Long-Time Asymptotics for the Wave Equation of Linear Elasticity in Cylindrical Waveguides

P. H. Lesky

Mathematisches Institut A der Universität Stuttgart, Pfaffenwaldring 57, D 70569 Stuttgart

Abstract

Let Ω = ℝ × D, D ⊂ ℝ^{2} bounded. Suppose that Ω is filled with a homogeneous and isotropic elastic medium, which is fixed at the boundary, and that a time-harmonic force f(x) e^{−iωt} acts in Ω, where f has bounded support. The resulting elastic motion and its behaviour as t → ∞ is studied. Depending on the choice of ω and f, two different types of time-asymptotic occur: Either the motion is unbounded as t → ∞ at almost every x ∈ Ω (resonance case), or the principle of limiting amplitude holds. The resonance is shown to occur for countable many frequencies of incitation. Even two-dimensional elastic motion in a domain Ω = ℝ × (0, 1) is considered; here the same phenomena happen. These results are proved using an explicit representation of the motion, which is obtained combinig spectral- and Fourier-transform. The method is presented in a general setting, so that it is applicable also to other translation-invariant wave equations.

1 Introduction

Let Ω ⊂ ℝ^{3} be filled with a homogeneous and isotropic elastic medium, which is fixed at the boundary ∂Ω. Suppose that a time-harmonic force F (x) e^{−iωt} (with F : Ω → ℝ^{3}) acts in Ω. Denote by s(t, x) ∈ ℝ^{3} the resulting motion of a point x ∈ Ω at time t ≥ 0. Then s is solution of (see e.g. [1] or [13])

\begin{matrix}μ Δs(t, x) + (λ + μ) grad div s(t, x) \\ + σF (t, x) e^{−iωt} & = & σ ∂_{t}^{2}s(t, x) & for t ≥ 0, x ∈ Ω, \\ s(t, x) & = & 0 & for t ≥ 0, x ∈ ∂Ω, \\ s(0, x) = s_{0}(x), ∂_{t}s(0, x) & = & s_{1}(x) & for x ∈ Ω, \end{matrix} \} (1.1)

where σ > 0 (density of mass) and μ > 0, 3λ + 2μ > 0 (Lamé-constants). More general, we study in a domain Ω ⊂ ℝ^{n} the solution u : [0, ∞) × Ω → ℝ^{n} of the initial-boundary-value problem

\begin{matrix}∂_{t}^{2}u(t, x) − (Δ + c_{0} grad div) u(t, x) & = & f(x) e^{−iωt} & for t ≥ 0, x ∈ Ω, \\ u(t, x) & = & 0 & for t ≥ 0, x ∈ ∂Ω, \\ u(0, x) = u_{0}(x), ∂_{t}u(0, x) & = & u_{1}(x) & for x ∈ Ω, \end{matrix} \} (1.2)

where c_{0} > −1 is supposed. The last condition c_{0} > −1 is needed to guarantee the coerciveness of the spatial operator. It is satisfied in all physically relevant situations, since c_{0} = 1 + \frac{λ}{μ}. Let Ω be given by

Ω = ℝ × D, D ⊂ ℝ^{n−1} bounded.  (1.3)

In the case n = 3, if D is connected, then Ω is an infinite tube with cross-section D. If n = 2 and D = (a, b) with −∞ < a < b < ∞, then Ω is a layer.

We are interested in the asymptotic behaviour as t → ∞ of u solving (1.2). In order to give a brief description of the results, we suppose that ∂D ∈ C^{∞}. It will be proved that there exists a set ω_{res} ⊂ ℝ of resonance frequencies, which is countable and has no finite accumulation point. If the frequency ω of incitation coincides with a resonance frequency, then resonance occurs: There exist N ∈ ℕ, α_{1}, …, α_{N−1} ∈ (0, 1) and f_{1}, …, f_{N}, u_{ω} ∈ C(\overline{Ω}), such that

u(t, x) = ∑_{j=1}^{N−1} e^{−iωt} t^{α_{j}} f_{j}(x) + e^{−iωt} ln t · f_{N}(x) + e^{−iωt} u_{ω}(x) + o(1)  (1.4)

as t → ∞, where the precise definition of the meaning of o(1) is given in Theorem 5.2. The functions f_{1}, …, f_{N} depend on f and can be computed together with the values of α_{1}, …, α_{N−1} ∈ (0, 1) solving a parameter-dependent eigenvalue problem in D. This will be done for a special example of D in a subsequent paper. The present article shows, that at least one of f_{1}, …, f_{N} does not vanish, if f is chosen suitable, and that the corresponding exponent α_{j} is element of [\frac{1}{2}, 1).

If ω ∈ [0, ∞) \ ω_{res}, then u satisfies the principle of limiting amplitude

u(t, x) = e^{−iωt} u_{ω}(x) + o(1) as t → ∞,  (1.5)

where u_{ω} ∈ C^{2}(\overline{Ω}) solves

\begin{matrix}(−Δ − c_{0}grad div − ω^{2})u_{ω}(x) & = & f(x) & in Ω, \\ u(x) & = & 0 & on ∂Ω. \end{matrix} \}  (1.6)

These results are obtained using a method, which was developed in [12] and used in [8]. It is based on an explicit representation of the spectral family of the spatial operator in (1.2), which is computed using Fourier-transform with repect to the unbounded variable. This article presents the method in a more general setting, so that it can be applied to other wave equations being translation invariant.

In order to be more precise, let H be a Hilbert-space given by

H := L_{2}(ℝ, H), H = separable Hilbert-space  (1.7)

(e.g. H = L_{2}(D)^{n}, H = L_{2}(Ω)^{n}). The Fourier-transform with respect to the first variable is defined by

F := F ⊗ Id : H = L_{2}(ℝ) ⊗ H → H (⊗ = tensor product),  (1.8)

with F : L_{2}(ℝ) → L_{2}(ℝ) and Id : H → H denoting respectively usual Fouriertransform and identity. Let A be an operator in H, given by

A = F^{−1} ∘ ∫_{ℝ}^{⊕} A(ξ) dξ ∘ F,  (1.9)

where {A(ξ)}_{ξ∈ℝ} denotes a family of self-adjoint operators in H (for the notation see section XIII.16 of [14]). Roughly spoken, (1.9) means that A(ξ) defines the Fourier-transform of A by

(F ∘ A ∘ F^{−1} f)(ξ) = A(ξ) f(ξ) for almost every ξ ∈ ℝ,

if f ∈ D(F ∘ A ∘ F^{−1}) ⊂ L_{2}(ℝ, H). The operator family is supposed to satisfy the following assumptions:

(A1) For every fixed ξ ∈ ℝ, A(ξ) is self-adjoint and positive in H.

(A2) For every fixed ξ ∈ ℝ, A(ξ) has an orthonormal system of eigenfunctions {v_{j}(ξ)}_{j∈ℕ} being complete in H. The corresponding eigenvalues are denoted by λ_{j}(ξ), where every eigenvalue eventually has to be counted multiple times according to his multiplicity, which is supposed to be finite.

(A3) For every fixed j ∈ ℕ, λ_{j}(ξ) and v_{j}(ξ) depend analytically on ξ.

(A4) For every λ ∈ ℝ, the set {(j, ξ) ∈ ℕ × ℝ : λ_{j}(ξ) = λ} is empty or finite, and for every fixed j ∈ ℕ, λ_{j}(ξ) → +∞ as ξ → ±∞.

Then A is self-adjoint in H (see [14], sec. XIII.16).

Section 2 studies the spectral family of A. The spectrum σ(A) of A consists of one infinite interval and is purely absolutely continuous. We say that λ ∈ σ(A) is a resonance point, if and only if

∃(j, ξ) ∈ ℕ × ℝ : λ = λ_{j}(ξ) ∧ \frac{dλ_{j}}{dξ} (ξ) = 0.  (1.10)

The set of all resonance points is countable and has no finite accumulation point. The spectral family {P_{λ}}_{λ∈ℝ} of A will be shown to have a Hölder-continuous derivative with respect to λ (in a certain sense) at every λ ∈ ℝ being no resonance point. If λ_{0} ∈ ℝ is a resonance point, then the derivative with respect to λ of the spectral family has a singularity of type |λ − λ_{0}|^{−α} with α ∈ (0, 1). For a more precise statement see (2.15).

Section 3 studies the solution of the initial value problem

\begin{matrix}u ∈ C^{2}([0, ∞), H), u(t) & ∈ & D(A) & for t ≥ 0. \\ u′′(t) + A u(t) & = & f e^{−iωt} & for t ≥ 0, \\ u(0) = u_{0}, u′(0) & = & u_{1}, \end{matrix} \}  (1.11)

where u′ denotes the derivative of u. The solution u is given by spectral integrals. Theorem 3.2 shows the connection between the long-time behaviour of u and the properties of the spectral family {P_{λ}f} at λ = ω^{2} and at λ = 0. If ω^{2} is a resonance point and if f is chosen suitable, then u shows resonance. If ω^{2} is no resonance point and if the behaviour of the spectral family near λ = 0 is not too bad, then the principle of limiting amplitude holds.

Section 4 studies spectral properties of the elastic spatial operator. It is shown that the theory developed in Sections 2 and 3 can be applied to problem (1.2). In section 5, the long-time behaviour of the solution u of (1.2) is computed. Application of the general theory yields estimates with respect to some weighted norms. Using elliptic regularity theory, pointwise estimates of u(t, x) as t → ∞ are proved.

Equation (1.1) with other (non homogeneous) boundary conditions in a domain Ω = ℝ^{2} × (0, h) is considered in the modelling of seismic waves. In this context, L. Brevdo obtained similar resonance effects in [3], [4] and [5]. He studies the solution using spatial Fourier-transform and Laplace-transform with respect to time. This method was developed by R. Briggs in [6]. The theory in this book leads to the same definition of resonance points as (1.10). The considerations there are not restricted to self-adjoint spatial operators. But one has to pay for this generality: It has to be distinguished between resonance points leading to convective instabilities and those leading to absolute instabilities. The result of the present paper shows in the case of a self-adjoint spatial operator, that at every resonance point an absolute instability occurs.

I want to thank P. Werner and J. Giannoulis for various discussions. Thanks to S. Müller, who gave me a hint how to optimize the underlying Sobolev-spaces.

2 Spectral properties of A

We study the spectral family {P_{λ}} of the self-adjoint operator A given by (1.9). The main result in this section consists of an explicit representation of the spectral family (see Theorem 2.3).

Suppose that the operator family {A(ξ)}_{ξ∈ℝ} satisfies Assumptions (A1) – (A4) given on page 4. Set

λ_{min} := min {λ_{j}(ξ) : j ∈ ℕ, ξ ∈ ℝ} .  (2.1)

From (A3) and (A4), one obtains that the given set has a minimum. Furthermore, (A1) implies that λ_{min} ≥ 0. Define the set of resonance points by

σ_{res}(A) := \{λ_{j}(ξ) : j ∈ ℕ ∧ ξ ∈ ℝ ∧ \frac{dλ_{j}}{dξ} (ξ) = 0\} .  (2.2)

By (A3) and (A4), σ_{res}(A) is countable infinite and has no finite accumulation point. Obviously λ_{min} ∈ σ_{res}(A). Write σ_{res}(A) as

σ_{res}(A) = {σ_{1}, σ_{2}, … } with λ_{min} = σ_{1} < σ_{2} < …, σ_{j} → ∞ as j → ∞. (2.3)

In order to define local inverses of the mapping ξ ↦ λ_{j}(ξ), let j ∈ ℕ be fixed. By the analycity of λ_{j}, the set

S_{j} := \{ξ ∈ ℝ : \frac{dλ_{j}}{dξ} (ξ) = 0\}

is finite or countable with no finite accumulation point. Hence one can write

S_{j} = {ρ_{jk} : k ∈ 𝕀_{j}} with ρ_{jk} < ρ_{j(k+1)} if k, k + 1 ∈ 𝕀_{j},

where 𝕀_{j} = {1, …, N} or 𝕀_{j} = ℕ or 𝕀_{j} = ℤ. Define intervals I_{jk} by

I_{jk} := \{ \begin{matrix}(−∞, ρ_{j(k+1)}] & if k ∉ 𝕀_{j}, k + 1 ∈ 𝕀_{j}, \\ [ρ_{jk}, ρ_{j(k+1)}] & if k, k + 1 ∈ 𝕀_{j}, \\ [ρ_{jk}, ∞) & if k ∈ 𝕀_{j}, k + 1 ∉ 𝕀_{j} \end{matrix}  (2.4)

for k ∈ 𝕀̃_{j}, where 𝕀̃_{j} := 𝕀_{j}∪{min{k ∈ 𝕀_{j}}−1} if 𝕀_{j} is bounded from below, and 𝕀̃_{j} := 𝕀_{j} otherwise. Then {I_{jk}}_{k∈𝕀̃_{j}} defines a decomposition of ℝ having the property, that the restriction λ_{j} : I_{jk} → λ_{j}(I_{jk}) is one-to-one for every k ∈ 𝕀̃_{j}. Let r_{jk} : λ_{j}(I_{jk}) → I_{jk} denote the inverse mapping. Note that the domain of definition of r_{jk} consists of a closed interval. By the analycity of λ_{j}, r_{jk} is C^{∞} in the interior of λ_{j}(I_{jk}). Define K(λ) by

K(λ) := {(j, k) ∈ ℕ × ℕ : λ ∈ λ_{j}(I_{jk})} for λ ∈ ℝ.  (2.5)

This set is empty, if λ < λ_{min}. Otherwise, K(λ) contains all pairs (j, k) ∈ ℕ × ℕ having the property, that λ is in the domain of definition of r_{jk}. According to (A4), K(λ) is finite for every λ ≥ λ_{min}. Remember that σ_{res}(A) = {σ_{1}, σ_{2}, … } and note that K(λ) is constant on (σ_{p}, σ_{p+1}) for every p ∈ ℕ. For every σ_{p} ∈ σ_{res}(A) set

K^{+}(σ_{p}) := \{(j, k) ∈ K(λ) with λ ∈ (σ_{p}, σ_{p+1}) : \frac{dλ_{j}}{dξ} (r_{jk}(σ_{p})) = 0\} ,
K^{−}(σ_{p}) := \{(j, k) ∈ K(λ) with λ ∈ (σ_{p−1}, σ_{p}) : \frac{dλ_{j}}{dξ} (r_{jk}(σ_{p})) = 0\} (p ≠ 1),
K^{−}(σ_{1}) := ∅,

(2.6)

N(j, k, p) := min \{l ≥ 2 : \frac{d^{l}λ_{j}}{dξ^{l}} (r_{jk}(σ_{p})) ≠ 0\} for (j, k) ∈ K^{+}(σ_{p}) ∪ K^{−}(σ_{p}). (2.7)

Lemma 2.1 1) If σ_{p}, σ_{p+1} ∈ σ_{res}(A) and (j, k) ∈ K(λ) for λ ∈ (σ_{p}, σ_{p+1}), then r_{jk} ∈ C^{∞}(σ_{p}, σ_{p+1}).

2) If σ_{p} ∈ σ_{res}(A), p ≥ 2, and if (j, k) ∈ K(σ_{p}) \ (K^{+}(σ_{p}) ∪ K^{−}(σ_{p})), then r_{jk} ∈ C^{∞}(σ_{p−1}, σ_{p+1}). Furthermore K(σ_{1}) = K^{+}(σ_{1}).

3) If σ_{p} ∈ σ_{res}(A) and (j, k) ∈ K^{+}(σ_{p}), then there exist constants c_{jkpq} ∈ ℝ, c_{jkp1} ≠ 0 and δ > 0 such that

\frac{dr_{jk}}{dλ} (λ) = ∑_{q=1}^{∞} \frac{c_{jkpq}}{|λ − σ_{p}|^{1−q/N(j,k,p)}}  (2.8)

for λ ∈ (σ_{p}, σ_{p} + δ). If (j, k) ∈ K^{−}(σ_{p}), then (2.8) holds for λ ∈ (σ_{p} − δ, σ_{p}) with suitable chosen constants c_{jkpq} ∈ ℝ, c_{jkp1} ≠ 0 and δ > 0.

4) If σ_{p} ∈ σ_{res}(A) and (j, k) ∈ K^{+}(σ_{p})∪K^{−}(σ_{p}), then either (j, k+1) ∈ K^{+}(σ_{p})∪ K^{−}(σ_{p}) or (j, k−1) ∈ K^{+}(σ_{p})∪K^{−}(σ_{p}). If (j, k), (j, k+1) ∈ K^{+}(σ_{p})∪K^{−}(σ_{p}), then the coefficients in (2.8) satisfy

c_{jkp1} = (−1)^{N(j,k,p)+1} c_{j(k+1)p1}.  (2.9)

Proof: Note that σ_{res}(A) = ⋃_{j=1}^{∞} λ_{j}(S_{j}). If (j, k) ∈ K(λ) for λ ∈ (σ_{p}, σ_{p+1}), then (σ_{p}, σ_{p+1}) is subset of the interior of λ_{j}(I_{jk}). This implies r_{jk} ∈ C^{∞}(σ_{p}, σ_{p+1}) by the considerations made before the lemma.

If σ_{p} ∈ σ_{res}(A), p ≥ 2, and if (j, k) ∈ K(σ_{p}) \ (K^{+}(σ_{p}) ∪ K^{−}(σ_{p})), then (σ_{p−1}, σ_{p+1}) is subset of the interior of λ_{j}(I_{jk}) and r_{jk} ∈ C^{∞}(σ_{p−1}, σ_{p+1}) follows as before. Assume that (j, k) ∈ K(σ_{1}). Since λ_{j}(r_{jk}(σ_{1})) = σ_{1} and λ_{j}(ξ) ≥ σ_{1} = λ_{min} for ξ ∈ ℝ, \frac{dλ_{j}}{dξ} (r_{jk}(σ_{1})) has to vanish. Hence (j, k) ∈ K^{+}(σ_{1}). On the other hand, K^{+}(σ_{1}) ⊂ K(σ_{1}) holds by definition.

If σ_{p} ∈ σ_{res}(A) and (j, k) ∈ K^{+}(σ_{p}) ∪ K^{−}(σ_{p}), then r_{jk}(σ_{p}) is a boundary point of I_{jk}. This implies that σ_{p} = λ_{j}(r_{jk}(σ_{p})) is a boundary point of λ_{j}(I_{jk}), the domain of definition of r_{jk}. According to [2], §21, r_{jk} can be represented by a puiseux-series

r_{jk}(λ) = r_{jk}(σ_{p}) + ∑_{q=1}^{∞} c̃_{jkpq} |λ − σ_{p}|^{q/N(j,k,p)}

for λ ∈ λ_{j}(I_{jk}) ∩ (σ_{p} − δ, σ_{p} + δ) with suitable chosen c̃_{jkpq} ∈ ℝ and δ > 0. Hence (2.8) holds. Since σ_{p} = λ_{j}(r_{jk}(σ_{p})) is a boundary point of λ_{j}(I_{jk}), σ_{p} is either a boundary point of λ_{j}(I_{j(k−1)}) or of λ_{j}(I_{j(k+1)}). This implies that either (j, k + 1) ∈ K^{+}(σ_{p}) ∪ K^{−}(σ_{p}) or (j, k − 1) ∈ K^{+}(σ_{p}) ∪ K^{−}(σ_{p}). Suppose that (j, k), (j, k + 1) ∈ K^{+}(σ_{p}) ∪ K^{−}(σ_{p}). Then r_{jk}(σ_{p}) = r_{j(k+1)}(σ_{p}) and N(j, k, p) = N(j, k + 1, p). Relation (2.9) follows inserting the puiseux-series representation of r_{jk} and r_{j(k+1)} into the Taylor expansion of λ_{j} at ξ = r_{jk}(σ_{p}) and using that λ_{j}(r_{jk}(λ)) = λ for λ ∈ λ_{j}(I_{jk}) ∪ λ_{j}(I_{j(k+1)}). □

For s ∈ ℝ define the Banach space H_{s} by

\begin{matrix}∥φ∥_{s} & := & (∫_{ℝ} (1 + |x|^{2})^{s} ∥φ(x)∥_{H}^{2} dx)^{1/2} , \\ H_{s} & := & L_{2,s}(ℝ, H) := \{φ ∈ L_{2,loc}(ℝ, H) : ∥φ∥_{s} < ∞\} \end{matrix} \}  (2.10)

Lemma 2.2 Suppose that s, s′ > \frac{1}{2} and f ∈ H_{s}. Set

ψ_{j}(ξ)(x) := e^{iξx} ⟨(F f)(ξ), v_{j}(ξ)⟩_{H} v_{j}(ξ)  for ξ, x ∈ ℝ, j ∈ ℕ.  (2.11)

Then ψ_{j}(ξ) ∈ H_{−s′} for ξ ∈ ℝ and ψ_{j} ∈ C^{k,α}(ℝ, H_{−s′}) with k ∈ ℕ_{0}, k < min\{s, s′\}− \frac{1}{2} and α := min \{s − k − \frac{1}{2}, s′ − k − \frac{1}{2}, 1\}.

Proof: If v ∈ H, then

⟨(Ff)(ξ), v⟩_{H} = F (⟨f(.), v⟩_{H})(ξ)

(see (1.8)). The mapping ξ ↦ F (⟨f(.), v⟩_{H})(ξ) is element of H^{s}(ℝ), since f ∈ H_{s}. By Sobolev’s Lemma, H^{s}(ℝ) ⊂ C^{k,α}(ℝ). Since v_{j} depends analytically on ξ by Assumption (A3), the mapping ξ ↦ ⟨(Ff)(ξ), v_{j}(ξ)⟩_{H} v_{j}(ξ) is element of C^{k,α}(ℝ, H).

Note that H_{−s′} is adjoint to H_{s′} with respect to the inner product in H. If g ∈ H_{s′}, then

⟨ψ_{j}(ξ), g⟩_{H} = ∫_{ℝ} e^{iξx} ⟨(Ff)(ξ), v_{j}(ξ)⟩_{H} ⟨v_{j}(ξ), g(x)⟩_{H} dx
= \sqrt{2π} ⟨(Ff)(ξ), v_{j}(ξ)⟩_{H} ⟨v_{j}(ξ), (Fg)(ξ)⟩_{H} .

Hence ψ_{j} ∈ C^{k,α}(ℝ, H_{−s′}) follows from the first part of the proof. □

Theorem 2.3 Let H be a Hilbert-space having the representation (1.7). Suppose that the self-adjoint operator A is given by (1.9) and that the associated operator family satisfies Assumptions (A1) – (A4) (see page 4). Denote by {P_{λ}} the (left-hand continuous) spectral family of A. Then:

1) The spectrum σ(A) of A is purely absolutely continuous and σ(A) = [λ_{min}, ∞), where λ_{min} is defined by (2.1). In particular, A has no eigenvalues and P_{λ} = 0 for λ ≤ λ_{min}.

2) Let s, s′ > \frac{1}{2} and f ∈ H_{s} be fixed (for the definition of H_{s} see (2.10)). Then

P_{λ}f = \frac{1}{\sqrt{2π}} ∫_{λ_{min}}^{λ} ∑_{(j,k)∈K(μ)} |\frac{dr_{jk}}{dμ} (μ)| ψ_{j}(r_{jk}(μ)) dμ  (2.12)

for λ > λ_{min}. (For the definition of ψ_{j}, K(μ) and r_{jk} see respectively (2.11), (2.5) and the text above (2.5)). Furthermore:

(a) Consider the mapping P_{.} f : ℝ → H_{−s′} : λ ↦ P_{λ}f. For every σ_{p}, σ_{p+1} ∈ σ_{res}(A) (see (2.3)),

P_{.} f ∈ C^{1,α}((σ_{p}, σ_{p+1}), H_{−s′}) with α := min\{s − \frac{1}{2}, s′ − \frac{1}{2}, 1\}. (2.13)

(b) If σ_{p} ∈ σ_{res}(A), set

N_{p} := max {N(j, k, p) : (j, k) ∈ K^{+}(σ_{p}) ∪ K^{−}(σ_{p})}  (2.14)

and suppose that s, s′ > N_{p} − \frac{1}{2} and f ∈ H_{s} are fixed. Then there exist bounded operators Q_{jkpl} : H_{s} → H_{−s′}, such that

∥\frac{dP_{λ}f}{dλ} (λ) − ∑_{(j,k)∈K^{+}(σ_{p})} ∑_{l=1}^{N(j,k,p)} \frac{1}{|λ − σ_{p}|^{1−l/N(j,k,p)}} Q_{jkpl}(f)∥_{−s′}
= O(|λ − σ_{p}|^{α/N})  (2.15)

as λ ↓ σ_{p} with α := min \{s − N_{p} + \frac{1}{2}, s′ − N_{p} + \frac{1}{2}, 1\}. The same estimate holds as λ ↑ σ_{p}, if K^{+}(σ_{p}) is replaced by K^{−}(σ_{p}). Furthermore,

Q_{jkp1}(f)(x)

= \frac{1}{\sqrt{2π}} |c_{jkp1}| e^{ix r_{jk}(σ_{p})} ⟨(Ff)(r_{jk}(σ_{p})), v_{j}(r_{jk}(σ_{p}))⟩_{H} v_{j}(r_{jk}(σ_{p})) (2.16)

for x ∈ ℝ, where c_{jkp1} denotes the constant of the first summand in (2.8).

Proof: Set

Â:= ∫_{ℝ}^{⊕} A(ξ) dξ.

According to Theorem XIII.85 in [14], Â is self-adjoint in H. Denote the (left-hand continuous) spectral family in of Â by {P̂_{λ}}_{λ∈ℝ}. Consider a fixed λ ∈ ℝ and set G(t) := 1 if t ≤ λ, G(t) := 0 if t > λ. Theorem XIII.85 in [14] shows that

P̂_{λ} = G(Â) = ∫_{ℝ}^{⊕} G(A(ξ)) dξ = ∫_{ℝ}^{⊕} P_{λ}^{(ξ)} dξ

with {P_{λ}^{(ξ)}} denoting the spectral family of A(ξ) for ξ ∈ ℝ. If g ∈ H, then

(P̂_{λ} g)(ξ) = P_{λ}^{(ξ)}g(ξ) = ∑_{j∈{j∈ℕ:λ_{j}(ξ)<λ}}⟨g(ξ), v_{j}(ξ)⟩_{H} v_{j}(ξ)

for every ξ ∈ ℝ according to Assumption (A2). Note that A = F^{−1}ÂF implies that P_{λ} = F^{−1}P̂_{λ}F for λ ∈ ℝ. Fix f ∈ H_{s}, where s > \frac{1}{2}. Then

(FP_{λ}f)(ξ) = (P̂_{λ} Ff)(ξ) = ∑_{j∈{j∈ℕ:λ_{j}(ξ)<λ}}⟨(Ff)(ξ), v_{j}(ξ)⟩_{H} v_{j}(ξ) .

The right-hand side consists of a finite sum, has bounded support with respect to ξ (see (A4)) and is a continuous mapping from ℝ into H (see the proof of Lemma 2.2). Set L(λ) := {j ∈ ℕ : λ_{j}(ξ) < λ for at least one ξ ∈ ℝ}. By (A4), L(λ) is empty or finite for every λ ∈ ℝ. Application of F^{−1} onto the last equation yields that

P_{λ}f = \frac{1}{\sqrt{2π}} ∑_{j∈L(λ)} ∫_{\{ξ∈ℝ:λ_{j}(ξ)≤λ\}} ψ_{j}(ξ) dξ,

where ψ_{j} depends continuously on ξ (see Lemma 2.2). Obviously P_{λ}f = 0 if λ ≤ λ_{min}. If λ > λ_{min} and j ∈ L(λ),

{ξ ∈ ℝ : λ_{j}(ξ) ≤ λ} = ⋃_{k∈𝕀̃_{j}} (I_{jk} ∩ {ξ ∈ ℝ : λ_{j}(ξ) ≤ λ})

with I_{jk} being defined by (2.4). Note that the sets on the right-hand side have disjoint interiors. Furthermore, I_{jk} ∩ {ξ ∈ ℝ : λ_{j}(ξ) ≤ λ} ≠ ∅ only for finitely many values of k. Substitution ξ = r_{jk}(μ) yields that

∫_{I_{jk}∩\{ξ∈ℝ:λ_{j}(ξ)≤λ\}} ψ_{j}(ξ) dξ = \{ \begin{matrix}∫_{λ_{j}(I_{jk})∩[λ_{min},λ]} ψ_{j}(r_{jk}(μ)) |\frac{dr_{jk}}{dμ} (μ)| dμ \\ if (j, k) ∈ K̃(λ) := ⋃_{μ≤λ} K(μ), \\ 0 & otherwise, \end{matrix}

since the integrand on the right-hand side is continuous in the interior of λ_{j}(I_{jk}) and has integrable singularities at the boundary points of λ_{j}(I_{jk}) according to (2.8). This implies that

P_{λ}f = \frac{1}{\sqrt{2π}} ∑_{j∈L(λ)} ∑_{k∈𝕀̃_{j}}∫_{I_{jk}∩\{ξ∈ℝ:λ_{j}(ξ)≤λ\}} ψ_{j}(ξ) dξ
= \frac{1}{\sqrt{2π}} ∑_{(j,k)∈K̃(λ)} ∫_{λ_{j}(I_{jk})∩[λ_{min},λ]} ψ_{j}(r_{jk}(μ)) |\frac{dr_{jk}}{dμ} (μ)| dμ

Now (2.12) follows. From (2.12) together with Lemmata 2.1 and 2.2 one obtains that (2.13) holds.

Representation (2.12) shows that σ(A) is purely absolutely continuous. For every λ ∈ [λ_{min}, ∞) \ σ_{res}(A), there exists f ∈ H_{s} such that \frac{dP_{λ}f}{dλ} (λ) ≠ 0. This together with P_{λ} = 0 for λ ≤ λ_{min} proves that σ(A) = [λ_{min}, ∞).

If the assumptions of case 2b) are satisfied, Lemma 2.2 implies that

∥ψ_{j}(ξ) − ∑_{l=0}^{N_{p}−1} \frac{1}{l!} \frac{d^{l}ψ_{j}}{dξ^{l}} (r_{jk}(σ_{p})) (ξ − r_{jk}(σ_{p}))^{l}∥_{−s′} = O(|ξ − r_{jk}(σ_{p})|^{N_{p}−1+α})

as ξ → r_{jk}(σ_{p}), where

f ↦ \frac{d^{l}ψ_{j}}{dξ^{l}} (r_{jk}(σ_{p})) : H_{s} → H_{−s′}

are bounded linear operators according to Lemma 2.2. Inserting (2.8) and the corresponding puiseux-series representation of r_{jk} at σ_{p} in the preceding estimate yields (2.15) and (2.16). □

3 Initial-value problems

We study the solution u of (1.11), where A denotes a positive self-adjoint operator in a Hilbert-space H. The long time behaviour of u is given by Theorem 3.2 and Corollary 3.3.

Denote by {P_{λ}}_{λ∈ℝ} the (left-hand continuous) spectral family of A. Then P_{λ} = 0 if λ ≤ 0. For r > 0, set

\begin{matrix}D(A^{r}) & := & \{φ ∈ H : ∫_{0}^{∞} λ^{2r} d(∥P_{λ}φ∥^{2}) < ∞\}, \\ A^{r} φ & := & ∫_{0}^{∞} λ^{r} d(P_{λ}φ) if φ ∈ D(A^{r}). \end{matrix} \}  (3.1)

Theorem 3.1 Let A be a self-adjoint positive operator in the Hilbert-space H. Then:

1) There is at most one solution u of the initial-value problem (1.11).

2) If ω ≥ 0 and u_{0} ∈ D(A^{k/2}), u_{1} ∈ D(A^{(k−1)/2}), f ∈ D(A^{(k−2)/2}) for some k ∈ ℕ, k ≥ 2, then (1.11) has a solution u. Furthermore,

\begin{matrix}A^{(k−j)/2} u & ∈ & C^{j}([0, ∞), H) & (j ≤ k), \\ \frac{d^{j}(A^{i/2}u)}{dt^{j}} & = & A^{i/2} (\frac{d^{j}u}{dt^{j}}) & (j + i ≤ k). \end{matrix} \}  (3.2)

Proof: If u solves (1.11) with vanishing data, then \frac{d}{dt} (∥u′(t)∥^{2} + ⟨Au(t), u(t)⟩) = 0 for t > 0. This implies that u = 0.

By functional calculus, the solution of (1.11) is given by

u(t) = ∫_{0}^{∞} ψ_{ω}(t, λ) d(P_{λ}f)
+ ∫_{0}^{∞} cos \sqrt{λ}t d(P_{λ}u_{0}) + ∫_{0}^{∞} \frac{sin \sqrt{λ}t}{\sqrt{λ}} d(P_{λ}u_{1}),  (3.3)

where in the case ω > 0

ψ_{ω}(t, λ) = \{ \begin{matrix}\frac{1}{λ − ω^{2}} (e^{−iωt} − cos \sqrt{λ}t + \frac{iω}{\sqrt{λ}} sin \sqrt{λ}t) & if λ ∈ ℝ \ \{ω^{2}, 0\}, \\ \frac{i}{2ω} (t e^{−iωt} − \frac{1}{ω} sin ωt) & if λ = ω^{2}, \\ \frac{1}{ω^{2}} (1 − e^{−iωt} − iωt) & if λ = 0, \end{matrix}(3.4)

and

ψ_{0}(t, λ) = \{ \begin{matrix}\frac{1}{λ} (1 − cos \sqrt{λ}t) & if λ ∈ ℝ \ \{0\}, \\ \frac{t^{2}}{2} & if λ = 0. \end{matrix}  (3.5)

From this (3.2) follows by standard calculations. □

Note that ψ_{ω}(t, ω^{2}) and ψ_{ω}(t, 0) are unbounded as t → ∞. The behaviour of u as t → ∞ depends crucially on the behaviour of P_{λ} as λ → ω^{2} and λ ↓ 0. We require, that the operator A satisfies the following assumption:

(P1) H is continuously imbedded in some Banach-space B and there is a subset B̃ ⊂ H, such that for every fixed g ∈ B̃ the mapping P_{·}g : ℝ ↦ B : λ ↦ P_{λ}g is differentiable almost everywhere with derivative \frac{dP_{λ}g}{dλ} ∈ L_{1}((0, M), B) for every M > 0.

If f ∈ B̃ and φ ∈ C([0, M]), then

∫_{0}^{M} φ(λ) d(P_{λ}f) = ∫_{0}^{M} φ(λ) \frac{dP_{λ}f}{dλ} (λ) dλ  (3.6)

(M > 0), where the right-hand side has to be read as a Bochner-integral, see eg. [16]. We will make use of the following theorem:

Theorem 3.2 Suppose that A is a positive and self-adjoint operator in the Hilbert space H and that A obeyes condition (P1) given above. Let ω ≥ 0, f ∈ B̃, u_{1} ∈ D(A^{1/2}) ∩ B̃, u_{2} ∈ D(A) ∩ B̃ be given. Furthermore suppose that the following two assumptions are satisfied:

(P2) There exist N_{f}, N_{u_{1}} ∈ ℕ_{0} and f_{1}^{(0)}, …, f_{[N_{f} /2]}^{(0)}, (u_{1})_{1}^{(0)}, …, (u_{1})_{[N_{u_{1}}/2]}^{(0)} ∈ B (with [\frac{N}{2} ] := max \{j ∈ ℕ_{0} : j ≤ \frac{N}{2} \}), such that for respectively g := f and g := u_{1}

∫_{0}^{1} \frac{1}{\sqrt{λ}} ∥\frac{dP_{λ}g}{dλ} (λ) − ∑_{j=1}^{[N_{g}/2]} \frac{g_{j}^{(0)}}{λ^{1−j/N_{g}}} ∥_{B} dλ < ∞.

(P3) There exist N ∈ ℕ_{0} and f_{1}^{±}, …, f_{N}^{±} ∈ B, such that

∫_{0}^{ω^{2}} \frac{1}{|λ − ω^{2}|} ∥\frac{dP_{λ}f}{dλ} (λ) − ∑_{j=1}^{N} \frac{f_{j}^{−}}{|λ − ω^{2}|^{1−j/N}} ∥_{B} dλ < ∞  (if ω^{2} > 0),
∫_{ω^{2}}^{ω^{2}+1} \frac{1}{λ − ω^{2}} ∥\frac{dP_{λ}f}{dλ} (λ) − ∑_{j=1}^{N} \frac{f_{j}^{+}}{(λ − ω^{2})^{1−j/N}} ∥_{B} dλ < ∞.

Then the solution u of (1.11) given by (3.3) has the following asymptotic behaviour:

1) If ω ≠ 0, then

lim_{t→∞} ∥u(t) − e^{−iωt}I_{1}(t) − I_{2}(t) − e^{−iωt}u_{ω}∥_{B} = 0,  (3.7)

where

I_{1}(t) := ∑_{j=1}^{N−1} t^{1−j/N} \frac{D_{1}(1 − \frac{j}{N})(f_{j}^{+} − f_{j}^{−}) + i D_{2}(1 − \frac{j}{N})(f_{j}^{+} + f_{j}^{−})}{(2ω)^{1−j/N}}
+ ln t · (f_{N}^{+} − f_{N}^{−}),  (3.8)
D_{1}(β) := \frac{π}{2Γ(β + 1) sin \frac{βπ}{2}}  for 0 < β < 2,  (3.9)
D_{2}(β) := \frac{π}{2Γ(β + 1) cos \frac{βπ}{2}}  for −1 < β < 1,  (3.10)
I_{2}(t) := − ∑_{j=1}^{[N_{f} /2]} t^{1−2j/N_{f}} \frac{2i D_{2}(1 − \frac{2j}{N_{f}})}{ω} f_{j}^{(0)}
+ ∑_{j=1}^{[N_{u_{1}}/2]} t^{1−2j/N_{u_{1}}} 2 D_{2}(1 − \frac{2j}{N_{u_{1}}} ) (u_{1})_{j}^{(0)},  (3.11)

u_{ω} := lim_{ε↓0} (∫_{|λ−ω^{2}|≥ε} \frac{1}{λ − ω^{2}} d(P_{λ}f)

− ∑_{j=1}^{N−1} \frac{1}{(1 − \frac{j}{N}) ε^{1−j/N}} (f_{j}^{+} − f_{j}^{−}) + ln ε · (f_{N}^{+} − f_{N}^{−}))

+ \frac{iπ}{2} (f_{N}^{+} + f_{N}^{−}) + (C_{e} − ln(2ω))(f_{N}^{+} − f_{N}^{−})  (3.12)

(C_{e} :=Euler-Mascheroni´ s constant, the limit has to be taken in B).

2) If ω = 0, then

lim_{t→∞} ∥u(t) − I_{3}(t) − u_{0}∥_{B} = 0,  (3.13)

where

I_{3}(t) := ∑_{j=1}^{N−1} t^{2−2j/N} 2 D_{1}(2 − \frac{2j}{N}) f_{j}^{+} + ln t · 2f_{N}^{+}
+ ∑_{j=1}^{[N_{u_{1}}/2]} t^{1−2j/N_{u_{1}}} 2 D_{2}(1 − \frac{j}{N_{u_{1}}} ) (u_{1})_{j}^{(0)},  (3.14)
u_{0} := lim_{ε↓0} (∫_{ε}^{∞} \frac{1}{λ} d(P_{λ}f) − ∑_{j=1}^{N−1} \frac{f_{j}^{+}}{(1 − \frac{j}{N}) δ^{1−j/N}} + ln δ · f_{N}^{+})
+ 2C_{e} f_{N}^{+}  (3.15)

(the limit has to be taken with respect to the norm in B). Proof: Note that

\begin{matrix}D_{1}(β) & = & ∫_{0}^{∞} \frac{1 − cos μ}{μ^{1+β}} dμ & for & 0 < β < 2, \\ D_{2}(β) & = & ∫_{0}^{∞} \frac{sin μ}{μ^{1+β}} dμ & for & −1 < β < 1. \end{matrix} \}  (3.16)

Let ε > 0 be given. The proof proceeds in several steps. Step 1: Since H is supposed to be continuously imbedded in B, we can choose M > ω^{2} + 1, such that

∥∫_{M}^{∞} ψ_{ω}(t, λ) d(P_{λ}f) + ∫_{M}^{∞} cos \sqrt{λ}t d(P_{λ}u_{0}) + ∫_{M}^{∞} \frac{sin \sqrt{λ}t}{\sqrt{λ}} d(P_{λ}u_{1})∥_{B}^{2}
≤ c ∥∫_{M}^{∞} ψ_{ω}(t, λ) d(P_{λ}f) + ∫_{M}^{∞} cos \sqrt{λ}t d(P_{λ}u_{0}) + ∫_{M}^{∞} \frac{sin \sqrt{λ}t}{\sqrt{λ}} d(P_{λ}u_{1})∥_{H}^{2}
≤ 3c (∫_{M}^{∞} 9 d(∥P_{λ}f∥_{H}^{2}) + ∫_{M}^{∞} 1 d(∥P_{λ}u_{0}∥_{H}^{2}) + ∫_{M}^{∞} 1 d(∥P_{λ}u_{1}∥_{H}^{2}))
< ε^{2}

and

∥∫_{M}^{∞} \frac{1}{λ − ω^{2}} d(P_{λ}f)∥_{B} < ε.

Step 2: By the Riemann-Lebesgue-Lemma (see eg. [9]), we have

∥∫_{0}^{M} cos \sqrt{λ}t d(P_{λ}f)∥_{B} = ∥Re ∫_{0}^{√\overline{M}} e^{iμt} \frac{dP_{λ}f}{dλ} (μ^{2}) 2μ dμ∥_{B} < ε

for t > T_{1}. Here and in the following, T_{1}, T_{2}, … denote suitable chosen positive real numbers.

Step 3: By (P2), we can choose δ > 0, such that

∥∫_{0}^{δ} \frac{sin \sqrt{λ}t}{\sqrt{λ}} (\frac{dP_{λ}u_{1}}{dλ} (λ) − ∑_{j=1}^{[N_{u_{1}}/2]} \frac{(u_{1})_{j}^{(0)}}{λ^{1−j/N_{u_{1}}}} ) dλ∥_{B}
≤ ∫_{0}^{δ} \frac{1}{\sqrt{λ}} ∥\frac{dP_{λ}u_{1}}{dλ} (λ) − ∑_{j=1}^{[N_{u_{1}}/2]} \frac{(u_{1})_{j}^{(0)}}{λ^{1−j/N_{u_{1}}}} ∥_{B} dλ
< ε.

Riemann-Lebesgue´ s Lemma yields as above

∥∫_{δ}^{M} \frac{sin \sqrt{λ}t}{\sqrt{λ}} d(P_{λ}f)∥_{B} < ε for t > T_{2}.

With (3.16), we obtain for j ∈ \{1, …, [\frac{N_{u_{1}}}{2} ]\}

∫_{0}^{δ} \frac{sin \sqrt{λ}t}{\sqrt{λ}} \frac{1}{λ^{1−j/N_{u_{1}}}} dλ = 2 t^{1−2j/N_{u_{1}}} ∫_{0}^{∞} \frac{sin μ}{μ^{2−2j/N_{u_{1}}}} dμ + O(\frac{1}{t})
= 2 t^{1−2j/N_{u_{1}}} D_{2}(1 − \frac{2j}{N_{u_{1}}} ) + O(\frac{1}{t})

as t → ∞. Combining all estimates of this step, we conclude that

∥∫_{0}^{M} \frac{sin \sqrt{λ}t}{\sqrt{λ}} d(P_{λ}u_{1}) − ∑_{j=1}^{[N_{u_{1}}/2]} t^{1−2j/N_{u_{1}}} 2D_{2}(1 − \frac{2j}{N_{u_{1}}} ) (u_{1})_{j}^{(0)}∥_{B} < 3ε

for t > T_{3}. Step 4: From here to Step 6, the case ω ≠ 0 will be considered. Note that

ψ_{ω}(t, λ) = e^{−iωt} \frac{1 − e^{−i(\sqrt{λ}−ω)t}}{λ − ω^{2}} − \frac{i sin \sqrt{λ}t}{ω \sqrt{λ}} + \frac{i sin \sqrt{λ}t}{ω (\sqrt{λ} + ω)}

The same argument used in Step 3 shows that

∥∫_{0}^{M} (−\frac{i sin \sqrt{λ}t}{ω \sqrt{λ}} + \frac{i sin \sqrt{λ}t}{ω (\sqrt{λ} + ω)}) d(P_{λ}f)

+ ∑_{j=1}^{[N_{f} /2]} t^{1−2j/N_{f}} \frac{2i D_{2}(1 − \frac{2j}{N_{f}})}{ω} f_{j}^{(0)}∥_{B} < ε

for t > T_{4}. Step 5: Assumption (P3) implies, that the limit

μ_{ω}^{−} := lim_{δ↓0} (∫_{0}^{ω^{2}−δ} \frac{1}{λ − ω^{2}} \frac{dP_{λ}f}{dλ} (λ) dλ + ∑_{j=1}^{N−1} \frac{f_{j}^{−}}{(1 − \frac{j}{N}) δ^{1−j/N}} − ln δ · f_{N}^{−})

exists in B. Choose δ > 0 so small, such that

∥∫_{0}^{ω^{2}−δ} \frac{1}{λ − ω^{2}} \frac{dP_{λ}f}{dλ} (λ) dλ + ∑_{j=1}^{N−1} \frac{f_{j}^{−}}{(1 − \frac{j}{N}) δ^{1−j/N}} − ln δ · f_{N}^{−} − μ_{ω}^{−}∥_{B} < ε

and

∫_{ω^{2}−δ}^{ω^{2}} \frac{|1 − e^{−i(\sqrt{λ}−ω)t}|}{|λ − ω^{2}|} ∥\frac{dP_{λ}f}{dλ} (λ) − ∑_{j=1}^{N} \frac{f_{j}^{−}}{|λ − ω^{2}|^{1−j/N}} ∥_{B} dλ < ε

(see (P3)). Let β ∈ (0, 1) be fixed. Standard calculations (see e.g. [11], (6.2)– (6.9) and the equation following (6.29)) show, that for every ε̃ > 0, there exists δ(ε̃) > 0, such that for every δ ∈ (0, δ(ε̃)) there is an T_{5}(ε̃, δ) > 0 with

|∫_{ω^{2}−δ}^{ω^{2}} \frac{1 − e^{−i(\sqrt{λ}−ω)t}}{(λ − ω^{2})|λ − ω^{2}|^{β}} dλ − \frac{1}{β δ^{β}} + \frac{t^{β}}{(2ω)^{β}} ∫_{0}^{∞} \frac{1 − e^{iμ}}{μ^{(1+β)}} dμ| < ε̃

and

|∫_{ω^{2}−δ}^{ω^{2}} \frac{1 − e^{−i(\sqrt{λ}−ω)t}}{λ − ω^{2}} dλ + ln δ + ln t + C_{e} − ln(2ω) − \frac{iπ}{2} | < ε̃

for t > T_{5}(ε̃, δ). By Riemann-Lebesgue,

∥∫_{0}^{ω^{2}−δ} \frac{e^{−i(\sqrt{λ}−ω)t}}{λ − ω^{2}} d(P_{λ}f)∥_{B} < ε for t > T_{6},

if δ > 0 is fixed. This, together with (3.16), proves

∥∫_{0}^{ω^{2}} \frac{1 − e^{−i(\sqrt{λ}−ω)t}}{λ − ω^{2}} d(P_{λ}f) − μ_{ω}^{−} + (C_{e} − ln(2ω) − \frac{iπ}{2} ) f_{N}^{−}

+ ∑_{j=1}^{N−1} t^{1−j/N} \frac{D_{1}(1 − \frac{j}{N}) − iD_{2}(1 − \frac{j}{N})}{(2ω)^{1−j/N}} f_{j}^{−} + ln t · f_{N}^{−}∥_{B} < 5ε

for t > T_{7}. Step 6: As above,

∥∫_{ω^{2}}^{M} \frac{1 − e^{−i(\sqrt{λ}−ω)t}}{λ − ω^{2}} d(P_{λ}f) − μ_{ω}^{+} − (C_{e} − ln(2ω) + \frac{iπ}{2} ) f_{N}^{+}

− ∑_{j=1}^{N−1} t^{1−j/N} \frac{D_{1}(1 − \frac{j}{N}) + iD_{2}(1 − \frac{j}{N})}{(2ω)^{1−j/N}}f_{j}^{+} + ln t · f_{N}^{+}∥_{B} < ε

for t > T_{8} with

μ_{ω}^{+} := lim_{δ↓0} (∫_{ω^{2}+δ}^{M} \frac{1}{λ − ω^{2}} \frac{dP_{λ}f}{dλ} (λ) dλ − ∑_{j=1}^{N−1} \frac{f_{j}^{+}}{(1 − \frac{j}{N}) δ^{1−j/N}} + ln δ · f_{N}^{+})

is proved. Combining the results of Step 1 to Step 6, we obtain that (3.7) holds.

The case ω = 0 is proved in the same way by

|∫_{0}^{δ} \frac{1 − cos \sqrt{λ}t}{λ^{1+β}} dλ − 2 t^{2β} ∫_{0}^{∞} \frac{1 − cos μ}{μ^{1+2β}} dμ + \frac{1}{β δ^{β}} | < ε̃ (β ∈ (0, 1)),
|∫_{0}^{δ} \frac{1 − cos \sqrt{λ}t}{λ} dλ − 2 ln t − ln δ − 2C_{e}| < ε̃

for t > T_{9}(ε̃, δ) (use (3.67) in [15]). □

Corollary 3.3 (Regularity estimates) Let all Assumptions of Theorem 3.2 be satisfied. In addition, suppose that u_{0} ∈ D(A^{k}), u_{1} ∈ D(A^{k−1/2}), f ∈ D(A^{k−1}) for a fixed k ∈ ℕ. Denote by u the solution of (1.11).

1) General case: If ω ≠ 0, then

lim_{t→∞} ∥A^{j}u(t) − e^{−iωt} ω^{2j} I_{1}(t) − e^{−iωt} u_{ω}^{(j)}∥_{B} = 0 for j = 1, 2, …, k

(3.17)

with I_{1}(t) being defined in Theorem 3.2 and

u_{ω}^{(j)} := lim_{ε↓0} (∫_{|λ−ω^{2}|≥ε} \frac{λ^{j}}{λ − ω^{2}} d(P_{λ}f)

− ∑_{l=1}^{N−1} \frac{ω^{2j}}{(1 − \frac{l}{N}) ε^{1−l/N}} (f_{l}^{+} + f_{l}^{−}) + ln ε · ω^{2j}(f_{N}^{+} − f_{N}^{−}))

+ \frac{iπ ω^{2j}}{2} (f_{N}^{+} + f_{N}^{−}) + ω^{2j} (C_{e} − ln(2ω)) (f_{N}^{−} − f_{N}^{+})  (3.18)

(j ≤ k), where the limit has to be taken in B. If ω = 0, then

lim_{t→∞} ∥A^{j}u(t) − A^{j−1}f∥_{B} = 0  for j = 1, 2, …, k.  (3.19)

2) Principle of limiting Amplitude: Suppose that additionally N_{f} = N_{u_{1}} = 0 in (P2) of Theorem 3.2 and that N = 1, f_{1}^{+} = f_{1}^{−} (= \frac{dP_{λ}}{dλ} (ω^{2})) in (P3). Then

lim_{t→∞} ∥A^{j} u(t) − e^{−iωt} u_{ω}^{(j)}∥_{B} = 0  for j = 0, 1, …, k  (3.20)

with

u_{ω}^{(j)} := lim_{ε↓0} ∫_{|λ−ω^{2}|≥ε} \frac{λ^{j}}{λ − ω^{2}} d(P_{λ}f) + iπ ω^{2j} f_{1}^{+}  (3.21)

(j ≤ k). The limit in (3.21) has to be taken in B.

Proof: For the first part, use the arguments of the proof of Theorem 3.2. The additional assumption is needed only in Step 1. From Step 2 on, replace \frac{dP_{λ}f}{dλ} (λ) by λ^{k} \frac{dP_{λ}f}{dλ} (λ).

The assertion of the second part follows directly from part 1, if j ≥ 1. The case j = 0 is obtained from Theorem 3.2. Note that f_{1}^{+} = f_{1}^{−} = 0 if ω = 0, since P_{λ}f = 0 for λ ≤ 0. □

The following theorem characterizes the limiting amplitude (3.21) by the principle of limiting absorption.

Theorem 3.4 Suppose that A is a positive and self-adjoint operator in the Hilbert space H and that A obeyes the condition (P1) given before Theorem 3.2. Let ω ≥ 0 and f ∈ B̃ be given and assume that (P3) of Theorem 3.2 is satisfied with N = 1 and f_{1}^{+} = f_{1}^{−}. Then u_{ω}^{(0)} defined by (3.21) exists in B and

lim_{τ↓0} ∥R_{ω^{2}+iτ}f − u_{ω}^{(0)}∥_{B} = 0.  (3.22)

Proof: Conclude from (P1) and (P3) (with N = 1 and f_{1}^{+} = f_{1}^{−}), that the limit u_{ω}^{(0)} exists in B. Recall that

R_{ω^{2}+iτ}f = ∫_{0}^{∞} \frac{1}{λ − ω^{2} − iτ} d(P_{λ}f).

Let ε > 0 be given. Choose δ > 0, such that

∥∫_{ω^{2}−δ}^{ω^{2}+δ} \frac{1}{λ − ω^{2} − iτ} (\frac{dP_{λ}f}{dλ} (λ) − f_{1}^{+}) dλ∥_{B}
≤ ∫_{ω^{2}−δ}^{ω^{2}+δ} \frac{1}{|λ − ω^{2}|} ∥\frac{dP_{λ}f}{dλ} (λ) − f_{1}^{+}∥_{B} dλ
< ε

and

∥∫_{|λ−ω^{2}|≥δ} \frac{1}{λ − ω^{2}} d(P_{λ}f) + iπf_{1}^{+} − u_{ω}^{(0)}∥_{B} < ε

with u_{ω}^{(0)} being defined by (3.21). There exists τ_{0} > 0, such that for τ ∈ (0, τ_{0})

∥∫_{|λ−ω^{2}|≥δ} \frac{1}{λ − ω^{2} − iτ} d(P_{λ}f) − ∫_{|λ−ω^{2}|≥δ} \frac{1}{λ − ω^{2}} d(P_{λ}f)∥_{B}
≤ c ∥∫_{|λ−ω^{2}|≥δ} \frac{τ}{(λ − ω^{2})(λ − ω^{2} − iτ)} d(P_{λ}f)∥_{H}
< ε

and

∥∫_{ω^{2}−δ}^{ω^{2}+δ} \frac{1}{λ − ω^{2} − iτ} f_{1}^{+} dλ − iπ f_{1}^{+}∥_{B} < ε.

This proves (3.22). □

4 Spectral properties of elastic operator

Let Ω ⊂ ℝ^{n} be given by (1.3) and set

\begin{matrix}D(A) & := & \{φ ∈ ^{∘} H^{1}(Ω)^{n} : (Δ + c_{0} grad div)φ ∈ L_{2}(Ω)^{n}\} , \\ A φ & := & −(Δ + c_{0} grad div)φ for φ ∈ D(A), \end{matrix} \}  (4.1)

where c_{0} > −1 is supposed, the derivatives have to be taken in distributional sense and ^{∘} H^{1}(Ω) denotes the closure of C_{0}^{∞}(Ω) in H^{1}(Ω). Standard calculations show that

⟨A φ, φ⟩ ≥ min{1, 1 + c_{0}} (∥φ∥_{1}^{2} − ∥φ∥_{0}^{2})  for φ ∈ D(A)  (4.2)

(∥.∥_{k} := norm in H^{k}(Ω)^{n}) and that A is self-adjoint in L_{2}(Ω)^{n}.

Note that L_{2}(Ω)^{n} = L_{2}(ℝ, L_{2}(D)^{n}), since Ω is given by (1.3). In this section we define an operator family {A(ξ)}_{ξ∈ℝ} in L_{2}(D), such that A can be represented by (1.9). We show that the theory developed in section 2 can be applied to obtain a representation of the spectral family of A.

Let the variable in D be denoted by (x_{2}, …, x_{n}) and set

grad_{D} := (∂_{2}, …, ∂_{n})^{T} , div_{D} := (∂_{2}, …, ∂_{n}), Δ_{D} := ∂_{2}^{2} + … + ∂_{n}^{2}.

Consider the dyadic product grad_{D} div_{D} defining a (n − 1) × (n − 1)-matrix. The n × n-matrix given by

(\begin{matrix}−ξ^{2} & iξdiv_{D} \\ iξgrad_{D} & grad_{D} div_{D} \end{matrix} )

contains (−ξ^{2}, iξ∂_{2}, iξ∂_{3}, …, iξ∂_{n}) in the first line and (−ξ^{2}, iξ∂_{2}, iξ∂_{3}, …, iξ∂_{n})^{T} in the first column. E.g. in the case n = 3,

(\begin{matrix}−ξ^{2} & iξdiv_{D} \\ iξgrad_{D} & grad_{D} div_{D} \end{matrix} ) = ( \begin{matrix}−ξ^{2} & iξ∂_{2} & iξ∂_{3} \\ iξ∂_{2} & iξ∂_{2}^{2} & iξ∂_{2}∂_{3} \\ iξ∂_{3} & iξ∂_{3}∂_{2} & iξ∂_{3}^{2} \end{matrix} ) .

Define the operator A(ξ) for ξ ∈ ℝ by

D(A(ξ)) := \{φ ∈ ^{∘} H^{1}(D)^{n} : (Δ_{D} + c_{0}(\begin{matrix}0 & 0 \\ 0 & grad_{D} div_{D} \end{matrix} ))φ ∈ L_{2}(D)^{n}\},
A(ξ)φ := −(Δ_{D} − ξ^{2} + c_{0}(\begin{matrix}−ξ^{2} & iξdiv_{D} \\ iξgrad_{D} & grad_{D} div_{D} \end{matrix} ))φ for φ ∈ D(A(ξ)).

(4.3)

Let ξ ∈ ℝ be fixed. As above,

⟨A(ξ)φ, φ⟩_{D} ≥ min{1, 1 + c_{0}} (∑_{j=1}^{n} ∥grad_{D} φ_{j}∥_{D}^{2} + ξ^{2} ∥φ∥_{D}^{2}) ≥ 0  (4.4)

for φ = (φ_{1}, …, φ_{n}) ∈ D(A(ξ)) is proved. Furthermore, A(ξ) is self-adjoint in L_{2}(D)^{n}.

Lemma 4.1 Suppose that D has the segment property and that A(ξ) is defined by (4.3). Then:

1) For every fixed ξ ∈ ℝ, A(ξ) has an orthonormal system of eigenfunctions {v_{j}(ξ)}_{j∈ℕ} being complete in L_{2}(D)^{n}. Eigenfunctions and associated eigenvalues {λ_{j}(ξ)}_{j∈ℕ} can be chosen in a way, such that for every fixed j ∈ ℕ the mappings ξ ↦ v_{j}(ξ), ξ ↦ λ_{j}(ξ) are analytic on ℝ.

2) For every ξ ∈ ℝ, λ_{j}(ξ) > 0 for j ∈ ℕ and λ_{j}(ξ) → ∞ as j → ∞.

3) For every j ∈ ℕ, there exists k ∈ ℕ, such that λ_{j}(ξ) = λ_{k}(−ξ) for ξ ∈ ℝ.

Proof: Let ξ ∈ ℝ be fixed. Using Rellich’s selection theorem, one obtains from (4.4) that (A(ξ) + Id)^{−1} : L_{2}(D)^{n} → L_{2}(D)^{n} is compact. Furthermore this operator is symmetric and positive. The theory of compact symmetric operators shows that (A(ξ) + Id)^{−1} has an orthonormal system {v_{1}(ξ), v_{2}(ξ), … } of eigenfunctions being complete in L_{2}(D)^{n}. Denote by μ_{j}(ξ) the eigenvalue of (A(ξ) + Id)^{−1} associated to v_{j}(ξ) (j ∈ ℕ). Then μ_{j}(ξ) ↓ 0 as j → ∞. The operator A(ξ) has the same eigenfunctions with associated eigenvalues λ_{j}(ξ) = \frac{1−μ_{j}(ξ)}{μ_{j}(ξ)} → ∞ as j → ∞.

Now let ξ ∈ ℝ vary. In the notation of [10], {A(ξ)} is a self-adjoint holomorphic operator family of type (A) with compact resolvent. According to Theorem 3.9 of Chapter VII there, eigenvalues and eigenfunctions can be chosen in a way, such that they depend analytically on ξ. Since the graphs of λ = λ_{j}(ξ) may intersect, enumeration has to be changed eventually. Note that this change doesnt’t touch the property λ_{j}(ξ) → ∞ as j → ∞ for fixed ξ ∈ ℝ. Finally, λ_{j}(ξ) > 0 is obtained from (4.4) and from D(A(ξ)) ⊂ ^{∘} H^{1}(D)^{n}.

Fix j_{0} ∈ ℕ and consider v_{j_{0}}(ξ) = \binom{u_{1}(ξ)}{u_{D}(ξ)}, where u_{D} := (v_{j_{0}2}, …, v_{j_{0}n})^{T}. From A(ξ)\binom{u_{1}(ξ)}{u_{D}(ξ)} = λ_{j_{0}}(ξ)\binom{u_{1}(ξ)}{u_{D}(ξ)} and the definition of A(ξ),

A(−ξ)\binom{−u_{1}(ξ)}{u_{D}(ξ)} = λ_{j_{0}}(ξ)\binom{−u_{1}(ξ)}{u_{D}(ξ)}

follows. In particular, for every ξ > 0, there exists j ∈ ℕ, such that λ_{j}(−ξ) = λ_{j_{0}}(ξ). This implies, that there is at least one j ∈ ℕ, such that λ_{j}(−ξ) = λ_{j_{0}}(ξ) for more than countable many values of ξ ∈ [0, 1]. Hence λ_{j}(−ξ) = λ_{j_{0}}(ξ) for ξ ∈ ℝ by analycity of λ_{j} and λ_{j_{0}}. □

Lemma 4.2 Suppose that ∂D has the segment property and let {λ_{j}(ξ)}_{j∈ℕ} denote the eigenvalues of A(ξ) given by Lemma 4.1. Then

λ_{min} := min {λ_{j}(ξ) : j ∈ ℕ, ξ ∈ ℝ} > 0.  (4.5)

Furthermore, for every λ ≥ λ_{min}, equation λ_{j}(ξ) = λ admits only finitely many solutions (j, ξ) ∈ ℕ × ℝ.

Proof: It is sufficient to consider λ_{j}(ξ) with ξ ≥ 0 according to Item 3 of Lemma 4.1. Note that

2Re ⟨A(ξ)v_{j}(ξ), \frac{d}{dξ} v_{j}(ξ)⟩_{D} = λ_{j}(ξ)\frac{d}{dξ} ∥v_{j}(ξ)∥_{D}^{2} = λ_{j}(ξ)\frac{d}{dξ} 1 = 0 for ξ ∈ ℝ.

From this, one obtains with v_{j}(ξ) = \binom{u_{1}(ξ)}{u_{D}(ξ)} and ξ ≥ 0 that

\frac{d}{dξ} λ_{j}(ξ) = \frac{d}{dξ} ⟨A(ξ)v_{j}(ξ), v_{j}(ξ)⟩_{D}
= 2Re ⟨A(ξ)v_{j}(ξ), \frac{d}{dξ} v_{j}(ξ)⟩_{D} + 2ξ ∥v_{j}(ξ)∥_{D}^{2}

−c_{0} ⟨\binom{−2ξu_{1}(ξ) + i div_{D} u_{D}(ξ)}{i grad_{D} u_{1}(ξ)}, \binom{u_{1}(ξ)}{u_{D}(ξ)}⟩_{D}

= 2ξ (∥v_{j}(ξ)∥_{D}^{2} + c_{0} ∥u_{1}(ξ)∥_{D}^{2}) + 2c_{0} Im ⟨grad_{D} u_{1}(ξ), u_{D}(ξ)⟩_{D}
≥ −|c_{0}| (∥grad_{D} u_{1}(ξ)∥_{D}^{2} + ∥u_{D}(ξ)∥_{D}^{2})
≥ −|c_{0}| (∥grad_{D} u_{1}(ξ)∥_{D}^{2} + 1) ,

since ∥u_{D}(ξ)∥_{D}^{2} ≤ ∥v_{j}(ξ)∥_{D}^{2} = 1. Inequality (4.4) implies that

∥grad_{D} u_{1}(ξ)∥_{D}^{2} ≤ \frac{1}{min\{1, 1 + c_{0}\}} ⟨A(ξ)v_{j}(ξ), v_{j}(ξ)⟩_{D} = \frac{λ_{j}(ξ)}{min\{1, 1 + c_{0}\}}.

Hence

\frac{d}{dξ} λ_{j}(ξ) ≥ −|c_{0}| − \frac{|c_{0}|}{min\{1, 1 + c_{0}\}} λ_{j}(ξ) =: −|c_{0}| − c_{1}λ_{j}(ξ)

follows. Application of Gronwall’s Lemma yields

λ_{j}(ξ) ≥ λ_{j}(0)e^{−c_{1}ξ} − |c_{0}| ∫_{0}^{ξ} e^{−c_{1}(ξ−s)} ds = (λ_{j}(0) + \frac{|c_{0}|}{c_{1}}) e^{−c_{1}ξ} − \frac{|c_{0}|}{c_{1}} (4.6)

for ξ ≥ 0. (Set φ(ξ) := e^{c_{1}ξ}λ_{j}(ξ) and integrate the resulting differential inequality.)

In order to prove Lemma 4.2, it is convenient to search for eigenvalues λ_{j}(ξ) being smaller than some given λ ≥ 0. From (4.4) one obtains that

λ_{j}(ξ) = ⟨A(ξ)v_{j}(ξ), v_{j}(ξ)⟩_{D} ≥ min{1, 1 + c_{0}} ξ^{2} ∥v_{j}(ξ)∥_{D}^{2} = min{1, 1 + c_{0}} ξ^{2} (4.7)

and hence

λ_{j}(ξ) > λ for j ∈ ℕ, ξ ≥ \sqrt{\frac{λ}{min\{1, 1 + c_{0}\}}} =: c_{2}.

If ξ ∈ [0, c_{2}], then (4.6) implies that

λ_{j}(ξ) ≥ (λ_{j}(0) + \frac{|c_{0}|}{c_{1}}) e^{−c_{1}c_{2}} − \frac{|c_{0}|}{c_{1}} > λ for j > J_{0}

with suitable chosen J_{0} ∈ ℕ, since λ_{j}(0) → ∞ as j → ∞. Hence equation λ_{j}(ξ) = λ admits only solutions (j, ξ) ∈ ℕ × ℝ with j ≤ J_{0} and ξ ∈ [0, c_{2}]. By the analycity of λ_{j}(.), both assertions of Lemma 4.2 follow. □

Lemma 4.3 Let Ω ⊂ ℝ^{n} be given by (1.3), where D is supposed to have the segment property. Set H := L_{2}(Ω)^{n} = L_{2}(ℝ, H) with H = L_{2}(D)^{n}) and denote by F : H → H the Fourier-transform (see (1.8)). Then

F ∘ A ∘ F^{−1} = ∫_{ℝ}^{⊕} A(ξ) dξ =: Â  (4.8)

with A and A(ξ) being defined by respectively (4.1) and (4.3).

Proof: For every φ, ψ ∈ L_{2}(D)^{n}, the mapping ξ ↦ ⟨φ, (A(ξ) + Id)^{−1}ψ⟩_{D} is continuous on ℝ by Lemma 4.1, and hence measurable. This shows that ∫_{ℝ}^{⊕} A(ξ) dξ is defined. It is sufficient to prove F ∘A∘F^{−1} ⊂ Â. This implies that F ∘A∘F^{−1} = Â, since both operators are self-adjoint. (For the case of Â see Theorem XIII.85 in [14].) Suppose that f ∈ D(F ∘ A ∘ F^{−1}), which means that F^{−1}f ∈ D(A). Since D(A) ⊂ ^{∘} H^{1}(Ω)^{n}, there exists a sequence {φ_{j}} in C_{0}^{∞}(Ω)^{n} with

∥φ_{j} − F^{−1}f∥_{H^{1}(Ω)^{n}} → 0 as j → ∞.

Since F is norm-invariant and commutes with derivatives with respect to x_{2}, …, x_{n}, this implies that

∫_{ℝ} ∥(Fφ_{j})(ξ) − f(ξ)∥_{H^{1}(D)^{n}}^{2} dξ → 0 as j → ∞.

There exists a subsequence of {φ_{j}}, again denoted by {φ_{j}}, such that

∥(Fφ_{j})(ξ) − f(ξ)∥_{H^{1}(D)^{n}}^{2} → 0 as j → ∞ a.e. on ℝ.

Note that (Ff)(ξ) ∈ C_{0}^{∞}(D)^{n} for every ξ ∈ ℝ. Hence f(ξ) ∈ ^{∘} H^{1}(D)^{n} a.e. for ξ ∈ ℝ. If φ ∈ C_{0}^{∞}(Ω)^{n}, then (F ∘ A ∘ F^{−1}φ)(ξ) = A(ξ)φ(ξ, .) and

⟨F ∘ A ∘ F^{−1}f, φ⟩_{Ω} = ⟨f, F ∘ A ∘ F^{−1}φ⟩_{Ω} = ∫_{ℝ} ⟨f(ξ), A(ξ)φ(ξ, .)⟩_{D} dξ

= ∫_{ℝ} ⟨( − Δ_{D} + ξ^{2} − c_{0}(\begin{matrix}−ξ^{2} & iξdiv_{D} \\ iξgrad_{D} & grad_{D} div_{D} \end{matrix} ))f(ξ), φ(ξ, .)⟩_{D} dξ

by the symmetry of F ∘ A ∘ F^{−1}, Fubini’s theorem and definition of weak derivative. Again using Fubini’s theorem, one obtains that

( − Δ_{D} + ξ^{2} − c_{0}(\begin{matrix}−ξ^{2} & iξdiv_{D} \\ iξgrad_{D} & grad_{D} div_{D} \end{matrix} ))f(ξ) = (F ∘ A ∘ F^{−1}f)(ξ) ∈ L_{2}(D)^{n}

a.e. for ξ ∈ ℝ, and hence that f(ξ) ∈ D(A(ξ)) and A(ξ)f(ξ) = (F ∘ A ∘ F^{−1}f)(ξ) a.e. for ξ ∈ ℝ. Furthermore,

∫_{ℝ} ∥A(ξ)f(ξ)∥_{D}^{2} dξ = ∫_{ℝ} ∥(F ∘ A ∘ F^{−1}f)(ξ)∥_{D}^{2} dξ < ∞

follows. This implies that f ∈ D(Â) and

(Âf)(ξ) = A(ξ)f(ξ) = (F ∘ A ∘ F^{−1}f)(ξ) a.e. for ξ ∈ ℝ,

which concludes the proof of F ∘ A ∘ F^{−1} ⊂ Â. □

The following corollary is obtained from Lemmata 4.1, 4.2, 4.3 and from (4.4), (4.7):

Corollary 4.4 Suppose that D ⊂ ℝ^{n−1} is bounded and has the segment property. Then:

1) The operator family {A(ξ)}_{ξ∈ℝ} given by (4.3) satisfies Assumptions (A1) – (A4) (see page 4).

2) All assertions of Theorem 2.3 are valid for spectrum and spectral family of the operator A given by (4.1).

5 Elastic wave equation

We study the long-time behaviour of u solving (1.2). L_{2}-estimates are given by Theorem 5.1, if u is a strong solution. Pointwise estimates of the classical solution are presented in Theorem 5.2.

In Section 2, Banach-spaces H_{s} = L_{2,s}(ℝ, H) with assiciated norm ∥.∥_{s} (s ∈ ℝ) were defined (see (2.10)). In the following application, H := L_{2}(D)^{n} has to be set, since Ω = ℝ × D. Note that

H_{s} = H_{s}^{0}(Ω)^{n}, ∥.∥_{s} is equivalent to ∥.∥_{0,s},

where

\begin{matrix}H_{s}^{k}(Ω)^{n} & := & \{φ ∈ H_{loc}^{k}(Ω)^{n} : ∥φ∥_{k,s} < ∞\} , \\ ∥φ∥_{k,s} & := & (∫_{Ω} (1 + |x|^{2})^{s} ( ∑_{|α|≤k} |D^{α}φ(x)|^{2}) dx)^{1/2} \end{matrix} \}  (5.1)

(k ∈ ℕ_{0}, s ≥ 0).

Theorem 5.1 Let Ω ⊂ ℝ^{n} be given by (1.3) whith D having the segment property, and let A be the elastic spatial operator defined by (4.1). Denote by u the solution of (1.11) (with H := L_{2}(Ω)^{n}), where

ω ≥ 0, u_{0} ∈ D(A), u_{1} ∈ D(A^{1/2}), f ∈ L_{2}(Ω)^{n}  (5.2)

is supposed.

1) Principle of limiting amplitude: If, in addition to (5.2), ω^{2} ∉ σ_{res}(A) (see (2.2)) and u_{0}, u_{1}, f ∈ H_{s}^{0}(Ω)^{n} for some s > \frac{1}{2}, then

u(t) = e^{−iωt} u_{ω} + r(t),  (5.3)

where

∥r(t)∥_{0,−s′} = o(1)  as t → ∞  (5.4)

for every s′ > \frac{1}{2} and u_{ω} := u_{ω}^{(0)} is given by (3.21) (with B := H_{−s′}^{0}(Ω)^{n}).

2) Resonance case: If, in addition to (5.2), ω^{2} = σ_{p} ∈ σ_{res}(A) and u_{0}, u_{1}, f ∈ H_{s}^{0}(Ω)^{n} for some s > N_{p} + \frac{1}{2} (see (2.14)), then resonance of order t^{1−1/N_{p}} occurs. In particular,

u(t) = e^{−iωt} ∑_{(j,k)∈K^{+}(σ_{p})} ( ∑_{l=1}^{N(j,k,p)−1} t^{1−l/N(j,k,p)}×

×\frac{D_{1}(1 − \frac{l}{N(j,k,p)}) + iD_{2}(1 − \frac{l}{N(j,k,p)})}{(2ω)^{1−l/N(j,k,p)}} Q_{jkpl}(f)

+ ln t · Q_{jkpN(j,k,p)}(f))

− e^{−iωt} ∑_{(j,k)∈K^{−}(σ_{p})} ( ∑_{l=1}^{N(j,k,p)−1} t^{1−l/N(j,k,p)} ×

×\frac{D_{1}(1 − \frac{l}{N(j,k,p)}) − iD_{2}(1 − \frac{l}{N(j,k,p)})}{(2ω)^{1−l/N(j,k,p)}} Q_{jkpl}(f)

+ ln t · Q_{jkpN(j,k,p)}(f)) + e^{−iωt} u_{ω} + r(t),  (5.5)

where r satisfies (5.4) for every s′ > N_{p} + \frac{1}{2} and where at least one term of order t^{1−1/N(j,k,p)} with N(j, k, p) = N_{p} does not vanish and is not cancelled out by other terms, if f is chosen suitable. Furthermore, u_{ω} is given by

u_{ω} = lim_{ε↓0} ( ∫_{|λ−σ_{p}|≥ε} \frac{1}{λ − σ_{p}} d(P_{λ}f)

− ∑_{(j,k)∈K^{+}(σ_{p})} ( ∑_{l=1}^{N(j,k,p)−1} \frac{1}{(1 − \frac{l}{N(j,k,p)}) ε^{1−l/N(j,k,p)}} Q_{jkpl}(f)

+ (\frac{iπ}{2} + C_{e} − ln(2ω)) Q_{jkpN(j,k,p)}(f))

+ ∑_{(j,k)∈K^{−}(σ_{p})} ( ∑_{l=1}^{N(j,k,p)−1} \frac{1}{(1 − \frac{l}{N(j,k,p)}) ε^{1−l/N(j,k,p)}} Q_{jkpl}(f)

+ (\frac{iπ}{2} − C_{e} + ln(2ω)) Q_{jkpN(j,k,p)}(f))) ,  (5.6)

where the limit has to be taken in H_{−s′}^{0}(Ω)^{n} for s′ > N_{p} + \frac{1}{2}.

Proof: According to Corollary 4.4, the spectral family of A and its properties are given by (2.12), (2.13) and (2.15). Application of Corollary 3.3, Part 2 (with k = 1 and B̃:= H_{s} = H_{s}^{0}(Ω)^{n}, B := H_{−s′} = H_{−s′}^{0}(Ω)^{n}) yields (5.3) and (5.4). From Item 1 of Theorem 3.2, (5.5) with (5.4) and (5.6) is obtained.

It remains to prove, that resonance of order t^{1−1/N_{p}} occurs, if f is chosen suitable. Note that every term on the right-hand side of (5.5) with N(j, k, p) < N_{p} or with l > 1 grows slower than t^{1−1/N_{p}}. Thus only summands with N(j, k, p) = N_{p} and l = 1 have to be studied. Choose one pair (j_{0}, k_{0}) ∈ K^{+}(σ_{p}) ∪ K^{−}(σ_{p}) such that N(j_{0}, k_{0}, p) = N_{p}. Furthermore, choose f ∈ H_{s}^{0}(Ω)^{n} such that ⟨(Ff)(r_{j_{0}k_{0}}(σ_{p})), v_{j_{0}}(r_{j_{0}k_{0}}(σ_{p}))⟩_{D} ≠ 0 (Note that it is possible to suppose f ∈ C_{0}^{∞}(Ω)^{n}). Then Q_{j_{0}k_{0}p1}(f) ≠ 0 according to (2.16). The associated resonanceterm is not cancelled out by the other terms on the right-hand side of (5.5), which will be shown in the following.

Consider another pair (j, k) ∈ K^{+}(σ_{p}) ∪ K^{−}(σ_{p}) with N(j, k, p) = N_{p}. If r_{jk}(σ_{p}) ≠ r_{j_{0}k_{0}}(σ_{p}), then Q_{jkp1}(f) and Q_{j_{0}k_{0}p1}(f) are linearly independent due to the factor e^{ixr_{jk}(σ_{p})} in the definition (2.16) of Q_{jkp1}(f). If r_{jk}(σ_{p}) = r_{j_{0}k_{0}}(σ_{p}) and j ≠ j_{0}, then Q_{jkp1}(f) and Q_{j_{0}k_{0}p1}(f) are again lineartly independent, since {v_{j}(r_{j_{0}k_{0}}(σ_{p}))}_{j∈ℕ} is supposed to be an orthonormal system. It remains to consider the case r_{jk}(σ_{p}) = r_{j_{0}k_{0}}(σ_{p}) and j = j_{0}. In this case, k = k_{0} ± 1 according to Item 4 of Lemma 2.1 and construction of r_{jk}. Furthermore Q_{jkp1}(f) = Q_{j_{0}k_{0}p1}(f) holds (see (2.9) and (2.16)). If (j, k), (j_{0}, k_{0}) are contained in the same set K^{+}(σ_{p}) (or respectively K^{−}(σ_{p})), then both associated summands in (5.5) lead to the same resonance term, so that the factor of t^{1−1/N_{p}}Q_{j_{0}k_{0}p1}(f) is multiplied by two. If e.g. (j, k) ∈ K^{+}(σ_{p}) and (j_{0}, k_{0}) ∈ K^{−}(σ_{p}), then the factor of t^{1−1/N_{p}}Q_{j_{0}k_{0}p1}(f) is given by

\frac{D_{1}(1 − \frac{1}{N_{p}}) + i D_{2}(1 − \frac{1}{N_{p}})}{(2ω)^{1−1/N_{p}}} − \frac{D_{1}(1 − \frac{1}{N_{p}}) − i D_{2}(1 − \frac{1}{N_{p}})}{(2ω)^{1−1/N_{p}}} = 2i \frac{D_{2}(1 − \frac{1}{N_{p}})}{(2ω)^{1−1/N_{p}}} ≠ 0.

The same happens if (j, k) ∈ K^{−}(σ_{p}) and (j_{0}, k_{0}) ∈ K^{+}(σ_{p}). □

Theorem 5.2 Let Ω and A be given by respectively (1.3) and (4.1). Suppose that ω ≥ 0 and that

∂D ∈ C^{K}, u_{0} ∈ D(A^{K′}), u_{1} ∈ D(A^{K′−1/2}), f ∈ D(A^{K′−1})  (5.7)

for some K > \frac{n}{2} + 2, K′ > \frac{n}{4} + 1. Then Problem (1.2) has a solution u ∈ C^{2}([0, ∞) × \overline{Ω}). It is the only solution of (1.2) satisfying

u ∈ C^{2}([0, ∞), L_{2}(Ω)), u(t) ∈ D(A) for t ≥ 0.  (5.8)

Furthermore the following asymptotic estimates hold:

1) Principle of limiting amplitude: If, in addition to (5.7), ω^{2} ∉ σ_{res}(A) and

u_{0}, u_{1}, f ∈ H_{s}^{0}(Ω)^{n} for some s > \frac{1}{2}, then

u(t, x) = e^{−iωt} u_{ω}(x) + r(t, x),  (5.9)

where u_{ω} ∈ C^{2}(\overline{Ω}) is solution of (1.6) and, for every s′ > \frac{1}{2},

\frac{1}{(1 + |x|^{2})^{s′/2}} r(t, x) = o(1) as t → ∞, uniformly with respect to x ∈ \overline{Ω}.

(5.10)

2) Resonance case: If, in addition to (5.8), ω^{2} ∈ σ_{res}(A) and u_{0}, u_{1}, f ∈ H_{s}^{0}(Ω)^{n} for some s > N_{p} + \frac{1}{2}, then u satisfies (5.5) pointwise with respect to x ∈ \overline{Ω}, where r(t, x) obeyes (5.10) for every s′ > N_{p} + \frac{1}{2} and u_{ω} ∈ C^{2}(\overline{Ω}) solves (1.6).

For the proof, the following lemma is needed. It can be found in [7], but without proof. For the idea of the proof see e.g. Hilfssatz 1.15 in [12]. Lemma 5.3 Set D := −(Δ + c_{0} grad div ) (in distributional sense) and ^{∘} H_{−s}^{1}(Ω)^{n} := completition of C_{0}^{∞}(Ω)^{n} in H_{−s}^{1}(Ω)^{n} (s ≥ 0). Suppose that Ω is given by (1.3) with ∂D ∈ C^{K} for some K ≥ 2 and that φ ∈ H_{−s}^{0}(Ω)^{n} satisfies

D^{j}φ ∈ ^{∘} H_{−s}^{1}(Ω)^{n} for j = 0, 1, …, K′ − 1, D^{K′}φ ∈ H_{−s}^{0}(Ω)^{n}

for some K′ ∈ ℕ and some s ≥ 0. Then φ ∈ H_{−s}^{min{K,2K′}}(Ω)^{n} and

∥φ∥_{min{K,2K′},−s} ≤ c(K, K′, s, Ω) (∥D^{K′}φ∥_{0,−s} + ∥φ∥_{0,−s})  (5.11)

with c(K, K′, s, Ω) > 0 not depending on φ.

Proof of Theorem 5.2: Denote by u the solution of (1.11) given by (3.3). From (5.7) and Theorem 3.1, one obtains that D^{K′−j} \frac{d^{2j}u}{dt^{2j}} ∈ C([0, ∞), L_{2}(Ω)^{n}), j ≤ K′. Application of Lemma 5.3 (with s = 0) and Sobolev’s imbedding theorem yields u ∈ C^{2}([0, ∞) × \overline{Ω}). By definition of A (see (4.1)), u solves (1.2). Uniqueness of the solution of (1.2) satisfying (5.8) follows from the uniqueness of the solution of (1.11) asserted by Theorem 3.1.

Suppose that all assumptions of Part 2 are satisfied and that s′ > N_{p} + \frac{1}{2}. Note that ω^{2} = σ_{p} > 0 by (4.5) and (2.3). According to (5.5) and Corollary 3.3, there exist constants N ∈ ℕ, α_{1}, …, α_{N} ∈ [0, 1), α_{1} < … < α_{N} and operators Q_{0}, …, Q_{N}, such that

∥D^{j}u(t) − e^{−iωt} ω^{2j} (∑_{k=1}^{N} t^{α_{k}} Q_{k}(f) + ln t · Q_{0}(f)) − e^{−iωt}u_{ω}^{(j)}∥_{0,−s′} → 0

as t → ∞ (j = 0, 1, …, K′). Set

v(t) := t^{−α_{N}} e^{iωt} u(t) for t ≥ 0.

Then v(t), Dv(t), …, D^{K′}v(t) converge as t → ∞ with respect to ∥.∥_{0,−s′}. Furthermore v(t), Dv(t), …, D^{K′−1}v(t) ∈ ^{∘} H_{−s′}^{1}(Ω)^{n} for every t ≥ 0 (since v(t) ∈ D(A^{K′})). Application of Lemma 5.3 yields

∥v(t) − v(t′)∥_{min{K,2K′},−s′} → 0 as t, t′ → ∞.

This implies that Q_{N}(f) = lim_{t→∞} v(t) ∈ H_{−s′}^{min{K,2K′}}(Ω)^{n}, D^{j}Q_{N}(f) = ω^{2j}Q_{N}(f) (j = 1, …, k) and D^{j}Q_{N}(f) ∈ ^{∘} H_{−s′}^{1}(Ω)^{n} for j = 0, 1, …, k − 1. Repeat the same argument with v(t) := t^{−α_{N−1}} e^{iωt} (u(t) − t^{α_{N}} Q_{N}(f)) and so on. This proves that

∥r(t)∥_{min{K,2K′},−s′}
= ∥u(t) − e^{−iωt} (∑_{k=1}^{N} t^{α_{k}}Q_{k}(f) + ln t · Q_{0}(f)) − e^{−iωt}u_{ω}^{(0)}∥_{min{K,2K′},−s′}
→ 0 as t → ∞.

Now use Sobolev’s inequality to obtain

|\frac{r(t, x)}{(1 + |x|^{2})^{s′/2}} | ≤ c_{1} ∥\frac{r(t)}{(1 + |.|^{2})^{s′/2}} ∥_{min\{K,2K′\},0}
≤ c_{2} ∥r(t)∥_{min{K,2K′},−s′}
→ 0 as t → ∞

uniformly with repect to x ∈ \overline{Ω}. Thus (5.10) is proved in the resonance case. Finally note that u_{ω}^{(0)} ∈ H_{−s′}^{min\{K,2K′\}} ⊂ C^{2}(\overline{Ω}) and

⟨(D − ω^{2})u_{ω}^{(0)}, φ⟩ = ⟨u_{ω}^{(0)}, (D − ω^{2})φ⟩
= lim_{t→∞} ⟨e^{iωt} u(t) − ∑_{k=1}^{N} t^{α_{k}} Q_{k}(f) − ln t · Q_{0}(f), (D − ω^{2})φ⟩
= lim_{t→∞} ⟨(D − ω^{2})e^{iωt} u(t), φ⟩
= ⟨f, φ⟩

for every φ ∈ C_{0}^{∞}(Ω)^{n}, since (D − ω^{2})Q_{k}(f) = 0 and

∥e^{iωt} (D − ω^{2})u(t) − f∥_{0,−s′} = ∥e^{iωt} (A − ω^{2})u(t) − f∥_{0,−s′} → 0 as t → ∞,

which follows by the same arguments used in the proof of Theorem 3.2. This shows that u_{ω} = u_{ω}^{(0)} solves (1.6). Part 1 of the theorem is proved in the same way. □

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