Asymptotically free property of the solutions of an abstract linear hyperbolic equation with time-dependent coefficients

Abstract

This paper is concerned with an abstract dissipative hyperbolic equation with time-dependent coefficient. Under an assumption which ensures that the energy does not decay, this paper provides a condition on the coefficient, which is necessary and sufficient so that the solutions tend to the solutions of the free wave equation.

Keywords

abstract linear hyperbolic equation asymptotic behavior asymptotically free property

1. Introduction

Let H be a separable complex Hilbert space H with inner product ${(\cdot, \cdot)}_{H}$ and norm $‖ \cdot ‖$ . Let A be a non-negative injective self-adjoint operator in H with domain $D (A)$ . Let $c (t)$ be a function which is of bounded variation and satisfies $\begin{matrix} (1.1) & inf_{t ⩾ 0} c (t) > 0 . \end{matrix}$ We consider the initial value problem of the abstract dissipative wave equation $\begin{array}{l} (1.2) & u^{″} (t) + c {(t)}^{2} A u (t) + b (t) u^{'} (t) = 0, t ⩾ 0, \\ (1.3) & u (0) = ϕ_{0}, u^{'} (0) = ψ_{0}, \end{array}$ with time-dependent coefficients. There are a number of results concerning (1.2)–(1.3) (see, for example, [1,6,8,10–12], [14, Section 2] and references therein).

In this paper, under the assumption that $b (t)$ is an integrable function on $[0, \infty)$ , we give a necessary and sufficient condition for the existence of a wave speed $c_{*}$ and a solution v of the free wave equation $\begin{array}{l} (1.4) & v^{″} (t) + c_{*}^{2} A v (t) = 0, \end{array}$ satisfying $\begin{matrix} (1.5) & lim_{t \to \infty} (‖ A^{1 / 2} (u (t) - v (t)) ‖ + ‖ u^{'} (t) - v^{'} (t) ‖) = 0 . \end{matrix}$

First, Arosio [1, Theorem 3] considered $\begin{array}{l} (1.6) & \frac{\partial^{2} u}{\partial t^{2}} (t, x) = a (t) Δ u (t, x) + G (x, t) + H (x, t) in [0, \infty) \times Ω, \\ (1.7) & u (t, x) = 0 on [0, \infty) \times \partial Ω, \\ (1.8) & u (0, x) = ϕ_{0} (x), \frac{\partial u}{\partial t} (0, x) = ψ_{0} (x) in Ω, \end{array}$ for a bounded open set Ω in $R^{n}$ , where $a (t) = c {(t)}^{2} + d (t)$ with $c {(t)}^{2} \in BV (0, \infty)$ and $d (t) \in L^{1} (0, \infty)$ satisfying $0 < ν ⩽ a (t)$ for almost every $t \in (0, \infty)$ , $G \in L^{1} ((0, \infty); L^{2} (Ω))$ , and $H \in BV ((0, \infty); H^{- 1} (Ω))$ with ${lim}_{t \to \infty} H (t) = 0$ in $H^{- 1} (Ω)$ . Then he showed the following.

If $\begin{matrix} (1.9) & lim_{t \to \infty} \int_{0}^{t} (c (s) - c_{\infty}) d s exists and is finite, \end{matrix}$ where $\begin{matrix} c_{\infty} = lim_{t \to \infty} c (t), \end{matrix}$ then for every weak solution $u \in C ([0, \infty); H_{0}^{1} (Ω)) \cap C^{1} ([0, \infty); L^{2} (Ω))$ of (1.6)–(1.7), there exists a solution $v \in C ([0, \infty); H_{0}^{1} (Ω)) \cap C^{1} ([0, \infty); L^{2} (Ω))$ of the free wave equation $\begin{array}{l} (1.10) & \frac{\partial^{2} v}{\partial t^{2}} (t, x) = c_{\infty}^{2} Δ v (t, x) in [0, \infty) \times Ω, \\ (1.11) & v (t, x) = 0 on [0, \infty) \times \partial Ω, \end{array}$ satisfying $\begin{matrix} (1.12) & lim_{t \to \infty} ({‖ u (t) - v (t) ‖}_{H_{0}^{1} (Ω)} + {‖ u^{'} (t) - v^{'} (t) ‖}_{L^{2} (Ω)}) = 0 . \end{matrix}$

Conversely, if there exists a weak solution $u (t) \in C ([0, \infty); H_{0}^{1} (Ω)) \cap C^{1} ([0, \infty); L^{2} (Ω))$ of (1.6)–(1.7) and a non-trivial solution $v (t)$ of the free wave equation (1.10)–(1.11) such that (1.12) holds, then (1.9) must hold.

If we take

H = L^{2} (Ω)

A = - Δ

with

D (A) = H^{2} (Ω) \cap H_{0}^{1} (Ω)

and

b (t) \equiv 0

, the abstract problem (1.2)–(1.3) becomes (1.6)–(1.7) above with

a (t) = c {(t)}^{2}

and

G (t) \equiv H (t) \equiv 0

. The method of [1] is applicable for positive self-adjoint operators A with compact resolvent. Here we note that if

c (t)

satisfies (1.1), then the assumptions

c^{2} (t) \in BV ([0, \infty))

and

c (t) \in BV ([0, \infty))

are equivalent.

Matsuyama [8, Theorem 2.1] considered the problem (1.6)–(1.7) for $Ω = R^{n}$ , where $a (t) = c {(t)}^{2}$ with $c (t)$ satisfying (1.1) and $\begin{matrix} (1.13) & c \in {Lip}_{loc} ([0, \infty)), c^{'} \in L^{1} (0, \infty), \end{matrix}$ $G (t) \equiv H (t) \equiv 0$ , that is, the problem (1.2)–(1.3) with $H = L^{2} (R^{n})$ , $A = - Δ$ with $D (A) = H^{2} (R^{n})$ and $b (t) \equiv 0$ , and showed the following: Assume that (1.9) holds. Then for every solution $u \in ⋂_{j = 0, 1, 2} C^{j} ([0, \infty); H^{s - j} (R^{n}))$ ( $s ⩾ 1$ ) of (1.6)–(1.7), there exists a solution v of the free wave equation (1.10)–(1.11) satisfying $\begin{matrix} (1.14) & lim_{t \to \infty} ({‖ \nabla (u (t) - v (t)) ‖}_{H^{s - 1} (R^{n})} + {‖ u^{'} (t) - v^{'} (t) ‖}_{H^{s - 1} (R^{n})}) = 0 . \end{matrix}$ On the other hand, he showed that if $\begin{matrix} (1.15) & lim_{t \to \infty} | \int_{0}^{t} (c (s) - c_{\infty}) d s | = \infty, \end{matrix}$ there exists a non-trivial free solution u of (1.6)–(1.7) such that no solution v of the free wave equation (1.10)–(1.11) satisfies (1.14). Then, applying the result to Kirchhoff equation, he proved in [9] the existence of a non-trivial small initial data such that the solution of Kirchoff equation is not asymptotically free.

Matsuyama and Ruzhansky [10, Theorem 1.1] considered the system $D_{t} U = A (t, D_{x}) U$ in $L^{2} {(R^{n})}^{m}$ , and generalized the results of [8]. Furthermore, in a case $m = 1$ and $A (t, D_{x}) = - c {(t)}^{2} Δ$ , this result is an improvement of the necessary condition for the asymptotically freeness of [8] as follows: Assume that c satisfies (1.1) and (1.13). If (1.15) holds, then for every non-trivial solutions of (1.6)–(1.7) with radially symmetric initial data, there exists no solution of the free wave equation (1.10)–(1.11) satisfying (1.14).

The purpose of this paper is to show a necessary and sufficient condition for asymptotically free property of (1.2)–(1.3) for general non-negative injective self-adjoint operator A (Theorem 1). Especially we are interested in the necessary condition. To obtain the necessary condition, Arosio [1, Theorem 3, (ii)] employed the discreteness of the spectrum corresponding to A, and Matsuyama and Ruzhansky [10, Theorem 1.1] employed the Riemann–Lebesgue theorem for the Fourier transform. In this paper, we use the property of continuous unitary group $e^{i t A^{1 / 2}}$ .

Another difference between the previous results and the result of this paper is that we do not assume $c_{*} = c_{\infty}$ in (1.4) a priori. We show that if there exists a non-trivial solution u of (1.2) which approaches to a solution of (1.4) with some wave speed $c_{*}$ , then $c_{*}$ coincides with $c_{\infty} = {lim}_{t \to \infty} c (t)$ (Theorem 1(ii)).

The result of this paper is applied to dissipative Kirchhoff equations in [15] to obtain the necessary decay condition on the dissipative term for the asymptotically free property. This condition is essentially stronger than that of linear dissipative wave equation.

2. Main result

Notation 1.
For every $α ⩾ 0$ , the domain $D (A^{α})$ of $A^{α}$ becomes a Hilbert space $H_{α}$ equipped with the inner product $\begin{matrix} {(f, g)}_{H_{α}} : = {(A^{α} f, A^{α} g)}_{H} + {(f, g)}_{H} . \end{matrix}$ The norm is denoted by $‖ f ‖_{H_{α}}^{2} = {(f, f)}_{H_{α}}$ . We note that $H_{0} = H$ . For every $α < 0$ , let $H_{α}$ denote the dual space of $H_{- α}$ with the dual norm, namely, $H_{α}$ is the completion of H by the norm $\begin{matrix} ‖ f ‖_{H_{α}} = sup {| {(f, g)}_{H} |; g \in H_{- α}, ‖ g ‖_{- α} = 1} . \end{matrix}$
Notation 2.
For every $α > 0$ , let $H_{α}$ denote the completion of $D (A^{α})$ by the norm $‖ A^{α} \cdot ‖$ . Let $A^{α}$ be extension of $A^{α}$ on $H_{α}$ . The fact that $A^{α}$ is an injective self-adjoint operator implies that the range $R (A^{α})$ is dense in H, and thus $A^{α} : H_{α} \to H$ is bijective. From this fact and the definition, it follows that $A^{α} : H_{α} \to H$ is an isometric isomorphism.
Example 1.
Let $H = L^{2} (R^{n})$ and $A = - Δ$ with $D (A) = H^{2} (R^{n})$ . For $α > 0$ , the space $H_{α} (R^{n})$ equals the homogeneous Sobolev space ${\dot{H}}^{2 α}$ , and $H_{- α}$ equals the negative Sobolev space $H^{- 2 α} (R^{n})$ .
Notation 3.
For a Banach space X, let $AC ([0, \infty); X)$ denote all of X valued absolutely continuous functions on $[0, \infty)$ , and ${AC}_{loc} ([0, \infty); X) = {f \in C ([0, \infty)); f \in AC ([0, T]) for every T > 0}$ .

We consider the equation (1.2)–(1.3) and a free wave equation (1.4) in a somewhat wide class as $\begin{array}{l} (2.1) & u^{″} (t) + c {(t)}^{2} A^{1 / 2} A^{1 / 2} u (t) + b (t) u^{'} (t) = 0, t ⩾ 0, \\ (2.2) & u (0) = ϕ_{0}, u^{'} (0) = ψ_{0}, \end{array}$ for $(ϕ_{0}, ψ_{0}) \in H_{1 / 2} \times H$ , and $\begin{array}{l} (2.3) & v^{″} (t) + c_{}^{2} A^{1 / 2} A^{1 / 2} v (t) = 0, t ⩾ 0 . \end{array}$
Definition 1.
We say that u is a weak solution of (2.1) if $u \in C ([0, \infty); H_{1 / 2})$ , $\begin{array}{l} u (t) - u (0) \in ⋂_{j = 0, 1} C^{j} ([0, \infty); H_{(1 - j) / 2}), \\ u^{'} (t) \in {AC}_{loc} ([0, \infty); H_{- 1 / 2}), \end{array}$ and (2.1) holds in the space $H_{- 1 / 2}$ for almost every $t \in (0, \infty)$ .

A weak solution of (2.3) is defined as a weak solution of (2.1) with $c (t) \equiv c^{}$ and $b (t) \equiv 0$ .

Here we note that if u is a weak solution of (2.1)–(2.2), then $x : = (A^{1 / 2} u, u^{'})$ is a weak solution of the following Cauchy problem: $\begin{array}{l} (2.4) & \frac{d}{d t} x (t) + (\begin{matrix} 0 & - A^{1 / 2} \\ c {(t)}^{2} A^{1 / 2} & b (t) \end{matrix}) x (t) = (\begin{matrix} 0 \\ 0 \end{matrix}), \\ (2.5) & x (0) = (\begin{matrix} A^{1 / 2} ϕ_{0} \\ ψ_{0} \end{matrix}) \in H \times H, \end{array}$ in the sense that $\begin{matrix} x (t) \in C ([0, \infty); H \times H) \cap {AC}_{loc} ([0, \infty); H_{- 1 / 2} \times H_{- 1 / 2}), \end{matrix}$ and that (2.4) holds in $H_{- 1 / 2} \times H_{- 1 / 2}$ for almost every $t \in (0, \infty)$ . Conversely, if $x = (w, z)$ is a weak solution of (2.4)–(2.5), then $u = A^{- 1 / 2} w$ is a weak solution of (2.1)–(2.2).

Our main result is the following: Theorem 1.
Let $c (t)$ be of bounded variation on $(0, \infty)$ satisfying (1.1), and put $c_{\infty} = {lim}_{t \to \infty} c (t)$ . Let $b (t)$ be an integrable function on $[0, \infty)$ . Then the following holds.
Suppose that (1.9) holds. Then for every weak solution u of (2.1), there exists a unique weak solution v of the free wave equation (2.3) with wave speed $c_{} = c_{\infty}$ such that* $\begin{matrix} (2.6) & lim_{t \to \infty} (‖ A^{1 / 2} (u (t) - v (t)) ‖ + ‖ u^{'} (t) - v^{'} (t) ‖) = 0 \end{matrix}$ holds.

Suppose that there exists a non-trivial weak solution u of (2.1), a positive constant $c_{}$ and a weak solution v of the free wave equation* (2.3) such that (2.6) holds. Then $c_{} = c_{\infty}$ and* (1.9) must hold.

Remark 1.
If $b (t)$ is integrable and of bounded variation as well, the Cauchy problem (2.1)–(2.2) is uniquely solvable. (See Proposition 5 in the Appendix.)
Remark 2.
Assume that the initial data $(A^{1 / 2} ϕ_{0}, ψ_{0})$ belongs to $D (A^{J / 2}) \times D (A^{J / 2})$ for $J ⩾ 1$ , and u is a solution of (2.1)–(2.2) in the sense that $\begin{matrix} (2.7) & (A^{1 / 2} u, u^{'}) \in C ([0, \infty); H_{J} \times H_{J}) \cap {AC}_{loc} ([0, \infty); H_{J - 1 / 2} \times H_{J - 1 / 2}), \end{matrix}$ and that (2.4) holds in $H_{(J - 1) / 2} \times H_{(J - 1) / 2}$ for almost every $t \in (0, \infty)$ . Then the solution v of (2.3) given by (i) of Theorem 1 satisfies (2.7) and $\begin{matrix} (2.8) & lim_{t \to \infty} ({‖ A^{1 / 2} (u (t) - v (t)) ‖}_{H_{J / 2}} + {‖ u^{'} (t) - v^{'} (t) ‖}_{H_{J / 2}}) = 0 . \end{matrix}$ In fact, since we see that (3.12) in Section 2 with $‖ \cdot ‖_{H \times H}$ replaced by $‖ \cdot ‖_{H_{J / 2} \times H_{J / 2}}$ holds, we can prove (2.8) in the same way as in the proof of Theorem 1(i).

3. Proof of Theorem 1

We first give a lemma, which is employed in the proof of the equality $c_{*} = c_{\infty}$ .

Lemma 2.

If $g (t)$ is of bounded variation on $[0, \infty)$ , then $\begin{matrix} lim_{T \to \infty} ‖ \frac{1}{T} \int_{0}^{T} g (t) exp (i G (t) A^{1 / 2}) u d t ‖ = 0 \end{matrix}$ for every $u \in H$ , where $G (t) = \int_{0}^{t} g (s) d s$ .

Proof.

Let w be an arbitrary element of $D (A^{1 / 2})$ . Then, $exp (i G (t) A^{1 / 2}) w$ is absolutely continuous on $[0, \infty)$ and differentiable almost everywhere on $(0, \infty)$ , and thus we have $\begin{matrix} (3.1) & \frac{d}{d t} exp (i G (t) A^{1 / 2}) w = i g (t) exp (i G (t) A^{1 / 2}) A^{1 / 2} w, \end{matrix}$ for almost every t in $(0, \infty)$ . Integrating (3.1) on $(0, T)$ , and dividing the equality by T, we have $\begin{matrix} \frac{1}{T} \int_{0}^{T} exp (i G (t) A^{1 / 2}) g (t) A^{1 / 2} w d t = \frac{exp (i G (T) A^{1 / 2}) w - w}{i T} . \end{matrix}$ Since $‖ exp (i τ (T) A^{1 / 2}) w ‖ = ‖ w ‖$ for every $T \in [0, \infty)$ , we obtain $\begin{matrix} lim_{T \to \infty} ‖ \frac{1}{T} \int_{0}^{T} exp (i G (t) A^{1 / 2}) g (t) A^{1 / 2} w d t ‖ = 0 . \end{matrix}$ Let $δ > 0$ be an arbitrary positive number. The assumption that A is an injective self-adjoint operator implies that the range of $A^{1 / 2}$ is dense in H. Thus, we can take $w \in D (A^{1 / 2})$ such that $‖ u - A^{1 / 2} w ‖ < δ$ , and therefore we have $\begin{array}{l} \underset{T \to \infty}{lim sup} ‖ \frac{1}{T} \int_{0}^{T} g (t) exp (i G (t) A^{1 / 2}) u d t ‖ \\ ⩽ \underset{T \to \infty}{lim sup} ‖ \frac{1}{T} \int_{0}^{T} g (t) exp (i G (t) A^{1 / 2}) (u - A^{1 / 2} w) d t ‖ \\ + lim_{T \to \infty} ‖ \frac{1}{T} \int_{0}^{T} g (t) exp (i G (t) A^{1 / 2}) A^{1 / 2} w d t ‖ \\ ⩽ sup_{t ⩾ 0} (| g (t) | ‖ exp (i G (t) A^{1 / 2}) (u - A^{1 / 2} w) ‖) \\ ⩽ δ sup_{t ⩾ 0} | g (t) | . \end{array}$ Since $δ > 0$ is arbitrary, we obtain $\begin{matrix} lim_{T \to \infty} ‖ \frac{1}{T} \int_{0}^{T} g (t) exp (i G (t) A^{1 / 2}) u d t ‖ = 0 . \end{matrix}$ □

Now we prove Theorem 1. We express the solution $x (t)$ of (2.4) by the method of ordinary differential equation by Wintner [13] (see also Coddington and Levinson [3], Hartman [7]), similarly to the proof of Matsuyama [8]. Let $\begin{matrix} Y (t) : = (\begin{matrix} e^{i τ (t) A^{1 / 2}} & e^{- i τ (t) A^{1 / 2}} \\ i c (t) e^{i τ (t) A^{1 / 2}} & - i c (t) e^{- i τ (t) A^{1 / 2}} \end{matrix}), \end{matrix}$ where $\begin{matrix} τ (t) = \int_{0}^{t} c (s) d s . \end{matrix}$ Then $\begin{matrix} Y {(t)}^{- 1} = \frac{1}{2} (\begin{matrix} e^{- i τ (t) A^{1 / 2}} & - \frac{i}{c (t)} e^{- i τ (t) A^{1 / 2}} \\ e^{i τ (t) A^{1 / 2}} & \frac{i}{c (t)} e^{i τ (t) A^{1 / 2}} \end{matrix}) . \end{matrix}$

In order to approximate c by $C^{1}$ class functions, we use the mollifier as in the proof of Arosio [1]. Let ρ be a $C_{0}^{\infty} (R)$ function with support contained in $[- 1, 1]$ and $\int_{R} ρ (t) d t = 1$ . Let δ be an arbitrary positive number. Put $ρ_{δ} = \frac{1}{δ} ρ (\frac{t}{δ})$ , and $c_{δ}$ be the mollification of c, that is, $\begin{matrix} c_{δ} (t) = \tilde{c} * ρ_{δ} (t) = \int_{R} \tilde{c} (t - s) ρ_{δ} (s) d s (\in C^{\infty} (R)), \end{matrix}$ where $\tilde{c}$ is a extension of c to $R$ such that $\tilde{c} (t) = c (0)$ for $t < 0$ . From the assumption that c is bounded variation on $[0, \infty)$ , it follows that $\begin{array}{l} (3.2) & \int_{S}^{T} | c (s) - c_{δ} (s) | d s ⩽ δ Var (c; [max {S - δ, 0}, T + δ]), \\ (3.3) & \int_{S}^{T} | c_{δ}^{'} (s) | d s ⩽ Var (c; [max {S - δ, 0}, T + δ]), \end{array}$ for every $S, T ⩾ 0$ with $S < T$ (see [4] and [1]). Inequality (3.2) with $δ = 1 / n$ implies ${lim}_{n \to \infty} c_{1 / n} = c$ in $L^{1} ((0, \infty))$ . Thus, we can take a subsequence ${n_{k}}_{k = 1}^{\infty}$ and a subset $N_{1} \subset (0, \infty)$ such that the Lebesgue measure of $N_{1}$ is 0 and that $\begin{matrix} (3.4) & lim_{k \to \infty} c_{1 / n_{k}} (t) = c (t) \end{matrix}$ for every $t \in (0, \infty) ∖ N_{1}$ . Let $\begin{matrix} Y_{k} (t) : = (\begin{matrix} e^{i τ (t) A^{1 / 2}} & e^{- i τ (t) A^{1 / 2}} \\ i c_{1 / n_{k}} (t) e^{i τ (t) A^{1 / 2}} & - i c_{1 / n_{k}} (t) e^{- i τ (t) A^{1 / 2}} \end{matrix}) . \end{matrix}$ Then $\begin{matrix} Y_{k} {(t)}^{- 1} = \frac{1}{2} (\begin{matrix} e^{- i τ (t) A^{1 / 2}} & - \frac{i}{c_{1 / n_{k}} (t)} e^{- i τ (t) A^{1 / 2}} \\ e^{i τ (t) A^{1 / 2}} & \frac{i}{c_{1 / n_{k}} (t)} e^{i τ (t) A^{1 / 2}} \end{matrix}), \end{matrix}$ and $\begin{matrix} \frac{d}{d t} Y_{k} (t) + (\begin{matrix} 0 & - \frac{c (t)}{c_{1 / n_{k}} (t)} A^{1 / 2} \\ c (t) c_{1 / n_{k}} (t) A^{1 / 2} & - \frac{c_{1 / n_{k}}^{'} (t)}{c_{1 / n_{k}} (t)} \end{matrix}) Y_{k} (t) = (\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}) . \end{matrix}$ From (1.1) and (3.4), it follows that $\begin{array}{l} (3.5) & lim_{k \to \infty} {‖ Y_{k} {(t)}^{- 1} - Y {(t)}^{- 1} ‖}_{L (H \times H)} = 0 for every t \in (0, \infty) ∖ N_{1} . \end{array}$

Let $x (t)$ be a weak solution of (2.4)–(2.5). By putting $\begin{array}{l} B_{k} (t) = Y_{k} {(t)}^{- 1} (\begin{matrix} 0 & (\frac{c (t)}{c_{1 / n_{k}} (t)} - 1) A^{1 / 2} \\ c (t) (c (t) - c_{1 / n_{k}} (t)) A^{1 / 2} & b (t) + \frac{c_{1 / n_{k}}^{'} (t)}{c_{1 / n_{k}} {(t)}^{2}} \end{matrix}) Y_{k} (t), \\ y (t) = (\begin{matrix} y_{1} (t) \\ y_{2} (t) \end{matrix}) : = Y {(t)}^{- 1} x (t), \end{array}$ and $\begin{matrix} y_{k} (t) = (\begin{matrix} y_{k}^{(1)} (t) \\ y_{k}^{(2)} (t) \end{matrix}) : = Y_{k} {(t)}^{- 1} x (t), \end{matrix}$ (2.4) is transformed into $\begin{matrix} (3.6) & \frac{d}{d t} y_{k} (t) + B_{k} (t) y_{k} (t) = 0 in H_{- 1 / 2} \times H_{- 1 / 2} . \end{matrix}$

Let ${E (λ)}$ be a spectral family associated with the self adjoint operator A. Then (3.6) yields $\begin{matrix} (3.7) & \frac{d}{d t} (E (λ) y_{k}) (t) + B_{λ, k} (t) (E (λ) y_{k}) (t) = 0 in H \times H, \end{matrix}$ for almost every $t \in (0, \infty)$ , where $\begin{array}{l} B_{λ, k} (t) = Y_{k} {(t)}^{- 1} (\begin{matrix} 0 & (\frac{c (t)}{c_{1 / n_{k}} (t)} - 1) A^{1 / 2} E (λ) \\ c (t) (c (t) - c_{1 / n_{k}} (t)) A^{1 / 2} E (λ) & b (t) + \frac{c_{1 / n_{k}}^{'} (t)}{c_{1 / n_{k}} {(t)}^{2}} \end{matrix}) Y_{k} (t) . \end{array}$ By (1.1) and the fact that $e^{\pm i s A^{1 / 2}}$ is unitary, the operators $Y_{k} (t)$ and $Y_{k} {(t)}^{- 1}$ are bounded on $H \times H$ uniformly in k and t. Thus, observing (1.1) again, we have a positive constant $K_{1}$ satisfying $\begin{matrix} (3.8) & {‖ B_{λ, k} (t) ‖}_{L (H \times H)} ⩽ K_{1} (λ^{1 / 2} | c (t) - c_{1 / n_{k}} (t) | + | c_{1 / n_{k}}^{'} (t) | + | b (t) |) \end{matrix}$ for every $λ, k > 0$ and every $t ⩾ 0$ .

We estimate $(E (λ) y_{k}) (t)$ . The definition of weak solution implies $x (t) \in {AC}_{loc} ([0, \infty); H_{- 1 / 2} \times H_{- 1 / 2})$ , and therefore, $(E (λ) y_{k}) (t) \in {AC}_{loc} ([0, \infty); H \times H)$ . Thus, it follows from (3.7) and (3.8) that $\begin{array}{l} {‖ (E (λ) y_{k}) (t) - (E (λ) y_{k}) (s) ‖}_{H \times H} \\ (3.9) & ⩽ K_{1} \int_{s}^{t} (λ^{1 / 2} | c (σ) - c_{1 / n_{k}} (σ) | + | c_{1 / n_{k}}^{'} (σ) | + | b (σ) |) {‖ (E (λ) y_{k}) (σ) ‖}_{H \times H} d σ, \end{array}$ for every $0 < s < t$ . Thus $\begin{array}{l} {‖ (E (λ) y_{k}) (t) ‖}_{H \times H} \\ ⩽ {‖ (E (λ) y_{k}) (0) ‖}_{H \times H} \\ + K_{1} \int_{0}^{t} (λ^{1 / 2} | c (σ) - c_{1 / n_{k}} (σ) | + | c_{1 / n_{k}}^{'} (σ) | + | b (σ) |) {‖ (E (λ) y_{k}) (σ) ‖}_{H \times H} d σ, \end{array}$ for every $t ⩾ 0$ . Hence by Gronwall’s inequality together with the assumption that $b \in L^{1} ((0, \infty))$ , (3.2) and (3.3), $\begin{array}{l} {‖ (E (λ) y_{k}) (t) ‖}_{H \times H} & ⩽ exp (K_{1} ((λ^{1 / 2} / n_{k} + 1) Var (c; [0, \infty)) + ‖ b ‖_{L^{1} (0, \infty)})) {‖ (E (λ) y_{k}) (0) ‖}_{H \times H} \\ ⩽ exp (K_{1} ((λ^{1 / 2} / n_{k} + 1) Var (c; [0, \infty)) + ‖ b ‖_{L^{1} (0, \infty)})) {‖ y_{k} (0) ‖}_{H \times H} \end{array}$ for every $t ⩾ 0$ . Substituting this inequality into (3.9), and observing (3.2) and (3.3) again, we obtain $\begin{array}{l} {‖ (E (λ) y_{k}) (t) - (E (λ) y_{k}) (s) ‖}_{H \times H} \\ ⩽ K_{1} ((λ^{1 / 2} / n_{k} + 1) Var (c; [max {s - (1 / n_{k}), 0}, t + (1 / n_{k})]) + ‖ b ‖_{L^{1} (s, t)}) \\ (3.10) & \times exp (K_{1} ((λ^{1 / 2} / n_{k} + 1) Var (c; [0, \infty)) + ‖ b ‖_{L^{1} (0, \infty)})) {‖ y_{k} (0) ‖}_{H \times H} \end{array}$ for every $0 ⩽ s ⩽ t$ . From (3.5), it follows that $\begin{array}{l} lim_{k \to \infty} {‖ y_{k} (t) - y (t) ‖}_{H \times H} = 0 for every t \in (0, \infty) ∖ N_{1}, \end{array}$ and therefore $\begin{array}{l} lim_{k \to \infty} {‖ (E (λ) y_{k}) (t) - (E (λ) y) (t) ‖}_{H \times H} = 0 \end{array}$ for every $s, t \in (0, \infty) ∖ N_{1}$ and $λ > 0$ . Thus, letting $k \to \infty$ in (3.10), we obtain $\begin{array}{rcl} {‖ (E (λ) y) (t) - (E (λ) y) (s) ‖}_{H \times H} & ⩽ & K_{1} (Var (c; [s -, t +]) + ‖ b ‖_{L^{1} (s, t)}) \\ \times exp (K_{1} Var (c; [0, \infty)) + ‖ b ‖_{L^{1} (0, \infty)}) {‖ y (0) ‖}_{H \times H} \end{array}$ for every $s, t \in (0, \infty) ∖ N_{1}$ and $λ > 0$ , where $Var (c; [s -, t +]) = {lim}_{δ \to 0 + 0} Var (c; [s - δ, t + δ])$ . Therefore we have $\begin{array}{rcl} {‖ y (t) - y (s) ‖}_{H \times H} & ⩽ & K_{1} (Var (c; [s -, t +]) + ‖ b ‖_{L^{1} (s, t)}) \\ (3.11) & \times exp (K_{1} Var (c; [0, \infty)) + ‖ b ‖_{L^{1} (0, \infty)}) {‖ y (0) ‖}_{H \times H} \end{array}$ for every $s, t \in (0, \infty) ∖ N_{1}$ . Since c is of bounded variation on $[0, \infty)$ , ${lim}_{s, t \to \infty} Var (c; [s -, t +]) = 0$ . Hence, letting $s, t$ $(\notin N_{1}) \to \infty$ in (3.11) implies the existence of the limit $\begin{matrix} lim_{t \notin N_{1}, t \to \infty} y (t) = y_{\infty} : = (\begin{matrix} y_{1, \infty} \\ y_{2, \infty} \end{matrix}) in H \times H . \end{matrix}$ Thus $y (t)$ is expressed as $\begin{matrix} y (t) = y_{\infty} + r (t), \end{matrix}$ with $\begin{matrix} (3.12) & lim_{t \notin N_{1}, t \to \infty} {‖ r (t) ‖}_{H \times H} = 0 . \end{matrix}$ Hence we obtain the expression of the solution of (1.2) $\begin{array}{l} (3.13) & \begin{matrix} (\begin{matrix} A^{1 / 2} u (t) \\ u^{'} (t) \end{matrix}) & = x (t) = Y (t) y (t) \\ = Y (t) y_{\infty} + Y (t) r (t) \end{matrix} \\ (3.14) & = (\begin{matrix} e^{i τ (t) A^{1 / 2}} y_{1, \infty} + e^{- i τ (t) A^{1 / 2}} y_{2, \infty} \\ i c (t) e^{i τ (t) A^{1 / 2}} y_{1, \infty} - i c (t) e^{- i τ (t) A^{1 / 2}} y_{2, \infty} \end{matrix}) + Y (t) r (t) . \end{array}$

Let v be a solution of (1.4). Then it is expressed as $\begin{matrix} (3.15) & z (t) = (\begin{matrix} A^{1 / 2} v (t) \\ v^{'} (t) \end{matrix}) = (\begin{matrix} e^{i c_{*} t A^{1 / 2}} ϕ + e^{- i c_{*} t A^{1 / 2}} ψ \\ i c_{*} e^{i c_{*} t A^{1 / 2}} ϕ - i c_{*} e^{- i c_{*} t A^{1 / 2}} ψ \end{matrix}), \end{matrix}$ where $\begin{array}{l} ϕ = \frac{1}{2} (A^{1 / 2} v (0) - \frac{i}{c_{*}} v^{'} (0)), ψ = \frac{1}{2} (A^{1 / 2} v (0) + \frac{i}{c_{*}} v^{'} (0)) . \end{array}$ Since $x, z \in C ([0, \infty); H \times H)$ , we easily see that ${lim}_{t \to \infty} ‖ x (t) - z (t) ‖_{H \times H} = 0$ if and only if $\begin{matrix} (3.16) & lim_{t \notin N_{1}, t \to \infty} {‖ x (t) - z (t) ‖}_{H \times H} = 0 . \end{matrix}$ Thus, the convergence (2.6) holds if and only if (3.16) holds. By the expressions (3.14) and (3.15), we see that (3.16) holds if and only if the following two convergences hold. $\begin{array}{l} (3.17) & lim_{t \notin N_{1}, t \to \infty} ‖ e^{i τ (t) A^{1 / 2}} y_{1, \infty} + e^{- i τ (t) A^{1 / 2}} y_{2, \infty} - e^{i c_{*} t A^{1 / 2}} ϕ - e^{- i c_{*} t A^{1 / 2}} ψ ‖ = 0, \\ (3.18) & lim_{t \notin N_{1}, t \to \infty} ‖ c (t) e^{i τ (t) A^{1 / 2}} y_{1, \infty} - c (t) e^{- i τ (t) A^{1 / 2}} y_{2, \infty} - c_{*} e^{i c_{*} t A^{1 / 2}} ϕ + c_{*} e^{- i c_{*} t A^{1 / 2}} ψ ‖ = 0 . \end{array}$

Here we prove the following lemma.

Lemma 3.

Assume that v is a weak solution of linear wave equation of (1.4) with $\begin{matrix} (3.19) & c_{*} = c_{\infty} (= lim_{t \to \infty} c (t)) . \end{matrix}$ Then the convergence (2.6) holds if and only if the following two convergences hold: $\begin{array}{l} (3.20) & lim_{t \notin N_{1}, t \to \infty} ‖ e^{i (τ (t) - c_{\infty} t) A^{1 / 2}} y_{1, \infty} - ϕ ‖ = 0, \\ (3.21) & lim_{t \notin N_{1}, t \to \infty} ‖ e^{- i (τ (t) - c_{\infty} t) A^{1 / 2}} y_{2, \infty} - ψ ‖ = 0 . \end{array}$

Proof.

By the argument above, the convergence (2.6) holds if and only if (3.17) and (3.18) hold. By the assumption (3.19) and the fact that $e^{i τ (t) A^{1 / 2}}$ is a $C^{0}$ unitary group on H, we see that (3.18) holds if and only if the following convergence holds. $\begin{array}{l} (3.22) & lim_{t \notin N_{1}, t \to \infty} ‖ e^{i τ (t) A^{1 / 2}} y_{1, \infty} - e^{- i τ (t) A^{1 / 2}} y_{2, \infty} - e^{i c_{\infty} t A^{1 / 2}} ϕ + e^{- i c_{\infty} t A^{1 / 2}} ψ ‖ = 0 . \end{array}$ Hence, (2.6) holds, if and only if (3.17) and (3.22) hold, equivalently, the following two convergences hold. $\begin{array}{l} lim_{t \notin N_{1}, t \to \infty} ‖ e^{i τ (t) A^{1 / 2}} y_{1, \infty} - e^{i c_{\infty} t A^{1 / 2}} ϕ ‖ = 0, \\ lim_{t \notin N_{1}, t \to \infty} ‖ e^{- i τ (t) A^{1 / 2}} y_{2, \infty} - e^{- i c_{\infty} t A^{1 / 2}} ψ ‖ = 0 . \end{array}$ Since $e^{i s A^{1 / 2}}$ is a unitary group on H, these convergences are equivalent to (3.20) and (3.21). □

Now we are ready to complete the proof of Theorem 1.

Proof of (i). Assume that (1.9) holds. We take $c_{*} = c_{\infty}$ ( $= {lim}_{t \to \infty} c (t)$ ), and $\begin{matrix} ϕ = e^{i {lim}_{t \to \infty} (τ (t) - c_{\infty} t) A^{1 / 2}} y_{1, \infty}, ψ = e^{- i {lim}_{t \to \infty} (τ (t) - c_{\infty} t) A^{1 / 2}} y_{2, \infty} . \end{matrix}$ Then by the strong continuity of the $e^{i s A^{1 / 2}}$ with respect to s on $[0, \infty)$ , the convergences (3.20) and (3.21) hold, and therefore (2.6) holds by Lemma 3.

Proof of (ii). Assume that there are a non-trivial solution u of (2.1), a positive number $c_{*}$ and a solution v of (2.3) such that (2.6) holds. Put $\begin{matrix} F (t) : = \frac{1}{2} {‖ u^{'} (t) ‖}^{2} + \frac{1}{2} c {(t)}^{2} {‖ A^{1 / 2} u (t) ‖}^{2} \end{matrix}$ for every $t ⩾ 0$ . Since u is non-trivial and $‖ u^{'} (t) ‖^{2} + ‖ A^{1 / 2} u (t) ‖^{2}$ is continuous, there is $S \in [0, \infty) ∖ N_{1}$ such that $‖ u^{'} (S) ‖^{2} + ‖ A^{1 / 2} u (S) ‖^{2} > 0$ . Then by (1.1), we have $\begin{matrix} (3.23) & F (S) > 0 . \end{matrix}$ For every $λ > 0$ and $δ > 0$ , we put $u_{λ} = E ([0, λ)) u$ , $\begin{array}{l} F_{λ} (t) : = \frac{1}{2} {‖ u_{λ}^{'} (t) ‖}^{2} + \frac{1}{2} c {(t)}^{2} {‖ A^{1 / 2} u_{λ} (t) ‖}^{2} for every t ⩾ 0, \\ F_{λ, k} (t) : = \frac{1}{2} {‖ u_{λ}^{'} (t) ‖}^{2} + \frac{1}{2} c_{1 / n_{k}} {(t)}^{2} {‖ A^{1 / 2} u_{λ} (t) ‖}^{2} for every t ⩾ 0 . \end{array}$ Since u satisfies (2.1) in $H_{- 1 / 2} \times H_{- 1 / 2}$ for almost every $t \in (0, \infty)$ , $u_{λ}$ satisfies (2.1) in $H \times H$ for almost every $t \in (0, \infty)$ . Thus we have $\begin{array}{rcl} F_{λ, k}^{'} (t) & = & (c_{1 / n_{k}} {(t)}^{2} - c {(t)}^{2}) {(u_{λ}^{'} (t), A^{1 / 2} A^{1 / 2} u_{λ} (t))}_{H} - b (t) {‖ u_{λ}^{'} (t) ‖}^{2} \\ + c_{1 / n_{k}} (t) c_{1 / n_{k}}^{'} (t) {‖ A^{1 / 2} u_{λ} (t) ‖}^{2} \\ ⩾ & - \frac{1}{c_{0}} | c_{1 / n_{k}} {(t)}^{2} - c {(t)}^{2} | \sqrt{λ} F_{λ, k} (t) - 2 (| b (t) | + \frac{| c_{1 / n_{k}}^{'} (t) |}{c_{1 / n_{k}} (t)}) F_{λ, k} (t) \\ ⩾ & - (\frac{2 \sqrt{λ}}{c_{0}} (sup_{t ⩾ 0} c (t)) | c_{1 / n_{k}} (t) - c (t) | - 2 | b (t) | - 2 \frac{| c_{1 / n_{k}}^{'} (t) |}{c_{0}}) F_{λ, k} (t), \end{array}$ for almost every $t \in (0, \infty)$ , where $c_{0} = {inf}_{t ⩾ 0} c (t)$ ( $> 0$ ). Hence, observing (3.2), (3.3) and the absolute continuity of $F_{λ, k} (t)$ with respect to t, we obtain $\begin{array}{rcl} F_{λ, k} (t) & ⩾ & F_{λ, k} (S) exp (- \frac{2 \sqrt{λ}}{c_{0} n_{k}} sup_{t ⩾ 0} c (t) Var (c; [0, \infty)) - 2 ‖ b ‖_{L^{1} (0, \infty)} - \frac{2}{c_{0}} Var (c; [0, \infty))) \end{array}$ for every $t ⩾ S$ . Letting $k \to \infty$ in the inequality above, and observing (3.4), we obtain $\begin{matrix} F_{λ} (t) ⩾ F_{λ} (S) exp (- 2 ‖ b ‖_{L^{1} (0, \infty)} - \frac{2}{c_{0}} Var (c; [0, \infty))), \end{matrix}$ for every $t ⩾ S$ satisfying $t \notin N_{1}$ . Letting $λ \to \infty$ in the above inequality yields $\begin{matrix} F (t) ⩾ F (S) exp (- 2 ‖ b ‖_{L^{1} (0, \infty)} - \frac{2}{c_{0}} Var (c; [0, \infty))), \end{matrix}$ for every $t ⩾ S$ satisfying $t \notin N_{1}$ , which together with (3.23) implies that $\begin{matrix} (3.24) & (y_{1, \infty}, y_{2, \infty}) \neq (0, 0) . \end{matrix}$

We next prove $\begin{matrix} (3.25) & c_{*} = c_{\infty} . \end{matrix}$ By the expression (3.15), we have $\begin{array}{l} (3.26) & {‖ A^{1 / 2} v (t) ‖}^{2} = ‖ ϕ ‖^{2} + ‖ ψ ‖^{2} + 2 Re {(e^{2 i c_{*} t A^{1 / 2}} ϕ, ψ)}_{H}, \\ (3.27) & {‖ v^{'} (t) ‖}^{2} = c_{*}^{2} (‖ ϕ ‖^{2} + ‖ ψ ‖^{2} - 2 Re {(e^{2 i c_{*} t A^{1 / 2}} ϕ, ψ)}_{H}) . \end{array}$ By Lemma 2 with $g (t) \equiv 2 c_{*}$ , we have $\begin{array}{rcl} | lim_{T \to \infty} \frac{1}{T} \int_{0}^{T} {(e^{2 i c_{*} t A^{1 / 2}} ϕ, ψ)}_{H} d t | & = & | {(lim_{T \to \infty} \frac{1}{T} \int_{0}^{T} e^{2 i c_{*} t A^{1 / 2}} ϕ d t, ψ)}_{H} | \\ (3.28) & ⩽ & ‖ lim_{T \to \infty} \frac{1}{T} \int_{0}^{T} e^{2 i c_{*} t A^{1 / 2}} ϕ d t ‖ ‖ ψ ‖ = 0 . \end{array}$ Thus, $\begin{matrix} lim_{t \to \infty} \frac{1}{T} \int_{0}^{T} {(e^{2 i c_{*} t A^{1 / 2}} ϕ, ψ)}_{H} d t = 0, \end{matrix}$ which together with (3.26) and (3.27) yields $\begin{matrix} (3.29) & c_{*}^{2} lim_{t \to \infty} \frac{1}{T} \int_{0}^{T} {‖ A^{1 / 2} v (t) ‖}^{2} d t = lim_{t \to \infty} \frac{1}{T} \int_{0}^{T} {‖ v^{'} (t) ‖}^{2} d t . \end{matrix}$

Put $\begin{matrix} \begin{matrix} (\begin{matrix} w_{1} (t) \\ w_{2} (t) \end{matrix}) & : = Y (t) y_{\infty} = (\begin{matrix} e^{i τ (t) A^{1 / 2}} y_{1, \infty} + e^{- i τ (t) A^{1 / 2}} y_{2, \infty} \\ i c (t) e^{i τ (t) A^{1 / 2}} y_{1, \infty} - i c (t) e^{- i τ (t) A^{1 / 2}} y_{2, \infty} \end{matrix}) . \end{matrix} \end{matrix}$ Then $\begin{array}{l} (3.30) & {‖ w_{1} (t) ‖}^{2} = ‖ y_{1, \infty} ‖^{2} + ‖ y_{2, \infty} ‖^{2} + 2 Re {(e^{2 i τ (t) A^{1 / 2}} y_{1, \infty}, y_{2, \infty})}_{H}, \\ (3.31) & {‖ w_{2} (t) ‖}^{2} = c {(t)}^{2} (‖ y_{1, \infty} ‖^{2} + ‖ y_{2, \infty} ‖^{2} - 2 Re {(e^{2 i τ (t) A^{1 / 2}} y_{1, \infty}, y_{2, \infty})}_{H}) . \end{array}$ Using Lemma 2 with $g (t) = 2 c (t)$ , we have in the same way as in (3.28), $\begin{matrix} lim_{t \to \infty} \frac{1}{T} \int_{0}^{T} {(e^{2 i τ (t) A^{1 / 2}} y_{1, \infty}, y_{2, \infty})}_{H} d t = 0 . \end{matrix}$ Thus (3.30), (3.31), (3.24) and the convergence $c_{\infty} = {lim}_{t \to \infty} c (t)$ yield $\begin{array}{rcl} c_{\infty}^{2} lim_{T \to \infty} \frac{1}{T} \int_{0}^{T} {‖ w_{1} (t) ‖}^{2} d t & = & lim_{T \to \infty} \frac{1}{T} \int_{0}^{T} {‖ w_{2} (t) ‖}^{2} d t \\ (3.32) & = & c_{\infty}^{2} (‖ y_{1, \infty} ‖^{2} + ‖ y_{2, \infty} ‖^{2}) \neq 0 . \end{array}$ From the expression (3.13) with (3.12) and the boundedness of the operator $Y (t)$ uniformly to $t ⩾ 0$ , it follows that $\begin{matrix} lim_{t \to \infty} ‖ A^{1 / 2} u (t) - w_{1} (t) ‖ + lim_{t \to \infty} ‖ u^{'} (t) - w_{2} (t) ‖ = 0, \end{matrix}$ which together with (3.32) yields $\begin{matrix} \begin{matrix} c_{\infty}^{2} lim_{T \to \infty} \frac{1}{T} \int_{0}^{T} {‖ A^{1 / 2} u (t) ‖}^{2} d t & = lim_{T \to \infty} \frac{1}{T} \int_{0}^{T} {‖ u^{'} (t) ‖}^{2} d t \neq 0 . \end{matrix} \end{matrix}$ The equality above and (2.6) imply $\begin{matrix} (3.33) & \begin{matrix} c_{\infty}^{2} lim_{T \to \infty} \frac{1}{T} \int_{0}^{T} {‖ A^{1 / 2} v (t) ‖}^{2} d t & = lim_{T \to \infty} \frac{1}{T} \int_{0}^{T} {‖ v^{'} (t) ‖}^{2} d t \neq 0 . \end{matrix} \end{matrix}$ Comparing (3.29) and (3.33), we obtain (3.25).

Now we prove (1.9) under the assumption $\begin{matrix} (3.34) & y_{1, \infty} \neq 0 . \end{matrix}$ The case $y_{1, \infty} = 0$ and $y_{2, \infty} \neq 0$ can be treated in the same way. Put $\begin{matrix} f (t) : = \int_{0}^{t} (c (s) - c_{\infty}) d s = τ (t) - c_{\infty} t for t ⩾ 0 . \end{matrix}$ Then $f \in C ([0, \infty))$ . We put $\begin{matrix} α = \underset{t \to \infty}{lim inf} f (t), β = \underset{t \to \infty}{lim sup} f (t) (\in [- \infty, \infty]) . \end{matrix}$ It suffices to show $\begin{matrix} (3.35) & α = β \in R . \end{matrix}$

First we show that $β < \infty$ . Suppose that $β = \infty$ . Since f is continuous and Lebesgue measure of $N_{1}$ is zero, we can take sequences ${t_{k}}_{k \in N}$ such that $\begin{matrix} \begin{matrix} t_{k} \notin N_{1}, lim_{k \to \infty} t_{k} = \infty, lim_{k \to \infty} f (t_{k}) = \infty . \end{matrix} \end{matrix}$ Let γ be an arbitrary positive number. For every $k \in N$ , since ${lim}_{n \to \infty} f (t_{k + n}) = \infty$ , the intermediate value theorem implies that there is $s_{k} > t_{k}$ satisfying $\begin{matrix} f (s_{k}) = f (t_{k}) + γ . \end{matrix}$ By using the continuity of f at $s_{k}$ and the fact that measure of $N_{1}$ is zero, we can take $r_{k}$ such that $\begin{matrix} (3.36) & r_{k} \notin N_{1}, r_{k} > t_{k}, | f (r_{k}) - f (t_{k}) - γ | = | f (r_{k}) - f (s_{k}) | < \frac{1}{k} . \end{matrix}$ By (3.25), Lemma 3 yields (3.20). This implies $\begin{matrix} lim_{t \notin N_{1}, t \to \infty} e^{- i f (t) A^{1 / 2}} ϕ = lim_{t \notin N_{1}, t \to \infty} e^{- i (τ (t) - c_{\infty} t) A^{1 / 2}} ϕ = y_{1, \infty} in H, \end{matrix}$ since $e^{i t A^{1 / 2}}$ is a unitary operator on H. Hence, letting $k \to \infty$ in the equality $\begin{matrix} e^{i (f (r_{k}) - f (t_{k}) - γ) A^{1 / 2}} e^{- i f (r_{k}) A^{1 / 2}} ϕ = e^{- i γ A^{1 / 2}} e^{- i f (t_{k}) A^{1 / 2}} ϕ, \end{matrix}$ and observing (3.36) and the continuity of the unitary operator $e^{i s A^{1 / 2}}$ with respect to s, we obtain $\begin{matrix} y_{1, \infty} = e^{- i γ A^{1 / 2}} y_{1, \infty} . \end{matrix}$ Thus, we have $\begin{matrix} {(I + A^{1 / 2})}^{- 1} y_{1, \infty} = e^{- i γ A^{1 / 2}} {(I + A^{1 / 2})}^{- 1} y_{1, \infty} . \end{matrix}$ Since $γ > 0$ is arbitrary, and since ${(I + A^{1 / 2})}^{- 1} y_{1, \infty} \in D (A^{1 / 2})$ , we differentiate the equality above with respect to γ to obtain $\begin{matrix} 0 = \frac{d}{d γ} e^{- i γ A^{1 / 2}} {(I + A^{1 / 2})}^{- 1} y_{1, \infty} = - i A^{1 / 2} e^{- i γ A^{1 / 2}} {(I + A^{1 / 2})}^{- 1} y_{1, \infty} \end{matrix}$ on $(0, \infty)$ . This implies that $y_{1, \infty} = 0$ by the injectivity of $A^{1 / 2}$ and $e^{- i γ A^{1 / 2}}$ , which contradicts (3.34).

The assumption $α = - \infty$ deduces contradiction in the same way.

We finally prove (3.35). The above facts imply that $α, β \in R$ . Suppose that (3.35) fails to hold. Then the interval $(α, β)$ is not empty. Let γ be an arbitrary number $γ \in (α, β)$ . For every $k \in N$ , the intermediate value theorem implies that there exists $s_{k} > k$ satisfying $f (s_{k}) = γ$ . Then by the same reason as (3.36), we can take $r_{k}$ such that $\begin{matrix} (3.37) & r_{k} \notin N_{1}, r_{k} > k, | f (r_{k}) - γ | = | f (r_{k}) - f (s_{k}) | < \frac{1}{k} . \end{matrix}$ Letting $k \to \infty$ in the equality $\begin{matrix} e^{- i (f (r_{k}) - γ) A^{1 / 2}} e^{i (τ (r_{k}) - c_{\infty} r_{k}) A^{1 / 2}} y_{1, \infty} = e^{i γ A^{1 / 2}} y_{1, \infty}, \end{matrix}$ and observing (3.20), (3.37) and the continuity of $e^{i t A^{1 / 2}}$ with respect to t, we obtain $\begin{matrix} ϕ = e^{i γ A^{1 / 2}} y_{1, \infty} . \end{matrix}$ Hence we have $\begin{matrix} (3.38) & {(I + A^{1 / 2})}^{- 1} ϕ = e^{i γ A^{1 / 2}} {(I + A^{1 / 2})}^{- 1} y_{1, \infty} . \end{matrix}$ Since $γ \in (α, β)$ is arbitrary and since ${(I + A^{1 / 2})}^{- 1} H \subset D (A^{1 / 2})$ , we differentiate (3.38) with respect to γ to obtain $\begin{matrix} i A^{1 / 2} e^{i γ A^{1 / 2}} {(I + A^{1 / 2})}^{- 1} y_{1, \infty} = \frac{d}{d γ} e^{i γ A^{1 / 2}} {(I + A^{1 / 2})}^{- 1} y_{1, \infty} = 0 \end{matrix}$ on $(α, β)$ . This implies that $y_{1, \infty} = 0$ by the injectivity of $A^{1 / 2}$ and $e^{i γ A^{1 / 2}}$ , which contradicts to (3.34).

Footnotes

In the case $b (t)$ is an integrable $C^{1}$ function and c is a $C^{1}$ function satisfying (1.1), it is clear that there exists a unique solution of initial value problem (2.4)–(2.5), equivalently, (2.1)–(2.2). Namely, the following proposition holds.

On the existence of solutions of the Cauchy problem (1.2)–(1.3) under the assumption that $c (t)$ is of bounded variation, there are some results. Colombini, De Giorgi and Spagnolo [4] showed the existence of solution $\begin{array}{l} \frac{\partial^{2} u}{\partial t^{2}} (t, x) - \sum_{i, j = 1}^{n} a_{i, j} (t) \frac{\partial^{2} u}{\partial x_{i} \partial x_{j}} = f (t, x) in [0, \infty) \times R^{n}, \end{array}$ in the class $u \in C ([0, T], H_{loc}^{s + 1})$ , $\partial u / \partial t \in L^{2} ([0, T], H_{loc}^{s})$ and $\partial^{2} u / \partial t^{2} \in L^{1} ([0, T], H_{loc}^{s - 1})$ , where $a_{i, j} (t)$ is of bounded variation and $\begin{matrix} a_{i, j} (t) = a_{j, i} (t), \sum_{i, j = 1}^{n} a_{i, j} (t) ξ_{i} ξ_{j} ⩾ λ_{0} | ξ |^{2} for all ξ \in R^{n}, \end{matrix}$ for $λ_{0} > 0$ . In the case A is a coercive self-adjoint operator, De Simon and Torelli [5] showed the unique existence of the solution of (1.2)–(1.3) in the class $u \in W^{1, 2} ([0, T], H)$ , $\partial u / \partial t \in L^{2} ([0, T], D (A^{1 / 2}))$ . Arosio [1] considered (1.6)–(1.8) with $(ϕ_{0}, ψ_{0}) \in H_{0}^{1} (Ω) \times L^{2} (Ω)$ for bounded domain Ω, and showed the unique existence of solution in the class $u \in C ([0, \infty), H_{0}^{1} (Ω)) \cap C^{1} ([0, \infty), L^{2} (Ω))$ . The results above ([4,5] and [1]) considered the solutions in the sense of distribution with respect to t. On the other hand, Bárta [2, Section 2] considered the hyperbolic equation $\begin{array}{l} \frac{\partial^{2} u}{\partial t^{2}} = \sum_{i, j = 1}^{n} \frac{\partial}{\partial x_{i}} (a_{i, j} (t, x) \frac{\partial u}{\partial x_{j}}) (t, x) + \sum_{i = 1}^{n} p_{i} (t, x) \frac{\partial u}{\partial x_{i}} (t, x) \\ (A.1) & + q (t, x) u (t, x) in (0, T) \times Ω, \\ (A.2) & u (0, x) = ϕ_{0} (x), \frac{\partial u}{\partial t} (0, x) = ψ_{0} (x) in Ω, \end{array}$ where Ω is a bounded domain in $R^{n}$ , and $a_{i, j}$ , $p_{i}$ and q are functions satisfying the following: $\begin{array}{l} a_{i, j} \in BV ([0, \infty), W^{1, \infty}) \cap L^{\infty} ([0, \infty), Lip (Ω)), \\ a_{i, j} (t, x) = a_{j . i} (t, x), \sum_{i, j = 1}^{n} a_{i, j} (t, x) ξ_{i} ξ_{j} ⩾ λ_{0} | ξ |^{2} for all ξ \in R^{n}, \\ p_{i}, q \in BV ([0, \infty), L^{\infty}) . \end{array}$ Then he showed the unique existence of the solution $u (t) \in C ([0, \infty); H_{0}^{1} (Ω)) \cap C^{1} ([0, \infty); L^{2} (Ω))$ of (A.1) with initial value in $(H^{2} (Ω) \cap H_{0}^{1} (Ω)) \times H_{0}^{1} (Ω)$ , such that for an at most countable subset N, $\begin{matrix} (u (t), u^{'} (t)) \in C ([0, \infty) ∖ N; (H^{2} (Ω) \cap H_{0}^{1} (Ω)) \times H_{0}^{1} (Ω)), \end{matrix}$ and $u^{'} (t)$ is differentiable with values in $L^{2} (Ω)$ at $t \in [0, \infty) ∖ N$ . Bárta [2] proved this by showing and applying an abstract theorem.

As is stated above, Bárta [2] applied Theorem B to the hyperbolic equation (A.1) to show the unique existence of solutions. Similarly, we can apply Theorem B to the Cauchy problem (2.1)–(2.2) to obtain the solution $u (t)$ . In the argument of this paper, we need the fact that $u^{'} (t)$ is absolutely continuous with value in $H_{- 1 / 2}$ . This fact is verified by the following lemma, which is proved at the end.

Now we state a proposition on the unique existence of the solution (2.1)–(2.2).

References

[1]

Arosio , Asymptotic behavior as

t \to + \infty

of the solutions of linear hyperbolic equations with coefficients discontinuous in time (on a bounded domain), J. Differential Equations 39 (1981), 291–309.

[2]

Bárta , A generation theorem for hyperbolic equations with coefficients of bounded variation in time, Riv. Mat. Univ. Parma (7) 9 (2008), 17–30.

[3]

E.A.

Coddington and

Levinson , Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.

[4]

Colombini ,

De Giorgi and

Spagnolo , Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps, Ann. Scuola Norm. Sup. Pisa 6 (1979), 511–559.

[5]

De Simon and

Torelli , Linear second order differential equations with discontinuous coefficients in Hilbert spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 1 (1974), 131–154.

[6]

M.R.

Ebert ,

Fitriana and

Hirosawa , On the energy estimates of the wave equation with time dependent propagation speed asymptotically monotone functions, J. Math. Anal. Appl. 432 (2015), 654–677.

[7]

Hartman , Ordinary Differential Equations, 2nd edn, SIAM, Philadelphia, 2002.

[8]

Matsuyama , Asymptotic behavior for wave equation with time-dependent coefficients, Annali dell’Universita di Ferrara 52 (2006), 383–393.

[9]

Matsuyama , Asymptotic profiles for the Kirchhoff equation, Rend. Lincei Mat. Appl. 17 (2006), 377–395.

10.

[10]

Matsuyama and

Ruzhansky , Scattering for strictly hyperbolic systems with time-dependent coefficients, Math. Nachr. 286 (2013), 1191–1207.

11.

[11]

Ruzhansky and

Wirth , Dispersive estimates for hyperbolic systems with time-dependent coefficients, J. Differential Equations 251 (2011), 941–969.

12.

[12]

Ruzhansky and

Wirth, Asymptotic behaviour of solutions to hyperbolic partial differential equations, in: Variable Lebesgue Spaces and Hyperbolic Systems, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser/Springer, Basel, 2014, pp. 91–169.

13.

[13]

Wintner , Asymptotic integrations of adiabatic oscillator, Amer. J. Math. 69 (1947), 251–272.

14.

[14]

Yamazaki , Hyperbolic–parabolic singular perturbation for quasilinear equations of Kirchhoff type with weak dissipation, Math. Methods Appl. Sci. 32 (2009), 1893–1918.

15.

[15]

Yamazaki , Necessary condition on the dissipative term for the asymptotically free property of dissipative Kirchhoff equations, Nonlinear Analysis 125 (2015), 90–112.