Journal of Mathematical Analysis and Applications

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1 J. Math. Anal. Appl. 366 ( Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications Weighted energy decay for 1D wave equation E.A. Kopylova 1 Institute for Information Transmission Problems AS, B. Karetnyi 19, Moscow 11447, GSP-4, ussia article info abstract Article history: eceived 1 June 29 Available online 2 February 21 Submitted by M. Nakao Keywords: Dispersion Wave equation esolvent Spectral representation Weighted spaces Continuous spectrum Born series Convolution Long time asymptotics We obtain a dispersive long time decay in weighted energy norms for solutions to the 1D wave equation with generic potential. The decay extends the results obtained by Murata for the 1D Schrödinger equation. 21 Elsevier Inc. All rights reserved. 1. Introduction In this paper, we establish a dispersive long time decay for the solutions to 1D wave equation ψ(x, t = Hψ(x, t := ψ (x, t V (xψ(x, t, x, (1.1 in weighted energy norms. In vectorial form, Eq. (1.1 reads i Ψ(t = HΨ(t, where ( ψ(t Ψ(t = ψ(t (, H = i i( d2 dx 2 V (1.2. (1.3 For s, σ, letusdenotebyh s σ = H s σ (3 the weighted Sobolev spaces introduced by Agmon [1], with the finite norms ψ H s σ = ( 1 + x 2 σ /2 ψ H s <. We assume that V (x is a real function, and V (x + V (x C ( 1 + x β, x, (1.4 for some β>4. Then the multiplication by V (x is bounded operator H 1 s H1 s+β for any s. address: elena.kopylova@univie.ac.at. 1 Supported partly by FWF grant P19138-N13, DFG grant 436 US 113/929/-1 and FB grants X/$ see front matter 21 Elsevier Inc. All rights reserved. doi:1.116/j.jmaa

2 E.A. Kopylova / J. Math. Anal. Appl. 366 ( We restrict ourselves to the following regular case in the terminology of [11] (or nonsingular case in [14] The point λ = is neither eigenvalue nor resonance for the operator H. (1.5 Then the truncated resolvent of the Schrödinger operator H = d2 + V (x is bounded at the end point λ = of the dx 2 continuous spectrum. It is known that the spectral condition holds for generic potentials [11,14]. Definition 1.1. i F is the complex Hilbert space Ḣ 1 L 2 of vector-functions Ψ = (ψ, π with the norm Ψ F = ψ L 2 + π L 2 <. ii F σ is the complex Hilbert space H 1 σ H σ of vector-functions Ψ = (ψ, π with the norm Ψ Fσ = ψ H 1 σ + π H σ <. Definition 1.2. For real α > 1denoteby α the number from N such that α < α 1 + α. Our main result is the following long time decay of the solutions to (1.2: in the regular case for initial data Ψ = Ψ( F σ with σ > 2wehave Pc Ψ(t = O ( t γ, γ = min { σ 1/2,σ 1, β/2 1/2,β/2 1 }, t ±. (1.6 Here P c is a iesz projector onto the continuous spectrum of the operator H. The decay is desirable for the study of asymptotic stability and scattering for the solutions to nonlinear hyperbolic equations. Let us comment on previous results in this direction. Local energy decay has been established first in the scattering theory for linear Schrödinger equation developed since 5s by Birman, Kato, Simon, and others. For wave equations with compactly supported potentials, and similar hyperbolic PDEs, Vainberg [17] established the decay in local energy norms for solutions with compactly supported initial data. However, applications to asymptotic stability of solutions to the nonlinear equations also require an exact characterization of the decay for the corresponding linearized equations in weighted norms (see e.g. [3 5,16]. The decay of type (1.6 in weighted norms has been established first by Jensen and Kato [11] for the Schrödinger equation in the dimension n = 3. The result has been extended to all other dimensions by Jensen and Nenciu [9,1,12], and to more general PDEs of the Schrödinger type by Murata [14]. The survey of the results can be found in [15]. For the free wave equations with V (x = weighted L p -norms estimates have been established in [2,6]. For 3D wave equations with potential the Strichartz type weighted estimate was obtained in [8], and some estimates in weighted L - norms have been established in [7]. In [13] the decay of type (1.6 in the weighted energy norms has been proved for the wave equation in the dimension n = 3. The approach develops the Jensen Kato techniques to make it applicable to the relativistic equations. Namely, the decay of the low energy component of the solution follows by the Jensen Kato techniques while the decay for the high energy component requires novel robust ideas. This problem has been resolved with a modified approach based on the Born series and convolution. Let us note that the decay rate in (1.6 corresponds to the spatial decay of the initial function Ψ( and potential V (x in contrast to the Schrödinger case [11], where the decay rate is t 3/2. This difference is related to the presenceofthelacunaforthefree3dwaveequation. Here we extend our approach [13] to the dimension n = 1. The extension is not straightforward since the decay (1.6 violates for the free 1D wave equation corresponding to V (x = when the solutions do not decay. Hence, the decay (1.6 cannot be deduced by perturbation arguments from the corresponding estimate for the free equation. This difficulty is well known, and it is caused by the zero resonance function ψ(x = const corresponding to the end point λ = ofthe continuous spectrum of the free 1D Schrödinger operator d 2 /dx 2. Main idea of our approach to n = 1 is a spectral analysis of the bad term, without decay. Namely, we show that the bad term does not contribute to the high energy component. Therefore, the decay t γ for the high energy component follows. On the other hand, for the low energy component, the decay t γ holds for the generic potentials by methods [11,14]. This decay implies the asymptotic completeness since γ > 1. Our paper is organized as follows. In Section 2 we obtain the time decay for the solution to the free wave equation and state the spectral properties of the free resolvent. In Section 3 we obtain spectral properties of the perturbed resolvent and prove the decay (1.6.

3 496 E.A. Kopylova / J. Math. Anal. Appl. 366 ( Free wave equation First, we consider the free wave equation: ψ(x, t = ψ (x, t, x. (2.1 In vectorial form Eq. (2.1 reads i Ψ(t = H Ψ(t, (2.2 where ( ( ψ(t i Ψ(t =, H =. (2.3 ψ(t i d2 dx Spectral properties We state spectral properties of the free wave dynamical group G(t. Fort > and Ψ = Ψ( F, there exists a unique solution Ψ(t C b (, F to the free wave equation (2.2. Hence, Ψ(t admits the spectral Fourier Laplace representation θ(tψ (t = 1 e i(ω+iεt (ω + iεψ dω, t, (2.4 with any ε > where θ(t is the Heavyside function, (ω = (H ω 1 for ω C + := {Im ω > } is the resolvent of the operator H. The representation follows from the stationary equation ω Ψ + (ω = H Ψ + (ω + iψ for the Fourier Laplace transform Ψ + (ω := θ(teiωt Ψ(t dt, ω C +.ThesolutionΨ(t is continuous bounded function of t with the values in F by the energy conservation for the free wave equation (2.2. Hence, Ψ + (ω = i (ωψ is analytic function of ω C + with the values in F, and bounded for ω + iε. Therefore, the integral (2.4 converges in the sense of distributions of t with the values in F. Similarly to (2.4, θ( tψ (t = 1 e i(ω iεt (ω iεψ dω, t. (2.5 For the resolvent (ω the following matrix representation holds ( ω (ω 2 i (ω 2 (ω =, (2.6 i(1 + ω 2 (ω 2 ω (ω 2 where (ζ stands for the free Schrödinger resolvent 1 (ζ, x y = ( d2 dx ζ = exp(i ζ x y 2 2i, ζ ζ C +, Im ζ 1/2 >. (2.7 Definition 2.1. Denote by L(B 1, B 2 the Banach space of bounded linear operators from a Banach space B 1 to a Banach space B 2. The explicit formula (2.7 implies the properties of (ζ which are obtained in [1,14]: i (ζ is strongly analytic function of ζ C \[, with the values in L(H 1, H1 ; ii For ζ>, the convergence holds (ζ ± iε (ζ ± i as ε + in L(Hσ 1, H1 σ with σ > 1/2, uniformly in ζ r for any r >. iii For any M the following asymptotic expansion holds (ζ = M k= 1 A k ζ k/2 + O ( ζ (M+1/2, ζ, ζ C \[,, (2.8 in the norm of L(Hσ 1; H1 σ with σ > 3/2+ M + 1. Here A k L(Hσ 1; H1 σ with σ > 3/2+k are the integral operators with the integral kernels A k (x, y, and A k (x, y = ik x y k+1, k = 1,, 1, 2,... (2.9 2(k + 1!

4 E.A. Kopylova / J. Math. Anal. Appl. 366 ( iv The asymptotics (2.8 can be differentiated M + 2times:for1 r M + 2, r ζ (ζ = r ζ ( M k= 1 in the norm of L(Hσ 1; H1 σ with σ > 3/2 + M + 1. The properties i iv and (2.6 imply the following lemma. A k ζ k/2 + O ( ζ M+1 r 2, ζ, ζ C \[,, (2.1 Lemma 2.2. i The resolvent (ω is strongly analytic function of ω C \ with the values in L(F, F. ii For ω \,theconvergenceholds (ω ± iε (ω ± i as ε + in L(F σ, F σ with σ > 1/2, uniformly in ω r for any r >. iii For any M, the following asymptotics hold (ω = M ω k A k + O ( ω M+1, ω, ω C \, (2.11 k= 1 in the norm of L(F σ ; F σ with σ > 3/2 + M + 1. HereA k L(F σ ; F σ with σ > 3/2 + k are the integral operators with the integral matrix kernels A k (x, y. iv The asymptotics (2.11 can be differentiated M + 1 times: for 1 r M + 1, ( M ω r (ω = ω r ω k A k + O ( ω M+1 r, ω, ω C \, (2.12 k= 1 in the norm of L(F σ, F σ with σ > 3/2 + M + 1. Finally, we state the asymptotics of (ω for large ω which follow from known Agmon Jensen Kato decay [1, (A.2 ] and [11, Theorem 8.1] of the resolvent. Proposition 2.3. For m =, 1,l= 1,, 1 and k =, 1, 2,...the asymptotics hold (k (ζ L(H m σ,h m+l σ = O( ζ 1 l+k 2, ζ,ζ C \ (,, (2.13 where σ > 1/2 + k. Then for (ω we obtain Corollary 2.4. For k =, 1, 2,...the asymptotics hold (k (ω L(Fσ = O(1, ω,,f σ ω C \, (2.14 where σ > 1/2 + k. Proof. The bounds follow from representation (2.6 for (ω and asymptotics (2.13 for (ζ with ζ = ω 2. Corollary 2.5. For t and Ψ F σ with σ > 1/2, the group G(t admits the integral representation G(tΨ = 1 e iωt[ (ω + i (ω i ] Ψ dω, (2.15 where the integral converges in the sense of distributions of t with the values in F σ. Proof. Summing up the representations (2.4 and (2.5, and sending ε +, we obtain (2.15 by the Cauchy theorem, Lemma 2.2 and Corollary 2.4.

5 498 E.A. Kopylova / J. Math. Anal. Appl. 366 ( Time decay The estimates (2.14 do not allow obtain the decay of G(t by partial integration in (2.15. We deduce the decay from explicit formulas. It suffices to consider t >. In this case the matrix kernel of the dynamical group G(t can be written as G(t, x y, where ( Ġ(t, z G(t, z G(t, z =, z, (2.16 G(t, z Ġ(t, z and G(t, z = 1 2 θ( t z. (2.17 Let us represent G(t, z as G(t, z = G + G r (t, z, where G = 1 ( 1. ( Evidently,thefreewavegroupG(t does not decay which not correspond to (1.6. On the other hand, G is only nondecreasing term. More exactly, in the next subsection we will prove the following basic proposition Proposition 2.6. Let σ > 1. Then for the operator G r (t = G(t G with the kernel G r (t, x y, the bound holds Gr (t L(Fσ ;F σ C(1 + t σ +1, t >. (2.19 The following key observation is that the bad term G does not contribute to the high energy component of the total group G(t since (2.18 contains just one zero frequency. This suggests that the high energy component of the group G(t decays like t σ +1. More precisely, let us introduce the following low energy and high energy components of G(t: G l (t = 1 G h (t = 1 e iωt l(ω [ (ω + i (ω i ] dω, (2.2 e iωt h(ω [ (ω + i (ω i ] dω, (2.21 where l(ω C ( is an even function, suppl [ 2ε, 2ε], l(ω = 1if ω ε with an ε >, and h(ω = 1 l(ω. Theorem 2.7. Let σ > 1. Then the following bound holds Gh (t L(Fσ ;F σ C(1 + t σ +1, t >. (2.22 Proof. I. First we consider the case of small t 1. Denote (ψ(t, ψ(t = G(tΨ (. Using energy conservation for the wave equation we obtain ψ (t L 2 + ψ(t L 2 = G(tΨ ( F = Ψ( F Ψ( F. (2.23 Further, the Hölder inequality and (2.23 imply ψ(t ( t 2 = L 2 ψ(x, s ds ψ(x, 2 dx 2 ψ( 2 L 2 + 2t t ψ(s 2 L 2 ds 2Ψ( 2. F Hence, G(tΨ ( F CΨ( F, t 1. (2.24 The integrand in (2.2 has finite support and belongs to L(F σ ; F σ with σ > 1/2. Hence for σ > 1/2 Gl (t L(Fσ C, t >. ;F σ (2.25 Then (2.22 for small t 1 follows, since G h (t = G(t G l (t.

6 E.A. Kopylova / J. Math. Anal. Appl. 366 ( Then II. Now we deduce asymptotics (2.22 for t 1 from Proposition 2.6. Step i Let Ψ( F σ.denote Ψ + (t = θ(tg(tψ (, Ψ + (t = θ(tg (tψ (, Ψ + h (t = θ(tg h(tψ (, Ψ + r (t = θ(tg r (tψ (. Ψ + h (t = 1 = 1 = Ψ + r (t + 1 e iωt h(ω (ω + iψ ( dω e iωt h(ω Ψ + (ω dω = 1 e iωt h(ω Ψ + (ω dω 1 e iωt h(ω [ Ψ + (ω + Ψ + r (ω ] dω e iωt l(ω Ψ + r (ω dω, (2.26 where for the first term Ψ + r (t the bound (2.22 holds by (2.19. Step ii Let us consider the second summand in the right-hand side of (2.26. By (2.18 the matrix function Ψ + (ω is a smooth function for ω > ε, and ω k Ψ + (ω = O(ω 1 k, k =, 1, 2..., ω. Hence partial integration implies that for any σ > 1/2 e iωt h(ω Ψ + (ω dω = O ( t N, N N, t. (2.27 Step iii Now we consider the third summand in the right-hand side of ( e iωt l(ω Ψ + r (ω dω = [ L Ψ + ] ( r (t = O t σ +1, L = l, (2.28 in the norm of F σ,since L(t C(1 + t N,foranyN N, and Ψ + r (t (1 + t σ +1 by (2.19. Finally, (2.26 (2.28 imply ( Proof of Proposition 2.6 Proof. I. First we consider the case of small t 1. Let Ψ( = Ψ = (ψ, π.bycauchyinequality ( (G Ψ 1 1 = π (x dx 2 C π 2 (x( 1 + x 2 1/2 ( σ dx since σ > 1/2. Therefore, 1/2 dx (1 + x 2 σ C π H C Ψ σ Fσ, G Ψ = (G Ψ 1 H σ C Ψ Fσ, t 1. (2.29 Then (2.19 for small t 1 follows from (2.24 and (2.29 since G r = G G. II. Now we consider an arbitrary t 1. Let us split the initial function Ψ in two terms, Ψ = Ψ 1,t + Ψ 2,t such that and Ψ 1,t (x = for x > t/2, and Ψ 2,t (x = for x < t/3, (2.3 Ψ 1,t Fσ + Ψ 2,t Fσ C Ψ Fσ, t 1. (2.31 We estimate G r (tψ 1,t and G r (tψ 2,t separately. Step i First we estimate G r (tψ 2,t = G(tΨ 2,t G Ψ 2,t.LetG(tΨ 2,t = (ψ 2 (t, ψ 2 (t. Using energy conservation for the wave equation and properties (2.3 and (2.31 we obtain ψ 2 (t H + ψ 2 (t σ H G(tΨ2,tF = Ψ 2,t F Ct σ Ψ 2,t Fσ Ct σ Ψ Fσ. (2.32 σ Further, the Hölder inequality, energy conservation and properties (2.3 (2.31 imply

7 5 E.A. Kopylova / J. Math. Anal. Appl. 366 ( ψ 2 (t ( t 2 H σ C ( ψ 2 (x, s ds ψ 2 (x, 2 dx 2 (Ψ 2,t 1 2 L 2 + 2t t 2σ (Ψ 2,t 1 2 H σ + t Ct 2 2σ Ψ 2 F σ. t t ψ 2 (s 2 L 2 ds Ψ 2,t 2 F ds ( C t 2σ Ψ 2 F σ + t 2 2σ Ψ 2 F σ (2.33 Hence, (2.32 and (2.33 imply G(tΨ2,t Ct σ +1 Ψ Fσ, t 1. (2.34 Second we estimate G Ψ 2,t.ByCauchyinequality ( (G Ψ 2,t 1 1 = π 2,t (x dx 2 C π 2,t (x 2 ( 1/2 1 + x 2 σ dx Ct σ +1/2 π 2,t H σ Ct σ +1/2 Ψ Fσ, where π 2,t is the second component of Ψ 2,t. Therefore, ( t/3 dx (1 + x 2 σ G Ψ 2,t = (G Ψ 2,t 1 H σ Ct σ +1/2 Ψ Fσ, t 1. (2.35 Finally, (2.34 (2.35 imply that Gr (tψ 2,t Ct σ +1 Ψ Fσ, t 1. (2.36 Step ii Now we consider G r (tψ 1,t = G(tΨ 1,t G Ψ 1,t. Formulas (2.16 (2.18, and (2.3 imply that [ Gr (tψ 1,t ] (x =, x < t/2. 1/2 Denote by χ(x the characteristic function of the domain x > t/2. Then Gr (tψ 1,t = χ(x ( G(tΨ 1,t G (tψ 1,t G(tΨ1,t + χ(xg (tψ 1,t L 2 + χ(xg (tψ 1,t. (2.37 Ct σ ( G(tΨ1,t F + ( G(tΨ 1,t 1 By energy conservation and (2.31, we obtain G(tΨ1,tF = Ψ 1,t F Ψ 1,t Fσ C Ψ Fσ, t 1. (2.38 Further, similarly (2.33, using energy conservation we obtain ( G(tΨ 1,t 1 2 L 2 2(Ψ 1,t L 2 2t t ( ( G(sΨ 1,t 2 L 2 ds C Ψ 1,t 2 F σ + t 2 t Ψ 1,t 2 F ds C ( Ψ 2 F σ + t 2 Ψ 1,t 2 F Ct 2 Ψ 2 F σ. (2.39 Finally, we estimate the last summand in the right-hand side of (2.37. Denote by π 1,t the second component of Ψ 1,t.By Cauchy inequality G 12 π 1 1,t = π 1,t (x dx 2 C π 1,t H σ C Ψ 1,t Fσ since σ > 1/2. Hence χ(xg (tψ 1,t 2 F σ = x >t/2 ( 1 + x 2 σ G 12 π 1,t 2 dx Ct 2σ +1 Ψ 1,t 2 F σ. Finally, the last estimate and (2.37 (2.39 imply Gr (tψ 1,t Ct σ +1 Ψ 1,t Fσ Ct σ +1 Ψ Fσ, t 1.

8 E.A. Kopylova / J. Math. Anal. Appl. 366 ( Perturbed wave equation To prove the long time decay for the perturbed wave equation, we first establish the spectral properties of the generator Spectral properties Let us collect the properties of the perturbed Schrödinger resolvent (ζ = (H ζ 1 obtained in [1,11,14] under conditions (1.4 and (1.5. Note, that in [11] there is considered 3D case, but corresponding properties can be proved in 1D case similarly. 1. (ζ is strongly meromorphic function of ζ C \[, with the values in L(H 1, H1 ;thepolesof(ζ are located at a finite set of eigenvalues ζ j <, j = 1,...,N, of the operator H with the corresponding eigenfunctions ψ j (x Hs 2 with any s. 2. For ζ>, the convergence holds (ζ ± iε (ζ ± i as ε + in L(Hσ 1, H1 σ with σ > 1/2, uniformly in ζ r for any r >. 3. Assume β>3 + 2M, M =, 1, 2... The expansion holds: (ζ = M B j ζ j/2 + O ( ζ (M+1/2, ζ, ζ C \ (,, (3.1 j= in the norm of L(Hσ 1, H1 σ with σ > 3/2 + M. The expansion (3.1 can be differentiated M + 1times. 4. Assume β>k + 1, σ > 1/2 + k, k =, 1, 2,...Thenform =, 1 and l = 1,, 1 the asymptotics hold (k (ζ L(H m σ ;H m+l σ = O( ζ (1 l+k/2, ζ,ζ C \ (,. (3.2 The resolvent (ω = (H ω 1 can be expressed similarly to (2.6: ( ω(ω 2 i(ω 2 (ω =. (3.3 i(1 + ω 2 (ω 2 ω(ω 2 Hence, the properties 1 4 imply the corresponding properties of (ω: Lemma 3.1. Let the potential V satisfy conditions (1.4 and (1.5. Then i (ω is strongly meromorphic function of ω C \ with the values in L(F, F ; ii The poles of (ω are located at a finite set of imaginary axis Σ = { ω ± j =± ζ j, j = 1,...,N } of eigenvalues of the operator H ( with the corresponding eigenfunctions ψ iω ± ; j j(x ψ iii For ω \ the convergence holds (ω ± iε (ω ± i, ε +, in the norm of L(F σ, F σ with σ > 1/2; iv Assume β>3, σ > 3/2 for k =,andβ>1 + 2k, σ > 1/2 + k for k = 1, 2,... Then the bounds hold (k (ω L(Fσ = O(1, ω,,f σ ω C \. (3.4 v Assume β>1 + k, σ > 1/2 + k, k =, 1, 2,... Then the bounds hold (k (ω L(Fσ = O(1, ω,,f σ ω C \. (3.5 Finally, let us denote by V the matrix ( V =. iv Then the vectorial equation (1.2 reads (3.6 i Ψ(t = (H + VΨ (t. The resolvents (ω and (ω are related by the Born perturbation series

9 52 E.A. Kopylova / J. Math. Anal. Appl. 366 ( (ω = (ω (ωv (ω + (ωv (ωv(ω, ω C \[ Σ], (3.7 which follows by iteration of (ω = (ω (ωv(ω. An important role in (3.7 plays the product W(ω := V (ωv. We obtain the asymptotics of W(ω for large ω. Lemma 3.2. Let k =, 1, 2,...and the potential V satisfy (1.4 with β>1/2 + k + σ where σ >. Then the asymptotics hold W (k (ω L(,F σ = O( ω 2, ω, ω C \. (3.8 Proof. Bounds (3.8 follow from the algebraic structure of the matrix ( W (k (ω = V (k (ωv = iv (k, (3.9 (ω2 V since (2.13 with m = 1 and l = 1 implies that V (k (ζ V f H σ C (k (ζ V f H σ β C(k ζ 1 k 2 Vf H 1 β σ C(k ζ 1 k 2 f H 1 σ (3.1 with 1/2 + k <β σ for k =, 1, 2, Time decay In this subsection we combine the spectral properties of the perturbed resolvent and time decay for the unperturbed dynamics using the (finite Born perturbation series. Our main result is the following. Theorem 3.3. Let conditions (1.4 and (1.5 hold. Then for σ > 2 e ith e iω jt P jl(fσ = O ( t γ, γ = { min σ 1/2,σ 1, β/2 1/2,β/2 } 1, (3.11 ω J Σ,F σ as t ±.HereP j are the iesz projectors onto the corresponding eigenspaces. Proof. Lemma 3.1 and bounds (3.5 with k = imply similarly to (2.15, that Ψ(t e iω jt P j Ψ = 1 e iωt[ (ω + i (ω i ] Ψ dω = Ψ 2π l (t + Ψ h (t, (3.12 i ω j Σ where P j stands for the corresponding iesz projector P j Ψ := 1 (ωψ dω with a small δ>, and Ψ l (t = 1 Ψ h (t = 1 ω ω j =δ l(ωe iωt[ (ω + i (ω i ] Ψ dω, (3.13 h(ωe iωt[ (ω + i (ω i ] Ψ dω, (3.14 where l(ω and h(ω are defined in Section 2.2. Further we analyze Ψ l (t and Ψ h (t separately Time decay of Ψ l (t Let σ > 3/2 and β>3. By Lemma 3.1iv we apply integration by parts γ times, with γ = min{ σ 1/2, β/2 1/2 } and obtain Ψ l (t C ( 1 + t γ Ψ Fσ, t. (3.15

10 E.A. Kopylova / J. Math. Anal. Appl. 366 ( Time decay of Ψ h Let us substitute the series (3.7 into the spectral representation (3.14 for Ψ h (t: Ψ h (t = e iωt h(ω [ (ω + i (ω i ] Ψ dω (3.16 e iωt h(ω [ (ω + iv (ω + i (ω iv (ω i ] Ψ dω e iωt h(ω [ V V(ω + i V V(ω i ] Ψ dω = Ψ h1 (t + Ψ h2 (t + Ψ h3 (t, t. (3.17 Further we analyze each term Ψ hk, k = 1, 2, 3, separately. Step i The first term Ψ h1 (t = G h (tψ by (2.21. Hence, Theorem 2.7 implies that Ψ h1 (t C ( 1 + t σ +1 Ψ Fσ, t. (3.18 Step ii Now we consider the second term Ψ h2 (t. Denoteh 1 (ω = h(ω and let e iωt h 1 (ω [ (ω + i (ω i ] Ψ dω. Φ h1 = 1 It is obvious that for Φ h1 the inequality (3.18 also holds. Namely, Φ h1 (t C ( 1 + t σ +1 Ψ Fσ, t. (3.19 Now the second term Ψ h2 (t can be rewritten as a convolution. Lemma 3.4. The convolution representation holds t Ψ h2 (t = i G h1 (t τ VΦ h1 (τ dτ, t, (3.2 where the integral converges in F σ with σ > 2. The group G h1 (t denoting in (2.21 with function h 1 instead h. Proof. Then the term Ψ h2 (t can be rewritten as Ψ h2 (t = 1 e iωt h 2 1 (ω[ (ω + iv (ω + i (ω iv (ω i ] Ψ dω. (3.21 Let us integrate the first term in the right-hand side of (3.21, denoting G ± h1 (t := θ(±tg h1(t, Φ ± h1 (t := θ(±tφ h1(t, t. We know that h 1 (ω (ω + iψ = i Φ + (ω, hence integrating the first term in the right-hand side of (3.21, we obtain h1 that Ψ + h2 (t = 1 2π = 1 2π = 1 2π (i t + i 2 e iωt h 1 (ω (ω + iv Φ + (ω dω h1 e iωt h 1 (ω (ω + iv [ e iωτ Φ + h1 (τ dτ ] dω e iωt [ (ω + i 2 h 1(ω (ω + iv e iωτ Φ + h1 (τ dτ ] dω. (3.22 The last double integral converges in F σ with σ > 2 by (3.19, Lemma 2.2ii, and (2.14 with k =. Hence, we can change the order of integration by the Fubini theorem. Then we obtain that

11 54 E.A. Kopylova / J. Math. Anal. Appl. 366 ( since Ψ + h2 (t = i G + h1 (t τ = 1 G + h1 (t τ VΦ+ h1 (τ dτ = { i t G h1(t τ VΦ h1 (τ dτ, t >,, t <, = 1 (i t + i 2 e iω(t τ h 1 (ω (ω + i dω e iω(t τ (ω + i 2 h 1(ω (ω + i dω by (2.4. Similarly, integrating the second term in the right-hand side of (3.21, we obtain {, t >, Ψ h2 (t = i G h1 (t τ VΦ (τ h1 dτ = t i G h1(t τ VΦ h1 (τ dτ, t <. Now (3.2 follows since Ψ h2 (t is the sum of two expressions (3.23 and (3.24. (3.23 (3.24 Lemma 3.5. Let β>4 and σ > 2.Then Ψ h2 (t C ( 1 + t γ Ψ Fσ, γ = min {σ 1, β2 } 1, t. (3.25 Proof. We apply Theorem 2.7 with h 1 instead h to the integrand in (3.2. For 2 < σ β/2 weobtain Gh1 (t τ VΦ h1 (τ C VΦ h1(τ Fσ (1 + t τ σ 1 C Φ h1(τ (1 + t τ σ 1 C Ψ Fσ (1 + t τ σ 1 (1 + τ σ 1, and for σ β/2 Gh1 (t τ VΦ h1 (τ Gh1 (t τ VΦ h1 (τ F β/2 C VΦ h1(τ Fβ/2 (1 + t τ β 2 1 Hence (3.2 implies (3.25. C Φ h1(τ F β/2 (1 + t τ β 2 1 C Ψ Fβ/2 (1 + t τ β 2 1 (1 + τ β 2 1 C Ψ Fσ (1 + t τ β 2 1 (1 + τ β 2 1. Step iii Finally, let us rewrite the last term Ψ h3 as Ψ h3 (t = 1 e iωt h(ωn (ωψ dω, (3.26 where N (ω := M(ω + i M(ω i for ω, and M(ω := (ωv (ωv(ω = (ωw(ω(ω, ω C \. (3.27 Now we obtain the asymptotics of N and its derivatives for large ω. Lemma 3.6. For k < min{β 1, σ 1/2} the bounds hold N (k (ω L(Fσ,F σ C(k ω 2, ω, ω 1. (3.28 Proof. We have M (k = k 1 +k 2 +k 3 =k Lemma 3.2 and bounds (2.14, (3.5 imply that k! k 1!k 2!k 3! (k 1 W (k2 (k3. (3.29

12 E.A. Kopylova / J. Math. Anal. Appl. 366 ( (k 1 W (k 2 (k 3 (ω f (k 1 W (k 2 (k 3 (ω f 1 C(k 1 W (k 2 (k 3 (ω f Fσ1 under the conditions C(k 1,k 2 (k 3 (ω ω 2 f 1 C(r,k 1,k 2,k 3 ω 2 f Fσ1 C(r,k 1,k 2,k 3 ω 2 f Fσ, ω 1, σ > σ 1 > 1/2 + max{k 1,k 3 }, β >1/2 + k 2 + σ 1, β >1 + k 3. All these inequalities hold if σ > 1/2 + k, β>1 + k, and 1/2 + max{k 1,k 3 } < σ 1 < min{σ,β 1/2 k 2 }. NowweprovethedesireddecayofΨ h3 (t from (3.26. Lemma 3.7. Ψ h3 (t ( C 1 + t γ Ψ Fσ, γ = { min σ 1/2,σ 1, β/2 1/2,β/2 } 1, t. (3.3 Proof. First, in the case 2 < σ β/2 thereexistsk 1 such that 1/2 + k < σ 3/2 + k. Thenβ>1 + 2k > 1 + k, and by Lemma 3.6 N (k (ω L 1( [1, ]; L(F σ, F σ. Then we can apply k times integration by parts in (3.26 to obtain Ψh3 (t C ( 1 + t k Ψ Fσ = C ( 1 + t σ 1/2 Ψ Fσ, t, (3.31 since k = σ 1/2 by Definition 1.2. Second, in the case 4 <β 2σ there exists k 1suchthatk + 1/2 <β/2 k + 3/2. Then σ > 1/2 + k and β>2k + 1 > 1 + k. Hence (3.31 holds by Lemma 3.6 and using k times integration by parts we obtain Ψ 3 (t C ( 1 + t β/2 1/2 Ψ Fσ, t. This completes the proof of the lemma and Theorem 3.3. Corollary 3.8. The asymptotics (3.11 imply (1.6 with the projector P c = 1 P j. ω j Σ (3.32 eferences [1] S. Agmon, Spectral properties of Schrödinger operator and scattering theory, Ann. Sc. Norm. Super. Pisa Ser. IV 2 ( [2] P. D Ancona, V. Georgiev, H. Kubo, Weighted decay estimates for the wave equation, J. Differential Equations 177 (1 ( [3] V.S. Buslaev, G. Perelman, On the stability of solitary waves for nonlinear Schrödinger equations, Trans. Amer. Math. Soc. 164 ( [4] V.S. Buslaev, C. Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (3 ( [5] S. Cuccagna, Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure Appl. Math. 54 (9 ( [6] V. Georgiev, Semilinear Hyperbolic Equations, MSJ Mem., vol. 7, Mathematical Society of Japan, Tokyo, 25. [7] V. Georgiev, C. Heiming, H. Kubo, Supercritical semilinear wave equation with non-negative potential, Comm. Partial Differential Equations 26 (11 12 ( [8] V. Georgiev, N. Visciglia, Decay estimates for the wave equation with potential, Comm. Partial Differential Equations 28 (7 8 ( [9] A. Jensen, Spectral properties of Schrödinger operators and time-decay of the wave function. esults in L 2 ( m, m 5, Duke Math. J. 47 ( [1] A. Jensen, Spectral properties of Schrödinger operators and time-decay of the wave function. esults in L 2 ( 4, J. Math. Anal. Appl. 11 ( [11] A. Jensen, T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J. 46 ( [12] A. Jensen, G. Nenciu, A unified approach to resolvent expansions at thresholds, ev. Math. Phys. 13 (6 ( [13] E. Kopylova, Weighed energy decay for 3D wave equation, Asymptot. Anal. 65 (1 2 ( [14] M. Murata, Asymptotic expansions in time for solutions of Schrödinger-type equations, J. Funct. Anal. 49 ( [15] W. Schlag, Dispersive estimates for Schrödinger operators, a survey, in: Jean Bourgain, et al. (Eds., Mathematical Aspects of Nonlinear Dispersive Equations. Lectures of the CMI/IAS Workshop on Mathematical Aspects of Nonlinear PDEs, Princeton, NJ, USA, 24, in: Ann. of Math. Stud., vol. 163, Princeton University Press, Princeton, NJ, 27, pp [16] A. Soffer, M.I. Weinstein, esonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math. 136 (1 ( [17] B.. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach, New York, 1989.