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Abstract and Applied Analysis Hyperbolic differentialoperator equations on a whole axis
Hyperbolic differentialoperator equations on a whole axis
Yakubov, YakovBạn thích cuốn sách này tới mức nào?
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Tập:
2004
Năm:
2004
Ngôn ngữ:
english
Tạp chí:
Abstract and Applied Analysis
DOI:
10.1155/s1085337504311103
File:
PDF, 1,89 MB
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HYPERBOLIC DIFFERENTIALOPERATOR EQUATIONS ON A WHOLE AXIS YAKOV YAKUBOV Received 24 August 2003 We give an abstract interpretation of initial boundary value problems for hyperbolic equations such that a part of initial boundary value conditions contains also a diﬀerentiation on the time t of the same order as equations. The case of stable solutions of abstract hyperbolic equations is treated. Then we show applications of obtained abstract results to hyperbolic diﬀerential equations which, in particular, may represent the longitudinal displacements of an inhomogeneous rod under the action of forces at the two ends which are proportional to the acceleration. 1. Introduction The first attempt to give an abstract interpretation of hyperbolic problems such that a part of boundary value conditions may contain the diﬀerentiation on the time was done in [8] for almost periodic solutions and oscillations decay cases, and in [5] for hyperbolic diﬀerentialoperator equations on a finite interval. In this paper, we continue this study to the case of diﬀerentialoperator equations on a whole axis. In particular, we find suﬃcient conditions for which the solution of the considered problems is stable. Let H and F be Hilbert spaces. The set H ⊕ F of all vectors of the form (u,v) where u ∈ H and v ∈ F, with usual coordinatewise linear operations and the norm (u,v) H ⊕F := u2H + v2F 1/2 (1.1) is a Hilbert space and called the orthogonal sum of Hilbert spaces H and F. For the operator A closed in a Hilbert space H, the domain D(A) is turned into a Hilbert space H(A) with respect to the norm 1/2 uH(A) := u2 + Au2 . (1.2) If H1 and H are two Hilbert spaces where H1 ⊂ H, then H1 can be represented as the domain D(S) = H1 of a suitable positive definite selfadjoint operator S in H (see, e.g., Copyright © 2004 Hindawi Publishing Corporation Abstract and Applied Analysis 2004:2 (2004) 99–113 2000 Mathematics Subject Classification: 34G10, 35L15 URL: http://dx.doi.org/10.1155/S1085337504311103 100 ; Hyperbolic diﬀerentialoperator equations on a whole axis [4, Remark 1.18.10/3]). Then, by [4, Theorem 1.18.10], the interpolation space H1 ,H θ,2 = H S1−θ . (1.3) 2. Hyperbolic diﬀerentialoperator equations We give, in this section, an abstract interpretation of initial boundary value problems for hyperbolic equations such that a part of boundary value conditions contains also the diﬀerentiation on the time t. Let H and H ν , ν = 1,...,s, be Hilbert spaces. Consider the following Cauchy problem (abstract “initial boundary value problem”): L Dt u := u (t) + Bu(t) = 0, Lν Dt u := Aν0 u(t) + Aν2 u(t) = 0, u(0) = u0 , (2.1a) ν = 1,...,s, u (0) = u1 , (2.1b) (2.1c) where t ∈ R; B is an operator in H; Aν0 and Aν2 are operators from a subspace of H into H ν ; and u(t) from R into H is an unknown function. Note that operators B, Aν0 , and Aν2 are, generally speaking, unbounded. A function u(t) is called a solution of problem (2.1) if the function t → (u(t),A10 u(t),..., As0 u(t)) from R into H ⊕ H 1 ⊕ · · · ⊕ H s is twice continuously diﬀerentiable, from R into H(B) ⊕ H 1 ⊕ · · · ⊕ H s is continuous, and u(t) satisfies (2.1). We say that problem (2.1) is stable if each of its solution u(t) with u0 ∈ H(B), u1 ∈ H(B), is bounded, that is, u(t) ≤ C, t ∈ R. (2.2) Consider a system of operator pencils corresponding to (2.1a) and (2.1b); L(λ) := λI + B, Lν (λ) := λAν0 + Aν2 , ν = 1,...,s, (2.3) where λ is a complex number. Theorem 2.1. Let the following conditions be satisfied: (1) B is a closed operator in a Hilbert space H with a dense domain D(B); the embedding H(B) ⊂ H is compact; (2) the operators Aν0 from (H(B),H)1/2,2 into H ν act boundedly and Aν2 , ν = 1,...,s, from H(B) into H ν act boundedly; (3) the linear manifold {v  v := (u,A10 u,...,As0 u), u ∈ D(B)} is dense in the Hilbert space H ⊕ H 1 ⊕ · · · ⊕ H s ; Yakov Yakubov 101 (4) for all u ∈ D(B), v ∈ D(B), (Bu,v)H + A12 u,A10 v H1 + · · · + As2 u,As0 v = (u,Bv)H + A10 u,A12 v H1 Hs + · · · + As0 u,As2 v Hs ; (2.4) (5) for all u ∈ D(B), 0 ≤ (Bu,u)H + A12 u,A10 u H1 + · · · + As2 u,As0 u Hs ≤ C u2(H(B),H)1/2,2 ; (2.5) (6) some real number λ0 is a regular point for the operator pencil L(λ): u → L(λ)u := (L(λ)u,L1 (λ)u,...,Ls (λ)u), which acts boundedly from H(B) onto H ⊕ H 1 ⊕· · ·⊕ H s ; (7) (u1 ,A10 u1 ,...,As0 u1 ) ∈ Im(B1/2 ), where D(B) := v  v := u,A10 u,...,As0 u , u ∈ D(B) , B u,A10 u,...,As0 u := Bu,A12 u,...,As2 u (2.6) is an operator in the Hilbert space Ᏼ := H ⊕ H 1 ⊕ · · · ⊕ H s and from condition (5), it follows that (Bv,v) ≥ 0, for all v ∈ D(B); (8) u0 ∈ H(B), u1 ∈ (H(B),H)1/2,2 . Then there exists a unique solution u(t) of problem (2.1) such that the function t → (u(t),A10 u(t),...,As0 u(t)) from R into H ⊕ H 1 ⊕ · · · ⊕ H s is twice continuously diﬀerentiable, and from R into H(B) ⊕ H 1 ⊕ · · · ⊕ H s is continuous, and for t ∈ R, the following estimate holds: s u(t) + u (t) + Aν0 u(t) ν =1 ≤ C Bu0 + u0 + u1 Hν + Bu(t) (H(B),H)1/2,2 (2.7) , consequently, problem (2.1) is stable. Proof. Consider, in the Hilbert space Ᏼ := H ⊕ H 1 ⊕ · · · ⊕ H s , the abovementioned operator B. Then the Cauchy problem v (t) + Bv(t) = 0, v(0) = v0 , v (0) = v1 , (2.8) is equivalent to the Cauchy problem (2.1), where v0 := (u0 ,A10 u0 ,...,As0 u0 ), v1 := (u1 , A10 u1 ,...,As0 u1 ). Indeed, let u(t) be a solution of problem (2.1). Then, v(t) = (u(t),A10 u(t), ...,As0 u(t)) is a solution of problem (2.8). Conversely, let v(t) be a solution of problem (2.8). Since v(t) ∈ D(B), then v(t) := (u(t),A10 u(t),...,As0 u(t)), where u(t) ∈ D(B), for all t ∈ R. Substituting v(t) into (2.8), we get that u(t) satisfies (2.1). 102 Hyperbolic diﬀerentialoperator equations on a whole axis By virtue of condition (3), D(B) is dense in Ᏼ. By virtue of condition (4), for ṽ1 = (ũ1 ,A10 ũ1 ,...,As0 ũ1 ) ∈ D(B), ṽ2 = (ũ2 ,A10 ũ2 ,...,As0 ũ2 ) ∈ D(B), Bṽ1 , ṽ2 Ᏼ = B ũ1 , ũ2 H + A12 ũ1 ,A10 ũ2 H 1 + · · · + As2 ũ1 ,As0 ũ2 H s = ũ1 ,B ũ2 H + A10 ũ1 ,A12 ũ2 H 1 + · · · + As0 ũ1 ,As2 ũ2 H s = ṽ1 , Bṽ2 Ᏼ . (2.9) Consequently, the operator B is symmetric. In turn, equation λ0 v + Bv = F, F := f , f1 ,..., fs , (2.10) where v = (u,A10 u,...,As0 u), is equivalent to the system L λ0 u = λ0 u + Bu = f , (2.11) ν = 1,...,s. Lν λ0 u = λ0 Aν0 u + Aν2 u = fν , By virtue of condition (6), problem (2.11) has a unique solution −1 u = L λ0 f , f1 ,..., fs . (2.12) So, a solution of (2.10) has the following form: −1 v = L λ0 −1 f , f1 ,..., fs ,A10 L λ0 −1 f , f1 ,..., fs ,...,As0 L λ0 f , f1 ,..., fs . (2.13) Hence, the operator B is closed and the image Im(λ0 I + B) = Ᏼ, where I is the identity operator in Ᏼ. By virtue of [3, Chapter Y, Section 3], the operator B is selfadjoint. From condition (5), it follows that (Bv,v) ≥ 0, v ∈ D(B). Consequently, the operator B is selfadjoint and nonnegative. By condition (7), B−1/2 v1 is well defined. Then, problem (2.8) has a unique solution v(t) ∈ C 2 (R;Ᏼ(B),Ᏼ) and v(t) = cos t B1/2 v0 + sin t B1/2 B−1/2 v1 , ∞ (2.14) ∞ where cos(t B1/2 )v = 0 cos(tλ1/2 )dE(λ)v, sin(t B1/2 )v = 0 sin(tλ1/2 )dE(λ)v, and E(λ) is the spectral decomposition of the selfadjoint operator B. Obviously, v0 ∈ Ᏼ(B). Show now that v1 ∈ Ᏼ(B1/2 ). From the definition of the operator B, it follows that Ᏼ(B) ⊂ H(B) ⊕ H 1 ⊕ · · · ⊕ H s . By Section 1, (Ᏼ(B),Ᏼ)1/2,2 = Ᏼ(B1/2 ). Then, Ᏼ B1/2 = Ᏼ(B),Ᏼ)1/2,2 ⊂ H(B),H 1/2,2 ⊕ H 1 ⊕ · · · ⊕ H s. (2.15) Yakov Yakubov 103 Assume, first, that v = (u,A10 u,...,As0 u) ∈ D(B), that is, u ∈ D(B). Then, by virtue of conditions (2) and (5) and the property of interpolation spaces: Ᏼ(B) ⊂ (Ᏼ(B),Ᏼ)1/2,2 ⊂ Ᏼ, we get 2 v 2Ᏼ(B1/2 ) = B1/2 v Ᏼ + v 2Ᏼ = B1/2 v, B1/2 v Ᏼ + (v,v)Ᏼ = (Bv,v)Ᏼ + (v,v)Ᏼ = (Bu,u)H + s ν =1 Aν2 u,Aν0 u Hν + (u,u)H + s ν =1 Aν0 u,Aν0 u Hν (2.16) ≤ C u2(H(B),H)1/2,2 . Now, v1 := (u1 ,A10 u1 ,...,As0 u1 ), where u1 ∈ (H(B),H)1/2,2 . Then, there exists a sequence un ∈ H(B) such that lim un − u1 (H(B),H)1/2,2 = 0, (2.17) n→∞ since the space H(B) is dense in (H(B),H)1/2,2 . Moreover, un is a fundamental sequence, that is, limn→∞ un − um (H(B),H)1/2,2 = 0. By (2.16), we get n v − v m Ᏼ(B1/2 ) ≤ C un − um (H(B),H)1/2,2 , (2.18) where vn = (un ,A10 un ,...,As0 un ). Therefore, the sequence vn is a fundamental in the Hilbert space Ᏼ(B1/2 ). Hence, vn converges in Ᏼ(B1/2 ), that is, there exists v = (u,A10 u,..., As0 u) ∈ Ᏼ(B1/2 ) such that limn→∞ vn − vᏴ(B1/2 ) = 0. In particular, lim un − u(H(B),H)1/2,2 = 0. (2.19) n→∞ Then, by virtue of (2.17), u = u1 and, therefore, v = v1 . Hence, v1 ∈ Ᏼ(B1/2 ). Moreover, writing (2.16) for vn and passing to the limit when n → ∞, we get that v Ᏼ(B1/2 ) ≤ C u(H(B),H)1/2,2 (2.20) is also true for v = (u,A10 u,...,As0 u), for all u ∈ (H(B),H)1/2,2 . Since v (t) = −B1/2 sin t B1/2 v0 + cos t B1/2 v1 , v (t) = −B cos t B1/2 v0 − B1/2 sin t B1/2 v1 , (2.21) then v(t) + v (t) + Bv(t) ≤ C v0 + v1 + Bv0 + B1/2 v1 ≤ C v0 + v1 Ᏼ(B1/2 ) + Bv0 , t ∈ R. From this, by (2.20) and condition (2), the statement of the theorem follows. (2.22) 104 Hyperbolic diﬀerentialoperator equations on a whole axis Consider now such a formulation of problem (2.1), which allows us to insert the firstorder derivative into (2.1a). Let H and H ν , ν = 1,...,s, be Hilbert spaces. Consider the following Cauchy problem (abstract initial boundary value problem): L Dt u := u (t) + Au (t) + Bu(t) = h(t), Lν Dt u := Aν0 u(t) ν = 1,...,s, + Aν2 u(t) = hν (t), (2.23) u(0) = u0 , u (0) = u1 , where t ≥ 0; A and B are operators in H; Aν0 and Aν2 are operators from a subspace of H into H ν ; u(t) from [0, ∞) into H is an unknown function; h(t) and hν (t) from [0, ∞) into H and H ν , respectively, are given functions. Note that operators A, B, Aν0 , and Aν2 are, generally speaking, unbounded. Consider, in the Hilbert space Ᏼ := H ⊕ H 1 ⊕ · · · ⊕ H s , operators A and B given by the equalities D(A) := D(A) ⊕ H 1 ⊕ · · · ⊕ H s , A u,v1 ,...,vs := (Au,0,...,0), (2.24) D(B) := v  v := u,A10 u,...,As0 u , u ∈ D(B) , B u,A10 u,...,As0 u := Bu,A12 u,...,As2 u . Theorem 2.2. Let the following conditions be satisfied: (1) B is an operator in a Hilbert space H with a dense domain D(B); A is an operator in H with D(A) ⊃ (H(B),H)1/2,2 ; the embedding H(B) ⊂ H is compact; (2) the operators Aν0 , ν = 1,...,s, from (H(B),H)1/2,2 into H ν act compactly, and the operators Aν2 , ν = 1,...,s, from H(B) into H ν act boundedly; (3) the linear manifold {v  v := (u,A10 u,...,As0 u), u ∈ D(B)} is dense in the Hilbert space H ⊕ H 1 ⊕ · · · ⊕ H s ; (4) for all u ∈ D(B), v ∈ D(B), (Bu,v)H + A12 u,A10 v H1 + · · · + As2 u,As0 v Hs = (u,Bv)H + A10 u,A12 v H 1 + · · · + As0 u,As2 v H s ; (2.25) (5) for all u ∈ D(B) and some C, c = 0, C u2(H(B),H)1/2,2 ≥ (Bu,u)H + A12 u,A10 u H1 + · · · + As2 u,As0 u 2 2 ≥ c2 u2H + A10 uH 1 + · · · + As0 uH s ; Hs (2.26) (6) some real number λ0 is a regular point for the operator pencil L(λ): u → L(λ)u := ((λI + B)u,(λA10 + A12 )u,...,(λAs0 + As2 )u), which acts boundedly from H(B) onto H ⊕ H 1 ⊕ · · · ⊕ H s; (7) A is a skewsymmetric operator in H, that is, A∗ u = −Au, u ∈ D(A), and A from (H(B),H)1/2,2 into H is bounded; Yakov Yakubov 105 (8) h ∈ W p1 ((0, ∞);H) ∩ L1 ((0, ∞);H), hν ∈ W p1 ((0, ∞);H ν ) ∩ L1 ((0, ∞);H ν ), ν = 1,2, for some p > 1; (9) u0 ∈ H(B), u1 ∈ (H(B),H)1/2,2 . Then there exists a unique solution u(t) of problem (2.23) such that the function t → (u(t),A10 u(t),...,As0 u(t)) from [0, ∞) into H ⊕ H 1 ⊕ · · · ⊕ H s is twice continuously diﬀerentiable and from [0, ∞) into H(B) ⊕ H 1 ⊕ · · · ⊕ H s is continuous, and for the solution the following estimate holds: u(t) (H(B),H)1/2,2 + u (t) + s Aν0 u(t) ν =1 Hν 2 h ν ≤ C u0 (H(B),H)1/2,2 + u1 (H(B),H)1/2,2 + hL1 ((0,∞);H) + L1 ((0,∞);H ν ) , ν =1 ∀t ≥ 0, (2.27) consequently (since H(B) ⊂ (H(B),H)1/2,2 ⊂ H), problem (2.23) is stable. Note that substituting t = −τ, τ ≥ 0, one can consider problem (2.23) for t ≤ 0 too. Therefore, in fact, Theorem 2.2 is true for t ∈ R. Proof. Consider, in the Hilbert space Ᏼ := H ⊕ H 1 ⊕ · · · ⊕ H s , the abovementioned operators A and B. Then the Cauchy problem v (t) + Av (t) + Bv(t) = f (t), v(0) = v0 , v (0) = v1 , (2.28) is equivalent to the Cauchy problem (2.23), where v0 := (u0 ,A10 u0 ,...,As0 u0 ), v1 := (u1 , A10 u1 ,...,As0 u1 ), f (t) := (h(t),h1 (t),... ,hs (t)), and v(t) := (u(t),A10 u(t),...,As0 u(t)) (for the proof, see the proof of Theorem 2.1). Apply Theorem A.1 (see the appendix) to problem (2.28), where Ã := A, B̃ := B. It was proved in Theorem 2.1 that the operator B is selfadjoint. From condition (5), it follows that (Bv,v) ≥ c2 (v,v), v ∈ D(B). Consequently, the operator B is selfadjoint and positivedefinite. So, by conditions (1), (2), and (3), conditions (1) and (2) of Theorem A.1 are fulfilled. From the proof of Theorem 2.1, it follows that Ᏼ(B1/2 ) ⊂ (H(B),H)1/2,2 ⊕ H 1 ⊕ · · · ⊕ s H . This implies, by conditions (1) and (7), D(A) ⊃ D(B1/2 ), the operator A from Ᏼ(B1/2 ) into Ᏼ is bounded and is skewsymmetric. Hence, condition (3) of Theorem A.1 is satisfied. From condition (8), it follows condition (4) of Theorem A.1. Similar arguments to those in the proof of Theorem 2.1 gives us that v0 ∈ Ᏼ(B), v1 ∈ Ᏼ(B1/2 ), that is, the last condition (5) of Theorem A.1 is satisfied too. So, on each interval [0,T], we have a unique solution v(t) ∈ C 2 ([0,T];Ᏼ(B),Ᏼ(B1/2 ),Ᏼ) of problem (2.28). In order to get the estimate of Theorem 2.2, one should use the proof of Theorem A.1 from the appendix. In particular, it follows from the proof that v(t) v0 + = etᏭ v1 v (t) t 0 e(t−τ)Ꮽ 0 dτ, f (τ) (2.29) 106 Hyperbolic diﬀerentialoperator equations on a whole axis where Ꮽ = 0 I −B −A v(t) . Moreover, Ᏼ(B1/2 ) + v (t)Ᏼ ≤ v0 Ᏼ(B1/2 ) + v1 Ᏼ + t 0 f (τ) dτ. Ᏼ (2.30) Using conditions (2), (8), and (9), and the inequality (2.20), we get the estimate of the theorem. 3. Initial boundary value problems for hyperbolic equations Consider, in the domain R × [0,1], an initial boundary value problem for the hyperbolic equation L Dt u := Dtt2 u(t,x) − Dx b(x)Dx u(t,x) = 0, (t,x) ∈ R × [0,1], L1 Dt u := αDtt2 u(t,0) + Dx u(t,0) = 0, t ∈ R, L2 Dt u := βDtt2 t ∈ R, u(0,x) = u0 (x), u(t,1) + Dx u(t,1) = 0, Dt u(0,x) = u1 (x), (3.1) x ∈ [0,1], where α, β are real numbers, Dt := ∂/∂t and Dx := ∂/∂x. This problem was considered, by a diﬀerent approach, in [1]. As it was mentioned in [1], “physically, such a problem may represent the longitudinal displacements of an inhomogeneous rod under the action of forces at the two ends which are proportional to the acceleration. In particular, this situation is realized if there are massive loads at the ends (see, e.g., [2, Chapter 12]) and in this case we have α < 0 and β > 0.” Theorem 3.1. Let the following conditions be satisfied: (1) b ∈ C 1 [0,1]; b(x) > 0 for x ∈ [0,1]; (2) α < 0, β > 0; (3) u0 ∈ W22 (0,1), u1 ∈ W21 (0,1); 1 (4) 0 u1 (x)dx − αb(0)u1 (0) + βb(1)u1 (1) = 0. Then there exists a unique solution u(t,x) of problem (3.1) such that the function t → (u(t,x),u(t,0),u(t,1)) from R into L2 (0,1) ⊕ C ⊕ C is twice continuously diﬀerentiable, and from R into W22 (0,1) ⊕ C ⊕ C is continuous, and for t ∈ R the following estimate holds: u(t, ·) L2 (0,1) + 2 D u(t, ·) tt 2 + Dxx u(t, ·)L2 (0,1) 2 D u(t,0) + D2 u(t,1) tt tt ≤ C u 0 2 + u 1 1 , L2 (0,1) + W2 (0,1) (3.2) W2 (0,1) consequently, problem (3.1) is stable. Proof. Apply Theorem 2.1. Consider, in the Hilbert space H := L2 (0,1), an operator B given by the equalities D(B) := W22 (0,1), Bu := − b(x)u (x) . (3.3) Yakov Yakubov 107 Taking H 1 := −(b(0)/α)C, H 2 := (b(1)/β)C, and A10 u := αu(0), A20 u := βu(1), (3.4) A22 u := u (1), A12 u := u (0), problem (3.1) can be rewritten in the form (2.1), where u(t) := u(t, ·) is a function with values in the Hilbert space H := L2 (0,1), and u0 := u0 (·), u1 := u1 (·). From [4, Section 3.2.5], it follows that condition (1) of Theorem 2.1 is satisfied. From [4, Section 4.3.1], it follows that (W22 (0,1),L2 (0,1))1/2,2 = W21 (0,1). Then condition (2) of Theorem 2.1 is satisfied too. Condition (3) of Theorem 2.1 follows from Theorem A.2 (see the appendix). We prove conditions (4) and (5) of Theorem 2.1. For u1 ∈ W22 (0,1), u2 ∈ W22 (0,1), we have Bu1 ,u2 L2 (0,1) + 1 =− 0 A12 u1 ,A10 u2 0 −(b(0)/α)C + A22 u1 ,A20 u2 (b(1)/β)C du (x) b(1) d b(0) b(x) 1 u2 (x)dx − u (0) · αu2 (0) + u (1)βu2 (1) dx dx α 1 β 1 1 =− u1 (x) 1 d du (x) b(x) 2 dx − b(x)u1 (x)u2 (x)0 dx dx (3.5) 1 + u1 (x) b(x)u2 (x) 0 − b(0)u1 (0)u2 (0) + b(1)u1 (1)u2 (1) = u1 ,Bu2 L2 (0,1) + A10 u1 ,A12 u2 −(b(0)/α)C + A20 u1 ,A22 u2 (b(1)/β)C , that is, condition (4) of Theorem 2.1 is satisfied. For u ∈ W22 (0,1), we have (Bu,u)L2 (0,1) + A12 u,A10 u = = 1 0 1 0 −(b(0)/α)C + A22 u,A20 u (b(1)/β)C 2 1 b(x)u (x) dx − b(x)u (x)u(x) − b(0)u (0)u(0) + b(1)u (1)u(1) 0 (3.6) 2 b(x)u (x) dx ≥ 0. 1 On the other hand, 0 b(x)u (x)2 dx ≤ C u2W21 (0,1) , that is, condition (5) of Theorem 2.1 is satisfied too. Denote L(λ)u := (λI + B)u = λu(x) − b(x)u (x) , L1 (λ)u := λA10 + A12 u = λαu(0) + u (0), (3.7) L2 (λ)u := λA20 + A22 u = λβu(1) + u (1). From Theorem A.3 (see the appendix), it follows that for any ε > 0, there exists Rε > 0 such that for all complex numbers λ which satisfy λ > Rε and lying inside the angle −π + ε < argλ < π − ε, (3.8) 108 Hyperbolic diﬀerentialoperator equations on a whole axis the operator L(λ) : u → L(λ)u := (L(λ)u,L1 (λ)u,L2 (λ)u) from W22 (0,1) onto L2 (0,1) ⊕ −(b(0)/α)C ⊕ (b(1)/β)C is an isomorphism, that is, condition (6) of Theorem 2.1 is satisfied. Condition (7) of Theorem 2.1 is fulfilled in view of condition (4) (see for details [1]). Condition (8) of Theorem 2.1 follows from condition (3). So, for problem (3.1), all conditions of Theorem 2.1 are fulfilled and the statement of Theorem 3.1 follows. We present now an application of Theorem 2.2. Consider, in the domain [0, ∞) × [0,1], an initial boundary value problem for the hyperbolic equation L Dt u := Dtt2 u(t,x) + ia(x)Dt u(t,x) − Dx b(x)Dx u(t,x) + c(x)u(t,x) = h(t,x), (t,x) ∈ [0, ∞) × [0,1], L1 Dt u := αDtt2 u(t,0) + Dx u(t,0) = h1 (t), L2 Dt u := βDtt2 u(t,1) + Dx u(t,1) = h2 (t), u(0,x) = u0 (x), Dt u(0,x) = u1 (x), t ∈ [0, ∞), (3.9) t ∈ [0, ∞), x ∈ [0,1], √ where α, β are real numbers, i = −1, Dt := ∂/∂t, Dx := ∂/∂x. Theorem 3.2. Let the following conditions be satisfied: (1) a ∈ C[0,1] and is realvalued; b ∈ C 1 [0,1], b(x) > 0 for x ∈ [0,1]; c ∈ C[0,1], c(x) > 0 for x ∈ [0,1]; (2) α < 0, β > 0; (3) h ∈ W p1 ((0, ∞);L2 (0,1)) ∩ L1 ((0, ∞);L2 (0,1)), hν ∈ W p1 (0, ∞) ∩ L1 (0, ∞), ν = 1,2, for some p > 1; (4) u0 ∈ W22 (0,1), u1 ∈ W21 (0,1). Then there exists a unique solution u(t,x) of problem (3.9) such that the function t → (u(t,x),u(t,0),u(t,1)) from [0, ∞) into L2 (0,1) ⊕ C ⊕ C is twice continuously diﬀerentiable, and from [0, ∞) into W22 (0,1) ⊕ C ⊕ C is continuous, and for the solution the following estimate holds: u(t, ·) L2 (0,1) + Dt u(t, ·) L2 (0,1) + Dt u(t,0) + Dt u(t,1) ≤ C u0 W21 (0,1) + u1 W21 (0,1) + ∞ 0 h(t, ·) L2 (0,1) dt + 2 hν (t)dt , ∞ ν =1 0 ∀t ≥ 0, (3.10) consequently, problem (3.9) is stable. Note that this theorem, as Theorem 2.2, is also true for t ≤ 0 and, therefore, for t ∈ R. Proof. Apply Theorem 2.2. Consider, in the Hilbert space H := L2 (0,1), operators A and B given by the equalities D(A) := L2 (0,1), D(B) := W22 (0,1), Au := ia(x)u(x), Bu := − b(x)u (x) + c(x)u(x). (3.11) Yakov Yakubov 109 Taking H 1 := −(b(0)/α)C, H 2 := (b(1)/β)C, and A10 u := αu(0), A20 u := βu(1), A12 u := u (0), A22 u := u (1), problem (3.9) can be rewritten in the form (2.23), where u(t) := u(t, ·) and h(t) := h(t, ·) are functions with values in the Hilbert space H := L2 (0,1), and ϕ0 := ϕ0 (·), ϕ1 := ϕ1 (·). From [4, Section 3.2.5], it follows that D(B) is dense in H, and the embedding H(B) ⊂ H is compact, and from [4, Section 4.3.1] it follows that (W22 (0,1),L2 (0,1))1/2,2 =W21 (0,1). Therefore, condition (1) of Theorem 2.2 is satisfied. It is well known that the embedding W2k (0,1) ⊂ C m [0,1], k > m ≥ 0, is compact (see, e.g., [4, Section 4.10.2, formula (15)]). Then condition (2) of Theorem 2.2 is satisfied too. Condition (3) of Theorem 2.2 follows from Theorem A.2 (see the appendix). We prove conditions (4) and (5) of Theorem 2.2. For u ∈ W22 (0,1), v ∈ W22 (0,1), we have (Bu,v)L2 (0,1) + A12 u,A10 v =− − =− 1 0 −(b(0)/α)C + A22 u,A20 v (b(1)/β)C 1 du(x) d b(x) v(x)dx + dx dx 0 c(x)u(x)v(x)dx b(1) b(0) u (0)αv(0) + u (1)βv(1) α β 1 0 1 d dv(x) u(x) b(x) dx + dx dx 0 1 c(x)u(x)v(x)dx − b(x)u (x)v(x) (3.12) 0 1 + u(x) b(x)v (x) 0 − b(0)u (0)v(0) + b(1)u (1)v(1) = (u,Bv)L2 (0,1) + A10 u,A12 v −(b(0)/α)C + A20 u,A22 v (b(1)/β)C , that is, condition (4) of Theorem 2.2 is satisfied. For u ∈ W22 (0,1), using conditions (1) and (2) and that W21 (0,1) ⊂ C[0,1] is bounded, we have (Bu,u)L2 (0,1) + A12 u,A10 u = 1 0 −(b(0)/α)C 2 b(x)u (x) dx + 1 0 + A22 u,A20 u (b(1)/β)C 2 1 c(x)u(x) dx − b(x)u (x)u(x)0 − b(0)u (0)u(0) + b(1)u (1)u(1) = 1 0 2 b(x)u (x) dx + 1 2 ≥ √ min 1 0 2 c(x)u(x) dx (3.13) min b(x), min c(x) u2W21 (0,1) x∈[0,1] x∈[0,1] 2 2 − b(0)αu(0) + b(1)βu(1) ≥ c2 u2L2 (0,1) 2 2 = c2 u2L2 (0,1) + A10 u−(b(0)/α)C + A20 u(b(1)/β)C , 1 1 ∃c = 0. On the other hand, 0 b(x)u (x)2 dx + 0 c(x)u(x)2 dx ≤ C u2W21 (0,1) , that is, condition (5) of Theorem 2.2 is satisfied too. Condition (6) of Theorem 2.2 is checked as in 110 Hyperbolic diﬀerentialoperator equations on a whole axis the proof of Theorem 3.1. We check condition (7) of Theorem 2.2. Take u,v ∈ D(A) = L2 (0,1). Then, (Au,v)L2 (0,1) = 1 0 ia(x)u(x)v(x)dx = − 1 0 u(x)ia(x)v(x)dx (3.14) = (u, −Av)L2 (0,1) . Conditions (8) and (9) of Theorem 2.2 are trivial. So, for problem (3.9), all conditions of Theorem 2.2 are fulfilled and the statement of Theorem 3.2 follows. Appendix Consider, in a Hilbert space H, the Cauchy problem for the secondorder hyperbolic diﬀerentialoperator equation L(D)u := u (t) + Ãu (t) + B̃u(t) = f (t), u(0) = g0 , t ∈ [0,T], u (0) = g1 , (A.1) and the characteristic operator pencil L(λ) := λ2 + λÃ + B̃. (A.2) Theorem A.1 (see [7, Theorem 6.4.3]). Let the following conditions be satisfied: (1) B̃ is a selfadjoint positivedefinite operator in a Hilbert space H; (2) the embedding H(B̃) ⊂ H is compact; (3) Ã is a skewsymmetric operator in H, that is, Ã∗ u = −Ãu, u ∈ D(Ã); the operator Ã from H(B̃ 1/2 ) into H is bounded; (4) f ∈ W p1 ((0,T);H), where p > 1; (5) g0 ∈ D(B̃), g1 ∈ D(B̃1/2 ). Then, problem (A.1) has a unique solution u ∈ C 2 ([0,T];H(B̃),H(B̃ 1/2 ),H), and the solution can be expanded to the series u(t) = ∞ e λk t 1/2 2 + λk 2 uk 2 k=1 B̃ uk × B̃g0 − λk g1 ,uk − λk t 0 e −λ k τ (A.3) f (τ),uk dτ uk , where λk are purely imaginary eigenvalues and uk are the corresponding eigenvectors of operator pencil (A.2), and the series converges in the sense of the space C 2 ([0,T];H(B̃),H(B̃ 1/2),H). Denote Aν0 u : = αν u(mν ) (0) + βν u(mν ) (1) + Nν j =1 δν j u(mν ) xν j + Tν u, ν = 1,...,m. (A.4) Yakov Yakubov 111 Theorem A.2 (see [7, Theorem 3.6.2]). Let the following conditions be satisfied: (1) m ≥ 1, mν ≥ 0, 0 ≤ s ≤ m; (2) a system of functionals (A.4) is pregular with respect to a system of numbers ω j := e2πi(( j −1)/m) , j = 1,...,m, that is, α1 ω1m1 . . . . . . α ωmm m 1 ··· .. . .. . ··· 1 α1 ωm p .. . 1 β1 ω m p+1 .. . ··· .. . m αm ωm p .. . m βm ω m p+1 .. . .. . ··· m1 β 1 ωm .. . = 0, .. . mm β m ωm (A.5) where p = m/2 if m is even, p = [m/2] or p = [m/2] + 1 if m is odd, xν j ∈ (0,1) and, for some q ∈ (1, ∞), functionals Tν in Wqmν (0,1) are continuous. Then, the linear manifold (u,v)  u ∈ C ∞ [0,1], Aν0 u = 0, ν = s + 1,...,m, v := A10 u,...,As0 u , (A.6) is dense in the space Wq (0,1) +̇ Cs , ≤ min{mν }. Consider a principally boundary value problem for an ordinary diﬀerential equation with a variable coeﬃcient in case when the spectral parameter appears linearly in the equation and can appear in boundaryfunctional conditions L(λ)u := λu(x) + a(x)u(m) (x) + Bux = f (x), Lν (λ)u := λ αν u(mν ) (0) + βν u(mν ) (1) + Nν x ∈ (0,1), δν j u(mν ) xν j + Tν u j =1 + Tν0 u = gν , Lν u := αν u(mν ) (0) + βν u(mν ) (1) + Nν (A.7a) (A.7b) ν = 1,...,s, δν j u(mν ) xν j + Tν u = 0, ν = s + 1,...,m, (A.7c) j =1 where m ≥ 1, mν ≤ m − 1, xν j ∈ (0,1), 0 ≤ s ≤ m, B is an operator in L2 (0,1), Tν and Tν0 are functionals in L2 (0,1). Theorem A.3 (see [6]). Let the following conditions be satisfied: (1) m ≥ 1; mν ≤ m − 1; 0 ≤ s ≤ m; (2) a ∈ C[0,1]; a(x) = 0; a(0) = a(1); supx∈[0,1] arg a(x) − inf x∈[0,1] arg a(x) < 2π if m is even; supx∈[0,1] arg a(x) − inf x∈[0,1] arga(x) < π if m is odd; (3) for all ε > 0, BuL2 (0,1) ≤ εuW2m (0,1) + C(ε)uL2 (0,1) , u ∈ W2m (0,1); (A.8) 112 Hyperbolic diﬀerentialoperator equations on a whole axis (4) functionals Tν in W2mν (0,1) and functionals Tν0 in W2m−ε (0,1), for some ε > 0, are continuous; (5) system (A.4) is pregular with respect to a system of numbers ω j = e2πi(( j −1)/m) , j = 1,...,m (see Theorem A.2). Then for any ε > 0, there exists Rε > 0 such that for all complex numbers λ which satisfy λ > Rε and for m = 2p lying inside the angle, πm πm − π + sup arga(x) + ε < argλ < + π + inf arga(x) − ε, 2 2 x∈[0,1] x∈[0,1] (A.9) for m = 2p + 1 lying inside the angle, πm πm + sup arg a(x) + ε < arg λ < + π + inf arg a(x) − ε, 2 x∈[0,1] 2 x∈[0,1] (A.10) and for m = 2p − 1 lying inside the angle, πm πm − π + sup arg a(x) + ε < arg λ < + inf arg a(x) − ε, 2 2 x∈[0,1] x∈[0,1] (A.11) the operator L(λ) : u → L(λ)u := (L(λ)u,L1 (λ)u,...,Ls (λ)u) from W2m ((0,1); Lν u = 0, ν = s + 1,...,m) onto L2 (0,1)+̇Cs is an isomorphism, and for these λ for a solution of problem (A.7), the estimate uW2m (0,1) + λ uL2 (0,1) + s s Aν0 u ≤ C(ε) f L (0,1) + g ν 2 ν =1 (A.12) ν =1 is valid, where Aν0 is defined by (A.4). Note that if boundaryfunctional conditions (A.7b) and (A.7c) are principally local, that is, αν = 0 or βν = 0 for all ν = 1,...,m, then the condition a(0) = a(1) should be omitted. Acknowledgment The author was supported by the Israel Ministry of Absorption. References [1] [2] [3] [4] [5] P. Lancaster, A. Shkalikov, and Q. Ye, Strongly definitizable linear pencils in Hilbert space, Integral Equations Operator Theory 17 (1993), no. 3, 338–360. C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences, Macmillan Publishing, New York, 1974. K. Maurin, Methods of Hilbert Spaces, Monografie Matematyczne, vol. 45, Państwowe Wydawnictwo Naukowe, Warsaw, 1967, translated from Polish by Andrzej Alexiewicz and Waclaw Zawadowski. H. Triebel, Interpolation Theory, Function Spaces, Diﬀerential Operators, NorthHolland Mathematical Library, vol. 18, NorthHolland, Amsterdam, 1978. S. Yakubov and Ya. Yakubov, An initial boundary value problem for hyperbolic diferentialoperator equations on a finite interval, accepted in Diﬀerential and Integral Equations. Yakov Yakubov 113 [6] [7] [8] , Abel basis of root functions of regular boundary value problems, Math. Nachr. 197 (1999), 157–187. , DiﬀerentialOperator Equations. Ordinary and Partial Diﬀerential Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 103, Chapman & Hall/CRC, Boca Raton, 2000. Ya. Yakubov, Almost periodic solutions and oscillations decay for hyperbolic diﬀerentialoperator equations, Funct. Diﬀer. Equ. 10 (2003), no. 12, 315–330. 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