Time Asymptotics for the Biharmonic Wave Equation in Exterior Domains

Peter H. Lesky Mathematisches Institut A

Universitt Stuttgart Pfaffenwaldring 57 D 70569 Stuttgart

Tel. 0711/685-5557

E-Mail lesky@mathematik.uni-stuttgart.de

Telefax 0711/685-5594

Time Asymptotics for the Biharmonic Wave Equation in Exterior Domains

P. H. Lesky

Math. Inst. A der Universitt Stuttgart, Pfaffenwaldring 57, D 70569 Stuttgart E-Mail: lesky@mathematik.uni-stuttgart.de

Abstract We consider initial-boundary value problems for the plate equation u′′(t, x)+Δ^{2}u(t, x) = f(x) in exterior domains of odd dimension. Especially the boundary condition Δu = ∂Δu/∂n = 0 is studied. We prove existence, uniqueness and time asymptotics for the solution. Furthermore, we study the plate equation with Dirichlet boundary condition. We give examples for right-hand sides f having unbounded support such that resonances occur.

1 Introduction

Consider in Ω = IR^{n} the initial value problem

\begin{matrix}∂_{t}^{2}u(t, x) + Δ^{2}u(t, x) = f(x) & in [0, ∞) × Ω, \\ u(x, 0) = ∂_{t}u(x, 0) = 0 & in Ω \end{matrix} \}  (1.1)

with time independent right-hand side f ∈ C_{0}^{∞}(Ω). We restrict our considerations to the case of odd n ≥ 3. It is well known (compare e.g. [2] and [5]) that (1.1) has a unique solution u ∈ ⋂_{j=0}^{2} C^{j}([0, ∞), H^{4−2j}(IR^{n})) and that u ∈ C^{∞}([0, ∞) × IR^{n}). Furthermore, if n = 3, then a resonance of the following form occurs:

u(t, x) = \frac{t^{1/2}}{(2π)^{3/2}} ∫_{IR_{3}} f(y) dy + u_{0}(x) + o(1) as t → ∞  (1.2)

uniformly in every bounded subset of IR^{3}. For n ≥ 5 the principle of limiting amplitude holds:

u(t, x) = u_{0}(x) + o(1) as t → ∞  (1.3)

uniformly in every bounded subset of IR^{n}, where u_{0} ∈ C^{∞}(IR^{n}) is a solution of the static equation

Δ^{2}u_{0} = f in IR^{n}. (1.4)

We are interested in the question, how the time asymptotics change if Ω is a proper subset of IR^{n}. Then, in addition to (1.1), some boundary condition has to be imposed on u to guarantee uniqueness. The time asymptotics depend essentially on this boundary condition. Let Ω ⊂ IR^{n} be an exterior domain with ∂Ω ∈ C^{∞}. We consider (1.1) combined alternatively with each of the boundary conditions

u(t, x) = \frac{∂u}{∂n}(t, x) = 0 on [0, ∞) × ∂Ω,  (1.5)

Biharmonic wave equation 3

u(t, x) = Δu(t, x) = 0 on [0, ∞) × ∂Ω,  (1.6)
\frac{∂u}{∂n}(t, x) = \frac{∂Δu}{∂n}(t, x) = 0 on [0, ∞) × ∂Ω,  (1.7)
Δu(t, x) = \frac{∂Δu}{∂n}(t, x) = 0 on [0, ∞) × ∂Ω;  (1.8)

here n denotes the unit normal vector on ∂Ω pointing in the exterior of Ω.

In the case of Dirichlet boundary condition (1.5) it is well known that u ∈ C^{∞}([0, ∞) × \overline{Ω}). Furthermore, it follows from results obtained by D.M. Eidus in [2] and [3] for a more general class of equations, that the principle of limiting amplitude holds for every odd n ≥ 3.

The boundary conditions (1.6) and (1.7) were studied by H. Mack and P. Werner in [7] and [8] under the assumption that IR^{n} \ \overline{Ω} is simple connected. They proved for n = 3 that (1.1), (1.6) has a classical solution u ∈ C^{∞}([0, ∞) × \overline{Ω}) and that

u(t, x) = \frac{t^{1/2}}{(2π)^{3/2}} ∫_{Ω} f(y)w_{1}(y) dy w_{1}(x) + u_{0}(x) + o_{M}(1) as t → ∞. (1.9)

Here o_{M}(1) means that the remaining terms are of order o(1) uniformly in every bounded subset M of \overline{Ω}. The zero-resonance in (1.9) is closely related to the “admissible standing wave” w_{1}, which is the unique solution w_{1} ∈ C^{∞}(\overline{Ω}) of

\begin{matrix}Δ^{2}w_{1} & = & 0 & in Ω, \\ w_{1} = Δw_{1} & = & 0 & on ∂Ω, \\ D^{α} (w_{1}(x) − 1) & = & O(|x|^{−|α|−1}) & as |x| → ∞, 0 ≤ |α| ≤ 3 \end{matrix} \}  (1.10)

(D^{α} := \frac{∂^{|α|}}{∂x_{1}^{α_{1}} ⋯ ∂x_{n}^{α_{n}}} for α ∈ IN_{0}^{n}). Note that w_{1} ∉ L_{2}(Ω). In the case of the boundary condition (1.7), estimate (1.9) remains valid with w_{1} ≡ 1. For n ≥ 5 the Biharmonic wave equation 4

solutions of (1.1), (1.6) or (1.1), (1.7), respectively, satisfy the principle of limiting amplitude (1.3).

The study of the solution of (1.1), (1.8) is the aim of this paper. In this case the null space N of the related spatial operator

D(A) := \{u ∈ C^{4}(\overline{Ω}) : Δu = \frac{∂Δu}{∂n} = 0 on ∂Ω, \begin{matrix}D^{α}u(x) = O (|x|^{−(n+1)/2}) \\ as |x| → ∞, 0 ≤ |α| ≤ 4\}, \end{matrix}
Au := Δ^{2}u for u ∈ D(A)

(1.11)

has infinite dimension, since it contains every potential function with sufficient decay as |x| → ∞. Let A denote the self-adjoint extension of A introduced in Section 2 and let P^{(N)} : L_{2}(Ω) → N be the orthogonal projection onto N. As a main result of this paper we shall prove that (1.1), (1.8) has a classical solution u ∈ C^{∞}([0, ∞) × \overline{Ω}) and that

u(t, x) = \frac{t^{2}}{2} (P^{(N)}f)(x) + ∑_{j=1}^{3} t^{2−j/2} (P^{(j)}f)(x) + u_{0}(x) + o_{M}(1) as t → ∞

(1.12)

if n = 3 or n = 5, that

u(t, x) = \frac{t^{2}}{2} (P^{(N)}f)(x) + t^{1/2} (P^{(1)}f)(x) + u_{0}(x) + o_{M}(1) as t → ∞ (1.13)

if n = 7 and that

u(t, x) = \frac{t^{2}}{2} (P^{(N)}f)(x) + u_{0}(x) + o_{M}(1) as t → ∞  (1.14)

for n = 9, 11, . . .. The linear operators P^{(j)} (j = 1, 2, 3) are closely connected with “admissible standing waves” satisfying

Δw = 0 in Ω.

Biharmonic wave equation 5

In particular, we shall prove in the case n = 3 that

P^{(1)}f = \frac{1}{9(2π)^{3/2}} (2 ∑_{j=1}^{3} ∫_{Ω} f(y)w_{x_{j}}(y) dy w_{x_{j}} + 3 ∫_{Ω} f(y)w_{|x|}(y) dy w_{|x|}) , (1.15)

where w_{p} denotes the “admissible standing wave” related to the function p (for the definition compare (9.7) and (8.3)). We note that w_{x_{j}} ∈ L_{2}(Ω) (j = 1, 2, 3) and w_{|x|} ∉ L_{2}(Ω).

The method used in the proof of (1.12) – (1.14) leads to some new observations concerning the asymptotic behaviour of the solution u in the Dirichlet case (1.1), (1.5). As mentioned above, u satisfies the principle of limiting amplitude (1.3) if the right-hand side f in (1.1) has bounded support. But resonances may occur if the support of f is not bounded. In the case n = 3 we prove that for every s ∈ (0, \frac{3}{4}) there exists a right-hand side f ∈ L_{2}(Ω), such that the solution of (1.1), (1.5) satisfies

u(t, x) = t^{s} (P̃^{(s)}f)(x) + o_{M}(t^{s}) as t → ∞  (1.16)

(compare Theorem 10.1).

In order to prove these results, we represent the solution by a spectral integral. This method is also used in the above mentioned papers [2], [7] and [8]. The time asymptotics of the solution are determined essentially by the behaviour of the spectral family {P_{λ}} of the corresponding spatial operator as λ ↓ 0, which can be obtained by estimating the resolvent R_{z} near z = 0. The resolvent and its behaviour as z → 0 is computed in [2] and [3] by an indirect method, while in [7] and [8] boundary integral equations are used. In this paper we reduce the resolvent of the operator A to the resolvent of the corresponding biharmonic Dirichlet operator, which has been studied by Eidus in [2] and [3].

Biharmonic wave equation 6

The plan of this paper is the following. In Section 2 the solution is given by a spectral integral. The resolvent R_{z} = (A − zI)^{−1} of the related self-adjoint operator A is studied in Section 3. The computation of R_{z} can be reduced to the resolvent R_{z}^{B} of the self-adjoint Dirichlet operator

\begin{matrix}D(B) & := & ∘ H^{2}(Ω) ∩ H^{4}(Ω), \\ Bu & := & Δ^{2}u for u ∈ D(B), \end{matrix} \}  (1.17)

where ∘ H^{2}(Ω) denotes the completion of C_{0}^{∞}(Ω) in the second Sobolev space H^{2}(Ω). In particular, we prove

R_{z}f = \frac{1}{z} (ΔR_{z}^{B}(Δf) − f)  for f ∈ H^{2}(Ω).  (1.18)

In Section 4 we study the spectral family {P_{λ}} of A. It follows from (1.18) that

∫_{μ_{1}}^{μ_{2}} φ(λ) d(P_{λ}f) = Δ ∫_{μ_{1}}^{μ_{2}} \frac{φ(λ)}{λ} d(P_{λ}^{B}(Δf))  (1.19)

for f ∈ H^{2}(Ω) and φ ∈ C([μ_{1}, μ_{2}]), where P_{λ}^{B} denotes the spectral family of B. This formula allows us to use well-known properties of the operator B to prove the regularity of our solution in Section 5 and the time asymptotics in Section 6. In Sections 7 – 9 we compute explicit representations of the coefficients in the expansion of R_{z}^{B} around z = 0. This gives us representations of the operators P^{(j)} used in (1.12). Section 10 is devoted to the study of the Dirichlet case (1.1), (1.5).

2 Spectral representation of the solution

In the following we assume:

Biharmonic wave equation 7

Assumption 2.1 Ω is a connected open subset of IR^{n}, n = 3, 5, …, and ∂Ω consists of a finite number of disjoint (n − 1)–dimensional surfaces Γ_{1}, …, Γ_{m} ∈ C^{∞}.

The operator A defined by (1.11) is symmetric, positive (⟨Au, u⟩ ≥ 0 for u ∈ D(A), where ⟨·, ·⟩ denotes the inner product in L_{2}(Ω)), and D(A) is dense in L_{2}(Ω). Hence A has a self-adjoint positive extension A in L_{2}(Ω) according to the theorem of Friedrichs. We note that A is not coercive. This means that there exist no constants c_{1}, c_{2} > 0 such that

⟨Aφ, φ⟩ ≥ c_{1}∥φ∥_{2}^{2} − c_{2}∥φ∥^{2} for every φ ∈ D(A),

where ∥.∥ and ∥.∥_{2} denote the norms in L_{2}(Ω) and H^{2}(Ω), respectively. This can be seen by inserting

φ(x) := \frac{∂}{∂x_{1}} \frac{1}{|x − (x_{0} + δn)|^{n−2}} for x ∈ \overline{Ω}

(x_{0} ∈ ∂Ω fixed, n: normal unit vector on ∂Ω pointing in the exterior of Ω) in this inequality and letting δ ↓ 0.

We denote the left-continuous spectral family of A by {P_{λ}}. Note that P_{λ} = 0 for λ ≤ 0, since A is positive. It is well known (compare [4]) that

u(t) := ∫_{0}^{∞} \frac{1}{λ} (1 − cos \sqrt{λ}t) d(P_{λ}f)  (2.1)

for f ∈ L_{2}(Ω) is the unique solution of

\begin{matrix}u′′(t) + Au(t) & = & f  for t ≥ 0, \\ u(0) = u′(0) & = & 0 \end{matrix} \}  (2.2)

with the property

u ∈ ⋂_{j=0}^{2} C^{j}([0, ∞), D(A^{1−j/2})) .  (2.3)

Biharmonic wave equation 8

It remains to show that u is a classical solution of (1.1), (1.8) under sufficient conditions on f. Here the difficulty occurs that the usual elliptic regularity theory does not apply to A, since A is not coercive. However, we shall prove in section 5 by regularity estimates for the Dirichlet operator B that u is a classical solution of (1.1), (1.8).

3 The resolvent

We start with some heuristic considerations. Let z ∈ C \IR and f ∈ C_{0}^{∞}(Ω). Then R_{z}f = (A − z I)^{−1}f satisfies

\begin{matrix}(Δ^{2} − z)R_{z}f & = & f & in Ω, \\ ΔR_{z}f = \frac{∂ΔR_{z}f}{∂n} & = & 0 & on ∂Ω. \end{matrix} \}  (3.1)

Formal differentiation yields

\begin{matrix}(Δ^{2} − z)ΔR_{z}f & = & Δf & in Ω, \\ ΔR_{z}f = \frac{∂ΔR_{z}f}{∂n} & = & 0 & on ∂Ω. \end{matrix} \}  (3.2)

Hence we obtain ΔR_{z}f = R_{z}^{B}(Δf), where R_{z}^{B} denotes the resolvent of the Dirichlet operator B defined by (1.17). By (3.1), we have

R_{z}f = \frac{1}{z}(Δ^{2}R_{z}f − f) = \frac{1}{z} (Δ(R_{z}^{B}Δf) − f) .

Now we show that this relation between the resolvents R_{z} = (A − z I)^{−1} and R_{z}^{B} = (B − z I)^{−1} holds for all f ∈ H^{2}(Ω).

Lemma 3.1 If z ∈ C \ IR and f ∈ H^{2}(Ω), then

R_{z}f = \frac{1}{z} (Δ(R_{z}^{B}Δf) − f) .  (3.3)

Biharmonic wave equation 9

Proof: Let f ∈ C^{∞}(\overline{Ω}) with bounded support and set

u_{z} = u_{z}[f] := \frac{1}{z} (Δ(R_{z}^{B}Δf) − f) .  (3.4)

From the elliptic theory for the operator B (compare [1]) and from Sobolev’s lemma we obtain R_{z}^{B}(Δf) ∈ C^{∞}(\overline{Ω}), and hence u_{z} ∈ C^{∞}(\overline{Ω}). Since (Δ^{2} − z)R_{z}^{B}Δf = Δf, we have

Δu_{z} = \frac{1}{z} (Δ^{2}(R_{z}^{B}Δf) − Δf) = R_{z}^{B}Δf.

This implies by (3.4)

Δ^{2}u_{z} = Δ(R_{z}^{B}Δf) = zu_{z} + f.  (3.5)

Furthermore, since R_{z}^{B}Δf ∈ D(B), we obtain

Δu_{z} = R_{z}^{B}Δf = 0, \frac{∂Δu_{z}}{∂n} = \frac{∂R_{z}^{B}Δf}{∂n} = 0 on ∂Ω. (3.6)

Thus u_{z} is a solution of (3.1). It remains to show that u_{z} ∈ D(A) ⊂ D(A).

Let G_{z} denote the fundamental solution of Δ^{2} − z in IR^{n} (compare e.g (2.5) in [5] for a representation). We have for every z ∈ C \ IR and every multi-index α ∈ IN_{0}^{n}

D_{x}^{α}G_{z}(|x − y|) = O(e^{−c(z)|x−y|}) as |x − y| → ∞

with suitable c(z) > 0. Furthermore, Greens formula implies

(R_{z}^{B}f)(x) = ∫_{Ω} f(y) G_{z}(|x − y|) dy

− ∫_{∂Ω} (\frac{∂Δ(R_{z}^{B}f)}{∂n}(y) G_{z}(|x − y|) − Δ(R_{z}^{B}f)(y) \frac{∂G_{z}(|x − y|)}{∂n_{y}}) ds_{y}.

(3.7)

Since f has bounded support, we obtain D^{α}(R_{z}^{B}Δf)(x) = O(e^{−c(z)|x|}) as |x| → ∞ Biharmonic wave equation 10

for every α ∈ IN_{0}^{n}. This, together with (3.6) and u_{z} ∈ C^{∞}(\overline{Ω}) shows that u_{z} ∈ D(A). Hence we have u_{z} = R_{z}f.

Now suppose that f ∈ H^{2}(Ω) and let \{f_{k}\} be a sequence in C^{∞}(\overline{Ω}) such that every f_{k} has bounded support and ∥f − f_{k}∥_{2} → 0 as k → ∞. Since

(B − z) (R_{z}^{B}Δf − R_{z}^{B}Δf_{k}) = Δf − Δf_{k}

we obtain by standard estimates that ∥R_{z}^{B}Δf − R_{z}^{B}Δf_{k}∥_{2} → 0 and hence by (3.4) ∥u_{z}[f] − u_{z}[f_{k}]∥ → 0 as k → ∞. This yields

Au_{z}[f_{k}] = f_{k} + zu_{z}[f_{k}] → f + zu_{z}[f] as k → ∞

in L_{2}(Ω). Since self-adjoint operators are closed, we obtain u_{z}[f] ∈ D(A) and Au_{z}[f] = f + zu_{z}[f]. This verifies u_{z} = R_{z}f for f ∈ H^{2}(Ω).

In order study R_{z}^{B} more closely, we introduce the weighted Sobolev spaces H_{s}^{k}(Ω) for k ∈ IN_{0}, s ∈ IR by

∥u∥_{k,s} := (∫_{Ω}(1 + |x|)^{s} ∑_{|α|≤k} |D^{α}u(x)|^{2} dx)^{1/2} ,  (3.8)
H_{s}^{k}(Ω) := {u ∈ H_{loc}^{k}(Ω) : ∥u∥_{k,s} < ∞}.  (3.9)

We set L_{2,s}(Ω) := H_{s}^{0}(Ω). Furthermore, we denote the Banach space of bounded linear operators from L_{2,s_{1}}(Ω) into H_{−s_{2}}^{4}(Ω) by B^{(s_{1},s_{2})}. B satisfies all assumptions of Theorem 1.1 in [3] with σ = ∞. Hence for every l ∈ IN there exists a constant ε_{l} > 0 such that for |κ| < ε_{l}, 0 < arg κ < \frac{π}{2}

R_{κ^{4}}^{B} = ∑_{j=0}^{l−1} κ^{j}F_{j} + κ^{l}F̃_{l}(κ)  (3.10)

whith F_{j} ∈ B^{(s_{1}(j),s_{2}(j))} and

∥F̃_{l}(κ)∥_{B^{(s_{1}(l−1),s_{2}(l−1))}} ≤ c  (3.11)

Biharmonic wave equation 11

for

s_{1}(j) > max{n, 10 − n + 2j},  s_{2}(j) > 10 − n + 2j, s_{2}(j) ≥ 0.  (3.12)

Furthermore, we have F_{j} = 0 for 1 ≤ j ≤ n − 5, \frac{j}{4} ∉ IN according to (4.10) in [3].

Lemma 3.2 For sufficiently small λ > 0, the limits

R_{λ±i0}^{B} := lim_{τ↓0} R_{λ±iτ}^{B}  (3.13)

exist with respect to the norm in B^{(s_{1}(0),s_{2}(0))}. Furthermore,

R_{λ±i0}^{B} = ∑_{j=0}^{l}(κ_{±})^{j}F_{j} + (κ_{±})^{l}F_{±}(λ),  (3.14)

where κ_{+} = λ^{1/4}, κ_{−} = iλ^{1/4}, F_{j} are the operators appearing in (3.10) and

∥F_{±}(λ)∥_{B^{(s_{1}(l),s_{2}(l))}} ≤ c,  (3.15)

where c > 0 does not depend on λ.

Proof: It follows from the argument preceding (3.15) in [3] that d^{l}R_{z}/dκ^{l} is uniformly continuous on {z ∈ C \ [0, ∞) : |z| < ε_{l}} with respect to the norm in B^{(s_{1}(l),s_{2}(l))} for every l ∈ IN. This proves Lemma 3.2.

The low frequency asymptotics of the resolvent R_{z} of A are given by the following lemma, which follows immediately from Lemma 3.1 and Lemma 3.2:

Lemma 3.3 Suppose that f ∈ H^{2}(Ω) and that Δf ∈ L_{2,s_{1}(l)}(Ω), where s_{1}(l) satisfies (3.12) for some l ∈ IN. Then

R_{κ^{4}}f = \frac{1}{κ^{4}} (ΔF_{0}Δf − f) + ∑_{j=1}^{l−1} κ^{j−4}ΔF_{j}Δf + ΔF̃_{l}(κ)Δf  (3.16)

Biharmonic wave equation 12

for 0 < arg κ < \frac{π}{2}, |κ| sufficiently small. Furthermore, the limits

R_{λ±i0}f := lim_{τ↓0} R_{λ±iτ}f

exist with respect to the norm in H_{−s_{2}(1)}^{2}(Ω) for sufficiently small λ > 0, and (3.16) holds also in the limit case κ^{4} = λ ± i0.

4 The spectral family

We study the spectral family of A using Stone’s formula

\frac{1}{2} ⟨(P_{λ} + P_{λ+0})f − (P_{μ} + P_{μ+0})f, g⟩ = lim_{τ↓0} \frac{1}{2πi} ∫_{μ}^{λ} ⟨(R_{ρ+iτ} − R_{ρ−iτ})f, g⟩ dρ (4.1)

for μ < λ and f, g ∈ L_{2}(Ω). Note that P_{λ} = 0 for λ ≤ 0 by the positivity of A.

Lemma 4.1 If f ∈ H^{2}(Ω) and φ ∈ C([μ_{1}, μ_{2}]) with 0 < μ_{1} < μ_{2} < ∞, then

∫_{μ_{1}}^{μ_{2}} φ(λ) d(P_{λ}f) = Δ ∫_{μ_{1}}^{μ_{2}} \frac{φ(λ)}{λ} d(P_{λ}^{B}Δf) .  (4.2)

Remarks: 1) The integral on the right-hand side of (4.2) belongs to D(B^{k}) for every k ∈ IN if 0 < μ_{1} < μ_{2} < ∞. By regularity theory for B, this implies that the right-hand side belongs to H^{4k−2}(Ω) for every k ∈ IN (compare Theorem 9.8 in [1] and the remark following this theorem; the case of bounded ∂Ω can be proved by a slight modification of the proof of Theorem 9.8).

2) If f ∈ H^{2}(Ω) and φ ∈ C([μ_{1}, ∞)) is bounded, then (4.2) also holds for μ_{2} = ∞.

Proof: At first we prove (4.2) for φ = 1. Consider a fixed τ > 0 and set

φ_{αβ} := Δ ∫_{α}^{β} \frac{1}{λ − iτ} d(P_{λ}^{B}Δf)  for 0 < α < β < ∞.  (4.3)

Biharmonic wave equation 13

Since

R_{z}^{B}(Δφ_{αβ}) = R_{z}^{B} (B ∫_{α}^{β} \frac{1}{λ − iτ} d(P_{λ}^{B}Δf)) = ∫_{α}^{β} \frac{λ}{(λ − iτ)(λ − z)} d(P_{λ}^{B}Δf) ,

we obtain by (3.3)

R_{z} φ_{αβ} = Δ ∫_{α}^{β} \frac{1}{(λ − iτ)(λ − z)} d(P_{λ}^{B}Δf) .

This implies that R_{z} φ_{αβ} is continuous in z for Re z < α and Re z > β. By (4.1), we conclude that P_{λ} φ_{αβ} is constant for λ < α and λ > β. Hence P_{λ} φ_{αβ} = 0 for λ ≤ α and P_{λ} φ_{αβ} = φ_{αβ} for λ > β. P_{λ}^{B}Δf is continuous in λ, since B has no eigenvalues. Therefore φ_{αβ} depends continuously on α and β for 0 < α < β with respect to the norm in L_{2}(Ω). Hence we have

P_{β} φ_{αβ} = lim_{γ↑β} P_{β} g_{αγ} = lim_{γ↑β} g_{αγ} = φ_{αβ}.

Now we obtain for 0 < α < μ_{1} < μ_{2} < β

(P_{μ_{2}} − P_{μ_{1}}) φ_{αβ} = (P_{μ_{2}} − P_{μ_{1}}) (g_{αμ_{1}} + g_{μ_{1}μ_{2}} + g_{μ_{2}β}) = g_{μ_{1}μ_{2}}.

Since φ_{αβ} → Δ(R_{iτ}^{B}Δf) in L_{2}(Ω) as α ↓ 0 and β → ∞, we conclude that

(P_{μ_{2}} − P_{μ_{1}}) (Δ(R_{iτ}^{B}Δf)) = φ_{μ_{1}μ_{2}}.  (4.4)

By (3.3), we obtain

(P_{μ_{2}} − P_{μ_{1}}) (Δ(R_{iτ}^{B}Δf)) = (P_{μ_{2}} − P_{μ_{1}}) (iτR_{iτ}f + f) =
= ∫_{μ_{1}}^{μ_{2}} \frac{λ}{λ − iτ} d(P_{λ}f) → ∫_{μ_{1}}^{μ_{2}} d(P_{λ}f) as τ ↓ 0  (4.5)

in L_{2}(Ω). Regularity theory for the operator B yields

g_{μ_{1}μ_{2}} → Δ ∫_{μ_{1}}^{μ_{2}} \frac{1}{λ} d(P_{λ}f) as τ ↓ 0  (4.6)

Biharmonic wave equation 14

in L_{2}(Ω). By (4.4) – (4.6), assertion (4.2) with φ = 1 follows. Approximating an arbitrary φ ∈ C([μ_{1}, μ_{2}]) by piecewise constant functions φ_{k} we conclude that (4.2) holds, since

∥∫_{μ_{1}}^{μ_{2}} \frac{φ(λ) − φ_{k}(λ)}{λ} d(P_{λ}^{B}Δf)∥_{4}^{2} ≤ \frac{c}{μ_{1}^{2}} ∥Δf∥^{2} max_{μ_{1}≤λ≤μ_{2}} |φ(λ) − φ_{k}(λ)|^{2}

by regularity theory for B. Lemma 4.2 If f ∈ H^{2}(Ω) with Δf ∈ L_{2,s_{1}(0)}(Ω), then the projection onto the nullspace of A is given by

P^{(N)}f = P_{0+}f = f − Δ(F_{0}Δf);  (4.7)

here F_{0} : L_{2,s_{1}(0)}(Ω) → H_{−s_{2}(0)}^{4}(Ω) denotes the first summand in (3.10). Remark: (4.7) implies Δ(F_{0}Δf) ∈ L_{2}(Ω).

Proof: From (3.3) and (3.10) we conclude that

⟨R_{ρ±iτ}f, g⟩ = \frac{1}{ρ ± iτ} ⟨(ΔR_{ρ±iτ}^{B}Δf) − f, g⟩
= \frac{1}{ρ ± iτ} ⟨(ΔF_{0}Δf) − f, g⟩ + r(ρ ± iτ),

where, by (3.11),

|r(ρ ± iτ)| = |\frac{1}{ρ ± iτ} ⟨κΔF̃_{1}(κ)(Δf), g⟩| ≤ \frac{c}{|ρ ± iτ|^{3/4}} ∥Δf∥_{0,s_{1}(0)} ∥g∥_{0,s_{2}(0)}

for every g ∈ C_{0}^{∞}(Ω), |ρ ± iτ| sufficiently small, κ^{2} = ρ ± iτ, 0 < arg κ < π/2. Inserting this in (4.1), we obtain

\frac{1}{2} ⟨(P_{λ} + P_{λ+0}) f − (P_{0} + P_{0+}) f, g⟩ =

= lim_{τ↓0} \frac{1}{2πi} ∫_{0}^{λ} (\frac{1}{ρ + iτ} − \frac{1}{ρ − iτ} ) dρ ⟨Δ(F_{0}Δf) − f, g⟩ + O(λ^{1/4})
= −\frac{1}{2} ⟨Δ(F_{0}Δf) − f, g⟩ + O(λ^{1/4})  as λ ↓ 0.

Biharmonic wave equation 15

Now (4.7) follows as λ ↓ 0, since P_{0}f = 0.

The following properties of the spectral family {P_{λ}^{B}} of B will be needed for the investigation of the regularity and the large time behaviour of the solution u of (1.1), (1.8). We set Ω_{R} := {x ∈ Ω : |x| < R}.

Lemma 4.3 Let g ∈ L_{2,s_{1}(0)}(Ω) and ε > 0 such that (3.10) holds for |z| < ε. Then P_{λ}^{B}g : IR → H_{−s_{2}(0)}^{4}(Ω) is continuous with respect to λ for 0 ≤ λ ≤ ε and differentiable for 0 < λ < ε. Moreover, there exists a c > 0 such that

∥\frac{dP_{λ}^{B}g}{dλ} ∥_{4,−s_{2}(0)} ≤ c λ^{1/4} ∥g∥_{0,s_{1}(0)}  for 0 < λ < ε, g ∈ L_{2,s_{1}(0)}(Ω)  (4.8)

holds. Furthermore, we have (d/dλ)P_{λ}^{B}g ∈ H^{k}(Ω_{R}) for every k ∈ IN, R > 0, and there exist constants c_{k,R} > 0 such that

∥\frac{dP_{λ}^{B}g}{dλ} ∥_{H^{k}(Ω_{R})} ≤ c_{k,R} λ^{1/4}∥g∥_{0,s_{1}(0)}  for 0 < λ < ε, g ∈ L_{2,s_{1}(0)}(Ω). (4.9)

If in addition g ∈ L_{2,s_{1}(4)}(Ω), then for k ∈ IN and R > 0

∥\frac{dP_{λ}^{B}g}{dλ} − ∑_{j=1}^{3} \frac{1 − i^{j}}{2πi} λ^{j/4}F_{j}g∥_{H^{k}(Ω_{R})} ≤ c_{k,R} λ^{5/4}∥g∥_{0,s_{1}(4)}  for 0 < λ < ε. (4.10)

Proof: According to the arguments preceding (3.15) in [3], the resolvent R_{z}^{B}g : C → H_{−s_{2}(0)}^{4}(Ω) is uniformly continuous with respect to z for 0 ≤ |z| ≤ ε, Im z ≥ 0 and also for 0 ≤ |z| ≤ ε, Im z ≤ 0. Hence (4.1) implies

\frac{dP_{λ}^{B}g}{dλ} = \frac{1}{2πi} (R_{λ+i0}^{B} g − R_{λ−i0}^{B} g) .  (4.11)

This, together with (3.14), implies (4.8). By

Δ^{2} \frac{dP_{λ}^{B}g}{dλ} = \frac{1}{2πi} Δ^{2}(R_{λ+i0}^{B} g − R_{λ−i0}^{B} g) = 0 in Ω

Biharmonic wave equation 16

and regularity theory for B, we obtain

∥\frac{dP_{λ}^{B}g}{dλ} ∥_{H^{4k}(Ω_{R})} ≤ c_{k} ∥\frac{dP_{λ}^{B}g}{dλ} ∥_{L_{2}(Ω_{R+1})} for k ∈ IN

with suitable constants c_{k} > 0 not depending on g. This yields (4.9) by (4.8). It follows from

(Δ^{2} − z) R_{z}^{B}g = g in Ω, R_{z}^{B}g = \frac{∂R_{z}^{B}g}{∂n} = 0 on ∂Ω

and (3.10) that

Δ^{2}F_{j}g = 0 in Ω,  F_{j}g = \frac{∂F_{j}g}{∂n} = 0 on ∂Ω for j = 1, 2, 3.  (4.12)

Hence (4.10) follows as above from (4.11) and (3.14).

In the same way we obtain from [2] (Theorems 3.2, 3.4 and 3.5):

Lemma 4.4 Let g ∈ L_{2}(Ω) such that g(x) = 0 outside some bounded set. Then (4.11) holds for λ > 0 . Furthermore,

∥ \frac{d}{dλ} P_{λ}^{B}g|_{λ=λ_{1}} − \frac{d}{dλ} P_{λ}^{B}g|_{λ=λ_{2}} ∥_{H^{k}(Ω(R))} ≤ c |λ_{1} − λ_{2}|  (4.13)

for R > 0, k ∈ IN and 0 < α ≤ λ_{1} < λ_{2} ≤ β < ∞, where c > 0 may depend on R, k, α, β and g.

5 Regularity of the solution

By (2.1) and (4.2), we have for every t ≥ 0

u(t) = \frac{t^{2}}{2} P_{0+}f + lim_{a↓0} Δ ∫_{a}^{∞} \frac{1}{λ^{2}} (1 − cos \sqrt{λ}t) d(P_{λ}^{B}Δf) .  (5.1)

Biharmonic wave equation 17

In the following we suppose that

f ∈ H^{4k+2}(Ω), Δf ∈ L_{2,s_{1}(0)}(Ω), Δ^{2j+1}f ∈ ∘ H^{2}(Ω) for j = 0, 1, …, k−1 (5.2)

with k ∈ IN. Note that the third condition in (5.2) is equivalent to Δ^{2j+1}f = (∂/∂n)Δ^{2j+1}f = 0 on ∂Ω for j = 0, 1, …, k − 1 (compare Theorem 11.5 in [6]). We need the following elliptic estimate, which is given without proof in [2].

Lemma 5.1 Let s ≥ 0 and k ∈ IN be given. If φ ∈ H_{−s}^{2}(Ω), φ = (∂/∂n)φ = 0 on ∂Ω and Δ^{2}φ ∈ H_{−s}^{k}(Ω), then φ ∈ H_{−s}^{k+4}(Ω) and there exists a constant c > 0 not depending on φ such that

∥φ∥_{k+4,−s} ≤ c (∥Δ^{2}φ∥_{k,−s} + ∥φ∥_{0,−s}) .  (5.3)

Proof: Let K(R, R′) := {x ∈ Ω : R < |x| < R′}. A modification of the proof of Theorem 9.8 in [1] yields

∥φ∥_{H^{k+4}(K(R−1,R+1))} ≤ c (∥Δ^{2}φ∥_{H^{k}(K(R−2,R+2))} + ∥φ∥_{L_{2}(K(R−2,R+2))}) ,

where c > 0 does not depend on φ and R. Multiplying this estimate with (R−1)^{−s} and taking the sum of the inequalities resulting for R = 2, 3, …, we obtain (5.3).

Consider the first term on the right-hand side of (5.1). In view of (4.7) we study the regularity of F_{0}Δf. Since F_{0}Δf ∈ H_{−s_{2}(0)}^{4}(Ω) and Δ^{2}(F_{0}Δf) = Δf, F_{0}Δf = (∂/∂n)(F_{0}Δf) = 0 on ∂Ω (compare (4.12)), we obtain by Lemma 5.1 that F_{0}Δf ∈ H_{−s_{2}(0)}^{4k+4}(Ω). This implies

P_{0+}f = f − Δ(F_{0}Δf) ∈ H_{−s_{2}(0)}^{4k+2}(Ω).  (5.4)

Biharmonic wave equation 18

We turn to the second term on the right-hand side of (5.1). In view of (5.2) we have Δf ∈ D(B^{k}). By the functional calculus and (5.3) with s = 0 we obtain, by regularity theory for B, for every fixed b > 0

∫_{b}^{∞} \frac{1}{λ^{2}} (1 − cos \sqrt{λ}t) d(P_{λ}^{B}Δf) ∈ D(B^{k+2}) ⊂ H^{4k+8}(Ω) ⊂ H_{−s_{2}(0)}^{4k+8}(Ω). (5.5)

For 0 < a < b < ε we conclude from Lemma 4.3 that

∥B^{j} ∫_{a}^{b} \frac{1}{λ^{2}} (1 − cos \sqrt{λ}t) d(P_{λ}^{B}Δf)∥_{−s_{2}(0)} =

= ∥∫_{a}^{b} \frac{1 − cos \sqrt{λ}t}{λ} λ^{j−1} \frac{dP_{λ}^{B}Δf}{dλ} dλ∥_{−s_{2}(0)} ≤ c t^{2} ∥Δf∥_{s_{1}(0)} (b^{j+1/4} − a^{j+1/4})

for j = 0, 1, …, where c > 0 may depend on j. With (5.3), we obtain

∥∫_{a}^{b} \frac{1 − cos \sqrt{λ}t}{λ^{2}} d(P_{λ}^{B}Δf)∥_{4j,−s_{2}(0)} ≤

≤ c_{j} ∑_{k=0}^{j} ∥B^{k} ∫_{a}^{b} \frac{1 − cos \sqrt{λ}t}{λ^{2}} d(P_{λ}^{B}Δf)∥_{0,−s_{2}(0)} ≤ c̃_{j} (b^{1/4} − a^{1/4})

for j ∈ IN with suitable constants c_{j}, c̃_{j} > 0. This, together with (5.5) proves

lim_{a↓0} Δ ∫_{a}^{∞} \frac{1}{λ^{2}} (1 − cos \sqrt{λ}t) d(P_{λ}^{B}Δf) ∈ H_{−s_{2}(0)}^{4k+6}(Ω) for every t ≥ 0. (5.6)

Now (5.4) and (5.6) imply u(t) ∈ H_{−s_{2}(0)}^{4k+2}(Ω) for t ≥ 0. In the same way we obtain u′(t) ∈ H_{−s_{2}(0)}^{4k+2}(Ω) and

u^{(j)}(t) ∈ H_{−s_{2}(0)}^{4k+6−2j}(Ω) for j = 2, 3, …, 2k + 3 and t ≥ 0  (5.7)

by using D(B^{1/2}) ⊂ H^{2}(Ω). With similar estimates for ∥u^{(j)}(t_{1})−u^{(j)}(t_{2})∥_{4k+6−2j,s_{2}(0)} we conclude that

u ∈ ⋂_{j=2}^{2k+3} C^{j} ([0, ∞), H_{−s_{2}(0)}^{4k+6−2j}(Ω)) .  (5.8)

Biharmonic wave equation 19

By the Lemma of Sobolev, this implies u ∈ C^{4}([0, ∞) × \overline{Ω}) if k ∈ IN, k > \frac{n+4}{8}. Hence u is a classical solution of (1.1). From ΔP_{0+}f = Δf − Δ^{2}F_{0}Δf = 0,

Δ lim_{a↓0} Δ ∫_{a}^{∞} \frac{1}{λ^{2}} (1 − cos \sqrt{λ}t) d(P_{λ}^{B}Δf) = ∫_{0}^{∞} \frac{1}{λ}(1 − cos \sqrt{λ}t) d(P_{λ}^{B}Δf) ∈ D(B)

and (5.1) we obtain that u satisfies (1.8). Thus we have proved:

Theorem 5.2 Let k ∈ IN, k > \frac{n+4}{8} and suppose that f satisfies (5.2). Then (1.1), (1.8) has a classical solution u ∈ C^{4}([0, ∞) × \overline{Ω}), which is given by (5.1). Furthermore, (5.8) holds.

Remark: This solution is the only solution of (1.1), (1.8) satisfying (2.3).

6 Time asymptotics

We rewrite (5.1) as

u(t) = \frac{t^{2}}{2} P^{(N)}f + I_{1}(t) + I_{2} + I_{3}(t),  (6.1)

where

I_{1}(t) := lim_{a↓0} Δ ∫_{a}^{δ} \frac{1}{λ^{2}} (1 − cos \sqrt{λ}t) d(P_{λ}^{B}Δf) ,  (6.2)
I_{2} := Δ ∫_{δ}^{∞} \frac{1}{λ^{2}} d(P_{λ}^{B}Δf) ,  (6.3)
I_{3}(t) := −Δ ∫_{δ}^{∞} \frac{1}{λ^{2}} cos \sqrt{λ}t d(P_{λ}^{B}Δf)  (6.4)

and δ > 0 is chosen so that (4.10) holds for 0 < λ ≤ δ.

Biharmonic wave equation 20

Lemma 6.1 Assume that f ∈ H^{2}(Ω), Δf ∈ L_{2,s_{1}(4)}(Ω) and that δ > 0 is sufficiently small. Then

I_{1}(t, x) = ∑_{j=1}^{3} (E_{j} t^{2−j/2} − \frac{1 − i^{j}}{2πi} \frac{1}{(1 − \frac{j}{4}) δ^{1−j/4}} ) Δ(F_{j}Δf)(x)
+ φ_{1}(t, x) + O_{M} (\frac{1}{t})  as t → ∞,  (6.5)

where

E_{j} = \frac{−i(1 − i^{j})}{(4 − j) Γ(\frac{4−j}{2} ) sin(\frac{4−j}{4} π)} (j = 1, 2, 3) (6.6)

and

|φ_{1}(t, x)| ≤ c_{M} δ^{1/4}∥Δf∥_{0,s_{1}(4)} for t ≥ 0, x ∈ M, (6.7)

if M is an arbitrary bounded subset of \overline{Ω}.

Remark: Recall that O_{M} (\frac{1}{t}) in (6.5) means that the remaining term is of order O(\frac{1}{t}) uniformly in every bounded subset M of \overline{Ω}.

Proof: With (4.9) and the Lemma of Sobolev we conclude that

I_{1}(t, x) = Δ ∫_{0}^{δ} \frac{1}{λ^{2}} (1 − cos \sqrt{λ}t) \frac{dP_{λ}^{B}f}{dλ} (x) dλ.

Note that by (2.16) and (2.19) in [5]

I_{4}(t, x) := Δ ∫_{0}^{δ} \frac{1}{λ^{2}} (1 − cos \sqrt{λ}t) (∑_{j=1}^{3} \frac{1 − i^{j}}{2πi} (F_{j}Δf)(x) λ^{j/4}) dλ
= ∑_{j=0}^{3} (E_{j} t^{2−j/2} − \frac{1 − i^{j}}{2πi} \frac{1}{(1 − \frac{j}{4}) δ^{1−j/4}} ) Δ(F_{j}Δf)(x) + O_{M} (\frac{1}{t})

as t → ∞. Hence we obtain by (4.10) and Sobolev’s Lemma that (6.7) holds with φ_{1}(t, x) = I_{4}(t, x) − I_{1}(t, x).

Biharmonic wave equation 21

Lemma 6.2 Suppose that Δf ∈ L_{2,s_{1}(4)}(Ω) and that δ > 0 is sufficiently small. Then

I_{2} = ∑_{j=1}^{3} \frac{1 − i^{j}}{2πi} \frac{1}{(1 − \frac{j}{4}) δ^{1−j/4}} Δ(F_{j}Δf) + Δ(F_{4}Δf) + φ_{2}  (6.8)

in every bounded subset M of \overline{Ω} with

|φ_{2}(x)| ≤ c_{M} δ^{1/4} ∥Δf∥_{s_{1}(4)}  for x ∈ M.  (6.9)

Proof: By (6.3) and (4.2), we have

I_{2} = ∫_{δ}^{∞} \frac{1}{λ} d(P_{λ}f) = lim_{τ↓0} ∫_{δ}^{∞} \frac{1}{λ − iτ} d(P_{λ}f) = lim_{τ↓0} (R_{iτ}f − ∫_{0}^{δ} \frac{1}{λ − iτ} d(P_{λ}f)) ,

(6.10)

where the limit has to be understood in the sense of L_{2}(Ω). In particular, this limit holds in L_{2,−s}(Ω) for every s ≥ 0. By (3.3) and (3.10), we have

R_{iτ}f = \frac{1}{iτ} (Δ(F_{0}Δf) − f)+ ∑_{j=1}^{4} \frac{−i e^{ijπ/8}}{τ^{1−j/4}} Δ(F_{j}Δf)−τ^{1/4}i e^{iπ/8}Δ(F̃_{5}(τ^{1/4}e^{iπ/8})Δf),

(6.11)

where

∥Δ(F̃_{5}(τ^{1/4}e^{iπ/8})Δf)∥_{2,−s_{2}(4)} ≤ c ∥Δf∥_{0,s_{1}(4)}.  (6.12)

Consider the second term on the right-hand side of (6.10). For j = 1, 2, 3 we have

∫_{0}^{δ} \frac{1}{(λ − iτ)λ^{1−j/4}} dλ = \frac{1}{τ^{1−j/4}} (∫_{0}^{∞} \frac{dξ}{(ξ − i)ξ^{1−j/4}} − ∫_{δ/τ}^{∞} \frac{dξ}{ξ^{2−j/4}} + O (τ^{2−j/4}))

= \frac{1}{τ^{1−j/4}} (\frac{π}{2 cos(\frac{jπ}{8} )} + \frac{iπ}{2 sin(\frac{jπ}{8} )}) − \frac{1}{(1 − \frac{j}{4}) δ^{1−j/4}} + O(τ)

as τ ↓ 0. Note that \frac{π}{2 cos(jπ/8)} + \frac{iπ}{2 sin(jπ/8)} = \frac{2π}{1−i^{j}} e^{ijπ/8} for j = 1, 2, 3. From this and (4.2) we obtain

∫_{0}^{δ} \frac{1}{λ − iτ} d(P_{λ}f) = \frac{1}{−iτ} P^{(N)}f + lim_{a↓0} Δ ∫_{a}^{δ} \frac{1}{λ(λ − iτ)} d(P_{λ}^{B}Δf)

= − \frac{1}{iτ} P^{(N)}f + ∑_{j=1}^{3} (−\frac{ie^{ijπ/8}}{τ^{1−j/4}} − \frac{1 − i^{j}}{2πi (1 − \frac{j}{4}) δ^{1−j/4}} ) Δ(F_{j}Δf)
+ φ_{2} + O(τ) as τ ↓ 0,  (6.13)

where

φ_{2} = Δ ∫_{0}^{δ} \frac{1}{λ} (\frac{dP_{λ}^{B}Δf}{dλ} − ∑_{j=1}^{3} \frac{1 − i^{j}}{2πi} λ^{j/4}F_{j}Δf) dλ.  (6.14)

From (6.10) – (6.14) we obtain by (4.10) that (6.8) and (6.9) hold.

Lemma 6.3 Let Δf ∈ D(B^{k}) for some k > \frac{n}{8}. Then we have for every fixed δ > 0

I_{3}(t, x) = o_{M}(1)  as t → ∞.  (6.15)

Proof: By

∥φ∥_{4k} ≤ c_{k} ∑_{j=0}^{k} ∥B^{j}φ∥ for φ ∈ D(B^{k}),  (6.16)

we conclude that there exists an A > 0 such that

∥∫_{A}^{∞} \frac{cos \sqrt{λ}t}{λ^{2}} d(P_{λ}^{B}Δf)∥_{4k} ≤ ε  (6.17)

for given ε > 0. Let g ∈ C_{0}^{∞}(Ω) such that ∥g − Δf∥ < ε. Then we obtain by (6.16) that

∥∫_{δ}^{A} \frac{cos \sqrt{λ}t}{λ^{2}} d(P_{λ}^{B}g) − ∫_{δ}^{A} \frac{cos \sqrt{λ}t}{λ^{2}} d(P_{λ}^{B}Δf)∥_{4k} ≤ \frac{c}{δ^{2}} ε.  (6.18)

It follows from Lemma 4.4 that

Δ ∫_{δ}^{A} \frac{cos \sqrt{λ}t}{λ^{2}} d(P_{λ}^{B}g) (x) = ∫_{δ}^{A} \frac{cos \sqrt{λ}t}{λ^{2}} Δ\frac{dP_{λ}^{B}g}{dλ} (x) dλ =: ∫_{δ}^{A} \frac{cos \sqrt{λ}t}{\sqrt{λ}} φ(x, λ) dλ.

(6.19)

We prove

|∫_{δ}^{A} \frac{cos \sqrt{λ}t}{\sqrt{λ}} φ(x, λ) dλ| < ε for t > T_{ε}, x ∈ M.  (6.20)

Biharmonic wave equation 23

Then (6.15) follows from (6.17) – (6.20) by Sobolev’s Lemma.

According to Lemma 4.4 we suppose that

|φ(x, λ_{1}) − φ(x, λ_{2})| < ε̃ for |λ_{1} − λ_{2}| < δ̃, x ∈ M.

We set λ_{n} := (\frac{π}{t} )^{2} (n + \frac{1}{2})^{2} and obtain for sufficiently large t and n

∫_{λ_{2n+1}}^{λ_{2n+3}} \frac{cos \sqrt{λ}t}{\sqrt{λ}} φ(x, λ) dλ ≤

≤ \frac{4}{t} ( max_{[λ_{2n+1},λ_{2n+2}]×M}\{φ(x, λ)\} − min_{[λ_{2n+2},λ_{2n+3}]×M}\{φ(x, λ)\}) ≤ \frac{4ε̃}{t}.

Taking the sum of these estimates for \frac{t\sqrt{δ}}{2π} − \frac{3}{4} ≤ n ≤ \frac{t\sqrt{A}}{2π} − \frac{7}{4}, n ∈ IN, we obtain

∫_{δ}^{A} \frac{cos \sqrt{λ}t}{\sqrt{λ}} φ(x, λ) dλ ≤ \frac{\sqrt{A} − \sqrt{δ}}{π} 4ε̃+ \frac{16}{t} ( max_{[δ,A]×M} |φ(λ, x)|) .

The same argument can be applied to estimate the integral from below. This proves (6.20).

From Lemmata 6.1 – 6.3 and the remark following (3.11) we obtain:

Theorem 6.4 Assume that k ∈ IN, k > \frac{n+4}{8} and that f satisfies (5.2) and Δf ∈ L_{2,s_{1}(4)}(Ω). Then (1.1), (1.8) has a unique solution u satisfying (2.3). Furthermore, we have u ∈ C^{4}([0, ∞) × \overline{Ω}) and

u(t, x) = \frac{t^{2}}{2} (P^{(N)}f)(x) + ∑_{j=1}^{3} t^{2−j/2}E_{j} Δ(F_{j}Δf)(x) + Δ(F_{4}Δf)(x) + o_{M}(1)

as t → ∞. (6.21)

F_{j} are the operators appearing in the expansion (3.10). Furthermore, F_{1} = F_{2} = 0 if n = 7 and F_{1} = F_{2} = F_{3} = 0 if n ≥ 9.

Biharmonic wave equation 24

7 Properties and implicit representations of F_{j}

Assume that f ∈ C^{∞}(\overline{Ω}) with bounded support. Inserting the expansion (3.10) in (B − κ^{4})R_{κ^{4}}^{B}f = f we obtain

Δ^{2}F_{j}f = \{ \begin{matrix}f & in Ω for j = 0, \\ 0 & in Ω for j = 1, 2, 3, \end{matrix}  (7.1)
F_{j}f = \frac{∂F_{j}f}{∂n} = 0 on ∂Ω for j = 0, 1, 2, 3.  (7.2)

The fundamental solution G_{z} of Δ^{2} in IR^{n} has the expansion

G_{κ^{4}}(ξ) = ∑_{j=0}^{∞} C_{j} \frac{κ^{4j}}{ξ^{n−4−4j}} + κ^{n−4} ∑_{j=0}^{∞} D_{j}κ^{2j}ξ^{2j}  for ξ ∈ IR^{n},  (7.3)

where

C_{j} = \frac{(−1)^{σ+1/2}}{π^{σ} 2^{4j+4} (2j + 1)! Γ(2j + 2 − σ)},  (7.4)
D_{j} = \frac{i (−1)^{j}}{π^{σ} 2^{2σ+2j+3} j! Γ(σ + j + 1)} (1 + (−1)^{j} i^{n})  (7.5)

with σ = \frac{n}{2} − 1 (compare e.g. (2.5) – (2.9) and (2.28) – (2.29) in [5]). Both series in (7.3) converge uniformly with respect to κ ∈ M for every bounded M ⊂ {κ ∈ C : 0 < arg κ < \frac{π}{2}\}.

In order to give implicit representations of F_{j} for j ≥ 0, we set

I(F_{j}f, φ) := ∫_{∂Ω} (\frac{∂ΔF_{j}f}{∂n} (y) φ(y) − (ΔF_{j}f)(y) \frac{∂φ}{∂n}(y)) ds_{y}.  (7.6)

Consider the case n = 3. We insert (3.10) and (7.3) into (3.7) and compare the coefficients of κ^{−1}, κ^{0}, κ^{1}, . . .. This yields

0 = ∫_{Ω} f(y) dy − ∫_{∂Ω} \frac{∂ΔF_{0}f}{∂n} ds,  (7.7)

F_{0}f(x) = C_{0} ∫_{Ω} f(y)|x − y| dy − C_{0} I(F_{0}f, |x − .|) − D_{0} ∫_{∂Ω} \frac{∂ΔF_{1}f}{∂n} ds, (7.8)
F_{1}f(x) = D_{1} ∫_{Ω} f(y)|x − y|^{2} dy − C_{0} I(F_{1}f, |x − .|)
− ∑_{j=0}^{1} D_{j} I(F_{2−2j}f, |x − .|^{2j}),  (7.9)
F_{2}f(x) = − C_{0} I(F_{2}f, |x − .|) − ∑_{j=0}^{1} D_{j} I(F_{3−2j}f, |x − .|^{2j}),  (7.10)
F_{3}f(x) = D_{2} ∫_{Ω} f(y)|x − y|^{4} dy − C_{0} I(F_{3}f, |x − .|)
− ∑_{j=0}^{2} D_{j} I(F_{4−2j}f, |x − .|^{2j}).  (7.11)

Note that

C_{0} = −\frac{1}{8π} , D_{0} = \frac{1 + i}{8π} , D_{1} = \frac{1 − i}{48π} , D_{3} = \frac{1 + i}{960π} (in the case n = 3).

(7.12)

We need some special properties of F_{0}f. Using

|x − y| = |x| (1 + \frac{|y|^{2} − 2xy}{|x|^{2}} )^{1/2} = |x| (1 − \frac{xy}{|x|^{2}} ) + O(\frac{1}{|x|}) as |x| → ∞,

(7.13)

we obtain from (7.7) and (7.8)

D^{α}(F_{0}f)(x) = D^{α} (c_{1} + ∑_{j=1}^{3} c_{2}^{(j)} \frac{x_{j}}{|x|}) + O(\frac{1}{|x|^{1+|α|}} ) as |x| → ∞, |α| ≥ 0

(7.14)

with

c_{1} = − \frac{1 + i}{8π} ∫_{∂Ω} \frac{∂ΔF_{1}f}{∂n} ds,  (7.15)
c_{2}^{(j)} = \frac{1}{8π} (∫_{Ω} f(y)y_{j} dy − ∫_{∂Ω} (\frac{∂ΔF_{0}f}{∂n} (y) y_{j} − (ΔF_{0}f)(y)\frac{∂y_{j}}{∂n_{y}} ) ds_{y}) . (7.16)

In the cases n = 5, 7, … representations similar to (7.8) – (7.11) hold. We note that, for f ∈ C^{∞}(\overline{Ω}) with bounded support, F_{0}f belongs to C^{∞}(\overline{Ω}) and F_{0}f is the unique solution of

\begin{matrix}Δ^{2}F_{0}f & = & f & in Ω, \\ F_{0}f = \frac{∂F_{0}f}{∂n} & = & 0 & on ∂Ω, \\ D^{α}(F_{0}f)(x) & = & O(\frac{1}{|x|^{L+|α|}} ) & as |x| → ∞, 0 ≤ |α| ≤ 3 \end{matrix} \}  (7.17)

with

L := \{ \begin{matrix}0 & if n = 3, \\ n − 4 & if n = 5, 7, …. \end{matrix}  (7.18)

(compare Theorem 3.6 in [2]).

8 Special solutions of Δ^{2}w = 0

We study solutions of Δ^{2}w = 0, which play an essential role by the description of the operators F_{j} appearing in (3.10). Throughout this section we denote by R_{0} a positive real number such that

∂Ω ⊂ {x ∈ IR^{n} : |x| < R_{0}}.  (8.1)

Furthermore, we set

Ω_{R} := {x ∈ Ω : |x| < R}.  (8.2)

By a (real) polynomial p we denote a function of the form

p(x) = ∑_{|α|≤k} c_{α} x^{α}

with c_{α} ∈ IR, k ∈ IN (x^{α} = x_{1}^{α_{1}} ⋯ x_{n}^{α_{n}}). We prove:

Lemma 8.1 Assume that p is a polynomial with Δ^{2}p = 0. Furthermore, in the case n = 3, p(x) = |x| is admitted. Then there exists a unique solution W_{p} ∈ C^{∞}(\overline{Ω}) of

\begin{matrix}Δ^{2}W_{p} & = & 0 & in Ω, \\ W_{p} = \frac{∂W_{p}}{∂n} & = & 0 & on ∂Ω, \\ D^{α}(W_{p}(x) − p(x)) & = & O(\frac{1}{|x|^{L+|α|}} ) & as |x| → ∞, 0 ≤ |α| ≤ 3, \end{matrix} \} (8.3)

where L is defined by (7.18).

Remark: If n = 3 and p(x) = const., then W_{p} = 0. But in all other cases W_{p} does not vanish.

Proof: Uniqueness is well known (compare Lemmata 3.9 and 3.10 in [2]). In order to prove existence let ψ ∈ C^{∞}(\overline{Ω}) such that ψ(x) = 1 for |x| ≤ R_{0}, ψ(x) = 0 for |x| ≥ R_{0} + 1. If p is a polynomial, then the assertion follows from (7.17) by setting

W_{p} := F_{0}(Δ^{2}(ψp)) + (1 − ψ)p.  (8.4)

If n = 3 and p(x) = |x|, we choose x_{0} ∈ IR^{3} \ \overline{Ω} and set p̃(x) := |x − x_{0}|. Then W_{|x|} is given by (8.4) with p being replaced by p̃.

The following two lemmata give essential properties of the functions W_{p}.

Lemma 8.2 Assume that p is a polynomial with Δ^{2}p = 0, and that p ≠ const if n = 3. If W_{p} is the associated solution of (8.3) and f ∈ C_{0}^{∞}(Ω), then

⟨f, W_{p}⟩ = ∫_{Ω} f p dx − ∫_{∂Ω} (\frac{∂ΔF_{0}f}{∂n} p − (ΔF_{0}f) \frac{∂p}{∂n}) ds.  (8.5)

Lemma 8.3 Suppose that n = 3 and W_{|x|} is the solution of (8.3) with p(x) = |x|. Then we have for f ∈ C_{0}^{∞}(Ω)

⟨f, W_{|x|}⟩ = −(1 + i) ∫_{∂Ω} \frac{∂ΔF_{1}f}{∂n} ds.  (8.6)

Biharmonic wave equation 28

Proof of Lemma 8.2: Let R > R_{0} such that f(x) = 0 for |x| > R. From (7.1) and Greens formula we obtain

∫_{Ω_{R}} f(W_{p} − p) dx = ∫_{Ω_{R}}(Δ^{2}F_{0}f)(W_{p} − p) dx

= ∫_{Ω_{R}}(F_{0}f)Δ^{2}(W_{p} − p) dx − ∫_{∂Ω} (\frac{∂ΔF_{0}f}{∂n} p − (ΔF_{0}f) \frac{∂p}{∂n}) ds + O(\frac{1}{R})

as R → ∞. This proves (8.5).

Proof of Lemma 8.3: By (7.1) and (8.3), we have for sufficiently large R > R_{0}

∫_{Ω_{R}} f W_{|x|} dx = ∫_{Ω_{R}}(Δ^{2}F_{0}f) W_{|x|} dx
= ∑_{j=0}^{1} ∫_{|x|=R} (\frac{∂Δ^{1−j}F_{0}f}{∂n} Δ^{j}W_{|x|} − (Δ^{1−j}F_{0}f) \frac{∂Δ^{j}W_{|x|}}{∂n} ) ds.

We insert (7.14) with (7.15), D^{α}W_{|x|} = D^{α}|x| + O(1/|x|^{|α|}) as |x| → ∞ and use that

∫_{|x|=R} (\frac{∂}{∂n}Δ^{j} \frac{x_{k}}{|x|}) Δ^{1−j}|x| ds = ∫_{|x|=R} (Δ^{j} \frac{x_{k}}{|x|}) \frac{∂Δ^{1−j}|x|}{∂n} ds = 0

(j = 0, 1 and k = 1, 2, 3) by symmetry. We conclude that

∫_{Ω_{R}} f W_{|x|} dx = \frac{i + 1}{8π} ∫_{∂Ω} \frac{∂ΔF_{1}f}{∂n} ds ∫_{|x|=R} \frac{∂Δ|x|}{∂n} ds + O(\frac{1}{R})

as R → ∞. This proves (8.6).

Remark: We can characterize the coefficients c_{1}, c_{2}^{(j)} in the expansion (7.14) of F_{0}f. In particular, we obtain from (7.15) – (7.16) with (8.6) and (8.5) that

c_{1} = \frac{1}{8π} ⟨f, W_{|x|}⟩ , c_{2}^{(j)} = \frac{1}{8π} ⟨f, W_{x_{j}} ⟩ . (8.7)

Assume that p, q are polynomials satisfying Δ^{2}p = Δ^{2}q = 0 or p = |.| or q = |.|. We define the constants C_{p,q} by

C_{q,p} := − \frac{1}{8π} ∫_{∂Ω} (\frac{∂ΔW_{q}}{∂n} p − ΔW_{q} \frac{∂p}{∂n}) ds for p ≠ |.|,  (8.8)

Biharmonic wave equation 29

C_{|x|,|x|} := C_{|x|,|x−x_{0}|} for arbitrary x_{0} ∈ IR^{3} \ \overline{Ω},  (8.9)
C_{q,|x|} := C_{|x|,q}.  (8.10)

Lemma 8.4 Suppose that n = 3 and that p is a polynomial satisfying Δ^{2}p = 0 or p = |.|. Then

W_{p}(x) = p(x) + C_{|x|,p} + ∑_{j=1}^{3} C_{x_{j},p} \frac{x_{j}}{|x|} + O(\frac{1}{|x|})  as |x| → ∞.  (8.11)

Proof: If p ≠ |.|, we apply (7.14) and (8.7) to (8.4). Greens formula yields (8.11). In the case p = |.| the coefficient C_{|x|,|x|} can be computed analogous. Furthermore, we obtain from (7.14), (8.4), (8.7) and an expansion of W_{x_{j}} around x_{0} for the coefficient of \frac{x_{j}}{|x|}

\frac{1}{8π} ∫_{Ω} Δ^{2}(ψ(x)|x − x_{0}|) W_{x_{j}} dx − (x_{0})_{j} =

= \frac{1}{8π} ∑_{j=0}^{1} ∫_{∂Ω} (\frac{∂Δ_{x}^{1−j}|x − x_{0}|}{∂n_{x}} Δ^{j}W_{x_{j}} − Δ_{x}^{1−j}|x − x_{0}| \frac{∂Δ^{j}W_{x_{j}}}{∂n} ) ds_{x} − (x_{0})_{j}
= − \frac{1}{8π} ∫_{|x−x_{0}|=R} \frac{∂Δ_{x}|x − x_{0}|}{∂n_{x}} (C_{|x|,x_{j}} + (x_{0})_{j}) ds_{x} − (x_{0})_{j} + O(\frac{1}{R})
= C_{|x|,x_{j}} + O(\frac{1}{R})  as R → ∞.

Remarks: 1) Note that C_{x_{j},x_{j}} ≠ 0 (j = 1, 2, 3), since

C_{x_{j},x_{j}} = −\frac{1}{8π} ∫_{∂Ω} (\frac{∂Δ(W_{x_{j}} − x_{j})}{∂n} (x_{j} − W_{x_{j}}) − Δ(W_{x_{j}} − x_{j}) \frac{∂x_{j} − W_{x_{j}}}{∂n} ) ds
= −\frac{1}{8π} ∫_{Ω} |ΔW_{x_{j}}|^{2}dx.

In the same way we obtain that C_{|x|,|x|} ≠ 0.

2) It would be possible to give an analogous expansion of W_{p} in the cases n = 5, 7, . . .. But this is not necessary for the proof of the representations of Biharmonic wave equation 30

F_{1}, F_{2}, F_{3}. We only note that C_{1,1} ≠ 0, since
C_{1,1} = C_{0} ∫_{∂Ω} (\frac{∂ΔW_{1}}{∂n} 1 − ΔW_{1} \frac{∂1}{∂n}) ds = C_{0} ∫_{Ω} |ΔW_{1}|^{2} dx. (8.12)

9 Representation of the operators F_{j} and P^{(j)}

Theorem 9.1 Let n = 3. For f ∈ C_{0}^{∞}(Ω), we have

F_{1}f = \frac{i − 1}{48π} (2 ∑_{j=1}^{3} ⟨f, W_{x_{j}} ⟩ W_{x_{j}} + 3 ⟨f, W_{|x|}⟩ W_{|x|}) ,  (9.1)
F_{2}f = −\frac{i}{144π} (3 ⟨f, W_{|x|}⟩ W_{|x|^{2}} + 2 ∑_{j=1}^{3} ⟨f, W_{x_{j}}^{(1)}⟩ W_{x_{j}}
+3 ⟨f, W_{|x|^{2}} + W_{|x|}^{(1)}⟩ W_{|x|}),  (9.2)
F_{3}f = −\frac{1 + i}{960π} (4 ∑_{j=1}^{3} ⟨f, W_{x_{j}} ⟩ W_{x_{j}|x|^{2}} − 4 ∑_{j,k=1}^{3} ⟨f, W_{x_{j}x_{k}}⟩ W_{x_{j}x_{k}}

−\frac{1}{3} ⟨f, 16W_{|x|^{2}} + 10W_{|x|}^{(1)}⟩ W_{|x|^{2}} + \frac{1}{9} ∑_{j=1}^{3} ⟨f, 20W_{x_{j}}^{(2)} − 36W_{x_{j}|x|^{2}}⟩ W_{x_{j}}

−\frac{10}{3} ⟨f, W_{|x|}^{(2)} + W_{|x|^{2}}^{(1)}⟩ W_{|x|}),  (9.3)

where W_{p} is given by Lemma 8.1 and

W_{p}^{(1)} = 2 ∑_{j=1}^{3} C_{x_{j},p}W_{x_{j}} + 3 C_{|x|,p}W_{|x|}  (9.4)
W_{p}^{(2)} = 3C_{|x|,p} (W_{|x|^{2}} + W_{|x|}^{(1)}) + 2 ∑_{j=1}^{3} C_{x_{j},p}W_{x_{j}}^{(1)} + 3C_{|x|^{2},p}W_{|x|}  (9.5)

Note that the operators F_{j} are bounded operators from L_{2,s_{1}(j)}(Ω) into H_{s_{2}(j)}^{4}(Ω) according to [3], if s_{1}(j), s_{2}(j) satisfy (3.12). Hence it follows by approximation that (9.1) – (9.3) hold for F_{j}f with f ∈ L_{2,s_{1}(j)}. Moreover, (9.1) – (9.3) imply: Corollary 9.2 If n = 3, then F_{j} (j = 1, 2, 3) can be extended by continuity to a bounded operator from L_{2,s_{1}(j)}(Ω) into H_{−s_{2}(j)}^{4}(Ω) for every

s_{1}(j), s_{2}(j) > 3 + 2j;  (9.6)

Biharmonic wave equation 31

F_{j}f is given for f ∈ L_{2,s_{1}(j)}(Ω) by (9.1) – (9.3).

We apply the representations of F_{1}, F_{2}, F_{3} to the time asymptotics in Theorem 6.4. We have for f ∈ H_{s}^{2}(Ω) with s > 9

⟨Δf, W_{p}⟩ = ⟨f, ΔW_{p}⟩

for all functions W_{p} appearing on the right-hand sides of (9.1) – (9.5). We set

w_{p} := ΔW_{p}.  (9.7)

Then obviously Δw_{p} = 0 in Ω. We obtain from Theorem 6.4:

Corollary 9.3 Suppose that n = 3, that all assumptions of Theorem 6.4 are satisfied and that in addition f ∈ H_{s}^{2}(Ω) for some s > 9. Then the time asymptotics of the solution u of (1.1), (1.8) are given by (1.12), where

P^{(1)}f = \frac{i − 1}{48π} E_{1} (2 ∑_{j=1}^{3} ⟨f, w_{x_{j}} ⟩ w_{x_{j}} + 3 ⟨f, w_{|x|}⟩ w_{|x|}) .  (9.8)

Analogous representations of \frac{1}{E_{j}} P^{(j)}f (j = 2, 3) are obtained from the representations (9.2) – (9.5) of F_{j}f by replacing W_{p} by w_{p}. The constants E_{j} are given by (6.6).

Remarks: 1) We note that by (8.3) w_{x_{j}} ∈ L_{2}(Ω) for j = 1, 2, 3, w_{|x|} ∉ L_{2}(Ω).

2) If n = 5, 7, …, then representations of P^{(j)} follow in a similar way from Theorem 9.5 below.

In order to prove Theorem 9.1, we use the following lemma, which is an immediate consequence of Lemma 8.1 and of Lemma 3.10 in [2].

Biharmonic wave equation 32

Lemma 9.4 Assume that p_{j} (j = 1, …, k) are polynomials with Δ^{2}p_{j} = 0, p_{j} not being constant. If v ∈ C^{4}(\overline{Ω}) is a solution of

Δ^{2}v = 0 in Ω,
v = \frac{∂v}{∂n} = 0 on ∂Ω,

with

v(x) = ∑_{j=1}^{k} c_{j} p_{j}(x) + c_{k+1} + \frac{1}{8π} I(v, |x − .|)

(c_{1}, …, c_{k+1} ∈ C), then

v = ∑_{j=1}^{k} c_{j} W_{p_{j}} + \frac{1}{8π} ∫_{∂Ω} \frac{∂Δv}{∂n} ds W_{|x|},

where W_{p_{j}} and W_{|x|} are given by Lemma 8.1.

Proof of Theorem 9.1: Using (7.7) and (8.5), we rewrite (7.9) as

F_{1}f(x) = \frac{1 − i}{48π} (∑_{j=1}^{3} ⟨f, W_{x_{j}} ⟩ (−2x_{j}) + ⟨f, W_{|x|^{2}}⟩) − \frac{1 + i}{8π} ∫_{∂Ω} \frac{∂ΔF_{2}f}{∂n} ds
+ \frac{1}{8π} I(F_{1}f, |x − .|).  (9.9)

By (7.1) – (7.2), Lemma 9.4 and (8.6), we conclude that (9.1) holds.

In the next step we prove that

∫_{∂Ω} \frac{∂ΔF_{2}f}{∂n} ds = −\frac{i}{6} ⟨f, W_{|x|^{2}} + W_{|x|}^{(1)}⟩  (9.10)

with W_{|x|}^{(1)} being given by (9.4). Using (9.1) and (8.11), we obtain

F_{1}f(x) − \frac{i − 1}{48π} (2 ∑_{j=1}^{3} ⟨f, W_{x_{j}} ⟩ x_{j} + 3 ⟨f, W_{|x|}⟩ |x|) =

= \frac{i − 1}{48π} (2 ∑_{j=1}^{3} ⟨f, W_{x_{j}} ⟩ (C_{|x|,x_{j}} + ∑_{k=1}^{3} C_{x_{k},x_{j}} \frac{x_{k}}{|x|})

+ 3 ⟨f, W_{|x|}⟩ (C_{|x|,|x|} + ∑_{k=1}^{3} C_{x_{k},|x|} \frac{x_{k}}{|x|})) + O(\frac{1}{|x|})

Biharmonic wave equation 33

as |x| → ∞. On the other hand we conclude from (9.9) and (7.13), (8.6) that

F_{1}f(x) − \frac{i − 1}{48π} (2 ∑_{j=1}^{3} ⟨f, W_{x_{j}} ⟩ x_{j} + 3 ⟨f, W_{|x|}⟩ |x|) =

= \frac{1 − i}{48π} ⟨f, W_{|x|^{2}}⟩ − \frac{1 + i}{8π} ∫_{∂Ω} \frac{∂ΔF_{2}f}{∂n} ds

− \frac{1}{8π} ∑_{j=1}^{3} \frac{x_{j}}{|x|} ∫_{∂Ω} (\frac{∂ΔF_{1}f}{∂n} (y) y_{j} − (ΔF_{1}f)(y) \frac{∂y_{j}}{∂n_{y}} ) ds_{y} + O(\frac{1}{|x|})

as |x| → ∞. Comparing the terms being independent of x, we conclude that (9.10) holds.

We conclude from (9.1), (9.4) and (8.8) that for every polynomial p with Δ^{2}p = 0

∫_{∂Ω} (\frac{∂ΔF_{1}f}{∂n} p − (ΔF_{1}f) \frac{∂p}{∂n}) ds = \frac{1 − i}{6} ⟨f, W_{p}^{(1)}⟩ .  (9.11)

Now we prove (9.2). Inserting (9.11) and (8.6) in (7.10), we obtain

F_{2}f(x) = −\frac{i}{48π} ⟨f, W_{|x|}⟩ |x|^{2} − \frac{i}{72π} ∑_{j=1}^{3} ⟨f, W_{x_{j}}^{(1)}⟩ x_{j} + \frac{i}{144π} ⟨f, W_{|x|^{2}}^{(1)}⟩
− \frac{1 + i}{8π} ∫_{∂Ω} \frac{∂ΔF_{3}f}{∂n} ds + \frac{1}{8π} I(F_{2}f, |x − .|).  (9.12)

It follows from Lemma 9.4 and (9.10) that (9.2) holds.

Now we study F_{3}f. By the argument used in the proof of (9.10) we obtain, by comparing the spatial expansions of (9.2) and (9.12), that

∫_{∂Ω} \frac{∂ΔF_{3}f}{∂n} ds = \frac{1 + i}{36} ⟨f, W_{|x|}^{(2)} + W_{|x|^{2}}^{(1)}⟩ .  (9.13)

It follows from (9.2), (8.8) and (9.5) that

∫_{∂Ω} (\frac{∂ΔF_{2}f}{∂n} p − (ΔF_{2}f)\frac{∂p}{∂n}) ds = \frac{i}{18} ⟨f, W_{p}^{(2)}⟩ .  (9.14)

Inserting (7.7), (8.5), (9.10) and (9.14) into (7.11), we obtain

F_{3}f(x) = \frac{1 + i}{960π} (−4 ∑_{j=1}^{3} ⟨f, W_{x_{j}} ⟩ x_{j}|x|^{2} + 4 ∑_{j,k=1}^{3} ⟨f, W_{x_{j},x_{k}}⟩ x_{j}x_{k}

+2 ⟨f, W_{|x|^{2}}⟩ |x|^{2} − 4 ∑_{j=1}^{3} ⟨f, W_{x_{j}|x|^{2}}⟩ x_{j})

+\frac{1 − i}{48π} (\frac{i}{6} ⟨f, W_{|x|^{2}} + W_{|x|}^{(1)}⟩ |x|^{2} + \frac{i}{9} ∑_{j=1}^{3} ⟨f, W_{x_{j}}^{(3)}⟩ x_{j})

+c + I(F_{3}f, |x − .|)

with suitable c ∈ C. From this we conclude by Lemma 9.4 and (9.13) that (9.3) holds.

If n = 5, 7, 9, …, we have F_{j} = 0 for \frac{j}{4} ∉ IN, j ≤ n − 5. The same method as above can be used to compute F_{n−4}, F_{n−3}, F_{n−2} (for n = 5, 9, 13, …) or F_{n−4}, F_{n−2}, F_{n−1} (for n = 7, 11, 15, …), respectively.

Theorem 9.5 If n = 5, 7, 9, … and f ∈ C_{0}^{∞}(Ω), then

F_{n−4}f = D_{0} ⟨f, W_{1}⟩ W_{1},  (9.15)
F_{n−2}f = D_{1} (⟨f, W_{1}⟩ W_{|x|^{2}} − 2 ∑_{j=1}^{n} ⟨f, W_{x_{j}} ⟩ W_{x_{j}} + ⟨f, W_{|x|^{2}}^{(3)}⟩ W_{1}) (9.16)

with

W_{|x|^{2}}^{(3)} = \{ \begin{matrix}W_{|x|^{2}} + \frac{D_{0}^{3} C_{1,1}^{2}}{C_{0}^{2}} W_{1} & if n = 5, \\ W_{|x|^{2}} & if n = 7, 9, …. \end{matrix}  (9.17)

Furthermore,

F_{n−3}f = \{ \begin{matrix}−\frac{D_{0}^{2} C_{1,1}}{C_{0}} ⟨f, W_{1}⟩ W_{1} & if n = 5, \\ 0 & if n = 9, 13, … \end{matrix}  (9.18)

and

F_{n−1}f = \{ \begin{matrix}−\frac{D_{0}^{2} C_{1,1}}{C_{0}} ⟨f, W_{1}⟩ W_{1} & if n = 7, \\ 0 & if n = 11, 15, …. \end{matrix}  (9.19)

Biharmonic wave equation 35

Remarks: 1) If n = 7, 11, 15, …, then Δ^{2}F_{n−3}f = F_{n−7}f ≠ 0 for f ≠ 0, since Δ^{(n−3)/2}F_{n−7}f = f.

2) Note that C_{1,1} ≠ 0 by (8.12).

3) We mention without proof, that it is possible to represent F_{j} with \frac{j}{4} ∉ IN for arbitrary j in a similar way by solutions W_{p} of equations of the form Δ^{2k}W_{p} = 0.

10 The Dirichlet problem

Consider the mixed problem (1.1), (1.5). The unique solution satisfying

u ∈ ⋂_{j=0}^{2} C^{j} ([0, ∞), D(B^{1−j/2}))  (10.1)

is given by

u(t) = ∫_{0}^{∞} \frac{1}{λ}(1 − cos \sqrt{λ}t) d(P_{λ}^{B}f) .  (10.2)

We suppose in the following that g ∈ C_{0}^{∞}(Ω) and set

f := ∫_{0}^{∞} \frac{1}{λ^{s}} d(P_{λ}^{B}g)  (10.3)

with 0 ≤ s < \frac{5}{8}. By (4.8), we have for 0 < δ < ε

∥∫_{δ}^{ε} \frac{1}{λ^{s}} d(P_{λ}^{B}g)∥^{2} = ∫_{δ}^{ε} \frac{1}{λ^{2s}} d(∥P_{λ}^{B}g∥^{2}) = ⟨∫_{δ}^{ε} \frac{1}{λ^{2s}} \frac{dP_{λ}^{B}g}{dλ} dλ, g⟩
≤ c_{g} ∫_{δ}^{ε} \frac{1}{λ^{2s−1/4}} dλ ∥g∥^{2},

since g has bounded support. Hence f ∈ L_{2}(Ω). By (10.3), we can rewrite (10.2) as

u(t) = ∫_{0}^{∞} \frac{1}{λ^{1+s}} (1 − cos \sqrt{λ}t) d(P_{λ}^{B}g) .  (10.4)

Biharmonic wave equation 36

It follows analogously to Theorem 6.4 (compare also (2.16) and (2.21) in [5]):

Theorem 10.1 Suppose that g ∈ C_{0}^{∞}(Ω). If 0 ≤ s < \frac{5}{8} and f ∈ L_{2}(Ω) is defined by (10.3), then (1.1), (1.5) has a classical solution u ∈ C^{∞}([0, ∞) × \overline{Ω}). Furthermore, the following estimates hold with suitable non-vanishing constants E_{1}(s), E_{2}(s) ∈ C:

1) If \frac{1}{2} < s < \frac{5}{8}, then

u(t, x) = t^{2s−1/2}E_{1}(s) F_{1}g(x) + t^{2s−1}E_{2}(s) F_{2}g(x) + u_{0}^{(s)}(x) + o_{M}(1)

as t → ∞. (10.5)

2) If s = \frac{1}{2}, then

u(t, x) = t^{1/2}E_{1}(\frac{1}{2}) F_{1}g(x)−(ln t) \frac{2i}{π} F_{2}g(x)+u_{0}^{(1/2)}(x)+o_{M}(1) as t → ∞.

(10.6)

3) If \frac{1}{4} < s < \frac{1}{2}, then

u(t, x) = t^{2s−1/2}E_{1}(s) F_{1}g(x) + u_{0}^{(s)}(x) + o_{M}(1)  as t → ∞. (10.7)

4) If s = \frac{1}{4}, then

u(t, x) = −(ln t) \frac{1 + i}{π} F_{1}g(x) + u_{0}^{(1/4)}(x) + o_{M}(1)  as t → ∞. (10.8)

5) If 0 ≤ s < \frac{1}{4}, then u satisfies the principle of limiting amplitude (compare (1.3)).

Remark: F_{1}g does not vanish in general if n = 3 or n = 5. This follows from the representation of F_{1} given in (9.1) or (9.15), respectively.

Biharmonic wave equation 37

References

[1] S. Agmon, Lectures on Elliptic Boundary Value Problems (Van Nostrand, New York, 1965).

[2] D. M. Eidus, The principle of limit amplitude, Russian Math. Surveys, 24 (1969) 97–167.

[3] D. M. Eidus, Asymptotical expansions of solutions of linear parabolic equations as t → ∞, J. Math. Anal. Appl. 130 (1988) 155–170.

[4] R. Leis, Initial Boundary Value Problems in Mathematical Physics (Teubner/Wiley, Stuttgart/Chichester et al., 1986).

[5] P. Lesky, A uniqueness condition for the polyharmonic equation in free space, Math. Meth. in the Appl. Sci. 12 (1990) 275–291.

[6] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, (Springer-Verlag, Berlin/Heidelberg/New York, 1972).

[7] H. Mack, Zur Zeitasymptotik eines Rand- und Anfangswertproblems höherer Ordnung mit Randbedingungen vom Neumannschen oder Dirichletschen Typ in Außenräumen, (Thesis at the University of Stuttgart, Stuttgart, 1993).

[8] H. Mack and P. Werner, Bemerkungen zur Anwendung von Integralgleichungsmethoden auf die Asymptotik partieller Differentialgleichungen, in: E. Schock, editor, Beiträge zur Angewandten Analysis und Informatik, Helmut Brakhage zu Ehren, (Shaker Verlag, Aachen, 1994) 225 – 247.