Conditional stability of multi-solitons for the 1D NLKG equation with double power nonlinearity
aa r X i v : . [ m a t h . A P ] A p r CONDITIONAL STABILITY OF MULTI-SOLITONS FOR THE 1DNLKG EQUATION WITH DOUBLE POWER NONLINEARITY
XU YUAN
Abstract.
We consider the one-dimensional nonlinear Klein-Gordon equationwith double power focusing-defocusing nonlinearity ∂ t u − ∂ x u + u − | u | p − u + | u | q − u = 0 , on R × R , with 1 < q < p < ∞ . The main result concerns the stability of the sum ofseveral solitary waves with different speeds in the energy space H ( R ) × L ( R ),up to the natural instabilities. The proof involves techniques developed byMartel, Merle and Tsai [12, 13] for the generalized KdV and NLS equations.In particular, we rely on an energy method and virial type estimates. Introduction
Main result.
We consider the one-dimensional nonlinear Klein-Gordon equa-tion with double power nonlinearity ( ∂ t u − ∂ x u + u − | u | p − u + | u | q − u = 0 , ( t, x ) ∈ [0 , ∞ ) × R ,u | t =0 = u ∈ H , ∂ t u | t =0 = u ∈ L , (1.1)where 1 < q < p < ∞ . This equation also rewrites as a first order system in timefor the function ~u = ( u , u ), ( ∂ t u = u ∂ t u = ∂ x u − u + f ( u ) , (1.2)where f ( u ) = | u | p − u − | u | q − u . Recall that the Cauchy problem for equa-tion (1.2) is locally well-posed in the energy space H × L . See e.g. [7]. Denote F ( u ) = p +1 | u | p +1 − q +1 | u | q +1 . For any H × L solution ~u = ( u , u ) of (1.2),the energy E ( ~u ) and momentum I ( ~u ) are conserved, where E ( ~u ) = Z R (cid:8) ( ∂ x u ) + u + u − F ( u ) (cid:9) d x, I ( ~u ) = 2 Z R ( ∂ x u ) u d x. Denote by Q the ground state , which is the unique positive even solution of theequation Q ′′ − Q + f ( Q ) = 0 on R . The existence and properties of this solution are studied in [1, Section 6] (see alsoRemark 1.4). It is well-known that Q , Q ′ , Q ′′ have exponential decay at infinity:there exists θ > Q ( x ) + | Q ′ ( x ) | + | Q ′′ ( x ) | . e − θ | x | . (1.3)The ground state generates the stationary solution ~Q = ( Q,
0) of (1.2).Using the Lorentz transformation on Q , one obtains traveling solitary waves or solitons : for ℓ ∈ R , with − < ℓ <
1, let Q ℓ ( x ) = Q (cid:18) x √ − ℓ (cid:19) , ~Q ℓ = (cid:18) Q ℓ − ℓ∂ x Q ℓ (cid:19) then ~u ( t, x ) = ~Q ℓ ( x − ℓt ) is a solution of (1.2). It is well-known that the operator L = − ∂ x + 1 − f ′ ( Q )appearing after linearization of equation (1.2) around ~Q = ( Q, − ν ( ν > Y .Set ~Y + = (cid:18) Yν Y (cid:19) and ~Z + = (cid:18) ν YY (cid:19) . From explicit computations, the function ~u + ( t, x ) = exp( ν t ) ~Y + ( x ) is solution ofthe linearized system (cid:26) ∂ t u = u ∂ t u = −L u . Since ν >
0, the solution ~u + illustrates the (one-dimensional) exponential instabil-ity of the solitary wave ~Q in positive time. An equivalent formulation of instabilityis obtained by observing that for any solution ~u of (1.1), it holdsdd t a + = ν a + where a + ( t ) = (cid:16) ~u ( t ) , ~Z + (cid:17) L . More generally, for − < ℓ <
1, set Y ℓ = Y (cid:18) x √ − ℓ (cid:19) and ~Z + ℓ = ( ℓ∂ x Y ℓ + ν √ − ℓ Y ℓ ) e ℓν √ − ℓ x Y ℓ e ± ℓν √ − ℓ x . The main purpose of this article is to study the conditional stability of multi-solitonswith different speeds for (1.1). More precisely, the main result is the following.
Theorem 1.1.
Let N ≥ . For all n ∈ { , · · · , N } , let σ n = ± , − < ℓ n < with − < ℓ < ℓ < · · · < ℓ N < . There exist L > , C > , γ > and δ > suchthat the following is true. Let ~ε ∈ H × L and y < · · · < y N be such that thereexist L > L and < δ < δ with k ~ε k H × L < δ and y n +1 − y n > L for all n = 1 , . . . , N − . Then, there exist h + = ( h + n ) n ∈{ , ··· ,N } satisfying N X n =1 | h + n | ≤ C (cid:0) δ + e − γ L (cid:1) , such that the solution ~u = ( u , u ) of (1.2) with initial data ~u = N X n =1 (cid:0) σ n ~Q ℓ n + h + n ~Z + ℓ n (cid:1) ( · − y n ) + ~ε is globally defined in H × L for t ≥ and, for all t ≥ , (cid:13)(cid:13)(cid:13)(cid:13) ~u ( t ) − N X n =1 σ n ~Q ℓ n ( · − y n ( t )) (cid:13)(cid:13)(cid:13)(cid:13) H × L ≤ C (cid:0) δ + e − γ L (cid:1) , where y ( t ) , · · · , y N ( t ) are C functions satisfying, for all n = 1 , · · · , N , t ≥ , | y n (0) − y n | ≤ C (cid:0) δ + e − γ L (cid:1) , | ˙ y n ( t ) − ℓ n | ≤ C (cid:0) δ + e − γ L (cid:1) . Remark 1.2.
As remarked before, each soliton ~Q ℓ n has exactly one exponentialinstability direction, which for a given perturbation ~ε , requires the choice of the N parameters ( h + n ) n ∈{ , ··· ,N } to control it. ONDITIONAL STABILITY FOR 1D NLKG EQUATION 3
Historically, multi-solitons were studied extensively for integrable equations, mainlyfor the Korteweg-de Vries equation and cubic nonlinear Schr¨odinger equation indimension one. In the nonintegrable cases, for dispersive and wave equations, thefirst result concerning stability and asymptotic stability of multi-soliton solutionswas given by Perelman [15], following Buslaev and Perelman [2] (single solitoncase) for the nonlinear Schr¨odinger equation (NLS). We refer to [12, 13] for resultson the stability and asymptotic stability of multi-solitons solutions (or sums ofseveral solitons) for the generalized Korteweg-de Vries equation (gKdV) and (NLS)equation, that inspired the present work. See also [8, 14] for the derivative (NLS)equation.Such stability results are closely related the existence of asymptotic pure multi-solitons for non-integrable dispersive and wave equations which have been estab-lished in several previous works, for both stable and unstable solitons, see [3, 5,9, 10, 11, 16] for (gKdV), (NLS), and the energy critical wave equation. For thenonlinear Klein-Gordon equation (1.2), the existence of asymptotic multi-solitonswas established by Cˆote and Mu˜noz [6].Our original motivation for studying multi-solitons problems for (1.1) was to providethe first statement of (conditional) stability of sums of solitons for a wave-typeequation. Observe that the Lorentz transform, used to propagate solitons withdifferent speeds has rather different properties than the Galilean transform for(NLS) or the natural propagation phenomenon related to the solitons of (gKdV).It was an interesting challenge to extend the methods of [12, 13] to this case.Second, the double power nonlinearity appears as a typical nonlinearity in dispersiveequations, especially for the (NLS) equation.
Remark 1.3.
The double power focusing-defocusing nonlinearity such as in (1.1)makes the nonlinearity defocusing for small value of u , which is important in ourproof. In the next remark, we give explicitely more general conditions on thenonlinearity. This issue is already present in [13] for (NLS) though the conditionimposed on the nonlinearity is weaker. Remark 1.4.
Let f be a real-valued C ,α function and F be the standard integralof f , i.e. F ( s ) = Z s f ( σ )d σ for s ∈ R . Theorem 1.1 can be extended to any nonlinearity f satisfying(i) f is odd, and f (0) = f ′ (0) = 0.(ii) There exists a smallest s > F ( s ) − s = 0, and f ( s ) − s > r > s ∈ ( − r , r ), sf ( s ) − F ( s ) ≤ N -soliton problem for (1.1): estimates of the non-linear interactions between solitons, decomposition by modulation and parameterestimates. Energy estimates and monotonicity properties are proved in Section 3.Finally, Theorem 1.1 is proved in Section 4.1.2. Notation.
We denote ( · , · ) L the L scalar product for real-valued functions u, v ∈ L , ( u, v ) L := Z R u ( x ) v ( x )d x. XU YUAN
For ~u = (cid:18) u u (cid:19) , ~v = (cid:18) v v (cid:19) , denote (cid:0) ~u, ~v ) L := X k =1 , (cid:0) u k , v k ) L , k ~u k H := k u k H + k u k L . For f ∈ L and ℓ ∈ ( − , f ℓ ( x ) = f ( x ℓ ) , where x ℓ = x √ − ℓ . Acknowledgements.
The author would like to thank his advisor, Professor YvanMartel, for his generous help, encouragement, and guidance related to this work.2.
Preliminaries
Spectral theory.
In this section, we recall the spectral properties of thelinearized operator around Q ℓ . First, for − < ℓ <
1, let L ℓ = − (1 − ℓ ) ∂ x + 1 − f ′ ( Q ℓ ) . We recall the following standard spectral properties for L and L ℓ (see e.g. [6,Lemma 1 and Corollary 1]). Lemma 2.1. (i) Spectral properties . The unbounded operator L on L with domain H is self-adjoint, its continuous spectrum is [1 , + ∞ ) , its kernel is spanned by Q ′ and it has a unique negative eigenvalue − ν ( ν > , with corresponding smoothradial eigenfunction Y . Moreover, on R , (cid:12)(cid:12) Y ( α ) ( x ) (cid:12)(cid:12) . e − √ ν | x | for any α ∈ N . (ii) Coercivity property of L . There exists ν > such that, for all v ∈ H , (cid:0) L v, v (cid:1) L ≥ ν k v k H − ν − (cid:0) ( v, Q ′ ) L + ( v, Y ) L (cid:1) . (iii) Coercivity property of L ℓ . There exists ν > such that, for all v ∈ H , (cid:0) L ℓ v, v (cid:1) L ≥ ν k v k H − ν − (cid:0) ( v, ∂ x Q ℓ ) L + ( v, Y ℓ ) L (cid:1) . Second, we define H ℓ = (cid:18) − ∂ x + 1 − f ′ ( Q ℓ ) − ℓ∂ x ℓ∂ x (cid:19) , J = (cid:18) − (cid:19) , and ~Z ℓ = (cid:18) ∂ x Q ℓ − ℓ∂ x Q ℓ (cid:19) , ~Z ± ℓ = ( ℓ∂ x Y ℓ ± ν √ − ℓ Y ℓ ) e ± ℓν √ − ℓ x Y ℓ e ± ℓν √ − ℓ x . We recall the following technical facts.
Lemma 2.2 ([6]) . (i) Properties of H ℓ and H ℓ J . It holds H ℓ ~Z ℓ = 0 , (cid:0) ~Z ℓ , ~Z ± ℓ (cid:1) L = 0 and H ℓ J (cid:0) ~Z ± ℓ (cid:1) = ∓ ν (1 − ℓ ) ~Z ± ℓ . (2.1)(ii) Coercivity property of H ℓ . There exists ν > such that, for all ~v = ( v, z ) ∈ H × L , (cid:0) H ℓ ~v, ~v (cid:1) L ≥ ν k ~v k H − ν − (cid:0) ( ~v, ~Z ℓ ) L + ( ~v, ~Z + ℓ ) L + ( ~v, ~Z − ℓ ) L (cid:1) . (2.2) Proof.
See the proof of Lemma 2 and Proposition 2 in [6]. (cid:3)
ONDITIONAL STABILITY FOR 1D NLKG EQUATION 5
Decomposition of the solution around N solitary waves. We recallgeneral results on solutions of (1.2) that are close to the sum of N ≥ n ∈ { , · · · , N } , let σ n = ± t y n ( t ) ∈ R be C -functions such that y n +1 − y n ≫ n = 1 , · · · N − . (2.3)For n ∈ { , · · · , N } , define Q n = σ n Q ℓ n ( · − y n ) , ~Q n = (cid:18) Q n − ℓ n ∂ x Q n (cid:19) . Similarly, ~Z n = σ n ~Z ℓ n ( · − y n ) , ~Z ± n = σ n ~Z ± ℓ n ( · − y n ) . We recall a decomposition result for solutions of (1.2).
Lemma 2.3.
There exist L > and < δ ≪ such that if ~u = ( u , u ) is asolution of (1.2) on [0 , T ] , where T > , such that for all t ∈ [0 , T ]inf z n +1 − z n >L k ~u ( t ) − N X n =1 σ n ~Q ℓ n ( · − z n ) k H < δ , (2.4) then there exist C -functions y = ( y n ) n ∈{ , ··· ,N } on [0 , T ] such that, ~ϕ being definedby ~ϕ = (cid:18) ϕ ϕ (cid:19) , ~u = N X n =1 ~Q n + ~ϕ, (2.5) satisfies (cid:0) ~ϕ, ~Z n (cid:1) L = 0 , for n = 1 , · · · , N , (2.6) and k ~ϕ k H . δ , y n +1 − y n ≥ L for n = 1 , · · · , N − . (2.7) Proof.
The proof of the decomposition lemma relies on a standard argument basedon the Implicit function Theorem (see e.g.
Lemma 3 in [4]) and we omit it. (cid:3)
Set ~U = (cid:18) U U (cid:19) = N X n =1 ~Q n and G = f ( U ) − N X n =1 f ( Q n ) . Lemma 2.4 (Equation of ~ϕ ) . The function ~ϕ satisfies ( ∂ t ϕ = ϕ + Mod ,∂ t ϕ = ∂ x ϕ − ϕ + f ( U + ϕ ) − f ( U ) + G + Mod , (2.8) where Mod = N X n =1 ( ˙ y n − ℓ n ) ∂ x Q n , Mod = − N X n =1 ( ˙ y n − ℓ n ) ℓ n ∂ x Q n . (2.9) Proof.
First, from the definition of ~ϕ = ( ϕ , ϕ ) in (2.5), ∂ t ϕ = ∂ t u − ∂ t U = ϕ + U − ∂ t U = ϕ + N X n =1 ( ˙ y n − ℓ n ) ∂ x Q n . Second, using (1.2), ∂ t ϕ = ∂ t u − ∂ t U = ∂ x u − u + f ( u ) − N X n =1 ˙ y n (cid:0) ℓ n ∂ x Q n (cid:1) . XU YUAN
We observe from (2.5) and − (1 − ℓ n ) ∂ x Q n + Q n − f ( Q n ) = 0, ∂ x u − u + f ( u ) = ∂ x ϕ − ϕ + f ( U + ϕ ) − N X n =1 f ( Q n ) + N X n =1 ℓ n ∂ x Q n . Therefore, from the definition of G and Mod , we obtain the second line of (2.8). (cid:3) First, we derive some preliminary estimates associated to the equation of ~ϕ and thenonlinear interaction term G from Taylor’s formula. Fix γ = 1100 min (1 , θ ) × min (cid:16) , (1 − ℓ ) − , · · · , (1 − ℓ n ) − (cid:17) × min (cid:18) , q − , q − (cid:19) × min ( ℓ , ℓ − ℓ , · · · , ℓ N − ℓ N − ) > , where θ is defined in (1.3). Lemma 2.5.
Assume (2.3) , for any n, n ′ ∈ { , · · · , N } , n = n ′ , the followingestimates hold. For n = n ′ , Z R (cid:0) | Q n Q n ′ | + | ∂ x Q n ∂ x Q n ′ | + (cid:12)(cid:12) ∂ x Q n ∂ x Q n ′ (cid:12)(cid:12)(cid:1) dx . e − γ | y n − y n ′ | , (2.10) and Z R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F ( U ) − N X n =1 F ( Q n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx . N − X n =1 e − γ ( y n +1 − y n ) , (2.11) k G k L + k f ′ ( U ) − N X n =1 f ′ ( Q n ) k L . N − X n =1 e − γ ( y n +1 − y n ) . (2.12) Proof. Proof of (2.10). First, by change of variable, for n = n ′ , Z R | Q n Q n ′ | d x = (1 − ℓ n ) Z R Q ( x ) Q ( y n − y n ′ ) + (1 − ℓ n ) x (1 − ℓ n ′ ) ! d x = H + H , where H = (1 − ℓ n ) Z I Q ( x ) Q ( y n − y n ′ ) + (1 − ℓ n ) x (1 − ℓ n ′ ) ! d x,H = (1 − ℓ n ) Z I Q ( x ) Q ( y n − y n ′ ) + (1 − ℓ n ) x (1 − ℓ n ′ ) ! d x, and I = (cid:26) x ∈ R : (cid:12)(cid:12)(cid:12) (1 − ℓ n ) x (cid:12)(cid:12)(cid:12) ≤ | y n − y n ′ | (cid:27) ,I = (cid:26) x ∈ R : (cid:12)(cid:12)(cid:12) (1 − ℓ n ) x (cid:12)(cid:12)(cid:12) ≥ | y n − y n ′ | (cid:27) . From the decay properties of Q and the definition of γ , we obtain H . e − γ | y n − y n ′ | Z I Q ( x )d x . e − γ | y n − y n ′ | ,H . e − γ | y n − y n ′ | Z I Q ( y n − y n ′ ) + (1 − ℓ n ) x (1 − ℓ n ′ ) ! d x . e − γ | y n − y n ′ | . This proves estimate (2.10) for Q n Q n ′ . The proof of (2.10) for ∂ x Q n ∂ x Q n ′ and ∂ x Q n ∂ x Q n ′ follows from similar arguments and it is omitted. ONDITIONAL STABILITY FOR 1D NLKG EQUATION 7
Proof of (2.11). From Taylor expansion and 1 < q < p < ∞ , we infer (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F ( U ) − N X n =1 F ( Q n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . X n = n ′ | Q n | q | Q n ′ | + X n = n ′ | Q n | p | Q n ′ | . X n = n ′ | Q n || Q n ′ | . Therefore, we conclude (2.11) from (2.10).
Proof of (2.12). First, from Taylor expansion and 1 < q < p < ∞ , we infer (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ′ ( U ) − N X n =1 f ′ ( Q n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . X n = n ′ | Q n | q − | Q n ′ | when 2 < q < ∞ , X n = n ′ | Q n | q − | Q n ′ | q − when 1 < q ≤ . Therefore, using the similar argument as in the proof of (2.10), we obtain (2.12) for f ′ ( U ) − P Nn =1 f ′ ( Q n ). The proof of (2.12) for G follows from similar argumentsand it is omitted. (cid:3) Second, we derive the control of y from the orthogonality conditions (2.6). Lemma 2.6 (Control of y ) . It holds N X n =1 (cid:12)(cid:12) ˙ y n − ℓ n (cid:12)(cid:12) . k ~ϕ k H + N − X n =1 e − γ ( y n +1 − y n ) . (2.13) Proof.
First, we rewrite the equation of ~ϕ = ( ϕ , ϕ ) as ∂ t ~ϕ = ~ L ~ϕ + ~ Mod + ~G + ~R + ~R , (2.14)where ~ L = (cid:18) ∂ x − P Nn =1 f ′ ( Q n ) 0 (cid:19) , ~ Mod = (cid:18)
Mod Mod (cid:19) ,~G = (cid:18) G (cid:19) , ~R = (cid:18) R (cid:19) = (cid:18) f ( U + ϕ ) − f ( U ) − f ′ ( U ) ϕ (cid:19) , and ~R = (cid:18) R (cid:19) = (cid:16) f ′ ( U ) − P Nn =1 f ′ ( Q n ) (cid:17) ϕ ! . Second, from the orthogonality conditions (2.6),0 = dd t (cid:16) ~ϕ, ~Z n (cid:17) L = (cid:16) ∂ t ~ϕ, ~Z n (cid:17) L + (cid:16) ~ϕ, ∂ t ~Z n (cid:17) L . Thus, using (2.14),0 = (cid:16) ~ L ~ϕ, ~Z n (cid:17) L + (cid:16) ~R , ~Z n (cid:17) L + (cid:16) ~R , ~Z n (cid:17) L + (cid:16) ~G, ~Z n (cid:17) L + (cid:16) ~ Mod , ~Z n (cid:17) L − ( ˙ y n − ℓ n ) (cid:16) ~ϕ, ∂ x ~Z n (cid:17) L − ℓ n (cid:16) ~ϕ, ∂ x ~Z n (cid:17) L . Since (cid:0) − (1 − ℓ n ) ∂ x + 1 − f ′ ( Q n ) (cid:1) ∂ x Q n = 0, the first term is (cid:16) ~ L ~ϕ, ~Z n (cid:17) L = − ℓ n X n ′ = n ( ϕ , f ′ ( Q n ′ ) ∂ x Q n ) L + ( ϕ , ∂ x Q n ) L = O ( k ~ϕ k H ) . Next, by Taylor expansion (as 1 < q < p < ∞ ), we infer R = f ( U + ϕ ) − f ( U ) − f ′ ( U ) ϕ = O ( | ϕ | + | ϕ | q + | ϕ | p ) . and by the Sobolev embedding theorem, we obtain (cid:12)(cid:12)(cid:12)(cid:16) ~R , ~Z n (cid:17) L (cid:12)(cid:12)(cid:12) . k ~ϕ k H + k ~ϕ k q H + k ~ϕ k p H . XU YUAN
Using the Cauchy-Schwartz inequality and (2.12), we obtain (cid:12)(cid:12)(cid:12)(cid:16) ~R , ~Z n (cid:17) L (cid:12)(cid:12)(cid:12) . k ~ϕ k H k f ′ ( U ) − N X n =1 f ′ ( Q n ) k L . k ~ϕ k H + N − X n =1 e − γ ( y n +1 − y n ) . Then, using again (2.12), we have (cid:12)(cid:12) (cid:16) ~G, ~Z n (cid:17) L (cid:12)(cid:12) . k G k L . N − X n =1 e − γ ( y n +1 − y n ) Next, using the expression of ~ Mod, we have (cid:16) ~ Mod , ~Z n (cid:17) L = ( ˙ y n − ℓ n ) (cid:16) ~Z n , ~Z n (cid:17) L + X n ′ = n ( ˙ y n ′ − ℓ n ′ ) (cid:16) ~Z n ′ , ~Z n (cid:17) L . Moreover, from (2.10), X n ′ = n ( ˙ y n ′ − ℓ n ′ ) (cid:16) ~Z n ′ , ~Z n (cid:17) L = O X n ′ = n | ˙ y n ′ − ℓ n ′ | N − X n =1 e − γ ( y n +1 − y n ) ! . Last, by the Cauchy-Schwartz inequality, (cid:12)(cid:12)(cid:12) ( ˙ y n − ℓ n ) (cid:16) ~ϕ, ∂ x ~Z n (cid:17) L (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ℓ n (cid:16) ~ϕ, ∂ x ~Z n (cid:17) L (cid:12)(cid:12)(cid:12) . k ~ϕ k H (1 + | ˙ y n − ℓ n | ) . Gathering above estimates, from the orthogonality condition (cid:16) ~ϕ, ~Z n (cid:17) L = 0, weobtain (cid:12)(cid:12) ˙ y n − ℓ n (cid:12)(cid:12) . k ~ϕ k H + N − X n =1 e − γ ( y n +1 − y n ) + O (cid:18) N X n ′ =1 | ˙ y n ′ − ℓ n ′ | (cid:19)(cid:18) N − X n ′ =1 e − γ ( y n ′ +1 − y n ′ ) + k ~ϕ k H (cid:19)! . Similarly, using the other orthogonality conditions, N X n =1 (cid:12)(cid:12) ˙ y n − ℓ n (cid:12)(cid:12) . k ~ϕ k H + N − X n =1 e − γ ( y n +1 − y n ) + O (cid:18) N X n ′ =1 | ˙ y n ′ − ℓ n ′ | (cid:19)(cid:18) N − X n ′ =1 e − γ ( y n ′ +1 − y n ′ ) + k ~ϕ k H (cid:19)! , which implies (2.13). (cid:3) Last, we consider the equation of the unstable directions.
Lemma 2.7 (Unstable direction) . Let a ± n = (cid:0) ~ϕ, ~Z ± n (cid:1) L . It holds (cid:12)(cid:12) ddt a ± n ∓ α n a ± n (cid:12)(cid:12) . k ~ϕ k H + k ~ϕ k q H + N − X n =1 e − γ ( y n +1 − y n ) , (2.15) where α n = ν (1 − ℓ n ) .Proof. Using (2.14), we calculate,dd t a ± = (cid:16) ∂ t ~ϕ, ~Z ± (cid:17) L + (cid:16) ~ϕ, ∂ t ~Z ± (cid:17) L = (cid:16) L ~ϕ, ~Z ± (cid:17) L − ℓ (cid:16) ~ϕ, ∂ x ~Z ± (cid:17) L + (cid:16) ~G, ~Z ± (cid:17) L + (cid:16) ~R , ~Z ± (cid:17) L + (cid:16) ~R , ~Z ± (cid:17) L + (cid:16) ~ Mod , ~Z ± (cid:17) L − ( ˙ y − ℓ ) (cid:16) ~ϕ, ∂ x ~Z ± (cid:17) L . ONDITIONAL STABILITY FOR 1D NLKG EQUATION 9
Observe that, (cid:16) ~ L ~ϕ, ~Z ± (cid:17) L − ℓ (cid:16) ~ϕ, ∂ x ~Z ± (cid:17) L = − (cid:16) ~ϕ, (cid:16) H ℓ J ~Z ± ℓ (cid:17) ( · − y ) (cid:17) L + N X n =2 (cid:0) ϕ , f ′ ( Q n ) Z ± (cid:1) L = ± α a ± + N X n =2 (cid:0) ϕ , f ′ ( Q n ) Z ± (cid:1) L . where Z ± = (cid:18) Y ℓ e ± ℓ ν √ − ℓ x (cid:19) ( · − y ) . By the decay properties of ~Z ± n and similar argument as (2.10) in Lemma 2.5, (cid:12)(cid:12)(cid:12)(cid:12) N X n =2 (cid:0) ϕ , f ′ ( Q n ) Z ± (cid:1) L (cid:12)(cid:12)(cid:12)(cid:12) . (cid:18) N − X n =1 e − γ ( y n +1 − y n ) (cid:19) k ϕ k L . k ϕ k L + N − X n =1 e − γ ( y n +1 − y n ) . Next, from (2.12), Taylor’s formula and Sobolev embedding theorem, (cid:12)(cid:12)(cid:12)(cid:16) ~R , ~Z ± (cid:17) L (cid:12)(cid:12)(cid:12) . Z R (cid:0) | ϕ | + | ϕ | q + | ϕ | p (cid:1) d x . k ~ϕ k H + k ~ϕ k q H + k ~ϕ k p H , (cid:12)(cid:12)(cid:12)(cid:16) ~R , ~Z ± (cid:17) L (cid:12)(cid:12)(cid:12) . k f ′ ( U ) − N X n =1 f ′ ( Q n ) k L k ϕ k L ≤ k ~ϕ k H + N − X n =1 e − γ ( y n +1 − y n ) , and (cid:12)(cid:12)(cid:12)(cid:16) ~G, ~Z ± n (cid:17) L (cid:12)(cid:12)(cid:12) . k G k L . N − X n =1 e − γ ( y n +1 − y n ) . Moreover, from the decay properties of ~Z ± , the similar argument of (2.10) andconcerning the term with ~ Mod, (cid:16) ~ Mod , ~Z ± (cid:17) L = N X n =2 ( ˙ y n − ℓ n ) (cid:16) ~Z n , ~Z ± (cid:17) L = O k ~ϕ k H + N − X n =1 e − γ ( y n +1 − y n ) ! . Finally, from (2.13), (cid:12)(cid:12) ( ˙ y − ℓ ) (cid:16) ~ϕ, ∂ x ~Z ± (cid:17) L (cid:12)(cid:12) . k ~ϕ k H + N − X n =1 e − γ ( y n +1 − y n ) . Gathering above estimates and proceeding similarly for ( a ± n ) Nn =2 , we obtain (2.15). (cid:3) Monotonicity property for the 1D Klein-Gordon equation
Bootstrap setting.
We introduce the following bootstrap estimates: for C to be chosen later, k ~ϕ ( t ) k H ≤ C (cid:0) δ + e − γ L (cid:1) , min n ( y n +1 − y n ) ≥ (1 − C − ) L + 2 γ t, N X n =1 | a + n ( t ) | ≤ C (cid:0) δ + e − γ L (cid:1) , N X n =1 | a − n ( t ) | ≤ C (cid:0) δ + e − γ L (cid:1) . (3.1)For ~u satisfies (2.4), set T ∗ ( ~u ) = sup { t ∈ [0 , ∞ ); ~u satisfies (2.4) and (3.1) holds on [0 , t ] } , (3.2)where ~u is the solution of (1.2) with the initial data ~u . Monotonicity property.
First, we choose suitable cutoff functions. Let χ ( x )be a C -function such that χ ′ ≥ , χ ( x ) = 0 , for x ≤ − , χ ( x ) = 1 , for x > . Set β n = ℓ n − + ℓ n , ¯ y n = y n − (0) + y n (0)2 , and χ = 1 , χ N +1 = 0 , χ n ( t, x ) = χ (cid:18) x − β n t − ¯ y n ( t + a ) α (cid:19) for n = 2 , · · · , N , where α and a are chosen so that12 < α <
47 and a = (cid:18) L (cid:19) α . Let ψ n ( t, x ) = χ n ( t, x ) − χ n +1 ( t, x ) for n = 1 , · · · , N . Note that ψ n ≡ Q n , and ψ n ≡ n ′ for n ′ = n . Moreover N X n =1 ψ n = 1 , and χ n = N X n ′ = n ψ n ′ , for n = 1 , · · · , N. (3.3)Set Ω n = (cid:8) x ∈ R : | x − β n t − ¯ y n | < ( t + a ) α (cid:9) Note that, from the definition of χ n and Ω n , we have the following estimates, ∂ x χ n = ( ∂ x χ ) (cid:18) x − β n t − ¯ y n ( t + a ) α (cid:19) t + a ) α = O (cid:18) Ω n ( t + a ) α (cid:19) , (3.4) (cid:12)(cid:12) ∂ x χ n (cid:12)(cid:12) + (cid:12)(cid:12) ∂ tx χ n (cid:12)(cid:12) . t + a ) α Ω n , (cid:12)(cid:12) ∂ x χ n (cid:12)(cid:12) . t + a ) α Ω n . (3.5)Second, let c = ℓ , c = ℓ − ℓ − β ℓ , c n = (cid:18) ℓ n − ℓ n − − β n ℓ n (cid:19) n − Y n ′ =2 (cid:18) − β n ′ ℓ n ′ − − β n ′ ℓ n ′ (cid:19) , (3.6)for n = 3 , · · · , N . Denote˜ c = 1 and ˜ c n = 1 + n X n ′ =2 c n ′ β n ′ for n = 2 , · · · , N. (3.7)By direct computation, we obtain the following Lemma. Lemma 3.1.
For n = 2 , · · · , N , we have n X n ′ =1 c n ′ = ˜ c n ℓ n , and ˜ c n = n Y n ′ =2 (cid:18) − β n ′ ℓ n ′ − − β n ′ ℓ n ′ (cid:19) . (3.8) Proof.
We prove (3.8) by induction.
Step 1.
For n = 2. By direct computation,˜ c = 1 + c β = 1 + β ℓ − β ℓ − β ℓ = 1 − β ℓ − β ℓ ,c + c = ℓ + ℓ − ℓ − β ℓ = ℓ − β ℓ ℓ − β ℓ = ˜ c ℓ , which implies (3.8) for n = 2. ONDITIONAL STABILITY FOR 1D NLKG EQUATION 11
Step 2.
We assume that (3.8) is true for n = k . Now, we prove that also true for n = k + 1. From the definition of c n for n ≥
3, (3.8) is true for n = k , we obtain˜ c k +1 = k Y n ′ =2 (cid:18) − β n ′ ℓ n ′ − − β n ′ ℓ n ′ (cid:19) + (cid:18) β k +1 ( ℓ k +1 − ℓ k )1 − β k +1 ℓ k +1 (cid:19) k Y n ′ =2 (cid:18) − β n ′ ℓ n ′ − − β n ′ ℓ n ′ (cid:19) = (cid:18) β k +1 ( ℓ k +1 − ℓ k )1 − β k +1 ℓ k +1 (cid:19) k Y n ′ =2 (cid:18) − β n ′ ℓ n ′ − − β n ′ ℓ n ′ (cid:19) = k +1 Y n ′ =2 (cid:18) − β n ′ ℓ n ′ − − β n ′ ℓ n ′ (cid:19) , and k +1 X n ′ =1 c n ′ = ˜ c k ℓ k + c k +1 = ℓ k k Y n ′ =2 (cid:18) − β n ′ ℓ n ′ − − β n ′ ℓ n ′ (cid:19) + (cid:18) ℓ k +1 − ℓ k − β k +1 ℓ k +1 (cid:19) k Y n ′ =2 (cid:18) − β n ′ ℓ n ′ − − β n ′ ℓ n ′ (cid:19) = ℓ k +1 k +1 Y n ′ =2 (cid:18) − β n ′ ℓ n ′ − − β n ′ ℓ n ′ (cid:19) = ˜ c k +1 ℓ k +1 . Therefore, (3.8) is also true for n = k + 1. By induction argument, we haveproved (3.8) for n = 2 , · · · , N . (cid:3) Third, we introduce the following modified virial elements J n ( ~u ) = I n ( ~u ) + β n E n ( ~u ) + β n F n ( ~u ) . (3.9)where I n ( ~u ) = 2 Z R (cid:18) χ n ∂ x u + 1 − β n (cid:0) ∂ x χ n (cid:1) u (cid:19) u d x, (3.10) E n ( ~u ) = Z R (cid:0) ( ∂ x u ) + u + u − F ( u ) (cid:1) χ n d x, (3.11) F n ( ~u ) = − α ( t + a ) − α Z R ( u u ) (cid:18) x − β n t − ¯ y n ( t + a ) α (cid:19) ∂ x χ n d x for n = 2 , · · · , N .Last, we set E ( ~u ) = E ( ~u ) + c I ( ~u ) + N X n =2 c n J n ( ~u )where we recall that E ( ~u ) = Z R (cid:0) ( ∂ x u ) + u + u − F ( u ) (cid:1) d x, I ( ~u ) = 2 Z R ( ∂ x u ) u d x. By expanding ~u ( t ) = P Nn =1 ~Q n ( t ) + ~ϕ ( t ), we obtain the following formula. Lemma 3.2.
The following holds, E ( ~u ) = N X n =1 ˜ c n (1 − ℓ n ) E ( ~Q ) + N X n =1 ˜ c n H n ( ~ϕ, ~ϕ )+ O (cid:18) k ~ϕ k H L + k ~ϕ k H + k ~ϕ k q +1 H + e − γ ( L + γ t ) (cid:19) . (3.12) where H n ( ~ϕ, ~ϕ ) = Z R (cid:0) ( ∂ x ϕ ) + ϕ + ϕ + 2 ℓ n ( ∂ x ϕ ) ϕ − f ′ ( Q n ) ϕ (cid:1) ψ n d x. for n = 1 , · · · , N . Proof.
Step 1.
Expansion of E ( ~u ). We prove the following estimate E ( ~u ) = N X n =1 (1 − ℓ n ) E ( ~Q ) + 2 N X n =1 Z R ( ℓ n ∂ x Q n ) ( ℓ n ∂ x Q n + ℓ n ∂ x ϕ − ϕ ) d x + Z R ( ∂ x ϕ ) + ϕ + ϕ − N X n =1 f ′ ( Q n ) ϕ ! d x + O (cid:16) k ~ϕ k H + k ~ϕ k q +1 H + e − γ ( L + γ t ) (cid:17) . (3.13)First, using the decomposition (2.5), the definition of G , the equation − (1 − ℓ n ) ∂ x Q n + Q n − f ( Q n ) = 0 and integration by parts, we find E ( ~u ) = E ( ~U ) + 2 N X n =1 Z R ( ℓ n ∂ x Q n ) (cid:0) ℓ n ∂ x ϕ − ϕ (cid:1) d x + Z R (cid:0) ( ∂ x ϕ ) + ϕ + ϕ − N X n =1 f ′ ( Q n ) ϕ (cid:1) d x + ˜ E + ˜ E + ˜ E , where ˜ E = − Z R (cid:0) F ( U + ϕ ) − F ( U ) − f ( U ) ϕ − f ′ ( U ) ϕ (cid:1) d x ˜ E = − Z R Gϕ d x and ˜ E = − Z R (cid:0) f ′ ( U ) − N X n =1 f ′ ( Q n ) (cid:1) ϕ d x. Note that by the Taylor formula, F ( U + ϕ ) − F ( U ) − f ( U ) ϕ − f ′ ( U ) ϕ = O (cid:0) | ϕ | + | ϕ | q +1 + | ϕ | p +1 (cid:1) . Therefore, using Sobolev embedding and 1 < q < p < ∞ , (cid:12)(cid:12)(cid:12) ˜ E (cid:12)(cid:12)(cid:12) . Z R (cid:0) | ϕ | p +1 + | ϕ | q +1 + | ϕ | (cid:1) d x . k ϕ k H + k ϕ k q +1 H . By (2.12), (cid:12)(cid:12)(cid:12) ˜ E (cid:12)(cid:12)(cid:12) . k G k L k ϕ k L . k ϕ k L + e − γ ( L + γ t ) , and then (3.1) and Sobolev embedding, (cid:12)(cid:12)(cid:12) ˜ E (cid:12)(cid:12)(cid:12) . (cid:13)(cid:13) f ′ ( U ) − N X n =1 f ′ ( Q n ) (cid:13)(cid:13) L k ϕ k H . k ~ϕ k H + e − γ ( L + γ t ) . Second, by direct computation, − (1 − ℓ n ) ∂ x Q n + Q n − f ( Q n ) = 0, E ( ~U ) = N X n =1 (1 − | ℓ n | ) E ( ~Q ) + 2 N X n =1 Z R (cid:0) ℓ n ∂ x Q n (cid:1) d x + X n = n ′ Z R (cid:2) (1 + ℓ n ℓ n ′ )( ∂ x Q n )( ∂ x Q n ′ ) + Q n Q n ′ (cid:3) d x − Z R (cid:0) F ( U ) − N X n =1 F ( Q n ) (cid:1) d x. Moreover, using (2.10), (2.11) and (3.1), X n = n ′ Z R (cid:0) | ∂ x Q n ∂ x Q n ′ | + | Q n Q n ′ | (cid:1) d x + Z R (cid:12)(cid:12) F ( U ) − N X n =1 F ( Q n ) (cid:12)(cid:12) d x . e − γ ( L + γ t ) . ONDITIONAL STABILITY FOR 1D NLKG EQUATION 13
We see that (3.13) follows from above estimates.
Step 2.
Expansion of I ( ~u ). We claim I ( ~u ) = − N X n =1 Z R ( ∂ x Q n ) ( ℓ n ∂ x Q n + ℓ n ∂ x ϕ − ϕ ) d x + 2 Z R ( ∂ x ϕ ) ϕ d x + O (cid:16) e − γ ( L + γ t ) (cid:17) . (3.14)By direct computation and (2.5) I ( ~u ) = − N X n =1 Z R ( ∂ x Q n ) ( ℓ n ∂ x Q n + ℓ n ∂ x ϕ − ϕ ) d x + 2 Z R ( ∂ x ϕ ) ϕ d x − X n = n ′ ℓ n ′ Z R ( ∂ x Q n ) ( ∂ x Q n ′ ) d x. From (2.10) and (3.1), we obtain (3.14).
Step 3 . Expansion of I n ( ~u ). We claim I n ( ~u ) = − N X n ′ = n Z R ( ∂ x Q n ′ ) ( ℓ n ′ ∂ x Q n ′ + ℓ n ′ ∂ x ϕ − ϕ ) d x + 2 Z R χ n ( ∂ x ϕ ) ϕ d x + O (cid:18) k ~ϕ k H L + k ~ϕ k H + e − γ ( L + γ t ) (cid:19) . (3.15)We decompose, I n ( ~u ) = I n ( ~u ) + (1 − β n ) I n ( ~u ) , where I n ( ~u ) = 2 Z R ( χ n ∂ x u ) u d x, I n ( ~u ) = Z R ( ∂ x χ n ) u u d x. Estimate on I n . We claim I n ( ~u ) = − N X n ′ = n Z R ( ∂ x Q n ′ ) ( ℓ n ′ ∂ x Q n ′ + ℓ n ′ ∂ x ϕ − ϕ ) d x + 2 Z R χ n ( ∂ x ϕ ) ϕ d x + O (cid:16) k ~ϕ k H + e − γ ( L + γ t ) (cid:17) . (3.16)By direct computation and (2.5), I n ( ~u ) = − N X n ′ = n Z R ( ∂ x Q n ′ ) ( ℓ n ′ ∂ x ϕ + ℓ n ′ ∂ x Q n ′ − ϕ ) d x + 2 Z R χ n ( ∂ x ϕ ) ϕ d x + I , n + I , n + I , n , where I , n = − X n ′ = n ′′ ℓ n ′ Z R χ n ( ∂ x Q n ′ ) ( ∂ x Q n ′′ ) d x, I , n = − n − X n ′ =1 Z R ( χ n ∂ x Q n ′ ) ( ℓ n ′ ∂ x ϕ + ℓ n ′ ∂ x Q n ′ − ϕ ) d x, I , n = − N X n ′ = n Z R ( χ n −
1) ( ∂ x Q n ′ ) ( ℓ n ′ ∂ x ϕ + ℓ n ′ ∂ x Q n ′ − ϕ ) d x. From (2.10) and (3.1), (cid:12)(cid:12) I , n (cid:12)(cid:12) . X n ′ = n ′′ Z R | ∂ x Q n ′ || ∂ x Q n ′′ | dx . e − γ ( L + γ t ) . By the decay properties of Q , the definition of χ n and (3.1), (cid:12)(cid:12) I , n (cid:12)(cid:12) . n − X n ′ =1 k χ n ∂ x Q n ′ k L ( k ~ϕ k H + k ∂ x Q n ′ k L ) . k ~ϕ k H + e − γ ( L + γ t ) , (cid:12)(cid:12) I , n (cid:12)(cid:12) . N X n ′ = n k ( χ n − ∂ x Q n ′ k L ( k ~ϕ k H + k ∂ x Q n ′ k L ) . k ~ϕ k H + e − γ ( L + γ t ) . We see that (3.16) follows from above estimates.
Estimate on I n . We decompose I n ( ~u ) = I , n + I , n + I , n , where I , n = − N X n ′ ,n ′′ =1 ℓ n ′ Z R ( ∂ x χ n )( ∂ x Q n ′ ) Q n ′′ d x, I , n = N X n ′ =1 Z R ( ∂ x χ n ) ( − ℓ n ′ ( ∂ x Q n ′ ) ϕ + Q n ′ ϕ ) d x, I , n = Z R ( ∂ x χ n ) ϕ ϕ d x. Note that, taking L large enough, for any x ∈ Ω n , (cid:12)(cid:12) x − ℓ n ′ t − y n ′ (cid:12)(cid:12) ≥ | ( ℓ n − β n ′ ) | t + (cid:12)(cid:12) y n − ¯ y n ′ (cid:12)(cid:12) − ( t + a ) α ≥ γ t + L
10 (3.17)for 1 ≤ n ′ ≤ N . Therefore, from the decay properties of Q , (3.4) and Cauchy-Schwartz inequality, (cid:12)(cid:12) I , n (cid:12)(cid:12) . N X n ′ ,n ′′ =1 Z R | ∂ x Q n ′ || Q n ′′ | Ω n d x . e − γ ( L + γ t ) , (cid:12)(cid:12) I , n (cid:12)(cid:12) . N X n ′ =1 ( k ∂ x Q n ′ Ω n k L + k Q n ′ Ω n k L ) k ~ϕ k H . k ~ϕ k H + e − γ ( L + γ t ) . Moreover, from the definition of a , (cid:12)(cid:12) I , n (cid:12)(cid:12) . k ~ϕ k H ( t + a ) α . k ~ϕ k H L .
From above estimates, we conclude, (cid:12)(cid:12) I n (cid:12)(cid:12) . k ~ϕ k H L + k ~ϕ k H + e − γ ( L + γ t ) . (3.18)We see that (3.15) follows from (3.16) and (3.18). Step 4.
Expansion of E n ( ~u ). We claim E n ( ~u ) = N X n ′ = n (1 − ℓ n ′ ) E ( ~Q ) + 2 N X n ′ = n Z R ( ℓ n ′ ∂ x Q n ′ ) ( ℓ n ′ ∂ x Q n ′ + ℓ n ′ ∂ x ϕ − ϕ ) d x + Z R ( ∂ x ϕ ) + ϕ + ϕ − N X n ′ = n f ′ ( Q n ′ ) ϕ ! χ n d x + O (cid:16) k ~ϕ k H + k ~ϕ k q +1 H + e − γ ( L + γ t ) (cid:17) . (3.19) ONDITIONAL STABILITY FOR 1D NLKG EQUATION 15
First, from (2.5), integration by parts and an elementary computation, E n ( ~u ) = E n ( ~U ) + 2 N X n ′ = n Z R ( ℓ n ′ ∂ x Q n ′ ) ( ℓ n ′ ∂ x ϕ − ϕ ) d x + Z R (cid:18) ( ∂ x ϕ ) + ϕ + ϕ − N X n ′ = n f ′ ( Q n ′ ) ϕ (cid:19) χ n d x + ˜ E n + ˜ E n + ˜ E n + ˜ E n + ˜ E n + ˜ E n , where ˜ E n = − Z R (cid:18) F ( U + ϕ ) − F ( U ) − f ( U ) ϕ − f ′ ( U ) ϕ (cid:19) χ n d x, ˜ E n = − Z R (cid:18) f ′ ( U ) − N X n ′ = n f ′ ( Q n ′ ) (cid:19) ϕ χ n d x, ˜ E n = − Z R Gϕ χ n d x, ˜ E n = − N X n ′ =1 (1 − ℓ n ′ ) Z R (cid:0) ∂ x Q n ′ (cid:1)(cid:0) ∂ x χ n (cid:1) ϕ d x, ˜ E n = 2 n − X n ′ =1 Z R (cid:0) ℓ n ′ ∂ x Q n ′ (cid:1)(cid:0) ℓ n ′ ∂ x ϕ − ϕ (cid:1) χ n d x, ˜ E n = 2 N X n ′ = n Z R (cid:0) ℓ n ′ ∂ x Q n ′ (cid:1)(cid:0) ℓ n ′ ∂ x ϕ − ϕ (cid:1) ( χ n − x. Using the similar argument as in step 1, we obtain (cid:12)(cid:12) ˜ E n (cid:12)(cid:12) + (cid:12)(cid:12) ˜ E n (cid:12)(cid:12) + (cid:12)(cid:12) ˜ E n (cid:12)(cid:12) . k ~ϕ k H + k ~ϕ k q +1 H + e − γ ( L + γ t ) . Next, by the decay properties of Q and the definition of χ n , (cid:12)(cid:12)(cid:12) ˜ E n (cid:12)(cid:12)(cid:12) . k ~ϕ k H N X n ′ =1 k ∂ x Q n ′ Ω n k L ! . k ~ϕ k H + e − γ ( L + γ t ) , (cid:12)(cid:12)(cid:12) ˜ E n (cid:12)(cid:12)(cid:12) . k ~ϕ k H (cid:18) n − X n ′ =1 k χ n ∂ x Q n ′ k L (cid:19) . k ~ϕ k H + e − γ ( L + γ t ) , (cid:12)(cid:12)(cid:12) ˜ E n (cid:12)(cid:12)(cid:12) . k ~ϕ k H (cid:18) N X n ′ = n k ( χ n − ∂ x Q n ′ k L (cid:19) . k ~ϕ k H + e − γ ( L + γ t ) . Second, by direct computation, − (1 − ℓ n ) ∂ x Q n + Q n − f ( Q n ) = 0, E n ( ~U ) = N X n ′ = n (1 − ℓ n ′ ) E ( ~Q ) + 2 N X n ′ = n Z R ( ℓ n ′ ∂ x Q n ′ ) d x + ˜ E n + ˜ E n + ˜ E n + ˜ E n , where ˜ E n = − Z R F ( U ) − N X n ′ =1 f ′ ( Q n ′ ) ! χ n d x, ˜ E n = n − X n ′ =1 Z R (cid:0) (1 + ℓ n ′ )( ∂ x Q n ′ ) + Q n ′ − F ( Q n ′ ) (cid:1) χ n d x, ˜ E n = N X n ′ = n Z R (cid:0) (1 + ℓ n ′ )( ∂ x Q n ′ ) + Q n ′ − F ( Q n ′ ) (cid:1) ( χ n −
1) d x, ˜ E n = X n ′ = n ′′ Z R ((1 + ℓ n ′ ℓ n ′′ )( ∂ x Q n ′ )( ∂ x Q n ′′ ) + Q n ′ Q n ′′ ) χ n d x. By (2.12) and (3.1) (cid:12)(cid:12)(cid:12) ˜ E n (cid:12)(cid:12)(cid:12) . Z R (cid:12)(cid:12) F ( U ) − N X n ′ =1 F ( Q n ′ ) (cid:12)(cid:12) d x . e − γ ( L + γ t ) . From the decay properties of Q , we obtain (cid:12)(cid:12)(cid:12) ˜ E n (cid:12)(cid:12)(cid:12) . n − X n ′ =1 Z R (cid:0) ( ∂ x Q n ′ ) + Q n ′ + | Q n ′ | p +1 + | Q n ′ | q +1 (cid:1) χ n d x . e − γ ( L + γ t ) , (cid:12)(cid:12)(cid:12) ˜ E n (cid:12)(cid:12)(cid:12) . N X n ′ = n Z R (cid:0) ( ∂ x Q n ′ ) + Q n ′ + | Q n ′ | p +1 + | Q n ′ | q +1 (cid:1) | χ n − | d x . e − γ ( L + γ t ) . Last, using again (2.10) and (3.1), we have (cid:12)(cid:12)(cid:12) ˜ E n (cid:12)(cid:12)(cid:12) . X n ′ = n ′′ Z R | (( ∂ x Q n ′ )( ∂ x Q n ′′ ) + Q n ′ Q n ′′ ) | d x . e − γ ( L + γ t ) . We see that (3.19) follows from above estimates.
Step 5.
Estimate of F n ( ~u ). We claim | F n ( ~u ) | . k ~ϕ k H L + k ~ϕ k H + e − γ ( L + γ t ) . (3.20)From (2.5), we decompose F n ( ~u ) = F n + F n + F n , where F n = − α ( t + a ) − α Z R ( ϕ ϕ ) (cid:18) x − β n t − ¯ y n ( t + a ) α (cid:19) ∂ x χ n d x,F n = α ( t + a ) − α N X n ′ ,n ′′ =1 ℓ n ′ Z R ( Q n ′ ∂ x Q n ′′ ) (cid:18) x − β n t − ¯ y n ( t + a ) α (cid:19) ∂ x χ n d x,F n = − α ( t + a ) − α N X n ′ =1 Z R ( ϕ Q n ′ − ℓ n ′ ϕ ∂ x Q n ′ ) (cid:18) x − β n t − ¯ y n ( t + a ) α (cid:19) ∂ x χ n d x. First, from (3.4), (cid:12)(cid:12) F n (cid:12)(cid:12) . t + a ) Z R | ϕ || ϕ | d x . k ~ϕ k H L .
Next, using again the decay properties of Q , (3.4) and the Cauchy-Schwarz inequal-ity, (cid:12)(cid:12) F n (cid:12)(cid:12) . t + a ) N X n ′ ,n ′′ =1 Z R | Q n ′ | | ∂ x Q n ′′ | Ω n d x . e − γ ( L + γ t ) , (cid:12)(cid:12) F n (cid:12)(cid:12) . t + a ) N X n ′ =1 Z R ( | ϕ Q n ′ | + | ϕ ∂ x Q n ′ | ) Ω n d x . k ~ϕ k H + e − γ ( L + γ t ) . Gathering above estimates, we obtain (3.20).
Step 6.
Conclude. First, we claim E ( ~u ) + N X n =2 c n β n E n ( ~u ) = E + E + E + O (cid:16) k ~ϕ k H + k ~ϕ k q +1 H + e − γ ( L + γ t ) (cid:17) . (3.21) ONDITIONAL STABILITY FOR 1D NLKG EQUATION 17 where E = N X n =1 ˜ c n (1 − ℓ n ) E ( ~Q ) , E = 2 N X n =1 ˜ c n ℓ n Z R ( ∂ x Q n ) ( ℓ n ∂ x Q n + ℓ n ∂ x ϕ − ϕ ) d x, E = N X n =1 ˜ c n Z R (cid:0) ( ∂ x ϕ ) + ϕ + ϕ − f ′ ( Q n ) ϕ (cid:1) ψ n d x. Indeed, from (3.13) and (3.19) and direct computation E ( ~u ) + N X n =2 c n β n E n ( ~u )= N X n =1 ˜ c n (1 − ℓ n ) E ( ~Q ) − N X n =1 (cid:20) Z R (cid:18) n X n ′ =2 c n ′ β n ′ χ n ′ (cid:19) f ′ ( Q n ) ϕ d x (cid:21) +2 N X n =1 ˜ c n ℓ n Z R ( ∂ x Q n )( ℓ n ∂ x Q n + ℓ n ∂ x ϕ − ϕ )d x + Z R (cid:18) N X n =1 c n β n χ n (cid:19)(cid:18) ( ∂ x ϕ ) + ϕ + ϕ (cid:19) d x + O (cid:16) k ~ϕ k H + k ~ϕ k q +1 H + e − γ ( L + γ t ) (cid:17) . Observe that, from (3.3) and the definition of ˜ c n , Z R (cid:18) N X n =1 c n β n χ n (cid:19)(cid:18) ( ∂ x ϕ ) + ϕ + ϕ (cid:19) d x = N X n =1 ˜ c n Z R (cid:18) ( ∂ x ϕ ) + ϕ + ϕ (cid:19) ψ n d x, and N X n =1 (cid:20) Z R (cid:0) n X n ′ =2 c n ′ β n ′ χ n ′ (cid:1) f ′ ( Q n ) ϕ d x (cid:21) = N X n =1 ˜ c n Z R f ′ ( Q n ) ϕ ψ n d x + N X n =1 X n ′ = n (cid:20) ( n ′ ) + X k =2 ( c k β k ) Z R f ′ ( Q n ) ϕ ψ n ′ d x (cid:21) , where ( n ′ ) + = min( n ′ − , n ). Note that, by the decay properties of Q and theCauchy-Schwarz inequality, for any n = n ′ , (cid:12)(cid:12)(cid:12)(cid:12) ( n ′ ) + X k =2 ( c k β k ) Z R f ′ ( Q n ) ϕ ψ n ′ d x (cid:12)(cid:12)(cid:12)(cid:12) . k ~ϕ k H + e − γ ( L + γ t ) . We see that (3.21) follows from combining these identities and estimates.Second, we claim c I + N X n =2 c n I n = − N X n =1 (cid:20)(cid:18) n X n ′ =1 c n ′ (cid:19) Z R ( ∂ x Q n ) ( ℓ n ∂ x Q n + ℓ n ∂ x ϕ − ϕ ) ψ n d x (cid:21) + 2 N X n =1 (cid:20)(cid:18) n X n ′ =1 c n ′ (cid:19) Z R ( ∂ x ϕ ) ϕ ψ n d x (cid:21) + O (cid:18) k ~ϕ k H L + k ~ϕ k H + e − γ ( L + γ t ) (cid:19) . (3.22) Indeed, from (3.14) and (3.15), c I + N X n =2 c n I n = − N X n =1 Z R (cid:18) n X n ′ =2 c n ′ χ n ′ (cid:19) ( ∂ x Q n )( ℓ n ∂ x Q n + ℓ n ∂ x ϕ − ϕ )d x + 2 Z R (cid:18) c + N X n =2 c n χ n (cid:19) ( ∂ x ϕ ) ϕ d x + O (cid:18) k ~ϕ k H L + k ~ϕ k H + e − γ ( L + γ t ) (cid:19) . Using again (3.3), we obtain (3.22).Last, combining (3.8), (3.21) and (3.22), we obtain (3.12). (cid:3)
Next, using the standard localized argument, we obtain the following coercivityresult.
Lemma 3.3 (Coercivity) . There exist < µ ≪ such that µ k ~ϕ k H − µ − (cid:0) N X n =1 ( a − n ) + N X n =1 ( a + n ) (cid:1) + O (cid:0) e − γ ( L + γ t ) (cid:1) ≤ N X n =1 ˜ c n H n ( ~ϕ, ~ϕ ) . (3.23) Proof.
First, from (3.8), we have ˜ c n is positive for n = 1 , · · · , N . Second, weobtain the localized coercivity of H n ( ~ϕ, ϕ ) from the coercivity property (2.2) aroundone solitary wave with the orthogonality properties (2.6). Last, we conclude thecoercivity (3.23) by an elementary localization argument. See e.g. [13, AppendixB]. (cid:3) Last, we prove the following almost monotonicity property for J n by virial argu-ment. Lemma 3.4.
There exist C such that for any n = 1 , · · · , N , t ∈ [0 , T ∗ ] , J n ( t ) − J n (0) ≤ C L α − sup s ∈ [0 ,t ] k ~ϕ ( s ) k H + C e − γ L . (3.24) Proof.
Step 1 . Time variation of I n . We claimdd t I n = − Z R ( ∂ x u + β n u ) ∂ x χ n d x + (1 − β n ) Z R (cid:0) u f ( u ) − F ( u ) (cid:1) ∂ x χ n d x + β n Z R (cid:0) ( ∂ x u ) + u + u (cid:1) ∂ x χ n d x + 2 β n Z R ( ∂ x u ) u ∂ x χ n d x − β n Z R F ( u ) ∂ x χ n d x − α ( t + a ) − α Z R (cid:0) ∂ x u ) u (cid:1) (cid:18) x − β n t − y n ( t + a ) α (cid:19) ∂ x χ n d x + O (cid:18) t + a ) α k ~ϕ k H + e − γ ( L + γ t ) (cid:19) . (3.25)First, using (1.2) and integrating by partsdd t Z R (cid:0) χ n ∂ x u (cid:1) u d x = − Z R (cid:0) ( ∂ x u ) − u + u + 2 F ( u ) (cid:1) ∂ x χ n d x + 2 Z R ( ∂ t χ n )( ∂ x u ) u d x. Observe that ∂ t χ n = − β n ∂ x χ n − α ( t + a ) − α (cid:18) x − β n t − y n ( t + a ) α (cid:19) ∂ x χ n . (3.26) ONDITIONAL STABILITY FOR 1D NLKG EQUATION 19
Therefore,dd t Z R (cid:0) χ n ∂ x u (cid:1) u d x = − Z R (cid:0) ( ∂ x u ) − u + u + 2 β n ( ∂ x u ) u (cid:1) ∂ x χ n d x − α ( t + a ) − α Z R (cid:0) ∂ x u ) u (cid:1) (cid:18) x − β n t − y n ( t + a ) α (cid:19) ∂ x χ n d x − Z R F ( u ) ∂ x χ n d x (3.27)Second, using again (1.2) and integrating by parts,dd t Z R u u ∂ x χ n d x = Z R (cid:0) − ( ∂ x u ) − u + u + u f ( u ) (cid:1) ∂ x χ n d x + Z R ( u u ) ∂ tx χ n d x + 12 Z R u ∂ x χ n d x. From (2.5), (3.5) and the decay properties of Q , (cid:12)(cid:12)(cid:12)(cid:12)Z R ( u u ) ∂ tx χ n d x (cid:12)(cid:12)(cid:12)(cid:12) . t + a ) α Z R N X n ′ =1 ( Q n ′ ) + N X n ′ =1 ( ∂ x Q n ′ ) + ϕ + ϕ ! Ω n d x . t + a ) α k ~ϕ k H + e − γ ( L + γ t ) , (cid:12)(cid:12)(cid:12)(cid:12)Z R u ∂ x χ n d x (cid:12)(cid:12)(cid:12)(cid:12) . t + a ) α Z R N X n ′ =1 ( Q n ′ ) + N X n ′ =1 ( ∂ x Q n ′ ) + ϕ ! Ω n d x . t + a ) α k ~ϕ k H + e − γ ( L + γ t ) . It follows that,dd t Z R u u ∂ x χ n d x = Z R (cid:0) − ( ∂ x u ) − u + u + u f ( u ) (cid:1) ∂ x χ n d x + O (cid:18) t + a ) α k ~ϕ k H + e − γ ( L + γ t ) (cid:19) (3.28)Gathering estimates (3.27) and (3.28), we obtain (3.25). Step 2.
Time variation of E n . We claimdd t E n = − Z R (cid:0) ( ∂ x u ) u (cid:1) ∂ x χ n d x − β n Z R (cid:0) ( ∂ x u ) + u + u (cid:1) ∂ x χ n d x + 2 β n Z R F ( u ) ∂ x χ n d x + 2 α ( t + a ) − α Z R F ( u ) (cid:18) x − β n t − y n ( t + a ) − α (cid:19) ∂ x χ n d x − α ( t + a ) − α Z R (cid:0) ( ∂ x u ) + u + u (cid:1) (cid:18) x − β n t − y n ( t + a ) − α (cid:19) ∂ x χ n d x. (3.29)First, from (1.2), ∂ t (cid:2) ( ∂ x u ) + u + u − F ( u ) (cid:3) = 2 ∂ x ( u ∂ x u ) = 2( ∂ x u ) ∂ x u + 2( ∂ x u ) u . Therefore, by integration by parts, Z R (cid:0) ∂ t (cid:2) ( ∂ x u ) + u + u − F ( u ) (cid:3)(cid:1) χ n d x = − Z R (cid:0) ( ∂ x u ) u (cid:1) ∂ x χ n d x. Second, from (3.26), Z R (cid:0) ( ∂ x u ) + u + u − F ( u ) (cid:1) ∂ t χ n d x = − β n Z R (cid:0) ( ∂ x u ) + u + u − F ( u ) (cid:1) ∂ x χ n d x − α ( t + a ) − α Z R (cid:0) ( ∂ x u ) + u + u − F ( u ) (cid:1) (cid:18) x − β n t − y n ( t + a ) α (cid:19) ∂ x χ n d x. Gathering these identities, we obtain (3.29).
Step 3.
Time variation of F n . We claimdd t F n = − α ( t + a ) − α Z R (cid:2) − ( ∂ x u ) − u + u + u f ( u ) (cid:3) (cid:18) x − β n t − y n ( t + a ) α (cid:19) ∂ x χ n d x + O (cid:18) t + a ) α k ~ϕ k H + e − γ ( L + γ t ) (cid:19) . (3.30)First, from (1.2), ∂ t ( u u ) = u − u + u f ( u ) + u ( ∂ x u ) . Therefore, by integration by parts, Z R (cid:2) ∂ t ( u u ) (cid:3) (cid:18) x − β n t − y n t + a (cid:19) ∂ x χ n d x = 1( t + a ) − α Z R (cid:0) − ( ∂ x u ) − u + u + u f ( u ) (cid:1) (cid:18) x − β n t − y n ( t + a ) α (cid:19) ∂ x χ n d x + 1 t + a Z R u ∂ x χ n d x + 12( t + a ) − α Z R u (cid:18) x − β n t − y n ( t + a ) α (cid:19) ∂ x χ n d x. Observe that, from (3.5) (cid:12)(cid:12)(cid:12)(cid:12) t + a Z R u ∂ x χ n d x (cid:12)(cid:12)(cid:12)(cid:12) . t + a ) α Z R N X n ′ =1 Q n ′ + ϕ ! Ω n d x . t + a ) α k ~ϕ k H + e − γ ( L + γ t ) , and (cid:12)(cid:12)(cid:12)(cid:12) t + a ) − α Z R u (cid:18) x − β n t − y n ( t + a ) α (cid:19) ∂ x χ n d x (cid:12)(cid:12)(cid:12)(cid:12) . t + a ) α Z R N X n ′ =1 Q n ′ + ϕ ! Ω n d x . t + a ) α k ~ϕ k H + e − γ ( L + γ t ) . It follows that Z R (cid:2) ∂ t ( u u ) (cid:3) (cid:18) x − β n t − y n t + a (cid:19) ∂ x χ n d x = 1( t + a ) − α Z R (cid:18) − ( ∂ x u ) − u + u + u f ( u ) (cid:19) (cid:18) x − β n t − y n ( t + a ) α (cid:19) ∂ x χ n d x + O (cid:18) t + a ) α k ~ϕ k H + e − γ ( L + γ t ) (cid:19) . (3.31) ONDITIONAL STABILITY FOR 1D NLKG EQUATION 21
Second, by direct computation and (3.26), ∂ t (cid:20)(cid:18) x − β n t − y n t + a (cid:19) ∂ x χ n (cid:21) = − β n ( t + a ) ∂ x χ n − t + a ) − α (cid:18) x − β n t − y n ( t + a ) α (cid:19) ∂ x χ n − β n ( t + a ) − α (cid:18) x − β n t − y n ( t + a ) α (cid:19) ∂ x χ n − α ( t + a ) − α (cid:18) x − β n t − y n ( t + a ) α (cid:19) ∂ x χ n . Therefore, from (3.4) and (3.5), (cid:12)(cid:12)(cid:12)(cid:12) ∂ t (cid:20)(cid:18) x − β n t − y n t + a (cid:19) ∂ x χ n (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) . t + a ) α Ω n . It follows that, (cid:12)(cid:12)(cid:12)(cid:12)Z R ( u u ) ∂ t (cid:20)(cid:18) x − β n t − y n t + a (cid:19) ∂ x χ n (cid:21) d x (cid:12)(cid:12)(cid:12)(cid:12) . t + a ) α Z R N X n ′ =1 Q n ′ + N X n ′ =1 ( ∂ x Q n ′ ) + ϕ + ϕ ! Ω n d x . t + a ) α k ~ϕ k H + e − γ ( L + γ t ) . (3.32)We see that (3.30) follows from (3.31) and (3.32). Step 4.
Conclude. Note that from (3.25), (3.29) and (3.30),dd t J n = dd t I n + β n dd t E n + β n dd t F n = F + F + O (cid:18) t + a ) α k ~ϕ k H + e − γ ( L + γ t ) (cid:19) . where F = − Z R ( ∂ x u + β n u ) ∂ x χ n d x − α ( t + a ) − α Z R u ( ∂ x u + β n u ) (cid:18) x − β n t − y n ( t + a ) α (cid:19) ∂ x χ n d x, and F = Z R ( u f ( u ) − F ( u )) (cid:20) − β n − αβ n ( t + a ) − α (cid:18) x − β n t − y n ( t + a ) α (cid:19)(cid:21) ∂ x χ n d x. Estimates of F . We claim F ≤ t + a ) α k ~ϕ k H + e − γ ( L + γ t ) . (3.33)Indeed, by the Cauchy-Schwarz inequality, (cid:12)(cid:12)(cid:12)(cid:12) α ( t + a ) − α Z R u ( ∂ x u + β n u ) (cid:18) x − β n t − y n ( t + a ) α (cid:19) ∂ x χ n d x (cid:12)(cid:12)(cid:12)(cid:12) . t + a ) − α Z R u ∂ x χ n d x + 1( t + a ) α Z R ( ∂ x u + β n u ) ∂ x χ n d x. Moreover, using (3.3), the decay properties of Q , (cid:12)(cid:12)(cid:12)(cid:12) t + a ) − α Z R u ∂ x χ n d x (cid:12)(cid:12)(cid:12)(cid:12) . t + a ) − α Z R N X n ′ =1 ( ∂ x Q n ′ ) + ϕ ! Ω n d x . t + a ) − α k ~ϕ k H + e − γ ( L + γ t ) . Thus, from < α < and taking L large enough, F ≤ − Z R ( ∂ x u + β n u ) ∂ x χ n d x + 1 L Z R ( ∂ x u + β n u ) ∂ x χ n d x + 1( t + a ) α k ~ϕ k H + e − γ ( L + γ t ) , which implies (3.33). Estimates of F . We claim F ≤ . (3.34)First, observe that, for L large enough,1 − β n − αβ n ( t + a ) − α (cid:18) x − β n t − y n ( t + a ) α (cid:19) ≥ − β n − (cid:12)(cid:12)(cid:12)(cid:12) αβ n L (cid:12)(cid:12)(cid:12)(cid:12) > Q and (3.1), for x ∈ Ω n , | u ( t, x ) | ≤ N X n =1 | Q n ( t, x ) | Ω n + | ϕ ( t, x ) | . e − γ ( L + γ t ) + k ~ϕ k H . e − γ L + δ. Therefore, for L large enough and δ small enough, we obtain for x ∈ Ω n , u f ( u ) − F ( u ) = − q − q + 1 | u | q +1 + p − p + 1 | u | p +1 ≤ , since 1 < q < p < ∞ . We conclude (3.34) from above estimates.Gathering estimates (3.33) and (3.34), we obtaindd t J n ( t ) ≤ t + a ) α k ~ϕ ( t ) k H + e − γ ( L + γ t ) . Integrating on [0 , t ] for any t ∈ [0 , T ∗ ], we obtain J n ( t ) − J n (0) . L α − max s ∈ [0 ,t ] k ~ϕ ( s ) k H + e − γ L , which implies (3.24). (cid:3) Proof of Theorem 1.1
In this section, we prove Theorem 1.1 using a bootstrap argument. We start with atechnical result that will allow us to adjust the initial value with N free parameters. Lemma 4.1 (Adjusting the initial unstable modes) . Let N ≥ . For n ∈ { , · · · , N } ,let σ n = ± and − < ℓ n < with − < ℓ < ℓ < · · · < ℓ N < . There exist L ≫ and < δ ≪ such that the following is true. Let y = ( y n ) n ∈{ , ··· ,N } ∈ R N besuch that L = min( y n +1 − y n , n = 1 , · · · , N − > L , and ~ε ∈ H × L , a + = ( a + n ) n ∈{ , ··· ,N } ∈ R N be such that k ~ε k H < δ < δ and a + ∈ ¯ B R N ( r ) where r = C (cid:0) δ + e − γ L (cid:1) ,C is defined in the bootstrap (3.1) and to be taken large enough. Then, there exist h + = ( h + n ) n ∈{ , ··· ,N } and ˜ y = (˜ y n ) n ∈{ , ··· ,N } satisfying N X n =1 (cid:0) | h + n | + | ˜ y n − y n | (cid:1) ≤ C ( δ + e − γ L ) (4.1) such that the initial value defined by ~u = N X n =1 (cid:16) σ n ~Q ℓ n + h + n ~Z + ℓ n (cid:17) ( · − y n ) + ~ε ONDITIONAL STABILITY FOR 1D NLKG EQUATION 23 rewrites as: ~u = N X n =1 σ n ~Q ℓ n ( · − ˜ y n ) + ~ϕ (0) (4.2) where ~ϕ (0) satisfies for all n = 1 , · · · , N , (cid:16) ~ϕ (0) , ~Z ℓ n ( · − ˜ y n ) (cid:17) L = 0 , a + n (0) = (cid:16) ~ϕ (0) , ~Z + ℓ n ( · − ˜ y n ) (cid:17) L = a + n . (4.3) Moreover, the initial data (4.2) is modulated in the sense of Lemma 2.3 with y n (0) =˜ y n , for all n = 1 , · · · , N .Proof. LetΓ = (cid:0) , y , · · · , y N (cid:1) ∈ R N , Γ = (cid:0) h +1 , · · · , h + N , ˜ y , · · · , ˜ y N (cid:1) ∈ R N . Consider the map Ψ : X → R N (cid:0) ~ε, a + , Γ (cid:1) (cid:0) Ψ a , · · · , Ψ aN , Ψ , · · · , Ψ N (cid:1) where X = (cid:0) H × L (cid:1) × R N × R N , and for n = 1 , · · · , N ,Ψ an = N X n ′ =1 σ n ′ (cid:16)(cid:16) ~Q ℓ n ′ ( · − y n ′ ) − ~Q ℓ n ′ ( · − ˜ y n ′ ) (cid:17) , ~Z + ℓ n ( · − ˜ y n ) (cid:17) L + N X n ′ =1 h + n ′ (cid:16) ~Z + ℓ n ′ ( · − y n ′ ) , ~Z + ℓ n ( · − ˜ y n ) (cid:17) L + (cid:16) ~ε, ~Z + ℓ n ( · − ˜ y n ) (cid:17) L − a + n , Ψ n = N X n ′ =1 σ n ′ (cid:16)(cid:16) ~Q ℓ n ′ ( · − y n ′ ) − ~Q ℓ n ′ ( · − ˜ y n ′ ) (cid:17) , ~Z ℓ n ( · − ˜ y n ) (cid:17) L + N X n ′ =1 h + n ′ (cid:16) ~Z + ℓ n ′ ( · − y n ′ ) , ~Z ℓ n ( · − ˜ y n ) (cid:17) L + (cid:16) ~ε, ~Z ℓ n ( · − ˜ y n ) (cid:17) L . From (4.2), ~ϕ (0) = N X n =1 σ n (cid:16) ~Q ℓ n ( · − y n ) − ~Q ℓ n ( · − ˜ y n ) (cid:17) + N X n =1 h + n ~Z + ℓ n ( · − y n ) + ~ε, (4.4)and thus the set of conditions in (4.3) is equivalent to Ψ ( ~ε, a + , Γ) = ∈ R N . Wesolve this nonlinear system by the Implicit Function Theorem. First, it is easy tocheck that Ψ( ~ , , Γ ) = 0 . Second, by direct computation and integration by parts, D Γ Ψ( ~ε, a + , Γ) = (cid:18)
A CB D (cid:19) , where A = (cid:18)(cid:18) ~Z + ℓ n ( · − y n ) , ~Z + ℓ n ′ ( · − y n ′ ) (cid:19) L (cid:19) n,n ′ ∈{ , ··· ,N } + O (cid:0) k ~ε k H + | Γ − Γ | (cid:1) ,B = (cid:18)(cid:18) ~Z ℓ n ( · − y n ) , ~Z + ℓ n ′ ( · − y n ′ ) (cid:19) L (cid:19) n,n ′ ∈{ , ··· ,N } + O (cid:0) k ~ε k H + | Γ − Γ | (cid:1) ,C = (cid:18)(cid:18) ~Z ℓ n ( · − y n ) , ~Z + ℓ n ′ ( · − y n ′ ) (cid:19) L (cid:19) n,n ′ ∈{ , ··· ,N } + O (cid:0) k ~ε k H + | Γ − Γ | (cid:1) ,D = (cid:18)(cid:18) ~Z ℓ n ( · − y n ) , ~Z ℓ n ′ ( · − y n ′ ) (cid:19) L (cid:19) n,n ′ ∈{ , ··· ,N } + O (cid:0) k ~ε k H + | Γ − Γ | (cid:1) . Moreover, from (cid:16) ~Z ℓ , ~Z + ℓ (cid:17) L = 0 and (2.10), we obtain B = O (cid:0) k ~ε k H + | Γ − Γ | + e − γ L (cid:1) , C = O (cid:0) k ~ε k H + | Γ − Γ | + e − γ L (cid:1) ,A = diag (cid:16)(cid:16) ~Z + ℓ , ~Z + ℓ (cid:17) L , · · · , (cid:16) ~Z + ℓ N , ~Z + ℓ N (cid:17) L (cid:17) + O (cid:0) k ~ε k H + | Γ − Γ | + e − γ L (cid:1) ,D = diag (cid:16)(cid:16) ~Z ℓ , ~Z ℓ (cid:17) L , · · · , (cid:16) ~Z ℓ N , ~Z ℓ N (cid:17) L (cid:17) + O (cid:0) k ~ε k H + | Γ − Γ | + e − γ L (cid:1) . Thus, D Γ Ψ( ~ , , Γ ) is an invertible matrix for L > L large enough, with a lowerbound uniform around (cid:16) ~ , , Γ (cid:17) . Therefore, by the uniform variant of the implicitfunction theorem, there exist 0 < δ ≪ < δ ≪ y = ( y n ) n ∈{ , ··· ,N } ) and continuous mapΠ : B H × L ( ~ , δ ) × B R N ( , δ ) → B R N (Γ , δ ) , such that for all ~ε ∈ B H × L ( ~ , δ ), a + ∈ B R N ( , δ ) and Γ ∈ B R N (Γ , δ ),Ψ (cid:0) ~ε, a + , Γ (cid:1) = 0 if and only if Γ = Π (cid:0) ~ε, a + (cid:1) . Moreover, taking 0 < δ < δ ≪ C and L > L large enough, (cid:12)(cid:12) Γ − Γ (cid:12)(cid:12) . k ~ε k H × L + k a + k . δ + C (cid:0) δ + e − γ L (cid:1) , which implies (4.1). The proof of Lemma 4.1 is complete. (cid:3) Now, we start the proof of Theorem 1.1.
Proof of Theorem 1.1.
Let ~ε ∈ H × L and y = ( y n ) n ∈{ , ··· ,N } as in the statementof the theorem. For all a + (0) = a + = ( a +1 , · · · , a + N ) ∈ ¯ B R ( r ), we consider thesolution ~u = ( u , u ) with the initial data as defined in Lemma 4.1 ~u = N X n =1 (cid:16) σ n ~Q ℓ n + h + n ~Z + ℓ n (cid:17) ( · − y n ) + ~ε = N X n =1 σ n ~Q ℓ n ( · − ˜ y n ) + ~ϕ (0) . Note that, for the proof of Theorem 1.1, we just need to prove the existence of ~u such that T ∗ ( ~u ) = ∞ . We start closing estimates except for the instable modes.Last, we will prove the existence of suitable parameters a + = ( a +1 , · · · , a + N ) bycontradiction and a topology argument. Step 1.
Closing the estimates in ~ϕ . First, from (4.1) and (4.4), k ~ϕ (0) k H . N X n =1 (cid:0) | h + n | + | ˜ y n − y n | (cid:1) . C ( δ + e − γ L ) . (4.5)Second, from (3.24) and the energy E ( ~u ( t )) and momentum I ( ~u ( t )) are conserved, E ( ~u ( t )) − E ( ~u (0)) ≤ C N X n =2 c n ! L α − sup s ∈ [0 ,t ] k ~ϕ ( s ) k H + e − γ ( L + γ t ) ! . Note that, from (3.12), (3.1) and the choose of initial data, N X n =1 ˜ c n H n ( ~ϕ ( t ) , ~ϕ ( t ))= E ( ~u ( t )) − E ( ~u (0)) + O (cid:18) k ~ϕ ( t ) k H L + k ~ϕ ( t ) k q +1 H + e − γ ( L + γ t ) + k ~ϕ (0) k H (cid:19) . ONDITIONAL STABILITY FOR 1D NLKG EQUATION 25 where q = min(2 , q ). Therefore, using (3.1), (3.23) and (4.1), we obtain µ k ~ϕ ( t ) k H ≤ N X n =1 ˜ c n H n ( ~ϕ ( t ) , ~ϕ ( t )) + µ − N X n =1 ( a + n ( t )) + N X n =1 ( a − n ( t )) ! + e − γ L . L α − sup s ∈ [0 ,t ] k ~ϕ ( s ) k H + 1 L k ~ϕ ( t ) k H + k ~ϕ ( t ) k q +1 H + C ( δ + e − γ L )+ C ( δ + e − γ L ) + k ~ϕ (0) k H . (cid:18) C L α − + C L + C (cid:19) (cid:0) δ + e − γ L (cid:1) , which strictly improves the estimate on ~ϕ in (3.1) for taking L , C large enoughand δ small enough. Step 2.
Closing the estimates in y = ( y n ) n ∈{ , ··· ,N } . Note that, from (2.13)and (3.1), for n = 1 , · · · , N − y n +1 − ˙ y n = ℓ n +1 − ℓ n + O (cid:0) δ + e − γ L (cid:1) ≥ γ , for δ small enough and L large enough. Integrating on [0 , t ] and using (4.1), weobtain y n +1 ( t ) − y n ( t ) ≥ γ t + y n +1 − y n + (˜ y n +1 − y n +1 ) − (˜ y n − y n ) ≥ γ t + (1 − C − ) L, which strictly improves the estimate on y = ( y ) n ∈{ , ··· ,N } in (3.1). Step 3.
Closing the estimates in a − = ( a − n ) n ∈{ , ··· ,N } . Note that, from (4.5), N X n =1 ( a − n (0)) . k ~ϕ (0) k H . C ( δ + e − γ L ) . By direct computation, (2.15) and (3.1), for n = 1 , · · · , N ,dd t (cid:0) e α n t ( a − n ( t )) (cid:1) = 2 e α n t a − n ( t ) (cid:18) dd t a − n ( t ) + α n a − n ( t ) (cid:19) = e α n t O | a − n | + k ~ϕ k H + N X n ′ =1 e − y n ′ +1 − y n ′ ) ! = e α n t O (cid:0) C (cid:0) δ + e − γ L (cid:1)(cid:1) . Integrating on [0 , t ], for any t ∈ [0 , T ∗ ] and any n = 1 , · · · , N , we obtain( a − n ( t )) − e − α n t ( a − n (0)) . C e − α n t Z t e α n s (cid:0) δ + e − γ L (cid:1) d s . C (cid:0) δ + e − γ L (cid:1) , which strictly improves the estimate on a − = ( a − n ) n ∈{ , ··· ,N } in (3.1). Step 4.
Final argument on the unstable parameters. Let b ( t ) = N X n =1 ( a + n ( t )) and ¯ α = min n α n . Observe that for any time t ∈ [0 , T ∗ ] where it holds b ( t ) = C (cid:0) δ + e − γ L (cid:1) , thefollowing transversality property holdsdd t b ( t ) = 2 N X n =1 α n ( a + n ( t )) + O N X n =1 | a + n | + k ~ϕ k H + e − γ L ! ≥ αC (cid:0) δ + e − γ L (cid:1) − δ − e − γ L > , (4.6)for δ small enough and L large enough. This transversality relation is enough tojustify the existence of at least a couple a + (0) = ( a +1 (0) , · · · , a + N (0)) ∈ ¯ B R N ( r ) suchthat T ∗ = ∞ where r = C (cid:0) δ + e − γ L (cid:1) . The proof is by contradiction, we assume that for all a + (0) ∈ ¯ B R N ( r ), it holds T ∗ < ∞ . Then, a construction follows from the following discussion (see for instancemore details in [5] and [6, Section 3.1]). Continuity of T ∗ . The above transversality condition (4.6) implies that the map a + (0) ∈ ¯ B R N ( r ) T ∗ ∈ [0 , ∞ )is continuous and T ∗ = 0 for a + (0) ∈ S R N ( r ) . Construction of a retraction . We define M : ¯ B R N ( r )
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Conclude. At this point, we have proved the existence of a + (0) ∈ ¯ B R N ( r ),associated with a global solution ~u = ( u , u ) of (1.2) with initial data defined inLemma 4.1, which also satisfies (3.1) for all t ∈ [0 , ∞ ). The proof of Theorem 1.1is complete. (cid:3) References [1] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state.
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CMLS, ´Ecole polytechnique, CNRS, Institut Polytechnique de Paris, F-91128 PalaiseauCedex, France.
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