embedding theorem and the AM-GM inequality, | E | . k ~ϕ k H + | ℓ | + | β | + b . N . Next, from integration by parts and − ∆ Q + Q − f ( Q ) = 0, we have E = Z R N ( − ∆ Q + Q − f ( Q )) ( V + ϕ )d x = 0 . Last, from the Sobolev embedding Theorem and the AM-GM inequality, we have | E | . Z R N | f ′′ ( Q ) | (cid:0) | ϕ | V + | V | ϕ + | ϕ | (cid:1) d x . a + N , | E | . Z R N (cid:0) | V + ϕ | + | V + ϕ | p +1 (cid:1) d x . a + | a | p +1 + N + N p +1 . Gathering above estimates, we find (2.42). (cid:3)
Time evolution analysis.
We introduce new parameters and functionals toanalyze the time evolution of solutions in the framework of Proposition 2.4. It isconvenient to consider together all the damped components, as follows: S = | ℓ | + | β | + K X k =1 ( a − k ) + b and F = E + µ − S . (2.43)We also define the unstable component: A = K X k =1 ( a + k ) . (2.44)Finally, for the analysis of the main ODE system in the degree-1 degenerate case,we will crucially relie on the following quantities R = a + b α and R = 2 α a + 12 α ab + a b. (2.45)Indeed, R and R enjoy monotonicity related properties as it is made explicit inthe following lemma, in which we also rewrite the estimates of Lemma 2.6, 2.7 and2.8, using our new notations. SYMPTOTICS OF DAMPED NLKG EQUATION 21
Lemma 2.10.
In the context of Proposition 2.4, there exist C > such that thefollowing hold. (i) Coercivity and bound. We have C − N F + µ − A C N . (2.46)(ii) Liapunov evolution of R . We have dd t R + 12 a C N , (2.47)dd t R + αa C N , (2.48) (cid:12)(cid:12)(cid:12)(cid:12) dd t R + a (cid:12)(cid:12)(cid:12)(cid:12) C (cid:0) N + | a |N + | a | ¯ p (cid:1) . (2.49)(iii) Damped evolution . There exists ν > µ such that dd t S + ν S C (cid:0) N + a N (cid:1) , (2.50)dd t F + 2 µ F C (cid:0) N + a N (cid:1) . (2.51)(iv) Exponential growth . There exists ν > ν > µ such that dd t A − ν A C (cid:0) N + a N (cid:1) , (2.52)dd t A − ν A > − C (cid:0) N + a N (cid:1) . (2.53) Proof.
Proof of (i). Due to the bound from below in (2.36), F + µ − A > µ k ~ϕ k H + µ − ( | ℓ | + | β | + b ) & N . Now | a ± k | = |h ~ϕ, ~Z ± k i| k ~ϕ k H N , so that using the bound from above in (2.36), F + µ − A . E + | ℓ | + | β | + b + K X k =1 ( a − k ) + ( a + k ) . N . Proof of (ii). First, from (2.31) and (2.32), we havedd t R = ˙ a + ˙ b α = − a + O ( N + | a |N + | a | + | a | p ) . Based on (2.24), the above estimate and the AM-GM inequality, we obtain (2.47)and (2.49) for C large enough. Second, using again (2.31), (2.32), we computedd t a = 3 a b + O (cid:0) a N + | a | N (cid:1) , dd t (cid:0) ab (cid:1) = − αab + O (cid:0) N + | a | N (cid:1) , dd t ( a b ) = − αa + 2 ab − αa b + O (cid:0) N + | a | N + | a | + | a | p +2 (cid:1) . Combining the above estimates, we inferdd t R = 2 α t a + 12 α dd t ( ab ) + dd t ( a b )= − αa + O (cid:0) N + | a | N + | a | + | a | p +2 (cid:1) , Using the AM-GM inequality (to see | a | N N / + | a | , a N N + | a | ), andas p + 2 >
4, this implies (2.48) for C large enough.Proof of (iii) and (iv). The estimates (2.50), (2.51), (2.52) and (2.53) are conse-quences of (2.30), (2.32), (2.34), (2.37) and taking 0 < µ < min k, ± (1 , α, ν ± k ) inLemma 2.8. (cid:3) Proof of Theorem 1.4 and Theorem 1.6
In this section, we prove Theorem 1.4 and Theorem 1.6. First, we prove the follow-ing trapping Proposition for packed solution.
Proposition 3.1.
There exists δ > such that the following holds. Let ~u =( u, ∂ t u ) be a packed solution of (1.1) , with cluster point ( q, where q is a non-degenerate state or degree-1 excited state. Due to Proposition 1.2, for any < δ <δ , there exists T δ > and ξ δ ∈ R N such that k u ( T δ ) − q ( · − ξ δ ) k H + k ∂ t u ( t ) k L + (cid:20) Z ∞ T δ k ∂ t u ( t ) k L d t (cid:21) < δ. (3.1) Then ~u admits a decomposition as in Proposition 2.4 for all t ∈ [ T δ , ∞ ) and satisfies ∀ t > T δ , k u ( t ) − q ( · − ξ δ ) k H + k ∂ t u ( t ) k L . δ . (3.2) Proof.
For δ < γ small enough, the existence of T δ is a direct consequence ofProposition 1.2: ~u ( T δ ) admits a decomposition as in Proposition 2.4 and satisfies | z ( T δ ) − ξ δ | + | θ ( T δ ) | + | a ( T δ ) | + N ( T δ ) . δ. (3.3)For a constant C > | z − ξ δ | + | θ | δ , | a | Cδ , N Cδ, A Cδ . (3.4)Set T ∗ = sup { t ∈ [ T δ , ∞ ) such that (2.22) and (3.4) holds on [ T δ , t ] } . We prove thqat T ∗ = ∞ by strictly improving the bootstrap assumption (3.4) on[ T δ , T ∗ ) (upon chosing C large enough). In the remainder of the proof, the impliedconstants in . or O do not depend on δ nor on the constant C appearing in thebootstrap assumption (3.4). Recall that, in the case when q is a non-degeneratestate, we denote a = b = 0. Step 1. Preliminary bounds.
Let t ∈ [ T δ , T ∗ ). Integrating (2.53) on [ T δ , t ], andusing (3.3), we have Z tT δ A ( s )d s . A ( t ) + Z tT δ N ( s )d s + Z tT δ | a | ( s )d s . Cδ + Cδ Z tT δ N ( s )d s + Cδ Z tT δ a ( s )d s, Similarly, integrating(2.51), we get Z tT δ F ( s )d s . |F ( t ) | + |F ( T δ ) | + Z tT δ N ( s )d s + Z tT δ | a | ( s )d s . C δ + Cδ Z tT δ N ( s )d s + Cδ Z tT δ a ( s )d s, and integrating (2.47), there hold12 Z tT δ a ( s )d s − C Z tT δ N ( s )d s R ( T δ ) − R ( t ) . Cδ + Cδ.
SYMPTOTICS OF DAMPED NLKG EQUATION 23
Now recall that due to the coercivity (2.46), F + µ − A > C − N , hence Z tT δ (cid:0) N ( s ) + a ( s ) (cid:1) d s Z tT δ a ( s )d s − C Z tT δ N ( s )d s + C ( C + 1) Z tT δ ( F ( s ) + µ − A ( s ))d s . Cδ + Cδ + C δ + Cδ Z tT δ N ( s )d s + Cδ Z tT δ a ( s )d s. Taking 0 < δ ≪ Z tT δ (cid:0) N ( s ) + a ( s ) (cid:1) d s . Cδ + Cδ + C δ . Cδ + C δ . Step 2. Estimate on z and θ . By (2.29) and (2.30), we have | z ( t ) − ξ δ | + | θ ( t ) | . (cid:12)(cid:12)(cid:12)(cid:12) z ( t ) − ξ δ + ℓ ( t )2 α (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) θ ( t ) + ℓ ( t )2 α (cid:12)(cid:12)(cid:12)(cid:12) + | ℓ ( t ) | + | β ( t ) | . N ( t ) + N ( T δ ) + | z ( T δ ) − ξ δ | + | θ ( T δ ) | + Z tT δ (cid:0) N ( s ) + a ( s ) (cid:1) d s . Cδ + Cδ + C δ , which strictly improves the estimate (3.4) of z and θ on [ T δ , T ∗ ) for δ small enough. Step 3. Estimate on a . Due to the energy dissipation (1.5) and the initial assump-tion (3.1), note that we have E ( q, E ( ~u ( t )) E ( q,
0) + O ( δ ) . Using the expansion of the energy (2.42) (and the bootstrap bounds (3.4)), wededuce that | a | . δ + N + a + | a | p +1 . C δ + C ¯ p +1 δ (¯ p +1) . It follows that | a | . δ + C δ + C ¯ p +13 δ (¯ p +1) . which strictly improves the estimate (3.4) of a on [ T δ , T ∗ ) for C large enough and δ small enough. Step 4. Estimate on N . Under the bootstrap assumption (3.4), estimate (2.51)writes dd t F + 2 µ F . C δ + C δ . C δ . Let t ∈ [ T δ , T ∗ ). Integrating the above estimate on [ T δ , t ], using also the initial timeassumption (3.3), yields F ( t ) . e − µ ( t − T δ ) F ( T δ ) + C Z tT δ e − µ ( t − s ) δ d s . δ + C δ . (3.5)We now use once more the coercivity (2.46), the (3.4) on A so that the estimate(3.5) above gives, for C large enough N ( t ) . F + A . δ + C δ + Cδ Cδ . This strictly improves the estimate (3.4) of N on [ T δ , T ∗ ). Step 5. Estimate on A . Let us first rewrite the estimates (2.50), (2.52), (2.53) inthe context of the bootstrap assumption (3.4): for all t ∈ [ T δ , T ∗ ),dd t S + ν S C (cid:16) C δ + C δ (cid:17) C C δ , (3.6)dd t A − ν A C (cid:16) C δ + C δ (cid:17) C C δ , (3.7)dd t A − ν A > − C (cid:16) C δ + C δ (cid:17) > − C C δ . (3.8)Then for t ∈ [ T δ , T ∗ ), we have (integrating (3.6) on [ T δ , t ]) S ( t ) . e − ν ( t − T δ ) S ( T δ ) + C Z tT δ e − µ ( t − s ) δ d s . δ + C δ . (3.9)We now prove by contradiction that for C large enough, it holds ∀ t ∈ [ T δ , T ∗ ) , A ( t ) < Cδ . (3.10)For the sake of contradiction, assume that there exists t ∈ [ T δ , T ∗ ) such that A ( t ) = 2 Cδ and for all t ∈ [ T δ , t ) , A ( t ) < Cδ . On the one hand, by continuity of A , there exists t ∈ [ T δ , t ] such that A ( t ) = Cδ and for all t ∈ ( t , t ) , Cδ < A ( t ) < Cδ . (3.11)For t ∈ [ t , t ], divide (3.7) and (3.8) by A (using (3.11)), and then integrate on[ t , t ]: it follows thatlog 2 ν + O ( C δ ) t − t log 2 ν + O ( C δ ) . (3.12)Therefore, using (3.11) again Z t t A ( t )d t > Cδ ( t − t ) > Cδ (cid:18) log 2 ν + O ( C δ ) (cid:19) . (3.13)On the other hand, by the definition of ϕ in (2.23) and the bound on S (3.9), forany t ∈ [ t , t ] k ϕ ( t ) k L . k ∂ t u ( t ) k L + S ( t ) . k ∂ t u ( t ) k L + δ + C δ . Thus, from (3.1) and (3.12), Z t t k ϕ ( t ) k L d t . Z t t (cid:16) k ∂ t u ( t ) k L + δ + C δ (cid:17) d t . δ + C δ . (3.14)By the definition of a ± k , we have a + k = ζ + k h ϕ , Υ k i + h ϕ , Υ k i , a − k = ζ − k h ϕ , Υ k i + h ϕ , Υ k i and thus for all k = 1 , · · · , Ka + k = ζ + k ζ − k a − k + ζ − k − ζ + k ζ − k h ϕ , Υ k i , from where we see that A . S + k ϕ k L . Gathering estimates (3.9) and (3.14), wefind Z t t A ( t )d t . Z t t S ( t )d t + Z t t k ϕ ( t ) k L d t . δ + C δ , which is a contradiction with (3.13) for C large enough and δ small enough. Thisproves (3.10), and this strictly improves the estimate on A Step 6. Conclusion . As a consequence of improving the bootstrap assumption (3.4)on z − ξ δ , θ , N and A , we conclude that T ∗ = ∞ . SYMPTOTICS OF DAMPED NLKG EQUATION 25
Finally, from (3.4), we know that, for all t ∈ [ T δ , + ∞ ), k u ( t ) − q ( · − ξ δ ) k H + k ∂ t u ( t ) k L . k u ( t ) − Q ( t ) k H + k Q ( t ) − q ( · − ξ δ ) k H + k ∂ t u ( t ) k L . N ( t ) + | a ( t ) | + | z ( t ) − ξ δ | + | θ ( t ) | . δ + Cδ + Cδ, which implies (3.2) for δ small enough. The proof of Proposition 3.1 is complete. (cid:3) Remark 3.2.
Observe that the proof of Proposition 3.1 actually proves that θ ( t ) , a ( t ) , N ( t ) , A ( t ) → t → + ∞ , a fact that we will use in the proofs of Theorem 1.4 and 1.6. End of the proof of Theorem 1.4.
We assume here that q is a non-degenerate boundstate, and prove exponential convergence. Recall that, in this context, we have set a = b = 0.First, from Proposition 3.1, we know that, for any 0 < δ < δ , there exists T ∗ δ ≫ ξ δ ∈ R N such that, ~u admits a decomposition as in Proposition 2.4 for all t ∈ [ T ∗ δ , ∞ ) and satisfies k u ( t ) − q ( · − ξ δ ) k H + k ∂ t u ( t ) k L + | z ( t ) − ξ δ | + | θ ( t ) | + N ( t ) < δ. (3.15) Step 1.
We claim that ∀ t > T ∗ δ , dd t (cid:0) F ( t ) + µ − A ( t ) (cid:1) − µ (cid:0) F ( t ) + µ − A ( t ) (cid:1) . (3.16)This will implies exponential decay of F + µ − A . So as to prove the above differen-tial inequality, it is convenient to introduce the auxiliary quantity ˜ A := A − δ F .First note that, from (2.46) and (3.15), N ( t ) δ N ( t ) δC (cid:0) F + µ − A (cid:1) . (3.17)As a consequence,˜ A = (1 + δ µ − ) A − δ (cid:0) F + µ − A (cid:1) (1 + δ µ − ) A . Therefore, using the evolution equations (2.51) and (2.53) for F and A , and in viewof (3.17),dd t ˜ A = dd t A − δ dd t F > (cid:16) ν − δ (cid:17) A + 2 µδ (cid:0) F + µ − A (cid:1) − (1 + δ ) C N > (cid:16) ν − δ (cid:17) A + (cid:16) µδ − δC (1 + δ ) (cid:17) (cid:0) F + µ − A (cid:1) > ν ˜ A , where δ > (cid:16) ν − δ (cid:17) > ν (1 + δ µ − ) , µδ − δµ − C (1 + δ ) > . Integrating on [ t, s ] ⊆ [ T ∗ δ , ∞ ), we get˜ A ( t ) e ν ( t − s ) ˜ A ( s ) . We now take the limit s → ∞ and using that ˜ A ( s ) is bounded due to (3.15), weobtain that ˜ A ( t )
0, that is ∀ t ∈ [ T ∗ δ , ∞ ) , A ( t ) δ F ( t ) δ (cid:0) F ( t ) + µ − A ( t ) (cid:1) . (3.18)Combining (2.51), (2.52), (3.17) and (3.18), we havedd t (cid:0) F + µ − A (cid:1) = dd t F + µ − dd t A − µ ( F + µ − A ) + ( µ − ν + 2) A + C (1 + µ − ) N ( − µ + δ ( µ − ν + 2) + δC (1 + µ − )) (cid:0) F + µ − A (cid:1) , which implies (3.16) for δ small enough. Step 2.
Integrating (3.16) on [ T ∗ δ , t ] and using (3.15), we have (cid:0) F + µ − A (cid:1) ( t ) e − µ ( t − T ∗ δ ) (cid:0) F + µ − A (cid:1) ( T ∗ δ ) . δ e − µ ( t − T ∗ δ ) . Recall once again (2.46): it follows that N ( t ) . ( F + µ − A )( t ) . e − µt . (3.19)Now we can integrate (2.29), using (3.19) and recalling that a = 0: this proves thatthere exists z ∞ ∈ R N such that z ( t ) → z ∞ as t → + ∞ , and ∀ t > T δ , | z ( t ) − z ∞ | + | θ ( t ) | . Z ∞ t N ( s ) d s . e − µt . (3.20)Gathering estimates (3.19) and (3.20), we obtain k u ( t ) − q ( · − z ∞ ) k H + k ∂ t u ( t ) k L . k u ( t ) − Q ( t ) k H + k Q ( t ) − q ( · − z ∞ ) k H + k ∂ t u ( t ) k L . N ( t ) + | z ( t ) − z ∞ | + | θ ( t ) | . e − µt . The proof of Theorem 1.4 is complete. (cid:3)
End of the proof of Theorem 1.6.
We here assume that q is a degree-1 excited state,and prove algebraic convergence of ~u to ( q, < δ < δ , there exists T ∗ δ ≫ T ∗ δ = T δ / ) and ξ δ ∈ R N such that, ~u admits a decomposition asin Proposition 2.4 for all t ∈ [ T ∗ δ , ∞ ) and satisfies k u ( t ) − q ( · − ξ δ ) k H + k ∂ t u ( t ) k L + | z ( t ) − ξ δ | + | a ( t ) | + | θ ( t ) | + N ( t ) < δ. (3.21) Step 1.
We claim that there exists C > δ ) such that ∀ t > T ∗ δ , A ( t ) C δ ( a ( t ) + N ( t ) ) . (3.22)Again, we consider an auxiliary quantity, which is here ˜ A := A − δ F − δ − R .From the evolution equations (2.48), (2.51) and (2.53) of A , F and R , we bounddd t ˜ A = dd t A − δ dd t F − δ − dd t R > µδ (cid:0) F + µ − A (cid:1) + ( ν − δ ) A + δ − αa − C (1 + δ − + δ ) N − C (cid:16) δ (cid:17) a N . > µ δ N + ( ν − δ ) A + δ − αa − C δ − N − C a N . Notice that from the AM-GM inequality, we have2 a N = 2( a δ − )( N δ ) a δ − + N δ , (3.23)so that, rearranging the terms in the preceding inequality, we getdd t ˜ A > (cid:16) µ δ − C δ − C δ (cid:17) N + ( ν − δ ) A + ( δ − α − δ − C ) a > , where δ is so small that2 µ δ − C δ − C δ > , ν − δ > δ − α − δ − C > . Now, from Proposition 3.1, we know that ˜ A ( t ) → t → ∞ . As we just showedthat ˜ A is non decreasing on [ T ∗ δ , ∞ ), we hence conclude that for all t > T ∗ δ , ˜ A ( t ) SYMPTOTICS OF DAMPED NLKG EQUATION 27 or equivalently (due to the definition of ˜ A ), that for all t > T ∗ δ , A ( t ) δ F ( t ) + δ − R ( t ) δ F ( t ) + δ (cid:18) α (cid:19) a + δ α (cid:0) a + N (cid:1) , As F . N , this implies (3.22) for δ small enough. Step 2.
We claim that there exists C > δ ) such that, ∀ t > T ∗ δ , | a ( t ) | C (cid:0) ( t − T ∗ δ + δ − ) − + N ( t ) (cid:1) . (3.24)We use yet another auxiliary quantity, namely e R := R + 2 C µ − (cid:0) F + µ − A (cid:1) .Note that, from the AM-GM inequality and (3.21), there exists C > e R R + O (cid:0) N (cid:1) C ( a + N ) . (3.25)Furthermore, using the evolution equation (2.51) and (2.52) for F and A , we havedd t (cid:0) F + µ − A (cid:1) − µ (cid:0) F + µ − A (cid:1) + ( ν + 2) A + C (cid:0) µ − (cid:1) (cid:0) N + a N (cid:1) − µ (cid:0) F + µ − A (cid:1) + O ( δ ( a + N ))where we also used (3.21) and (3.22). Therefore, using the evolution equation (2.47)for R , we deduce that,dd t e R = dd t R + 2 C µ − dd t (cid:0) F + µ − A (cid:1) − a + C N + 2 C µ − (cid:0) − µ ( F + µ − A ) (cid:1) + O ( δ ( a + N )) − a − C N + O ( δ ( a + N )) . (We used the coercivity estimate (2.46): F + µ − A > C − N , and also that | b | . N ).For δ small enough, we infer thatdd t e R − a − C N − c ˜ R . for some universal constant c >
0. Recall now that from Proposition 3.1, ˜ R ( t ) → t → + ∞ , and therefore, after dividing by ˜ R , and integrating on [ T ∗ δ , t ], weobtain ∀ t > T ∗ δ , ˜ R ( t ) . ( t − T ∗ δ + δ − ) − . (Notice that ˜ R ( T ∗ δ ) . δ , in view of (3.21)). It can be rewritten as − C µ − (cid:0) F ( t ) + µ − A ( t ) (cid:1) a ( t ) + b ( t )2 α . ( t − T ∗ δ + δ − ) − . As | b | , F + µ − A . N , this implies (3.24) for C large enough. Step 3. Conclusion.
Now we prove the algebraic decay rate by a bootstrap argu-ment. For C > | a ( t ) | C ( t − T ∗ δ + δ − ) , N ( t ) C ( t − T ∗ δ + δ − ) . (3.26)Let T ∗∗ = sup { t ∈ [ T ∗ δ , ∞ ) such that (3.26) holds on [ T ∗ δ , t ] } . We will prove that T ∗∗ = + ∞ by strictly improving the bootstrap assumption (3.26)on [ T ∗ δ , T ∗∗ ), upon choosing δ small enough. In this bootstrap, the implied constantsdo not depend on δ and C , but can depend on C , C , C . As | a ( T ∗ δ ) | δ and N ( T ∗ δ ) δ due to (3.21), the bootstrap estimate (3.26) holds(strictly) at t = T ∗ δ , and so T ∗∗ > T ∗ δ . Estimate on a . From (3.24) and the boostrap estimate (3.26) on N , we have | a ( t ) | C (cid:18) t − T ∗ δ + δ − + N ( t ) (cid:19) C ( C + 1)( t − T ∗ δ + δ − ) , which strictly improves the estimate (3.26) on a for taking C large enough (de-pended on C ). Estimate on N . First, we claim that ∀ t ∈ [ T ∗ δ , T ∗∗ ) , F ( t ) . t − T ∗ δ + δ − ) . (3.27)Indeed, using (3.21) and the bootstrap bound (3.26), the evolution equation (2.51)on F writes ∀ t ∈ [ T ∗ δ , T ∗∗ ) , dd t F + 2 µ F C N ( a + N ) . δC ( C + 1)( t − T ∗ δ + δ − ) . Let t ∈ [ T ∗ δ , T ∗∗ ), and integrate on [ T ∗ δ , t ]: for δ small enough (dependent on C ),this gives F ( t ) . e − µ ( t − T ∗ δ ) F ( T ∗ δ ) + δ Z tT ∗ δ e − µ ( t − s ) C ( C + 1)d s ( s − T ∗ δ + δ − ) . δ e − µ ( t − T ∗ δ ) + δC ( C + 1)( t − T ∗ δ + δ − ) . t − T ∗ δ + δ − ) , which means (3.27). Therefore, from the coercivity estimate (2.46), combining thebound (3.22) and (3.27) on F and A that we just obtained and the boostrap bounds(3.26), we get N . F + µ − A . t − T ∗ δ + δ − ) + δ C ( C + 1)( t − T ∗ δ + δ − ) . t − T ∗ δ + δ − ) , which strictly improves the estimate (3.26) of N , upon δ small enough.As a consequence of improving the bootstrap assumption on a and N , we concludethat T ∗∗ = ∞ .Finally, it suffices to bound the geometric parameters. First, recall the equation(2.30) on ℓ and β : proceeding as for F (using (3.26) for t > T ∗ δ ), we get that ∀ t > T ∗ δ , | ℓ ( t ) | + | β ( t ) | . t − T ∗ δ + 1) . Then we consider z and θ : using the above estimate and (3.26), the equation (2.29)now writes | ˙ z | | ˙ z − ℓ | + | ℓ | . | ℓ | + N + | a |N . t − T ∗ δ + 1) , | ˙ θ | | ˙ θ − β | + | β | . | β | + N + | a |N . t − T ∗ δ + 1) . This proves that z ( t ) → z ∞ as t → + ∞ , and that ∀ t > T ∗ δ , | z ( t ) − z ∞ | . t − T ∗ δ + 1 . From Proposition 3.1, we already know that θ ( t ) → t → + ∞ , and we obtainas for z the convergence rate ∀ t > T ∗ δ , | θ ( t ) | . t − T ∗ δ + 1 . SYMPTOTICS OF DAMPED NLKG EQUATION 29
Finally, gathering the above estimates, we conclude that for all t > T , k u ( t ) − q ( · − z ∞ ) k H + k ∂ t u ( t ) k L k u ( t ) − Q ( t ) k H + k Q ( t ) − q ( · − z ∞ ) k H + k ∂ t u ( t ) k L . N ( t ) + | a ( t ) | + | z ( t ) − z ∞ | + | θ ( t ) | . t − T ∗ δ + 1 . The proof of Theorem 1.6 is complete. (cid:3) Proof of Theorem 1.7
In this section, we prove Theorem 1.7. Let q be a degree-1 excited state. Proof of Theorem 1.7.
Let 0 < δ ≪ a + = ( a + k ) k =1 , ··· ,K ∈ ¯ B R K ( δ ), we consider the solution ~u of (1.1) with initial data ~u (0) = ( q,
0) + ( δφ,
0) + ~W ( a + k ) , (4.1)where ~W ( a + ) = K X k =1 a + k ( ζ + k ) + 1 (cid:18) ζ + k Y k Y k (cid:19) . (4.2) Step 1. Decomposition.
For any t > ~u is defined and satisfies (2.22),we consider its decomposition according to Proposition 2.4: this gives the functions z, ℓ, θ, β, a, b and ~ϕ . Note that by the definition of ~W ( a + ), the initial data ~u (0) ismodulated in the sense that ~ϕ (0) = ~W ( a + ) , z (0) = ℓ (0) = 0 , θ (0) = β (0) = 0 , b (0) = 0 , a (0) = δ. (4.3)Also, by. (2.4), and as ~Z ± k (0) = (cid:18) ζ ± k Y k Y k (cid:19) , we have ∀ k = 1 , . . . , K, a + k (0) = h ~W ( a ) , ~Z k (0) i = a + k . (4.4)We introduce the following bootstrap setting, ( (cid:12)(cid:12) a ( t ) − ( t + δ − ) − (cid:12)(cid:12) ( t + δ − ) − + ( t + δ − ) − ¯ p , | θ ( t ) | δ , N ( t ) ( t + δ − ) − , A ( t ) ( t + δ − ) − . (4.5)Let T ∗ = T ∗ ( a + ) be the supremum of times t > ~u is defined on[0 , t ], satisfies the assumption (2.22) of Proposition 2.4 on [0 , t ], and such thatthe bootstrap estimates (4.5) hold on [0 , t ].Our goal is to prove that there exists at least one choice of a + ∈ ¯ B R K ( δ ) such that T ∗ ( a + ) = ∞ . For this, we start by closing all bootstrap estimates except the onefor the instable modes, A , given any a + ∈ ¯ B R K ( δ ). Then we prove the existence ofsuitable parameters a + = ( a + k ) k =1 , ··· ,K for which A is controlled, using a topologicalargument. Before we proceed with the actual proof, let us emphasize a majordifference between the argument here and the previous boostraps. In Theorem 1.7the goal is to construct a solution, and our only choice is a good guess for theinitial data as in (4.1); hence, we do not have any long time a priori knowledge, inparticular, we have no way to ensure a priori that ∂ t u ∈ L ([0 , + ∞ ) , L ) (a boundwhich plays a key role in controlling the unstable modes): it will be consequence ofour construction.In the remainder of the proof, the implied constants in . or O do not depend onthe small parameter δ > a + ∈ ¯ B R K ( δ ). Step 2. Closing the estimates in (4.5) except for A . Fix a + ∈ ¯ B R K ( δ ). For simplicity of notations, we drop any references to a + inthis step. Estimate of a . From (4.5) and the definition of R , we have, for any t ∈ [0 , T ∗ ), (cid:12)(cid:12) R ( t ) − a ( t ) (cid:12)(cid:12) . | a ( t ) |N ( t ) + N ( t ) . ( t + δ − ) − . (4.6)Also, (cid:12)(cid:12) R ( t ) − ( t + δ − ) − (cid:12)(cid:12) ( t + δ − ) − + ( t + δ − ) − ¯ p + ( t + δ − ) − , (4.7)and so R ( t ) > δ > (cid:12)(cid:12)(cid:12)(cid:12) dd t R + R (cid:12)(cid:12)(cid:12)(cid:12) . (cid:12)(cid:12) R ( t ) − a ( t ) (cid:12)(cid:12) + N ( t ) + | a ( t ) |N ( t ) + | a ( t ) | ¯ p . ( t + δ − ) − + ( t + δ − ) − ¯ p . Therefore, from (4.7), we have (cid:12)(cid:12)(cid:12)(cid:12) dd t (cid:0) R − (cid:1) ( t ) − (cid:12)(cid:12)(cid:12)(cid:12) . ( t + δ − ) − + ( t + δ − ) − ¯ p +2 . Integrating above estimate on [0 , t ], and using (4.3), we obtain (cid:12)(cid:12) R − ( t ) − ( t + δ − ) (cid:12)(cid:12) . ( t + δ − ) + ( t + δ − ) − ¯ p +3 , so that in particular R ( t ) − > ( t + δ − ) (for δ small enough) and (cid:12)(cid:12)(cid:12)(cid:12) R ( t ) − t + δ − (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12) R − ( t ) − ( t + δ − ) (cid:12)(cid:12) R ( t ) − ( t + δ − ) . ( t + δ − ) − + ( t + δ − ) − ¯ p +1 . From there, we infer (cid:12)(cid:12) a ( t ) − ( t + δ − ) − (cid:12)(cid:12) . (cid:12)(cid:12) R ( t ) − ( t + δ − ) − (cid:12)(cid:12) + N ( t ) . ( t + δ − ) − + ( t + δ − ) − ¯ p +1 , which strictly improves the estimate (4.5) of a for δ small enough (recall that ¯ p > Estimate of θ . We use the evolution equations (2.29) and (2.30) for θ and β , underthe bootstrap assumption (4.5): this gives, for all t ∈ [0 , T ∗ ), (cid:12)(cid:12)(cid:12)(cid:12) dd t (cid:18) θ + β α (cid:19) ( t ) (cid:12)(cid:12)(cid:12)(cid:12) . N ( t ) + a ( t ) . ( t + δ − ) − . Fix t ∈ [0 , T ∗ ) and integrate the above estimate on [0 , t ], using (4.3) and (4.5): ityields | θ ( t ) | . N ( t ) + (cid:12)(cid:12)(cid:12)(cid:12) θ ( t ) + β ( t )2 α (cid:12)(cid:12)(cid:12)(cid:12) . ( t + δ − ) − + Z t ( s + δ − ) − d s . δ + δ . δ, which strictly improves the estimate (4.5) of θ , for δ small enough. Estimate of N . We first derive a bound on F using its evolution equation (2.51):together with the boostrap assumption (4.5), we havedd t F + 2 µ F C (cid:16) ( t + δ − ) − + 2( t + δ − ) − (cid:17) . ( t + δ − ) − . Fix t ∈ [0 , T ∗ ) and integrate this on [0 , t ] and using the initial bounds (4.2) and(4.3) to infer that F ( t ) . e − µt F (0) + e − µt Z t e µs ( t + δ − ) − d s . δ e − µt + ( t + δ − ) − . ( t + δ − ) − . SYMPTOTICS OF DAMPED NLKG EQUATION 31
To get a bound on N , we now recall the coercivity bound (2.46) again and thebootstrap assumption (4.5) on A , and conclude that N ( t ) . F ( t ) + µ − A ( t ) . ( t + δ − ) − , which strictly improves the boostrap estimate (4.5) of N for δ small enough. Step 3. Transversality and choice of a + . Observe that for any time t where thebootstrap bounds (4.5) holds together with the equality A ( t ) = ( t + δ − ) − , theevolution equation (2.53) on A yields the following transversality property:dd t (cid:0) ( t + δ − ) A ( t ) (cid:1) > (cid:0) ν ( t + δ − ) + 3( t + δ − ) (cid:1) A ( t ) − C ( t + δ − ) (cid:0) N ( t ) + a ( t ) N ( t ) (cid:1) > ν + O (cid:16) ( t + δ − ) − + ( t + δ − ) − + ( t + δ − ) − (cid:17) > ν , (4.8)for δ > K -uple a + ∈ ¯ B R K ( δ ) such that T ∗ ( a + ) = ∞ .The proof is by contradiction: assume for its sake that for all a + ∈ ¯ B R K ( δ ), itholds T ∗ ( a + ) < ∞ . Then, a contradiction follows from the following discussion (seefor instance more details in [8] and [11, Section 3.1]). Continuity of the map a + T ∗ ( a + ) . Let a + ∈ ¯ B R K ( δ ): as T ∗ ( a + ) < + ∞ , then as we improved all other estimates in(4.5) in the previous step, necessarily, the equality A ( t ) = ( t + δ − ) − holds for t = T ∗ ( a + ), and so (4.8) holds at t = T ∗ ( a + ).By continuity of the flow of (1.1) (and of the modulation technique), the abovetransversality property implies that the map¯ B R K ( δ ) → [0 , + ∞ ) , a + T ∗ ( a + )is continuous and there is instantaneous exit for initial data on the boundary: T ∗ ( a + ) = 0 for all a + ∈ S R K (cid:16) δ (cid:17) . Construction of a retraction . As a consequence, the map giving the exit point (onthe boundary) M : ¯ B R K ( δ ) → S R K ( δ ) a + δ (cid:0) T ∗ ( a + ) + δ − (cid:1) a + (cid:0) T ∗ ( a + ) (cid:1) is well defined, continuous, and moreover, the restriction of M to S R K ( δ ) is theidentity.The existence of such a map M contradicts Brouwer’s no retraction theorem forcontinuous maps from the ball to the sphere. We conclude that there exists at leastone a + ∈ ¯ B R K ( δ ) such that T ∗ ( a + ) = ∞ . Step 4. Conclusion.
At this point, we have proved the existence of a + ∈ B R K ( δ ),associated with a global solution ~u ∈ C ([0 , + ∞ ) , H × L ) of (1.1) with initialdata defined in (4.1), which also can be modulated (in the sense of (2.22)) andsatisfies (4.5) for all t ∈ [0 , ∞ ). Let us now control the convergence of z and θ :using their evolution equations (3.20) (and (2.30)) under the (4.5) regime, we see that there exist z ( t ) → z ∞ and θ ( t ) → θ ∞ as t → + ∞ and moreover | z ( t ) − z ∞ | + | θ ( t ) − θ ∞ | (cid:12)(cid:12)(cid:12)(cid:12) z ( t ) + ℓ α ( t ) − z ∞ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) θ ( t ) + β α ( t ) − θ ∞ (cid:12)(cid:12)(cid:12)(cid:12) + 12 α N ( t ) α N ( t ) + Z ∞ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˙ z + ˙ ℓ α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˙ θ + ˙ β α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)! ds . N ( t ) + Z ∞ t (cid:0) N ( s ) + | a ( s ) |N ( s ) + | a | ¯ p ( s ) (cid:1) d s . ( t + δ − ) − + ( t + δ − ) − + ( t + δ − ) − ¯ p +1 . (4.9)We now use the invariance of the equation to get a solution to (1.1) where z ∞ = 0and θ ∞ = 0 by setting ~u ∗ ( t, x ) = ~u ( t, R − θ ∞ ( x + z ∞ )) for all ( t, x ) ∈ [0 , ∞ ) × R N . Then, from (4.5) and (4.9), ~u ∗ ∈ C ([0 , + ∞ ) , H × L ) is a solution of (1.1) whichenjoys requested properties in the conclusion of Theorem 1.7, and more precisely, ∀ t > , k ~u ∗ ( t ) − ( q, − ( t + δ − ) − ( φ, k H × L . ( t + δ − ) − + ( t + δ − ) − ¯ p +1 . The proof of Theorem 1.7 is complete. (cid:3)
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