SCATTERING FOR QUASILINEAR HYPERBOLIC EQUATIONS OF KIRCHHOFF TYPE WITH PERTURBATION. Osaka Journal of Mathematics. 54(2) P.287-P.

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1 Title Authors SCATTERING FOR QUASILINEAR HYPERBOLIC EQUATIONS OF KIRCHHOFF TYPE WITH PERTURBATION Yamazaki, Taeko Citation Osaka Journal of Mathematics. 542 P.287-P.322 Issue Date Text Version publisher URL DOI /6194 rights

2 Yamazaki, T. Osaka J. Math , SCATTERING FOR QUASILINEAR HYPERBOLIC EQUATIONS OF KIRCHHOFF TYPE WITH PERTURBATION Taeko YAMAZAKI Received September 2, 215, revised March 29, 216 Abstract This paper is concerned with the abstract quasilinear hyperbolic equations of Kirchhoff type with perturbation. We show the existence of the wave operators and the scattering operator for small data, and that these operators are homeomorphic with respect to a suitable metric in a neighborhood of the origin. Introduction Let H be a separable complex Hilbert space H with the inner product, H and the norm.leta be a non-negative injective self-adjoint operator with domain DA, and let m be a function satisfying m C 2 [, ; [m,, with a positive constant m.letbt beac 1 function on R. We consider the initial value problem of the abstract quasilinear hyperbolic equations of Kirchhoff type with perturbation.1.2 u t + btu t + m A 1/2 ut 2 2 Aut =, u = φ, u = ψ. The asymptotic behavior of the solution of the equation above depends on the integrability of b with respect to t. If bt = 1 + t p with p 1, it is known that the global solution of.1-.2 exists uniquely and behaves like solutions of a corresponding parabolic equation, for small initial data φ,ψ DA DA 1/2 see Yamazaki [14] for p < 1 and Ghisi and Gobbino [7] for p = 1, and see Ghisi [6] for a mildly degenerate case mλ = λ γ with γ 1and p 1. There are no result about the global solvability for large initial data in Sobolev spaces, even for the constant dissipation term. On the other hand, if b satisfies the assumptions.3 lim bt =, t ±.4 t p tb t L 1 R, where p is a constant and t := 1 + t 2 1/2, the author [15] showed the global existence of a solution for small data in some class and showed the following see Theorems B and C: 21 Mathematics Subject Classification. Primary 35L72; Secondary 35L9.

3 288 T. Yamazaki i The solution of the Cauchy problem for the perturbed Kirchhoff equation has the same asymptotic behavior of a function obtained by a transformation of time variable from a solution of the free wave equation with an appropriate wave speed. ii Conversely, there exists a solution of the Kirchhoff equation which has the same asymptotic behavior of a function obtained by a transformation of the time variable from the solution of the Cauchy problem of the free wave equation with an appropriate wave speed. However, these facts do not imply that solutions of the perturbed Kirchhoff equation are not asymptotically free, in the sense that it has the same asymptotic behavior as that of solutions of free wave equations, since the transformation of time variable is used. The author [16] showed the following: If b satisfies.3 and.4 with p = and does not change sign for sufficient large t, and there exists a solution with small data which is asymptotically free, then tbt must be integrable see Theorem D. The integrability of tbt is equivalent to the condition.3 and.4 with p = 1ifbis monotone for sufficiently large t see remark 3. Hence, in order to show that the solutions are asymptotically free, it is necessary to assume that p 1 in the assumption.4, if b is monotone for sufficiently large t see remark 3. The purpose of this paper is to show the following assertions under the assumption that b satisfies.3 and.4 with p 1. iii The solutions of the perturbed Kirchhoff equation with small data in some class are asymptotically free. iv For small initial data in some class and the solution of the free wave equation with an appropriate wave speed, there exists a solution of perturbed Kirchhoff equation which approaches to the solution. v The wave operators exist and they are homeomorphic in a neighborhood of the origin with respect to a metric given in Notation 5 in Section 1. It follows that the scattering operator is also homeomorphic in a neighborhood of the origin. Last we list previous results on the asymptotic behavior or scattering, in the case b and A = Δ on L 2 R n. Ghisi [5] first showed iii above. The author [11] showed iii and iv. However [11] did not prove v, but proved the existence of the wave and its inverse operators for initial data in some class, and the continuity of the scattering operator only at the origin with respect to a metric somewhat different from the one in this paper, together with the continuity from a neighborhood of the origin in this topology to a Sobolev space with weaker norm. Kajitani [8, 9] considered the Kirchhoff equation of the type mλ = 1 + ελ, where A is an elliptic differential operator with coefficients depending on space variables and he showed that, for every initial data in a class, there exists a positive constant ε > such that for every ε,ε the property i and ii above hold. Hence, even in the case b anda = Δ on L 2 R n, our result v is new. The outline of this paper is as follows: In section 1, we state notations and results. In section 2, we state some examples of Sobolev spaces included in the spaces we consider. In section 3, we transform original problem. In section 4, we prove propositions. In section 5, we prove Theorem 1. In section 6, we prove Theorems 2 and Results For a closed operator B in a Hilbert space H, DB andrb denote the domain of B and 1. Results

4 Scattering for Perturbed Kirchhoff Equation 289 the range of B, respectively. For a non-negative number J, the domain of A J/2 becomes a Hilbert space DA J/2 equipped with the inner product f,g J := A J/2 f, A J/2 g H + f,g H. and the norm f 2 J = f, f J. We note that f = f. Notation 1. Let H denote the space of all bounded linear operator on Hilbert space H equipped with the operator norm H. We state notations and some classes of functions as in [15]. Notation 2. Let p be a non-negative number. For a continuous function f on R, put f p = r p f r dr. R Notation 3. For every α>, let α denote the completion of DA α by the norm A α. Let α be the extension of A α on α. The fact that A α is an injective self-adjoint operator implies that the range RA α is dense in H, and thus α : α H is surjective. From this fact and the definition, it follows that α : α H is an isometric isomorphism. For example, if H = L 2 R n anda = Δ with DA = H 2 R n, then α is equal to the homogeneous Sobolev space Ḣ 2α. and Notation 4. Let k and p be non-negative numbers and let ε be a positive number. We put We abbreviate k if k = 1as We put X k,p :={ f,g 1/2 H; 1/2 f,g DA k/4 DA k/4, f,g 2 X k,p := A k/4 e 2irA1/2 1/2 f, A k/4 1/2 f H p + A k/4 e 2irA1/2 g, A k/4 g H p + 2 A k/4 e 2irA1/2 1/2 f, A k/4 g H p < }, Y k,p := X k,p H H. X p := X 1,p, Y p := Y 1,p. Ỹ p :={ f,g Y p ;f,g satisfies the following 1.1}, 1.1 lim eita1/2 A 1/2 f, A 1/2 f H = lim eita1/2 g, g H t ± t ± = lim eita1/2 A 1/2 f,g H =. t ± Remark 1. Kajitani [9] introduced notation Y k,p and noted Y p Y,p = Y 1,p Y,p Ỹ p,

5 29 T. Yamazaki since the functions in 1.1 and their derivatives with respect to t are integrable. Remark 2. In the case H = L 2 R n anda = Δ with DA = H 2 R n, we can prove that 1.1 holds for every f,g 1/2 H, in the same way as in the proof of Riemann-Lebesgue Theorem. Notation 5. For J, we put and put Xp J = { f,g X p ; 1/2 f,g DA J/2 DA J/2 }, Yp J = Y p DA J+1/2 DA J/2, Ỹ J p = Ỹ p DA J+1/2 DA J/2, f,g X J p := f,g Xp + 1/2 f J + g J for f,g X J p f,g Y J p := f,g X J p + f = f,g Xp + f J+1 + g J for f,g Y J p. We define a metric on X J p as d J p f 1,g 1, f 2,g 2 := d p f 1,g 1, f 2,g 2 + 1/2 f 1 f 2 J + g 1 g 2 J for f 1,g 1, f 2,g 2 X J p, where d p f 1,g 1, f 2,g 2 2 := A 1/4 e 2irA1/2 1/2 f 1, 3/4 f 1 H A 1/4 e 2irA1/2 1/2 f 2, 3/4 f 2 H p We define a metric on Y J p as for f 1,g 1, f 2,g 2 Y J p. + A 1/4 e 2irA1/2 g 1, A 1/4 g 1 H A 1/4 e 2irA1/2 g 2, A 1/4 g 2 H p + 2 A 1/4 e 2irA1/2 1/2 f 1, A 1/4 g 1 H A 1/4 e 2irA1/2 1/2 f 2, A 1/4 g 2 H p. d J Y p f 1,g 1, f 2,g 2 := d J p f 1,g 1, f 2,g 2 + f 1 f 2 It is easy to see that d J p and d J Y p become metrics on X J p and Y J p respectively, and thus, X J p and Y J p become metric spaces with the metrics d J p and d J Y p respectively. Notation 6. For J andε>, we define a subset X J pεofx J p by X J pε :={ f,g X J p; f,g Xp + 1/2 f + g ε}. We define subsets Y J pεandỹ J pε ofy J p by Y J pε := X J pε Y p = X J pε H H, Ỹ J pε := X J pε Ỹ p, In [15], we extended the space considered in problem.1.2 to a larger one, namely, we consider the problem 1.2 u t + btu t + m 1/2 ut 2 2 A 1/2 1/2 ut =, t >,

6 Scattering for Perturbed Kirchhoff Equation u = φ, u = ψ, for φ,ψ 1/2 DA 1/2 satisfying 1/2 φ DA 1/2. Notation 7. Let I be one of the interval R,, or, and let bt CI. We say that u is a solution of 1.4 u t + btu t + m 1/2 ut 2 2 A 1/2 1/2 ut =, t I, if and 1.4 holds in H. u C 1 I; 1/2, 1/2 u, u C j I; DA 1 j/2 DA 1 j/2, j=,1 The author showed the following global existence of solution. Theorem A Theorem 1 of [15]. Let J 1 and p. Assume that b satisfies.3 and.4. Then there exist a positive number ε 1 such that the following holds. For every φ,ψ Xpε J 1, perturbed Kirchhoff equation has a unique global solution u C R; 1/2 satisfying 1.5 1/2 u, u C j R; DA 1 j/2 DA 1 j/2. Furthermore, the following holds. j=,1 u, u C R; X J p, 1/2 u, u C 1 R; DA J 1/2 DA J 1/2. If φ,ψ Y J pε 1,thenu j=,1,2 C j R; DA J+1 j/2. The global solutions of Kirchhoff equation given by Theorem A has the same asymptotic behavior as time-transformed function of the solution of a free equation: Theorem B Theorem 2 of [15]. Let J 1 and p. Assume that b satisfies.3 and.4. Letε 1 be the positive constant in Theorem A. i There exists a positive constant ε 2 ε 1 such that the following holds. Suppose that φ,ψ Xpε J 2, and that ut C 1 R; 1/2 is the solution of satisfying 1/2 u, u j=,1 C j R; DA 1 j/2 DA 1 j/2. Then the limits exist and satisfy C ± = lim t ± m A1/2 ut 2 lim t ± t p m A 1/2 ut 2 C ± =, and there uniquely exist global solutions v +,v C 1 R; 1/2 of v +t + C+ A 2 1/2 1/2 v + =, t R, v t + CA 2 1/2 1/2 v =, t R,

7 292 T. Yamazaki satisfying and where 1/2 v ±,v ± C j R; DA 1 j/2 DA 1 j/2 j=,1 lim 1/2 ut 1/2 v ± C± τt 1 + u t v ±C± τt 1 =, t ± Furthermore, it holds that τt := t m A 1/2 us 2 ds for t R. v ±,v ± C 1.6 R; Xp J, 1.7 1/2 v ±,v ± C 1 R; DA J 1/2 DA J 1/2, lim t ± t p 1/2 ut 1/2 v ± C± τt 1 J + u t v ±C± τt 1 J =. ii Assume moreover that φ,ψ satisfies 1.1. Then C ± = m 2 1/2 φ ± ψ 2C± 2 ± 2, and the limits lim t ± 1/2 ut 2 and lim t ± u t 2 exist and satisfy the equality lim t ± u t 2 = m lim t ± 1/2 ut 2 2 lim t ± 1/2 ut 2. Conversely to Theorem B, [15] showed the existence of the solution of perturbed Kirchhoff equation which has similar asymptotic behavior as that of the solution of the free equation. By 1.8, it is necessary to assume the relation 1.9 below. Theorem C Theorem 3 of [15]. Let J 1 and p. Assume that b satisfies.3 and.4. Then there exist a positive constant ε 3 such that the following holds. Suppose that φ, ψ Xpε J 3 satisfies 1.1. Then there uniquely exists a positive number c satisfying the equality c = m 2 1/2 φ ψ 2. 2c 2 Let v be the solution of v t + c 2 A 1/2 1/2 v = t R with v = φ, v = ψ. Then there uniquely exist global solutions u ± t C 1 R; 1/2 of 1.2 satisfying 1/2 u ±, u ± C j R; DA 1 j/2 DA 1 j/2, j=,1 lim 1/2 u ± t 1/2 vc± τt 1 + u ±t v C± τt 1 =, t ±

8 Scattering for Perturbed Kirchhoff Equation 293 and that d dt 1/2 u ± t 2 is integrable on R. Furthermore, u ± satisfy the following. u ±, u ± C R; X J p, 1/2 u ±, u ± C 1 R; DA J 1/2 DA J 1/2. lim t p 1/2 u ± t 1/2 vc± τt 1 J + u ±t v C± τt 1 J =, t ± t p d dt 1/2 u ± t 2 L 1 R. Theorem C does not mean that the non-trivial solutions of Kirchhoff equation which is given by Theorem A are asymptotically free, since the transformation τt is used. Different from the linear wave equation, we need more decay condition on b for the asymptotically free property, in view of the next theorem. Theorem D Theorem, its proof and Remark 2 of [16]. Assume that m satisfies m x for every x >. Assume that b satisfies.3 and.4 with p = and bt does not change sign for every t T, and bt does not change sign for every t T, for a positive constant T.Letε 2 be a positive constant given by Theorem B. If there exists asolutionu C R; 1/2 of perturbed Kirchhoff equation with initial value φ,ψ, X 1 ε 2 X, which is asymptotically free as t ±, in the sense that there exist positive constants c +, c and global solutions v +,v of such that v +t + c 2 + A 1/2 1/2 v + =, t R, v t + c 2 A 1/2 1/2 v =, t R, then lim 1/2 ut 1/2 v ± t + u t v ±t =, t ± tbt L 1 R. Remark 3. If b satisfies.3 and.4, then Lemma 3 in section 4 with f = b and α ± = implies 1.1 t p bt L 1 R. This relation is also seen in the proof of Lemma 5 in [15]. If b is monotone for sufficiently large t, then the converse holds, that is, 1.1 implies.3 and.4. In fact, if bt is monotone-decreasing on [T, forsufficiently large T and satisfies 1.1, then it follows that b satisfies.3, bt on[t, and

9 294 T. Yamazaki R R t p+1 b t dt = t p+1 b t dt T T = [ t p+1 bt ] R R + p + 1 t p 1 tbt dt t=t T R T p+1 bt + p + 1 t p 1 tbt dt <, T for every R T. Hence, t p tb L 1,. In the same way, we can treat the case bt is monotone-increasing on [T, forsufficiently large T. In the same way, we can prove that the integrability of t p bt on, implies t p tb t L 1,, under the assumption that b is monotone on, T forsufficiently large T, which completes the proof. In view of Theorem D and Remark 3, it is natural to assume that b satisfies the decay condition.4 with p 1, for solutions of perturbed Kirchhoff equation to be asymptotic free. On the other hand, let us consider the case H = L 2 R n, A = Δ with DA = H 2 R n. For every p [, 1, we see from Matsuyama [1] that there exists a small initial data φ, in Y p such that Kirchhoff equation.1.2 with b has a solution whose asymptotic behavior is different from that of solutions of the corresponding free equations. In view of this fact, it is natural to assume that initial data belong to Y p with p 1, for solutions of Kirchhoff equation to be asymptotic free. Hence assuming p 1, we show that the global solutions of perturbed Kirchhoff equation given by Theorem A has the same asymptotic behavior of that of the solutions of free equations. Theorem 1. Let J 1 and p 1. Assume that b satisfies.3 and.4. Then there exist positive constants ε 4 and K ± 2 1 such that the following holds for every ε,ε 4]. Suppose that φ,ψ Y J pε, and that ut be the solution of.1.2. Then the limit C ± = lim t ± m A1/2 ut 2, whose existence is guaranteed by Theorem B, satisfies decay estimates t p 1 m A 1/2 ut 2 C dt <, t p 1 m A 1/2 ut 2 C + dt <, and there uniquely exist global solutions v +,v C j R; DA J+1 i/2 of LE:C + LE:C j=,1,2 v +t + C+ Av 2 + =, t R, v t + CAv 2 =, t R,

10 Scattering for Perturbed Kirchhoff Equation 295 such that 1.13 lim A 1/2 ut A 1/2 v ± t + u t v ±t =. t ± Furthermore it holds that 1.14 lim t ± ut v± t J+1 + u t v ±t J =, 1.15 lim t ± t p 1 ut v ± t J + u t v ±t J 1 =. The operators 1.16 Υ ± :u, u = φ,ψ φ ±,ψ ± = v ±,v ± are continuous from Y J pε to Y J pk ± 2 ε for every ε,ε 4]. If p 2, then 1.17 t p 2 ut v ± t J + u t v ±t J 1 dt <, 1.18 t p 2 ut v ± t J + u t v ±t J 1 dt <. Remark 4. Assume that b is monotone for sufficiently large t. Then in view of Theorems D and 1 together with Remark 3, the assumption.3 and.4 with p = 1 is necessary and sufficient condition for solutions of perturbed Kirchhoff equation with small initial data in Y1 1 to be asymptotic free. Remark 5. The operator Υ ± is the inverse of the wave operator W ±,asisseeninthenext theorem. Conversely, we show the existence of the solution of perturbed Kirchhoff equation which converges to the solution of the free equation. For this purpose, it is necessary to assume the relation 1.9 in view of 1.8 as well as Theorem C. Here we note that 1/2 φ = A 1/2 φ in 1.8 and 1.9 if φ, ψ Y 1 p. Theorem 2. Let J 1 and p 1. Assume that b satisfies.3 and.4. Then there exist positive constants ε 5 and K ± 3 1 such that the following holds for every ε,ε 5]: Suppose that φ, ψ Ỹ J pε. Letc be a uniquely determined positive number satisfying the equality 1.9 in Theorem C, and let v be the solution of LE:c v t + c 2 Av = t R, v = φ, v = ψ. Then there uniquely exist global solutions u +, u 2 C i R; DA J+1 j/2 of.1 such that 1.19 lim A 1/2 u ± t A 1/2 vt + u ±t v t =, t ± j=

11 296 T. Yamazaki and that d 1.2 dt A1/2 u ± t 2 L 1 R n. Furthermore, the following holds: 1.21 lim t ± m A1/2 u ± t 2 = c, lim u± t vt J+1 + u ±t v 1.22 t J =, t ± lim t ± t p 1 u ± t vt J + u ±t v 1.23 t J 1 =, and the wave operators W ± :φ, ψ = v,v φ,±,ψ,± = u ±, u ± are continuous from Ỹ J pε to Ỹ J pk ± 3 ε. If p 2, then 1.24 t p 2 A 1/2 u ± t A 1/2 vt J 1 + u ±t v t J 1 dt <, 1.25 t p 2 A 1/2 u ± t A 1/2 vt J 1 + u ±t v t J 1 dt <. Assume furthermore that ε min{ε 4 /K 3 ±,ε 5/K 2 ± }. Then W ± is a homeomorphism from Ỹpε J to W ± Ỹpε J ỸpK J 3 ± ε Ỹ pε J 4 with respect to the metric dy J p. Let Υ ± be the mapping defined by ThenW ± Υ ± and Υ ± W ± are the identity mappings on Ỹpε. J The existence and the continuity of the scattering operator immediately follows from Theorems 1 and 2. Theorem 3. Let J 1/2 and p 1. Letε 4 and K 2 + be the constants given by Theorem 1. Let ε 5 and K3 be the constants given by Theorem 2. Then for every ε min{ε 4/K3,ε 5}, the scattering operator S = W+ 1 W is a homeomorphism from Ỹpε J to S Ỹpε J ỸpK J 2 + K 3 ε with respect to the metric dy J p. 2. Examples of Sobolev spaces included in the set Ỹ p Let Ω be the whole space R n or a non-trapping exterior domain with smooth boundary. In 2. this Examples section, we of give Sobolev some spaces examples included of Sobolev in the spaces set Ỹ p which are included in Ỹ p,inthe case H = L 2 R n, A = Δ with domain DA = {u H 2 Ω; ut, x = on Ω}. Here DA = H 2 R n ifω=r n. As is noted in Remark 1, these follows from examples included in Y k,p for k =, 1, which were shown in the previous papers [11, 12, 13]. Before describing sufficient conditions, we prepare some notations. Notation 8. Let 1 < p < and s. Let with the norm f W s p = 1 ξ s ˆ f L p.let W s pr n = { f R n 1 ξ s ˆ f L p R n }, W s pω = { f g W s p R n such that g Ω = f },

12 Scattering for Perturbed Kirchhoff Equation 297 { } with the norm f W s p Ω = inf g W s p R n g Ω = f.letw s p, Ω be the completion of C Ω with respect to the norm W s p Ω. When p = 2, W s 2 Rn andw s 2 Ω are denoted by Hs R n and H s Ω, respectively. Notation 9. For non-negative numbers s and p, H s,p = { f ; x p f H s R n }= { f ; D s x p f L 2 R n }. Notation 1. For non-negative numbers k and q, we put φ, ψ 2 Z k,q := sup r q A k/4 1/2 e 2irA1/2 φ, A k/4 1/2 φ H + A k/4 e 2irA1/2 ψ, A k/4 ψ H r R + 2 A k/4 e 2irA1/2 1/2 φ, k/4 ψ H. Here we note that φ, ψ Xk,p C φ, ψ Zk,q,if1 p + 1 < q. We first give examples of weighted Sololev spaces included in Y k,p. Although [11] and [12] consider about the sufficient conditions for belonging to Z 1,q, it follows from the proof that these sufficient conditions are also sufficient conditions for belonging to Z 1,q Z,q : Example 1 see [3, Lemma A] for integer p, and [11, Lemma 1.1] for general p. Let H = L 2 R n, A = Δ with domain DA = H 2 R n. Let p inthecasen = 1and p < n in the case n 2. Let q > p + 1, and assume furthermore that q n + 1inthe case n 2. Then the following inclusion holds. H 3/2,q R n H 1/2,q R n Z 1,q Z,q H H Y 1,p Y,p Ỹ 1. We give examples of non-weighted Sobolev spaces included in Y k,p : Example 2 see [12, Theorem 2]. Assume that n 4. Let 2n 1/n 3 < q <,and let q be the dual exponent of q, that is, 1/q+1/q = 1, and let p < n 11/2 1/q 1. Then H max{n1/2 1/q+1,3} R n W n+11/2 1/q+1 q R n H max{n1/2 1/q,2} R n W n+11/2 1/q q R n Y 1,p Y,p Ỹ p. Example 3 see [12, Theorem 4]. Assume that n 4 and Ω is non-trapping exterior domain with smooth boundary. Let 2n 1/n 3 < q <, and let q be the dual exponent of q, that is, 1/q + 1/q = 1, and let p < n 11/2 1/q 1. Let M be the smallest integer such that M n + 11/2 1/q. Then W 2M+1 q, Ω W 2M q, Ω Y 1,p Y,p Ỹ p. In the above examples, the sufficient condition for belonging to Y p,k follows from the condition for belonging to Z q,k. In the space dimension 3, we [13] showed that the following unweighted Sobolev spaces are included in Y = Y 1, directly, although we did not use the notation Y 1,. It follows from the proof that these sufficient conditions are also sufficient

13 298 T. Yamazaki conditions for belonging to Y Y, : Example 4 see [13, Theorem 3]. W 2 1 R 3 H 3 R 3 2 Y1, Y, Ỹ. Example 5 see [13, Theorem 5]. Let Ω be a non-trapping exterior domain in R 3 with smooth boundary. Let p, p, q and q be real numbers such that 1 < q < 3/2 < q 2, 3 < p. Then there exists a positive constant C such that W 8 p, Ω W9 q, Ω W9 q, Ω W2 1 Ω 2 Y1, Y, Ỹ. 3. Transformed equation Greenberg-Hu [4] introduced a transformation from a solution u of Kirchhoff equation 3. Transformed equation into a pair of unknown functions expressed by using Fourier decomposition. D Ancona- Spagnolo [1, 2, 3] improved their transformation in R n. [15] used a similar transformation by using spectral decomposition of the operator A instead of the Fourier transform. We first state some classes and metrics which are transformed from X p, Y p, d p and d Yp in Section 1. Notation 11. Let J, p be a non-negative number and let ε be a positive number. We define sets p and p as where p := {V, W DA 1/4 DA 1/4 ; V, W p < }, p := {V, W p ; V W RA 1/2 }, V, W 2 p := A 1/4 e 2irA1/2 V, A 1/4 V H p + A 1/4 e 2irA1/2 W, A 1/4 W H p + 2 A 1/4 e 2irA1/2 V, A 1/4 W H p, and subsets p ε of p and p ε of p as p ε := {V, W p ; V, W p + V + W ε}, p ε := p ε p. We put p J := p DA J/2 DA J/2, pε J := p ε DA J/2 DA J/2, p J := p DA J/2 DA J/2, pε J := p ε DA J/2 DA J/2, and V, W J p = V, W p + V J + W J, V, W J p = V, W J p + A 1/2 V W. For V 1, W 1, V 2, W 2 p,wedefine

14 Scattering for Perturbed Kirchhoff Equation 299 d p V 1, W 1, V 2, W 2 2 := A 1/4 e 2irA1/2 V 1, A 1/4 V 1 H A 1/4 e 2irA1/2 V 2, A 1/4 V 2 H p + A 1/4 e 2irA1/2 W 1, A 1/4 W 1 H A 1/4 e 2irA1/2 W 2, A 1/4 W 2 H p + 2 A 1/4 e 2irA1/2 V 1, A 1/4 W 1 H A 1/4 e 2irA1/2 V 2, A 1/4 W 2 H p, and put d pv J 1, W 1, V 2, W 2 := d p V 1, W 1, V 2, W 2 + V 1 V 2 J + W 1 W 2 J, d Y J p V 1, W 1, V 2, W 2 := d pv J 1, W 1, V 2, W 2 + A 1/2 V 1 W 1 A 1/2 V 2 W 2. We see that d J p and d J Y p become metrics on J p and J p, respectively. We equip J p and a subset J pεof J p with the metric d J p,and J p and a subset J pεof J p with the metric d J Y p Notation 12. Let be the operator defined by G :, DA 3/4 DA 1/4 DA 1/4 DA 1/4 G λ, φ, ψ := λ 1/2 ψ iλa 1/2 φ,λ 1/2 ψ + iλa 1/2 φ. Notation 13. Let :, {V, W DA 1/4 DA 1/4 ; V W RA 1/2 } DA 3/4 DA 1/4 be the operator defined by λ, V, W := i 2 λ 1/2 A 1/2 V W, 1 2 λ1/2 V + W. Lemma A Lemma 2 of [15]. i For every positive numbers λ, p,ε, the following holds. G λ,, Xpε J p2ε J, λ,, pε J Xpε J, where ε = max{λ, 1/λ}ε ii d pg J λ 1, φ 1,ψ 1, G λ 2, φ 2,ψ 2 2max j=1,2 max{λ j, 1/λ j } + φ j,ψ j Jp λ 1 λ 2 + d J pφ 1,ψ 1, φ 2,ψ 2 for every and λ j, φ j,ψ j, X J p j = 1, 2, and

15 3 T. Yamazaki dp J λ 1, V 1, W 1, λ 2, V 2, W 2 max max{λ j, 1/λ j } + V J, W j Jp j=1,2 for every λ j, V j, W j, J p j = 1, 2. λ 1 λ 2 + d J pv 1, W 1, V 2, W 2 Lemma 1. i For every positive numbers λ, p,ε, the following holds. G λ,, Ypε J p2ε J, λ,, pε J Ypε J, where ε = max{λ, 1/λ}ε. ii d Y J p G λ 1, φ 1,ψ 1, G λ 2, φ 2,ψ 2 2max j=1,2 max{λ j, 1/λ j } + φ j,ψ j Jp λ 1 λ 2 + d J Y p φ 1,ψ 1, φ 2,ψ 2 for every and λ j, φ j,ψ j, Y J p j = 1, 2, and dy J p λ 1, V 1, W 1, λ 2, V 2, W 2 max max{λ j, 1/λ j } + V j, W j Jp j=1,2 for every λ j, V j, W j, J p j = 1, 2. λ 1 λ 2 + d J Y p V 1, W 1, V 2, W 2 Proof. Let V, W = G λ, φ, ψ. Then by definition, we have A 1/2 V W = 2λ 1/2 φ, with which Lemma A implies the conclusion. Lemma B Lemma 3 of [15]. Let p, ε>, J 1/2 and a, b [m,. i If V, W pε, J then e iaa1/2 V, e iaa1/2 W p2 J p/4 a p/2 ε. ii The mapping is continuous from R J p to J p. Φ :a, V, W e iaa1/2 V, e iaa1/2 W Lemma 2. Let p, ε>, J 1/2 and a, b [m,. i If V, W pε, J then e iaa1/2 V, e iaa1/2 W p2 J p/4 a p/2 ε. ii The mapping is continuous from R J p to J p. Φ :a, V, W e iaa1/2 V, e iaa1/2 W

16 Scattering for Perturbed Kirchhoff Equation 31 Proof. i Since 3.1 e iaa1/2 V e iaa1/2 W = e iaa1/2 V W + 2iA 1/2 A 1/2 sinaa 1/2 W RA 1/2, sinaλ 1/2 de λ 1/2 λ H is meaningful by the spectral decomposi- where A 1/2 sinaa 1/2 = tion of A with 3.2 A 1/2 sinaa 1/2 H = a sinaλ 1/2 aλ 1/2 de λ a. H Then Lemma B i and 3.1 imply the conclusion. ii Let a 1, V 1, W 1 be an arbitrary point of R p, J and put U 1 = A 1/2 V 1 W 1. By Lemma B, we only have to prove the continuity of the mapping a, V, W A 1/2 e iaa1/2 V e iaa1/2 W from p to H at the point a 1, V 1, W 1. Let a, V, W R p and put U = A 1/2 V W, U 1 = A 1/2 V 1 W 1. Since d/dse ±isa1/2 f = ±ia 1/2 e ±isa1/2 f for f = V 1, W 1,wehave e iaa1/2 V e iaa1/2 W e ia 1A 1/2 V 1 e ia 1A 1/2 W 1 = e iaa1/2 V V 1 W W 1 + e iaa1/2 e iaa1/2 W W 1 + e iaa1/2 e ia 1A 1/2 V 1 e iaa1/2 e ia 1A 1/2 W 1 = e ia 1A 1/2 A 1/2 U U 1 + 2i sinaa 1/2 W W 1 a + ia 1/2 e isa1/2 V 1 + e isa1/2 W 1 ds. a 1 Hence by using 3.2 and e iaa1/2 H 1, we obtain A 1/2 e iaa1/2 V e iaa1/2 W e ia 1A 1/2 V 1 e ia 1A 1/2 W 1 U U a W W a a 1 V 1 + W 1, which implies the mapping a, V, W A 1/2 e iaa1/2 V e iaa1/2 W is continuous from p to H. Now we transform the equation.1. Assume that ut is a solution of.1. Put 3.3 ct := m A 1/2 ut 2, τt := t csds, for t R. Then, τt is a strictly increasing function on R. Lettτ be the inverse of τt, and put for every τ R, where Cτ := ctτ, Vτ := Cτ 1/2 e iτa1/2 +Btτ/2 u tτ icτa 1/2 utτ, Wτ := Cτ 1/2 e iτa1/2 +Btτ/2 u tτ + icτa 1/2 utτ,

17 32 T. Yamazaki Bt := t bsds for t R. By this transformation, we easily see that the equation.1.2 is transformed into V τ = qτe 2iτA1/2 Wτ, W τ = qτe 2iτA1/2 Vτ, e Btτ Cτ = m e iτa1/2 Vτ e iτa1/2 Wτ 2, 4Cτ V, W = G m A 1/2 φ 2,φ,ψ, where qτ := C τ + btτ 2Cτ The Hamiltonian Hu, t is defined as where Then we have = d dτ. Hu, t := M A 1/2 ut 2 + u t 2, Mρ = ρ Hu, t = Hu, T 2 and thus, we have by Gronwall s inequality that ms 2 ds ρ. t T bs u s 2 ds, 3.6 Hu, t exp2 b L 1Hu, T for every t R. Since Mρ m 2 ρ, inequality 3.6 implies 3.7 A 1/2 ut 2 exp2 b L 1Hu, T/m 2 for every t R. Put Then 3.7 implies if A 1/2 ut + ut 1. Observing the fact above, we put Notation 14. Let L := exp2 b L 1M1 + 1/m 2. A 1/2 ut 2 L for every t R, m 1 = sup{mx; x L}, m 2 = sup{ m x ; x L} m 3 := sup{ m x ; x L}. p = p R := {C BC 1 R; C p < }. Here, BC 1 R is the set of all bounded C 1 functions with bounded derivatives on R. The space p R becomes a Banach space with the norm

18 For a positive number δ,let Scattering for Perturbed Kirchhoff Equation 33 C p = C p R := C L R + C L R + C p. p,δ R := {C p R; m Cτ m 1 for every τ R, C p δ}, which is a closed subset of p R. Let T [, ]. Taking the aforementioned fact into account, we consider the following equation: V τ = qτe 2iτA1/2 Wτ, W τ = qτe 2iτA1/2 Vτ, e Btτ Cτ = m e iτa1/2 Vτ e iτa1/2 Wτ 2, 4Cτ VT = Ṽ, WT = W, where τ tτ = dσ for τ R, Cσ qτ := C τ + btτ = d t Cτ dτ, Bt := Here, V± = Ṽ and W± = W mean that bsds. lim Vt = Ṽ and lim Wt = W in DA J/2. t ± t ± Lemma C Lemma 4 of [15]. Let C, C j p R j = 1, 2. Then the functions t defined by 3.11 and t j defined by 3.11 with C = C j for j = 1, 2 satisfy the following τ tτ τ for τ R, m 1 m t 1 t 2 τ τ C m 2 1 C 2 L for every τ R. Lemma D Lemma 5 of [15]. Let T [, ] and fix it. Let J and p. Let C j p, and let t j and q j be the functions defined by 3.11 and 3.12 with C = C j for j = 1, 2. t p q j L 1 R and q j p 1 C j 2m p + m 1 p b p m for j = 1, 2 and q 1 q 2 p 1 2m + m 1 m 1 p 2m 2 m 1 tb p m + b p + C 2 p C 2m 2 1 C 2 p. Proposition A Propositions 3 and 5, Proof of Proposition 5 and Lemma 3 of [15]. Let T [, ]. Let J 1 and p. Then there exist positive constants ε 6 and K 5 such that for every Ṽ, W J pε with <ε ε 6, a solution C, V, W C 1 R j=,1 C j R; DA J j/2 DA J j/2 of exists and satisfies

19 34 T. Yamazaki C p,k5 εr, V, W CR; pk J 5 ε, Vτ J + Wτ J exp q L 1 R Ṽ J + W J. Furthermore, the limits 3.16 V ± = lim ± = lim τ ± τ ± DAJ/2, 3.17 C ± = lim Cτ R, τ ± exist and satisfy the following: 3.18 lim τ ± τ p Vτ V ± J = lim τ ± τ p Wτ W ± J =, 3.19 lim τ ± τ p Cτ C ± =. Assume furthermore that 3.2 lim e2ira1/2ṽ, Ṽ H = lim e2ira1/2 W, W H r ± r ± = lim e2ira1/2ṽ, W H =, r ± which condition we denote as Ṽ, W. Then e B ± 3.21 C ± = m V ± 2 + W ± 2, 4C ± where B ± = ± bsds, and V ±, W ±. Proposition B Proposition 4 of [15]. Let J 1, p. Let ε 6 be a constant given by Proposition A. There exist positive constants ε 7 ε 6 and K 6 such that the following assertion holds. Let Ṽ j, W j p,ε J 7, and assume that C j, V j, W j C 1 R j=,1 C j R; DA J j/2 DA J j/2 are solutions of satisfying 3.33 and Ṽ, W = Ṽ j, W j for j = 1, 2. Assume moreover that t p C L 1 R if T = ±. Then C 1 C 2 p K 6 d p Ṽ 1, W 1, Ṽ 2, W 2, d pv J 1 τ, W 1 τ, V 2 τ, W 2 τ K 6 d pṽ J 1, W 1, Ṽ 2, W 2 for every τ [, ]. If initial data belongs to p for p 1, then the following propositions hold. Proposition 1. Assume that A is injective. Let T [, ]. LetJ 1 and p 1. Letε 6 and K 5 be the constants in Proposition A. Let Ṽ, W J pε with <ε ε 6. Then there exists a solution C, V, W BC 1 R C 1 R; DA J/2 DA J/2 of such that C p,k5 ε, Vτ, Wτ pk J 5 ε for every τ R, V, W CR; p. J The uniqueness holds in the class C, V, W BC 1 R C 1 R; DA J/2 DA J/2 if T ±,

20 Scattering for Perturbed Kirchhoff Equation 35 C, V, W BC 1 R C 1 R; DA J/2 DA J/2, C L 1 R if T = ±. Furthermore the following hold. Put Then U C 1 R; DA J+1/2, and the limit Uτ := A 1/2 Vτ Wτ for τ R. U ± = lim Uτ in τ ± DAJ+1/2 exists. The limits C ±,V ± and W ± shown by Proposition A and U ± satisfy the following: lim τ ± τ p 1 Uτ U ± J+1 =, ± τ p 1 Vτ V ± J dτ ± + τ p 1 Wτ W ± J dτ <, ± τ p 2 Uτ U ± J+1 dτ < if p 2, V ± W ± = A 1/2 U ±. V ±, W ± pk J 5 ε, τ p 1 Cτ C + dτ + τ p 1 Cτ C dτ <. Proposition 2. Let J 1, p 1 and ε>. Let ε 7 be a positive constant given by Propositions B. There exist a positive constant K 7 such that the following assertion holds. Let Ṽ j, W j J pε 7, and assume that C j, V j, W j C 1 R j=,1 C j R; DA J j/2 DA J j/2 are solutions of with Ṽ, W = Ṽ j, W j for j = 1, 2, satisfying 3.33 m Cτ m 1 for every τ R. Assume moreover that C p if T = ±. Then 3.34 d J Y p V 1 τ, W 1 τ, V 2 τ, W 2 τ K 7 d J Y p Ṽ 1, W 1, Ṽ 2, W 2 for every τ [, ]. 4. Proof of Propositions Since we sometimes use the following calculations, we describe them as a lemma. 4. Proof of Propositions Lemma 3. Let Z be a Banach space with norm Z. Let p be an nonnegative number. If f C 1 R; Z satisfies t p f t Z L 1 R, then the limit lim t ± f t = α ± exists in Z and satisfies the following. 4.1 lim t ± t p f t α ± Z =. Furthermore, if p 1 and t p 1 t f t Z L 1 R, then 4.2 t σ p 1 f σ α + Z dσ t σ p 1 σ f σ Z dσ<

21 36 T. Yamazaki for every t, and for every t. t σ p 1 f σ α Z dσ t σ p 1 σ f σ Z dσ< Proof. We only prove lemma in the case t. The case t is calculated in the same way. Let S be an arbitrary nonnegative number of R. Then S S τ p f S f τ Z τ p f σ Z dσ σ p f σ Z dσ τ for every τ S <. Hence, there exists a limit lim t + f t inz satisfying 4.1. For every t, we have by Fubini s Thoerem τ p 1 f τ α + Z dτ τ p 1 f σ Z dσdτ = t σ t t τ p 1 f σ Z dτdσ t t τ τ σ p 1 σ f σ Z dσ<, which implies 4.2. Proof of Proposition 1. Step 1 By Propositions A and.4, we see that t p q L 1 R and sup τ R Vτ J + Wτ J <. Hence, using the fact that V, W satisfies 3.8, we have τ p V τ J + W τ J L 1 R. Thus by Lemma 3 with Z = DA J/2, we obtain Step 2 We show that there is Uτ DA 1/2 such that 4.3 Vτ Wτ = A 1/2 Uτ RA 1/2 for every τ R, which satisfies U C 1 R; H. Kajitani [9, Proposition 2.4] proved this fact for a second order differential operator A which defines non-negative definite self-adjoint operator on L 2 R n, by using the differential equation of Vτ Wτ together with the expression e iτa1/2 = cosτa 1/2 + i sinτa 1/2 and the boundedness of τa 1/2 1 sinτa 1/2. By using this idea, we prove 4.3. From 3.8, it follows that 4.4 for every τ R. Hence, d Vτ Wτ = qτe2iτa1/2 Vτ Wτ 2iqτ sin2τa 1/2 Vτ dτ d dτ exp = 2i exp τ T τ qσe 2iσA1/2 dσ T qσe 2iσA1/2 dσ Vτ Wτ qτ sin2τa 1/2 Vτ for every τ R. Here, exp τ qσe2iσa1/2 dσ is meaningful by spectral decomposition. T Thus,

22 4.5 τ Vτ Wτ = exp Scattering for Perturbed Kirchhoff Equation 37 2i T τ T exp qσe 2iσA1/2 dσ τ s VT WT qσe 2iσA1/2 dσ qs sin2sa 1/2 Vsds. By the assumption that VT, WT = Ṽ, W p, there is Ũ DA 1/2 such that VT WT = A 1/2 Ũ. By 3.2, the assumption qss L 1 R and the closeness of the operator A 1/2, we see that the second term of the right-hand side of 4.5 belongs to RA 1/2. Hence, Vτ Wτisexpressedas Vτ Wτ = A 1/2 Uτ, where 4.6 τ Uτ := exp 4i T τ T DA 1/2 qσe 2iσA1/2 dσ τ exp s Ũ qσe 2iσA1/2 dσ qss 2sA 1/2 1 sin2sa 1/2 Vsds for every τ R,andU L R; H. We show that U C 1 R; H. We easily see that the first term of 4.6 belongs to C 1 R; H. The inequality A 1/2 sin2s + ha 1/2 f A 1/2 sin2sa 1/2 H = 2 cos2s + ha 1/2 sinha 1/2 A 1/2 H 2 h for h R implies that A 1/2 sin2sa 1/2 is a norm continuous operator on H with respect to s. Using this fact together with the facts that t q L 1 R andv CR; H, we see that the second term of the right-hand side of 4.6 also belongs to C 1 R; H, and thus we conclude that U C 1 R; H. This fact together with 3.14 and 3.15 implies 3.25 and Step 3 Last we consider the asymptotic behavior. Equation 4.4 yields d Uτ = qτe2iτa1/2 Uτ 4iqττ2τA 1/2 1 sin2τa 1/2 Vτ dτ for every τ R. Then, by the uniform boundedness of Vτ anduτ, the fact that τ τ p 1 qτ L 1 R and the boundedness of the operator 2τA 1/2 1 sin2τa 1/2 onh uniformly to τ, we can use Lemma 3 with Z = H to obtain the existence of U ± = lim τ ± Uτ in H satisfying 3.27 and 3.29 with J =. This fact, 3.18 and 3.28 imply 3.27 and 3.29 for general J. By the closeness of the operator A 1/2 in H, we see that U ± DA 1/2 and satisfies 3.3. Since 3.14 and 3.16 hold, Fatou s lemma implies V, W pk J 5 ε, which together with 3.3 yields Lemma 3 together with 3.24 implies Proof of Proposition 2. By using the expression 4.6, we easily see that

23 38 T. Yamazaki U 1 τ U 2 τ J e q 1 L 1 R Ũ1 Ũ J 2 + e max{ q 1 L 1 R, q 2 L 1 } R q 1 q 2 L 1 R Ũ J 2 + 4e max{ q 1 L 1 R, q 2 L 1 } R q 1 q 2 L R sq 1 1 L 1 R sup V 1 s J s R + 4e q 2 L 1 R sq1 q 2 L 1 R sup V 1 s J s R + 4e q 2 L 1 R sq2 L 1 R sup V 1 s V 2 s J s R for every τ R, where U j = A 1/2 V j W j andũ j = A 1/2 Ṽ j W j. This inequality, 3.23 and 3.25 yield U 1 τ U 2 τ J e max{ q 1 L 1 R, q 2 L 1 R } Ũ1 Ũ J 2 + q 1 q 2 L 1 R Ũ J 2 + 4K 5 ε 7 q1 q 2 L R sq 1 1 L 1 R + sq 1 q 2 L 1 R + 4K 6 sq 2 d L 1 R pṽ J 1, W 1, Ṽ 2, W 2, which together with Lemma D, 3.22, 3.23 and 3.24 yields Proof of Theorem 1 Proof of Theorem 1. Let ε 1 and ε 7 be positive constants given by Theorem A and 5. Proof of Theorem 1 Proposition B, respectively. Let K denote various constants independent of t. Assume that ε 4 min{1,ε 1,ε 7 /2K 1 }, where K 1 = max{, m 1 } 1. m Step 1 We give a solution u of.1. Let φ,ψ Y p ε forε,ε 4 ]. By Lemma 1, we have 5.2 Ṽ, W := G m A 1/2 φ 2,φ,ψ J p2k 1 ε J pε 7. Then by Propositions A and 1, the equation with Ṽ, W defined by 5.2 has a solution 5.3 C, V, W p,1 C 1 R; DA J/2 DA J/2, satisfying Furtheremore, the limits V, W CR, p 2K 5 K 1 ε U := A 1/2 V W C 1 R, DA J+1/ C ± = lim Cτ R, τ ± V± = lim Vτ and W ± = lim Wτ in τ ± τ ± DAJ/2, U ± = lim Uτ τ ± indaj+1/2, exist and satisfy 3.18, 3.19, ,

24 Scattering for Perturbed Kirchhoff Equation V ±, W ± J p2k 5 K 1 ε, and Let tτ be the function defined by By the fact that C p,1, it holds that dt m 1 dτ τ = 1 Cτ 1. m Then, tτ is a strictly increasing function on R. Letτt be the inverse of tτ, and define 5.7 ct = Cτt. Here we note that 5.8 τt = t csds, for t R, and the assumption that C p, 3.13 and 5.6 yield 5.9 c p. Since we can define ut by 5.1 e iτta1/2 Vτt e iτta1/2 Wτt = e iτta1/2 Vτt Wτt + e iτta1/2 e iτta1/2 Wτt = e iτta1/2 A 1/2 Uτt 2i sinτta 1/2 Wτt RA 1/2, ut := i 2 Cτt e Bt/2 A 1/2 e iτta1/2 Vτt e iτta1/2 Wτt Then we have i 5.11 ut = 2 Cτt e Bt/2 e iτta1/2 Uτt 2iA 1/2 sinτta 1/2 Wτt. Since C, V, W is the solution of with T =, we see that Cτt u t = e Bt/2 e iτta1/2 Vτt + e iτta1/2 Wτt, 2 ct = Cτt = m A 1/2 ut, and ut j=,1,2 C 2 j R : DA 2 j/2 is a solution of.1. By the uniqueness of the solution which is stated in Theorem A, the solution given by Theorem A equals u given by the formula Step 2 We express the solution v of LE : C ±. The formulas 5.3, 5.4 and 5.6 together with Lemma 3 with Z = R imply 1.11 and 1.12, and we can define 5.12 a ± := ± ct C ± dt = ± Cτ C ± 1 dτ R, Cτ

25 31 T. Yamazaki with a ± 1 ± 5.13 Cτ C ± dτ m 1 ± σc σ dσ m 1 C 1 1. m m Here we used the assumption that C 1,1 R in the last inequality above. Using Lemma 3 with Z = R again, we have t 5.14 τt C t + a = t cs C ds sc s ds, for every t <. Thus, t p 1 τt C t + a for every t <, which together with 5.9 implies t σ p c σ dσ 5.15 lim t t p 1 τt C t + a =. In the case p 2, it follows from 5.14, 5.16 = t p 2 τt C t + a dt Hence by 5.9, we have 5.17 s t p 2 sc s dtds 1 p 1 By 3.3 of Proposition 1, we have t t p 2 sc s dsdt s p 1 sc s ds. t p 2 τt C t + a dt < e ic ± t+a ± A 1/2 V ± e ic ± t+a ± A 1/2 W ± = e ic ± t+a ± A 1/2 A 1/2 U ± 2iA 1/2 A 1/2 sinc ± t + a ± A 1/2 W ± RA 1/2. with U ± DA 1/2. Thus, we can define function v ± t by 5.19 Then we have 5.2 v ± t = v ± t := 1 2C 1/2 ± i 2C 1/2 ± e B ± /2 A 1/2 e ic ± t+a ± A 1/2 V ± e ic ± t+a ± A 1/2 W ± e B ± /2 ie ic ± t+a ± A 1/2 U ± 2iA 1/2 sinc ± t + a ± A 1/2 W ±, where B ± = ± bsds. Then, we easily see that

26 Scattering for Perturbed Kirchhoff Equation 311 v ± t C j R; DA J+1 j/2, j=,1,2 and satisfies LE:C ±. Step 3 We prove formulas 1.14 and 1.15 for t and Formula 1.13 follows from The formulas 1.14 and 1.15 for t + and 1.18 are proved in the same way. By definitions 5.1 and 5.19, A 1/2 ut v t = i 2 ct e Bt/2 e iτta1/2 Vτt V + e iτta1/2 e ic t+a A 1/2 V e iτta1/2 Wτt W e iτta1/2 e ic t+a A 1/2 W + 1 ie Bt/2 ie B /2 e ic t+a A 1/2 V e ic t+a A 1/2 W. 2 ct C Let k be a nonnegative integer such that k J. Sincect m, we have by the intermediate theorem that e Bt/2 e B / ct C e Bt/2 1 ct 1 C + 1 C e Bt/2 e B /2 e Bt/2 ct C e B t/2 t = C ct C + ct + bsds t, t C e b L 1 /2 t bs ds + 1 ct C m 1/2. m Thus, we have 5.22 A 1/2 ut v t k e b L 1 /2 [ Vτt V 2m 1/2 k + Wτt W k + e iτta1/2 e ic t+a A 1/2 k V + e iτta1/2 e ic t+a A 1/2 k W t + bs ds + 1 ] Cτt C V m k + W k. We shall estimate the term e iτta1/2 e ic t+a A 1/2 k V in the inequality above. Let k [, J 1] and let f be an arbitrary element of DA k+1/2. Using the equality d/dse isa1/2 f = ie isa1/2 A 1/2 f and that the operator e iaa1/2 is unitary on DA k/2, we have 5.23 e iτta1/2 e ic t+a A 1/2 f = k τt C t+a e isa1/2 A 1/2 fds k τt C t + a A 1/2 f k.

27 312 T. Yamazaki Hence 5.15 implies 5.24 lim t t p 1 e iτta1/2 e ic t+a A 1/2 f =. k Since operators e iτta1/2 and e ic t+a A 1/2 are bounded in DA k/2, and since DA k+1/2 is dense in DA k/2, it follows that 5.24 holds also for every f DA k/2 in the case p = 1. Thus, by the fact that V DA J/2, we see that lim e iτta1/2 e ic t+a A 1/2 J V =, t lim t p 1 e iτta1/2 e ic t+a A 1/2 J 1 V =. t In the same way, we have lim e iτta1/2 e ic t+a A 1/2 J W =, t lim t t p 1 e iτta1/2 e ic t+a A 1/2 J 1 W =. By using these convergences, 3.18, 3.19 together with 3.13 and 1.1, we derive the following estimates from 5.22 lim t A1/2 ut v t J =, lim t t p 1 A 1/2 ut v t J 1 =. In the same way, we can prove that lim t u t v t J = lim t t p 1 u t v t J 1 =. Thus, the proof of 1.14 and 1.15 for t is complete if we show that 5.25 lim t t p 1 ut v t =. From equalities 5.11 and 5.2, it follows that ut v t = i 2 Cτt e Bt/2 [ e iτta1/2 Uτt U + e iτta1/2 e ic t+a A 1/2 U 2i sinτta 1/2 A 1/2 Wτt W + 4iA 1/2 sin τt C t + a A 1/2 /2 cos τt + C t + a A 1/2 /2 ] W ie Bt/2 + 2 ct ie B/2 2 C e ic t+a A 1/2 U 2i sinc t + a ± A 1/2 A 1/2 W. Hence, by 5.21, 5.23, 3.2 and cossa 1/2 H 1, we obtain

28 ut v t e b L 1 /2 2m 1/2 Scattering for Perturbed Kirchhoff Equation 313 [ Uτt U + τt C t + a A 1/2 U + 2 τt Wτt W + 2 τt C t + a W t + bs ds + 1 U ct C + 2 C t + a ± W ]. m Thus, by using 3.13, we have 5.26 t k ut v t [ K τt k Uτt U + τt k+1 Wτt W ± + t k τt C t + a A 1/2 U + W t U + t k+1 bs ds + τt k+1 Cτt C + W ]. By 1.1, 3.18, 3.19, 3.27 and 5.15, we see that the right-hand side of 5.26 with k = p 1tendstoast. Thus, 5.25 holds, which completes the proof of 1.14 and Assume that p 2. By Fubini s Theorem together with 1.1, we have t 5.27 t p 1 bs dsdt = t p 1 bs dtds Inequalities 5.23 and 5.17 imply In the same way, we have s s s p 1 bs ds <. t p 2 e iτta1/2 e ic t+a A 1/2 V J 1 dt <. t p 2 e iτta1/2 e ic t+a A 1/2 W J 1 dt <. By these inequalities, 5.27, 3.28 and 3.32 together with 5.6, we derive the following from 5.22; 5.28 In the same way, we can prove 5.29 t p 2 A 1/2 ut v t J 1 dt <. t p 2 u t v t J 1 dt <. By using 3.28, 3.29, 3.32, 5.17 and 5.27, we derive the following from 5.26; t p 2 ut v t dt <,

29 314 T. Yamazaki which together with 5.28 and 5.29 implies Uniqueness of the solution v ± is clear. Step 4 We prove the continuity of the mapping Υ ±. Formula 5.19 implies Υ ± φ,ψ :=φ ±,ψ ± = v ±, dv ± dt = exp B ± /2 C±, e ia ± A 1/2 V ±, e ia ± A 1/2 W ±. By Lemma 2 i together with 5.5 and 5.13, we have e ia ± A 1/2 V ±, e ia ± A 1/2 W ± J p K 11 ε with K 11 := 2 p/4+1 1 m p/2 K 5 K 1. Hence by i of Lemma 1, we have φ ±,ψ ± Y J pk ± 2 ε with K± 2 = exp B ± /2K 1 K 11, and therefore we obtain Υ ± Y J pε Y J pk ± 2 ε. We show the continuity of the mapping Υ ±. By the definition of Ṽ, W in 5.2, Lemma 1 ii together with the continuity of m, we easily see that the mapping φ,ψ Ṽ, W is continuous from Y J pεto J p2k 1 ε J pε 7. By Propositions 1 and 2, Ṽ, W C, V ±, W ± is continuous from J pε 7 to p,k5 ε 7 J pk 5 ε 7. Thus, if we prove the continuity of the mapping Φ : C a ± from p,k5 ε 7 to R, Lemma 2 ii guarantees the continuity of the mapping C, V ±, W ± C ±, e ia ± A 1/2 V ±, e ia ± A 1/2 W ± from p,k5 ε 7 J pk 5 ε 7 to[m, m 1 ] J pk 11 ε 7. Then Lemma 1 ii guarantees the continuity of the mapping C±, e ia ± A 1/2 V ±, e ia ± A 1/2 W ± φ±,ψ ± from [m, m 1 ] pk J 11 ε 7 to pk J 2 ± ε 7, and therefore the continuity of Υ ± follows. Now we prove the continuity of the mapping Φ. LetC j p,k5 ε 7, and let C j, and a j be the numbers defined by 5.4 and 5.12 with C = C j j = 1, 2. Then a j,± is expressed as 5.3 a j,± = = = = ± ± 1 C j,± C j τ ± σ ± dτ ± 1 C j,± τ C j σ C j,± C j σ dτdσ C j σ 2 σc j,± C j σ C j σ 2 dσ, dσ dτ

30 Scattering for Perturbed Kirchhoff Equation 315 where we used Fubini s Theorem in the third step. Thus, we have ± C 1 a 1,± a 2,± = σ C σ 1,± C 1 σ C C 2 σ 2 2,± dσ. C 2 σ 2 Since C 1 C σ 1, C 1 σ C C 2 σ 2 2, C 2 σ 2 C 1 σ C 1 σ C 2 1, C 2, + C 2, C 1 σ 2 C 1 σ C 2 σ + C 2, C 2 σ C 1 σ 2 C 2 σ C 1σ + C 2 2 σ C 1 σ C 2 σ, and C j 1,1,wehave 5.31 a 1, a 2, C 1 1 C m 2 1 C 2 L + m 1 C 1 C m2 1 C 1 C 2 L C 2 1 m m 1 + 2m2 1 C m 2 m 4 1 C 2 p. In the same way, we see that a 1,+ a 2,+ is dominated by the right-hand side of Hence we have proved the continuity of Φ, and thus we conclude that the continuity of the mapping Υ ± holds. 6. Proof of Theorems 2 and 3 Proof of Theorem 2. Let K 5, ε 7 and K 7 be positive constants given by Propositions A, B 6. Proof of Theorems 2 and 3 and 2, respectively, and K 1 be the number defined by 5.1. Put p/2 6.1 K 12 ± = 2eB ± /2 K 1, K 13 = 2 p/4 m1, K 3 ± = K 5K 1 K 12 ± K 13, where B ± = ± bsds.letε 5 be a positive constant satisfying 6.2 K 1 ε 5 < min{ m Lm, } and max{k 12 +, K 12 }K 13ε 5 ε 7. m2 Assume that ε ε 5,andletφ, ψ Ỹ J pε. We prove the statements of Theorem 2 in the case. The case + isprovedinthesameway. Step 1 In Proof Step 1 of [15, Theorem 3] see also Proof of [11, Theorem 1.3], we proved the existence of c satisfying 1.9 and the Lipschitz continuity of the mapping φ, ψ c from DA 1/4 H R. Especially c is determined uniquely, and satisfies m m c m 1. Step 2 We express the solution of LE : c. Put 6.4 V, W := G c, φ, ψ, U := 2ic 1/2 φ = A 1/2 V W. Then by 6.3 and Lemma 1, we have 6.5 V, W J p2k 1 ε. m 4

31 316 T. Yamazaki Since V W = A 1/2 U RA 1/2, we can define 6.6 vt := i 2c 1/2 A 1/2 e ic ta 1/2 V e ic ta 1/2 W, e ic ta 1/2 U 2iA 1/2 sinc ta 1/2 W. = i 2c 1/2 Then we easily see that vt is the unique solution of LE : c with initial value v,v = φ, ψ. Step 3 We show the existence of a solution u, u / t of.1 satisfying 1.22 and Convergence 1.19 follows from In view of 6.1 and 6.2 and 6.5, Propositions A and 1 with T = imply that there uniquely exists a solution of 3.8, 3.9 and Cτ, Vτ, Wτ p,1 C 1 R; DA J/2 DA J/2 CR; J p 6.7 V, W = lim Vτ, Wτ = τ eb /2 V, W pk J 12 ε pε J 7, where p J = p J and the limit is taken in DA J/2 DA J/2, and the solution satisfies the following: 6.8 Vτ, Wτ pk J 5 K12 ε for every τ [, ], and the limit C = lim τ Cτ m exists and satisfies Then in the same way as in the proof of [15, Theorem 3], we can prove 6.9 c = C = lim Cτ, τ by observing that both c and C are unique solution of V 2 + W 2 λ = m. 4λ Let tτ andct be the functions defined by 3.11 and 5.7. In the same way as in the definition of a ± see 5.12 and 5.3, together with using 6.9, we can define T = T C := cs c ds c 1 = Cτ 1 τc τ dτ = c Cτ dτ. 2 From the last equality, 5.8 and the fact that C p,1, it follows that 6.11 T 1 m 2 τ T = Equalities 5.8 and 6.1 yield τc τ dτ 1 C m 2 1 1, m 2 csds T m 1 T m 1. m 2

32 Scattering for Perturbed Kirchhoff Equation 317 τt T c t = τt T c t T c T = t T Thus in the same way as in the proof of 5.15 and 5.17, we obtain lim t t p 1 τt T c t =, Let u be a function defined by 5.1, that is, 6.14 u t := = t p 2 τt T c t dt < if p 2. cs c ds. i 2 ct e Bt/2 A 1/2 e iτta1/2 Vτt e iτta1/2 Wτt i 2 Cτt e Bt/2 e iτta1/2 Uτt 2iA 1/2 sinτta 1/2 Wτt, which is a solution of.1 satisfying u 1 j= C i R; DA J+1 j/2.put 6.15 u t = u t T. Then it is easy to see that u is a solution of.1. Since Cτt = m A 1/2 u t 2, the equality 6.9 implies lim t m A1/2 ut 2 = lim t m A1/2 u t 2 = c, that is 1.21 for t holds. The fact C p implies 1.2. By definition 6.15 and expressions 6.6 and 6.14, we have u t vt = i 2 Cτt T e Bt T /2 [ e iτt T A 1/2 Uτt T e B /2 U + + e iτt T A 1/2 e ic ta 1/2 e B /2 U 2iA 1/2 sinτt T A 1/2 Wτt e B /2 W + 4iA 1/2 sin τt T C ta 1/2 /2 cos τt T + C ta 1/2 /2 ] e B /2 W ie Bt T /2 2 ct T ie B/2 2 C e B /2 e ic ta 1/2 U 2iA 1/2 sinc ta 1/2 W, Then using 6.12 and 6.13, we can prove for t, in the same way as in the proof of Theorem 1. Step 4 We prove the uniqueness of the solution satisfying 1.19 and 1.2 in the case -. Let u be the solution constructed above and put u 1 = u.letct, τt, Cτ, Vτand Wτ be the functions defined in Step 3. Let u 2 C 1 R; 1/2 be an arbitrary solution of.1 satisfying

33 318 T. Yamazaki 1 1/2 u 2, u 2 C j R; DA 1 j/2 DA 1 j/2, j= 1.19 for t and 1.2. We show that u 1 = u 2.Letc j t, τ j t, t j τ, C j τ, V j τ, W j τ be the functions defined by 3.3, 3.4 and 3.5 for u = u j j = 1, 2. Then C j τ = m 1/2 u j t j τ 2 C 1 R, V j, W j C 1 R; DA 1/2 DA 1/2, and C j, V j, W j satisfies and C j τ L1 R. Since V, W satisfies 3.5 with u = u and 6.16 we see that 6.17 Cτt T = ct T = m A 1/2 u t T 2 τt T τ T = = Vτt T = m A 1/2 u 1 t 2 = c 1 t = C 1 τ 1 t, t T T t m A 1/2 u s 2 ds m A 1/2 u 1 s 2 ds = τ 1 t, = ct T 1/2 e iτt T A 1/2 +Bt T /2 u t T ict T A 1/2 u t T In the same way, we have = c 1 t 1/2 e iτ 1tA 1/2 +Bt/2 e iτ T A 1/2 + t T bsds/2 t u 1 t ic 1tA 1/2 u 1 t = e t t T bsds/2 e iτ T A 1/2 V 1 τ 1 t Wτt T = e t t T bsds/2 e iτ T A 1/2 W 1 τ 1 t. By 6.7 and the assumption b L 1 R, letting t in the equations 6.17 and 6.18, we see that the limits 6.19 V 1,, W 1, := lim V 1τ, W 1 τ τ exists in DA J/2 DA J/2 with 6.2 V 1,, W 1, = e iτ T A 1/2 V, e iτ T A 1/2 W. Hence, by Lemma 2 i together with 6.7 and 6.11, we have 6.21 V 1,, W 1, J pk 12 K 13ε, where K12 and K 13 are constants defined by 6.1. Equality 1.19 with u = u j j = 1, 2 for the same v implies 6.22 lim A 1/2 u 1 t u 2 t + u t 1 t u 2 t =. Since lim τ Cτ exists, 6.16 implies that lim τ C 1 τ exists. Hence by 6.22,

34 Scattering for Perturbed Kirchhoff Equation lim C 1τ = lim τ τ m A1/2 u 1 t 1 τ 2 = lim t m A1/2 u 1 t 2 = lim t m A1/2 u 2 t 2 = lim τ m A1/2 u 2 t 2 τ 2 = lim C 2τ. τ By the equality lim τ Bt 1 τ = B = lim τ Bt 2 τ, 6.19, , we see that the limit lim τ V 2 τ, W 2 τ := V 2,, W 2, exists in DA J/2 DA J/2 with V 2,, W 2, = V 1,, W 1, J pk 12 K 13ε J pε 7. Hence, C 1 τ, V 1 τ, W 1 τ and C 2 τ, V 2 τ, W 2 τ are solutions of with the same data at T =. Hence by Proposition 2, we have C 1 τ, V 1 τ, W 1 τ = C 2 τ, V 2 τ, W 2 τ for every τ R, and therefore u 1 t = u 2 t for every t R. Step 5 We prove the continuity of the wave operator. By the construction above, W φ, ψ = φ,ψ = u T, du dt T = Cτ T, e iτ T A 1/2 Vτ T, e iτ T A 1/2 Wτ T, where u, τ, T, V, W are functions and numbers defined in Step 3. By the procedure above together with 6.8, 6.11 and Lemmas 1 i and 2 i, we see that the operator W is decomposed into the following four operators: with W φ, ψ = Φ 3 Φ 2 Φ 1 φ, ψ, Φ 1 :φ, ψ c, V, W := c, e B /2 G c, φ, ψ Ỹpε J [m, m 1 ] pk J 12 ε, Φ 2 :c, V, W C, V, W [m, m 1 ] pk J 12 ε p,k 5 K12 ε BCR : pk J 5 K12 ε, Φ 3 :C,V, W Cτ T, e iτ T A 1/2 Vτ T, e iτ T A 1/2 Wτ T p,k5 K12 ε BCR : pk J 5 K12 ε [m, m 1 ] pk J 5 K12 K 13ε, :[m, m 1 ] pk J 5 K12 K 13ε YpK J 3 ε. Here BCR : pk J 5 K12 ε denotes a metric space of all pk J 5 K12 ε valued bounded continuous functions, where the distance of two element V j, W j for j = 1, 2isdefinedby sup d J p V 1 τ, W 1 τ, V 2 τ, W 2 τ. τ R The function τ is the inverse of tτ which is defined by 3.11, and T is the number defined by 6.1, and both depend on C. We write τ = τ C and T = T C, if we need to represent the dependence on C. By Lemma 1 ii together with the continuity of the mapping φ, ψ c which is shown in Step 1, we easily see that the mapping Φ 1 is continuous. Propositions 2 implies the continuity of the mapping Φ 2.