Asymptotic expansion of the L 2 -norm of a solution of the strongly damped wave equation in space dimension 1 and 2

Abstract

In previous work the authors found the asymptotic expansion of the $L^{2}$ -norm of the solution $u (t, x)$ of the strongly damped wave equation $u_{t t} - Δ u_{t} - Δ u = 0$ and also of the $L^{2}$ -norm of the difference between $u (t, x)$ and its asymptotic approximation $ν (t, x)$ . This was done in space dimension $N ⩾ 3$ . In the present work results are extended to the exceptional cases $N = 1$ and $N = 2$ . This extension is achieved by deriving new lemmas on the asymptotic expansion of some parameter dependent integrals.

Keywords

Asymptotic analysis wave equation strong damping Fourier analysis

1. Introduction

We begin by considering the strongly damped wave equation in $R^{N}$ for $N = 1, 2, \dots$ $\begin{matrix} (1.1) & \begin{array}{l} u_{t t} (t, x) - Δ u_{t} (t, x) - Δ u (t, x) = 0, (t, x) \in (0, \infty) \times R^{N}, \\ u (0, \cdot) = u_{0} \in H^{1} (R^{N}), u_{t} (0, \cdot) = u_{1} \in L^{2} (R^{N}) . \end{array} \end{matrix}$ (1.1) admits a unique weak solution $u \in C ([0, \infty); H^{1} (R^{N})) \cap C^{1} ([0, \infty); L^{2} (R^{N}))$ as determined by Ikehata–Todorova–Yordanov [12]. The squared $L^{2}$ -norm $‖ u (t, \cdot) ‖_{2}^{2}$ of this weak solution of (1.1) has been studied extensively by Ikehata [8] and, more recently, by Barrera–Volkmer [1] in space dimension $N ⩾ 3$ . Ikehata [8] found sharp decay estimates $‖ u (t, \cdot) ‖_{2}^{2} = O (t^{- N / 2 + 1})$ as $t \to \infty$ , while Barrera–Volkmer [1] followed up with the asymptotic expansion of $‖ u (t, \cdot) ‖_{2}^{2}$ under slightly different assumptions. Despite the differing assumptions, the common idea in obtaining the estimates and expansions is the use of weighted $L^{1}$ -initial data (to be defined). This technique has been used in the past to study both the linear [6,7,14] and strongly [1,8] damped wave equations. Furthermore, the use of weighted $L^{1}$ -data has been employed by Ikehata–Natsume [9] in their generalization of [12] to find energy decay estimates of the wave equation with fractional damping, and also by Ikehata–Soga [11] in their study of a strongly damped plate equation.

Regarding the weak solution u of (1.1), the low space dimension cases $N = 1, 2$ were known to Ikehata [8] to be exceptional, but sharp estimates on the long-term behavior of $‖ u (t, \cdot) ‖_{2}^{2}$ were unknown until Ikehata–Onodera [10]. They determined that as $t \to \infty$ , $‖ u (t, \cdot) ‖_{2}^{2} = O (t)$ in space dimension $N = 1$ and $‖ u (t, \cdot) ‖_{2}^{2} = O (log (t))$ in dimension $N = 2$ . One of the two main goals of this paper is to find the asymptotic expansion of $‖ u (t, \cdot) ‖_{2}^{2}$ in space dimensions $N = 1, 2$ . In doing so, we will see that these low dimension cases are exceptional because more intricate integrals appear in the expansions, and as such, must be treated using different methods. Nonetheless this paper will carry over many preliminary results from the recent paper of Barrera–Volkmer [1].

The weak solution $u (t, x)$ of (1.1) is found using the Fourier transform. In the Fourier space, Ikehata [8] found an asymptotic profile $ν (t, ξ)$ for $\hat{u} (t, ξ)$ , which exhibits a diffusion-like property in all space dimensions when $‖ \hat{u} (t, \cdot) - ν (t, \cdot) ‖_{2}^{2}$ is considered. He did so using a mean value theorem argument, which, along with the paper [14] of Volkmer, formed the motivation for [1] of Barrera–Volkmer. However due to the exceptional nature of the problem in space dimensions $N = 1, 2$ , the asymptotic expansion of $‖ \hat{u} (t, \cdot) - ν (t, \cdot) ‖_{2}^{2}$ in these dimensions is the other main goal of this paper.

Notation.
Throughout this paper, $N$ will represent the positive integers and $N_{0}$ will represent the non-negative integers. The $L^{q} (R^{N})$ -norm is denoted by $‖ \cdot ‖_{q}$ . For each $ϵ > 0$ we let $B_{ϵ}$ denote the closed ball centered at the origin in $R^{N}$ , whereby we denote the $L^{q} (B_{ϵ})$ -norm by $‖ \cdot ‖_{q, ϵ}$ .

For all $θ ⩾ 0$ we define the weighted $L^{1}$ -space $\begin{matrix} L^{1, θ} (R^{N}) : = {ϕ \in L^{1} (R^{N}) | \int_{R^{N}} {(1 + | x |)}^{θ} | ϕ (x) | d x < \infty}, \end{matrix}$ with norm $‖ ϕ ‖_{L^{1, θ} (R^{N})} : = \int_{R^{N}} {(1 + | x |)}^{θ} | ϕ (x) | d x$ . For all $ϕ \in L^{1} (R^{N})$ we define the Fourier transform $\begin{matrix} F (ϕ) (ξ) = \hat{ϕ} (ξ) : = \frac{1}{{(2 π)}^{N / 2}} \int_{R^{N}} ϕ (x) e^{- i x \cdot ξ} d x, \end{matrix}$ and the Fourier inverse $\begin{matrix} F^{- 1} (ϕ) (x) : = \frac{1}{{(2 π)}^{N / 2}} \int_{R^{N}} ϕ (ξ) e^{i ξ \cdot x} d ξ . \end{matrix}$ To extend $F$ to $ϕ \in L^{2} (R^{N})$ , consider functions $ϕ_{j} \in L^{1} (R^{N}) \cap L^{2} (R^{N})$ such that $‖ ϕ - ϕ_{j} ‖_{2} \to 0$ as $j \to \infty$ . Then $\hat{ϕ}$ is defined to be the unique $L^{2} (R^{N})$ -limit of ${\hat{ϕ}}_{j}$ as $j \to \infty$ . See [4] for more details.

Finally, whenever a sum of the form $\sum_{j = 1}^{0}$ or $\sum_{n = 2}^{1}$ is encountered, it is taken to be the empty sum and evaluates to 0.

1.1. Assumptions

We assume that the space dimension is $N \in N$ and that there exists $K \in N$ such that $u_{0}, u_{1} \in L^{1, 2 K} (R^{N})$ , where the functions $u_{0} \in H^{1} (R^{N})$ and $u_{1} \in L^{2} (R^{N})$ are the initial data of (1.1). Under these assumptions both ${\hat{u}}_{0}$ and ${\hat{u}}_{1}$ are $2 K$ -times continuously differentiable on $R^{N}$ , and in particular at $ξ = 0$ . Thus we may choose $0 < δ < 1$ such that the Taylor approximations $\begin{array}{l} (1.2) & \begin{aligned} {\hat{u}}_{1} (ξ) = \sum_{| σ | ⩽ 2 K - 1} a_{σ} ξ^{σ} + O (| ξ |^{2 K}), \\ {\hat{u}}_{0} (ξ) = \sum_{| σ | ⩽ 2 K - 1} b_{σ} ξ^{σ} + O (| ξ |^{2 K}), \end{aligned} \end{array}$ hold in the closed δ-neighborhood of $ξ = 0$ , where $σ \in N_{0}^{N}$ is a multi-index of order $| σ | = σ_{1} + \dots + σ_{N}$ (see [4] for further details), and for all $0 ⩽ | σ | ⩽ 2 K - 1$ $\begin{array}{l} a_{σ} = \frac{{(- i)}^{| σ |}}{σ! {(2 π)}^{N / 2}} \int_{R^{N}} x^{σ} u_{1} (x) d x, \\ b_{σ} = \frac{{(- i)}^{| σ |}}{σ! {(2 π)}^{N / 2}} \int_{R^{N}} x^{σ} u_{0} (x) d x . \end{array}$

Remark.
We fix this chosen value of $0 < δ < 1$ , and whenever δ is hereafter referred to, we mean this fixed value.

In this fixed closed δ-neighborhood of $ξ = 0$ , we also have the Taylor approximations $\begin{matrix} \frac{| ξ |^{2}}{2} {\hat{u}}_{0} (ξ) = \sum_{| σ | ⩽ 2 K - 1} b_{σ}^{'} ξ^{σ} + O (| ξ |^{2 K}), \end{matrix}$ where $\begin{matrix} b_{σ}^{'} = \{\begin{matrix} 0 & all entries of σ are ⩽ 1 \\ \sum_{{j | σ_{j} ⩾ 2}} \frac{1}{2} b_{σ - 2 e_{j}} & otherwise, \end{matrix} \end{matrix}$ and $e_{j} = {(δ_{i j})}_{i = 1}^{N}$ denotes the unit multi-index in the j-direction. Moreover, $\begin{array}{l} (1.3) & {\hat{u}}_{1} (ξ) + \frac{| ξ |^{2}}{2} {\hat{u}}_{0} (ξ) = \sum_{| σ | ⩽ 2 K - 1} c_{σ} ξ^{σ} + O (| ξ |^{2 K}), with c_{σ} = a_{σ} + b_{σ}^{'}, \\ (1.4) & {| {\hat{u}}_{1} (ξ) + \frac{| ξ |^{2}}{2} {\hat{u}}_{0} (ξ) |}^{2} = \sum_{| σ | ⩽ 2 K - 1} d_{σ} ξ^{σ} + O (| ξ |^{2 K}), with d_{σ} = \sum_{ψ + ω = σ} c_{ψ} {\bar{c}}_{ω}, \\ (1.5) & {| {\hat{u}}_{0} (ξ) |}^{2} = \sum_{| σ | ⩽ 2 K - 1} f_{σ} ξ^{σ} + O (| ξ |^{2 K}), with f_{σ} = \sum_{ψ + ω = σ} b_{ψ} {\bar{b}}_{ω}, \\ (1.6) & ({\hat{u}}_{1} (ξ) + \frac{| ξ |^{2}}{2} {\hat{u}}_{0} (ξ)) {\bar{\hat{u}}}_{0} (ξ) = \sum_{| σ | ⩽ 2 K - 1} l_{σ} ξ^{σ} + O (| ξ |^{2 K}), with l_{σ} = \sum_{ψ + ω = σ} c_{ψ} {\bar{b}}_{ω} . \end{array}$

With the definitions $P_{0} : = {(2 π)}^{- N / 2} \int_{R^{N}} u_{0} (x) d x$ and $P_{1} : = {(2 π)}^{- N / 2} \int_{R^{N}} u_{1} (x) d x$ , it is easy to see that $a_{0} = P_{1}$ and $b_{0} = P_{0}$ . Hence $c_{0} = P_{1}$ , $d_{0} = | P_{1} |^{2}$ , $f_{0} = | P_{0} |^{2}$ , and $l_{0} = P_{1} {\bar{P}}_{0}$ .
1.2. Preliminary work

Applying the Fourier transform to (1.1) we obtain the ordinary differential equation $\begin{matrix} (1.7) & \begin{array}{l} {\hat{u}}_{t t} (t, ξ) + | ξ |^{2} {\hat{u}}_{t} (t, ξ) + | ξ |^{2} \hat{u} (t, ξ) = 0, (t, ξ) \in (0, \infty) \times R^{N}, \\ \hat{u} (0, \cdot) = {\hat{u}}_{0}, {\hat{u}}_{t} (0, \cdot) = {\hat{u}}_{1} . \end{array} \end{matrix}$ The solution to (1.7) is $\begin{matrix} \hat{u} (t, ξ) = ({\hat{u}}_{1} (ξ) + \frac{| ξ |^{2}}{2} {\hat{u}}_{0} (ξ)) h_{1} (t, ξ) + {\hat{u}}_{0} (ξ) h_{2} (t, ξ), \end{matrix}$ where $\begin{matrix} (1.8) & h_{1} (t, ξ) = e^{- t | ξ |^{2} / 2} \frac{sin (t | ξ | \sqrt{1 - | ξ |^{2} / 4})}{| ξ | \sqrt{1 - | ξ |^{2} / 4}} \end{matrix}$ and $\begin{matrix} (1.9) & h_{2} (t, ξ) = e^{- t | ξ |^{2} / 2} cos (t | ξ | \sqrt{1 - | ξ |^{2} / 4}) . \end{matrix}$

We define $\begin{array}{l} μ_{1} (t, ξ) : = ({\hat{u}}_{1} (ξ) + \frac{| ξ |^{2}}{2} {\hat{u}}_{0} (ξ)) h_{1} (t, ξ), \\ μ_{2} (t, ξ) : = {\hat{u}}_{0} (ξ) h_{2} (t, ξ), \\ ν_{1} (t, ξ) : = P_{1} e^{- t | ξ |^{2} / 2} \frac{sin (t | ξ |)}{| ξ |}, \\ ν_{2} (t, ξ) : = P_{0} e^{- t | ξ |^{2} / 2} cos (t | ξ |) . \end{array}$ Then $\hat{u} (t, ξ) = μ_{1} (t, ξ) + μ_{2} (t, ξ)$ . With the definition $ν (t, ξ) : = ν_{1} (t, ξ) + ν_{2} (t, ξ)$ , we wish to find the asymptotic expansions of $\begin{matrix} {‖ u (t, \cdot) ‖}_{2}^{2} and {‖ \hat{u} (t, \cdot) - ν (t, \cdot) ‖}_{2}^{2} as t \to \infty . \end{matrix}$ Indeed $ν (t, ξ)$ is the profile determined by Ikehata in his paper [8].

1.3. Main results

It was determined by Ikehata [8] and Barrera–Volkmer [1] that in every space dimension $N \in N$ , for all $ϵ > 0$ there is some $η > 0$ such that $\begin{array}{l} {‖ u (t, \cdot) ‖}_{2}^{2} = {‖ \hat{u} (t, \cdot) ‖}_{2}^{2} = {‖ \hat{u} (t, \cdot) ‖}_{2, ϵ}^{2} + O (e^{- η t}) (t \to \infty), \\ {‖ \hat{u} (t, \cdot) - ν (t, \cdot) ‖}_{2}^{2} = {‖ \hat{u} (t, \cdot) - ν (t, \cdot) ‖}_{2, ϵ}^{2} + O (e^{- η t}) (t \to \infty) . \end{array}$ Thus all the interesting asymptotic behaviors of $‖ u (t, \cdot) ‖_{2}^{2}$ and $‖ \hat{u} (t, \cdot) - ν (t, \cdot) ‖_{2}^{2}$ are captured in any ϵ-neighborhood of the origin. We let ϵ equal the fixed value of $0 < δ < 1$ , and hence we are interested in the expansions of $\begin{matrix} {‖ \hat{u} (t, \cdot) ‖}_{2, δ}^{2} and {‖ \hat{u} (t, \cdot) - ν (t, \cdot) ‖}_{2, δ}^{2} as t \to \infty . \end{matrix}$ We will compute these expansions by first noting $‖ \hat{u} (t, \cdot) - ν (t, \cdot) ‖_{2, δ}^{2} = X (t) + Y (t) + Z (t)$ , where $\begin{array}{l} (1.10) & X (t) & = {‖ μ_{1} (t, \cdot) - ν_{1} (t, \cdot) ‖}_{2, δ}^{2}, \\ (1.11) & Y (t) & = {‖ μ_{2} (t, \cdot) - ν_{2} (t, \cdot) ‖}_{2, δ}^{2}, \\ (1.12) & Z (t) & = 2 ℜ {⟨ μ_{1} (t, \cdot) - ν_{1} (t, \cdot), μ_{2} (t, \cdot) - ν_{2} (t, \cdot) ⟩}_{2, δ}, \end{array}$ and with ${⟨ \cdot, \cdot ⟩}_{2, δ}$ denoting the $L^{2}$ -inner product over the closed ball of radius δ about the origin. Then $X (t) = X_{1} (t) + X_{2} (t) + X_{3} (t)$ , $Y (t) = Y_{1} (t) + Y_{2} (t) + Y_{3} (t)$ , and $Z (t) = Z_{1} (t) + Z_{2} (t) + Z_{3} (t) + Z_{4} (t)$ , where $\begin{array}{l} X_{1} (t) = {‖ μ_{1} (t, \cdot) ‖}_{2, δ}^{2}, X_{2} (t) = {‖ ν_{1} (t, \cdot) ‖}_{2, δ}^{2}, X_{3} (t) = - 2 ℜ {⟨ μ_{1} (t, \cdot), ν_{1} (t, \cdot) ⟩}_{2, δ}; \\ Y_{1} (t) = {‖ μ_{2} (t, \cdot) ‖}_{2, δ}^{2}, Y_{2} (t) = {‖ ν_{2} (t, \cdot) ‖}_{2, δ}^{2}, Y_{3} (t) = - 2 ℜ {⟨ μ_{2} (t, \cdot), ν_{2} (t, \cdot) ⟩}_{2, δ}; \\ Z_{1} (t) = 2 ℜ {⟨ μ_{1} (t, \cdot), μ_{2} (t, \cdot) ⟩}_{2, δ}, Z_{2} (t) = - 2 ℜ {⟨ μ_{1} (t, \cdot), ν_{2} (t, \cdot) ⟩}_{2, δ}, \\ Z_{3} (t) = - 2 ℜ {⟨ ν_{1} (t, \cdot), μ_{2} (t, \cdot) ⟩}_{2, δ}, Z_{4} (t) = 2 ℜ {⟨ ν_{1} (t, \cdot), ν_{2} (t, \cdot) ⟩}_{2, δ} . \end{array}$ Furthermore it is apparent that $‖ \hat{u} (t, \cdot) ‖_{2, δ}^{2} = X_{1} (t) + Y_{1} (t) + Z_{1} (t)$ .

Remark 1.
I accordance with the Notation section, the sums in the following theorems vanish when $K = 1$ , leaving only the terms outside the sum and the O-term.
Theorem 1.1.
Let $N \in {1, 2}$ and $K \in N$ . Let $u (t, x)$ be the weak solution of ( 1.1 ) under the Fourier transform with the complex-valued initial data $u_{0} (x)$ and $u_{1} (x)$ satisfying $u_{0} \in H^{1} (R^{N}) \cap L^{1, 2 K} (R^{N})$ and $u_{1} \in L^{2} (R^{N}) \cap L^{1, 2 K} (R^{N})$ .
For $N = 2$ $\begin{matrix} {‖ u (t, \cdot) ‖}_{2}^{2} = \frac{π | P_{1} |^{2}}{2} (log (4 t) + γ) + \sum_{j = 1}^{K - 1}_{2} W_{j} t^{- j} + O (t^{- K}) \end{matrix}$ as $t \to \infty$ , where γ is the Euler–Mascheroni constant.

For $N = 1$ $\begin{matrix} {‖ u (t, \cdot) ‖}_{2}^{2} = | P_{1} |^{2} (π t - \sqrt{π t}) + π ℜ (P_{1} {\bar{P}}_{0}) + \sum_{j = 1}^{K - 1}_{1} W_{j} t^{- j + 1 / 2} + O (t^{- K + 1 / 2}) \end{matrix}$ as $t \to \infty$ .

In both cases the $_{N} W_{j}$ are coefficients dependent on the space dimension N and the initial data $u_{0} (x)$ and $u_{1} (x)$ .
Theorem 1.2.
Let $N \in {1, 2}$ and $K \in N$ . Let $u (t, x)$ be the weak solution of ( 1.1 ) under the Fourier transform with the complex-valued initial data $u_{0} (x)$ and $u_{1} (x)$ satisfying $u_{0} \in H^{1} (R^{N}) \cap L^{1, 2 K} (R^{N})$ and $u_{1} \in L^{2} (R^{N}) \cap L^{1, 2 K} (R^{N})$ . Consider the asymptotic profile of $\hat{u} (t, ξ)$ found by Ikehata in [ 8 ]: $\begin{matrix} ν (t, ξ) = P_{1} e^{- t | ξ |^{2} / 2} \frac{sin (t | ξ |)}{| ξ |} + P_{0} e^{- t | ξ |^{2} / 2} cos (t | ξ |), \end{matrix}$ where $P_{0} = {(2 π)}^{- N / 2} \int_{R^{N}} u_{0} (x) d x$ and $P_{1} = {(2 π)}^{- N / 2} \int_{R^{N}} u_{1} (x) d x$ . Then as $t \to \infty$ $\begin{matrix} {‖ \hat{u} (t, \cdot) - ν (t, \cdot) ‖}_{2}^{2} = t^{- N / 2 + 1} (\sum_{j = 1}^{K - 1}_{N} V_{j} t^{- j} + O (t^{- K})), \end{matrix}$ where the $_{N} V_{j}$ are coefficients dependent on the space dimension N and the initial data $u_{0} (x)$ and $u_{1} (x)$ .

In the process of proving Theorem 1.2 we will find the asymptotic expansions of $X_{2} (t)$ , $Y_{2} (t)$ , and $Z_{4} (t)$ . Since $‖ ν (t, \cdot) ‖_{2}^{2} = ‖ ν (t, \cdot) ‖_{2, δ}^{2} + O (e^{- t δ^{2} / 2}) = X_{2} (t) + Y_{2} (t) + Z_{4} (t) + O (e^{- t δ^{2} / 2})$ as $t \to \infty$ , we obtain the expansion of the squared $L^{2}$ -norm of the asymptotic profile $ν (t, ξ)$ . Corollary 1.3.
Let $N \in {1, 2}$ and $K \in N$ . Let $ν (t, ξ)$ be the asymptotic profile of $\hat{u} (t, ξ)$ found by Ikehata in [ 8 ].
For $N = 2$ , as $t \to \infty$ , $\begin{matrix} {‖ ν (t, \cdot) ‖}_{2}^{2} = \frac{π | P_{1} |^{2}}{2} (log (4 t) + γ) + \sum_{j = 1}^{K - 1}_{2} U_{j} t^{- j} + O (t^{- K}) . \end{matrix}$

For $N = 1$ , as $t \to \infty$ , $\begin{matrix} {‖ ν (t, \cdot) ‖}_{2}^{2} = | P_{1} |^{2} (π t - \sqrt{π t}) + π ℜ (P_{1} {\bar{P}}_{0}) + \sum_{j = 1}^{K - 1}_{1} U_{j} t^{- j + 1 / 2} + O (t^{- K + 1 / 2}) . \end{matrix}$
In both cases the $_{N} U_{j}$ are coefficients dependent on the space dimension N and the initial data $u_{0} (x) \in H^{1} (R^{N}) \cap L^{1, 2 K} (R^{N})$ and $u_{1} \in L^{2} (R^{N}) \cap L^{1, 2 K} (R^{N})$ of ( 1.1 ).
Remark 2.
Our proofs of Theorem 1.1, Theorem 1.2 and Corollary 1.3 provide explicit formulas for the coefficients $_{N} W_{j}$ , $_{N} V_{j}$ , $_{N} U_{j}$ . For example, if $K ⩾ 2$ , we have $\begin{array}{l} _{2} W_{1} = - \frac{π | P_{1} |^{2}}{8} + \frac{π | P_{0} |^{2}}{2} + π ℜ (P_{1} {\bar{P}}_{0}) + \frac{π}{4} \sum_{j = 1}^{2} d_{2 e_{j}}, \\ _{2} U_{1} = - \frac{| P_{1} |^{2} π}{4} + \frac{| P_{0} |^{2} π}{2} + ℜ (P_{1} {\bar{P}}_{0}) π, \\ _{2} V_{1} = \frac{π | P_{1} |^{2}}{64} + \frac{π}{4} \sum_{j = 1}^{2} d_{2 e_{j}} - \frac{π}{2} \sum_{j = 1}^{2} ℜ ({\bar{P}}_{1} c_{2 e_{j}}), \\ _{1} W_{1} = \frac{\sqrt{π} | P_{1} |^{2}}{8} + \frac{\sqrt{π} | P_{0} |^{2}}{2} + \frac{\sqrt{π} d_{2}}{2}, \\ _{1} U_{1} = \frac{\sqrt{π} | P_{0} |^{2}}{2}, \\ _{1} V_{1} = \frac{3 \sqrt{π} | P_{1} |^{2}}{512} + \frac{\sqrt{π} d_{2}}{2} - \sqrt{π} ℜ ({\bar{P}}_{1} c_{2}) . \end{array}$

2. Proof of Theorem 1.1

We start by recalling some results from [1]. In the following all asymptotic statements for functions of t hold as $t \to \infty$ .

Lemma 2.1.

For $0 < ϵ < 2$ , $t > 0$ and $m \in N_{0}$ , define $\begin{matrix} F_{m}^{ϵ} (t) : = \int_{0}^{ϵ} r^{m} e^{- t r^{2} - 2 i r t} d r . \end{matrix}$ Then, for $Q \in N_{0}$ , $\begin{matrix} (2.1) & F_{m}^{ϵ} (t) = \sum_{n = 0}^{Q - 1} A_{m, n} t^{- m - n - 1} + O (t^{- m - Q - 1}), \end{matrix}$ where $\begin{matrix} (2.2) & A_{m, n} = \frac{m!}{{(2 i)}^{m + 1}} \frac{{(\frac{m + 1}{2})}_{n} {(\frac{m + 2}{2})}_{n}}{n!} \end{matrix}$ with the Pochhammer symbol ${(a)}_{n}$ denoting the rising factorial.

For $0 < ϵ < 2$ , $t > 0$ and $m \in N_{0}$ , define $\begin{matrix} G_{1, m}^{ϵ} (t) : = \int_{0}^{ϵ} r^{m} e^{- t r^{2} - i r t \sqrt{4 - r^{2}}} d r . \end{matrix}$ Then, for $Q \in N_{0}$ , $\begin{matrix} G_{1, m}^{ϵ} (t) = \sum_{n = 0}^{Q - 1} B_{m, n} t^{- m - n - 1} + O (t^{- m - Q - 1}), \end{matrix}$ where $\begin{matrix} B_{m, n} \in \{\begin{matrix} R & m \in N odd \\ i R & m \in N_{0} even . \end{matrix} \end{matrix}$

The numbers $B_{m, n}$ can be expressed in terms of $A_{m, n}$ ; see [1, (2.13)] Lemma 2.2.
For $0 < ϵ < 2$ , $t > 0$ and $m \in N_{0}$ , define $\begin{matrix} (2.3) & I_{1, m}^{ϵ} (t) : = \int_{0}^{ϵ} r^{m} e^{- t r^{2}} {sin}^{2} (t r \sqrt{1 - r^{2} / 4}) d r . \end{matrix}$
If $m \in N_{0}$ is even and $Q \in N_{0}$ , then $\begin{matrix} I_{1, m}^{ϵ} (t) = \frac{1}{4} Γ (\frac{m + 1}{2}) t^{- m / 2 - 1 / 2} + O (t^{- m - Q - 1}) . \end{matrix}$

If $m \in N$ is odd and $Q \in N_{0}$ , then $\begin{matrix} I_{1, m}^{ϵ} (t) = \frac{1}{4} Γ (\frac{m + 1}{2}) t^{- m / 2 - 1 / 2} - \sum_{n = 0}^{Q - 1} \frac{1}{2} B_{m, n} t^{- m - n - 1} + O (t^{- m - Q - 1}) . \end{matrix}$

Lemma 2.3.
For $0 < ϵ < 2$ and $m \in N_{0}$ define $\begin{matrix} I_{3, m}^{ϵ} (t) : = \int_{0}^{ϵ} r^{m} e^{- t r^{2}} sin (t r \sqrt{1 - r^{2} / 4}) cos (t r \sqrt{1 - r^{2} / 4}) d r . \end{matrix}$
If $m \in N_{0}$ is even and $Q \in N_{0}$ , then $\begin{matrix} I_{3, m}^{ϵ} (t) = - \sum_{n = 0}^{Q - 1} \frac{1}{2} ℑ (B_{m, n}) t^{- m - n - 1} + O (t^{- m - Q - 1}) . \end{matrix}$

If $m \in N$ is odd and $Q \in N_{0}$ , then $\begin{matrix} I_{3, m}^{ϵ} (t) = O (t^{- m - Q - 1}) . \end{matrix}$

2.1. The space dimension $N = 2$ case

2.1.1. Auxiliary results

In this section we need Lemma 2.2 for $m = - 1$ .

Lemma 2.4.
For $0 < ϵ < 2$ and $t > 0$ we define $\begin{matrix} I_{1, - 1}^{ϵ} (t) = \int_{0}^{ϵ} r^{- 1} e^{- t r^{2}} {sin}^{2} (t r \sqrt{1 - r^{2} / 4}) d r . \end{matrix}$ If $M \in N$ , then $\begin{array}{rcl} I_{1, - 1}^{ϵ} (t) & = & \frac{1}{4} (log (4 t) + γ) - \sum_{n = 0}^{M - 2} \frac{1}{2 (n + 1)} B_{1, n} t^{- n - 1} \\ + \sum_{1 ⩽ 2 k + n ⩽ M - 1} \frac{β_{k}}{2 k + n} ℑ (B_{2 k, n}) t^{- 2 k - n} + O (t^{- M}), \end{array}$ where $\begin{matrix} (2.4) & β_{k} = \{\begin{matrix} 1 & k = 0 \\ - \frac{(2 k - 3)!!}{k! 8^{k}} & k \in N . \end{matrix} \end{matrix}$
Proof.
For every $t > 0$ , it is a routine argument to verify (using Lebesgue’s dominated convergence theorem and the mean value theorem) that $\frac{d}{d t} I_{1, - 1}^{ϵ} (t) = - I_{1, 1}^{ϵ} (t) + I_{1} (t)$ , where $- I_{1, 1}^{ϵ} (t)$ is as in (2.3) and $\begin{matrix} I_{1} (t) = \int_{0}^{ϵ} \sqrt{1 - r^{2} / 4} e^{- t r^{2}} sin (t r \sqrt{4 - r^{2}}) d r . \end{matrix}$ By Lemma 2.2(b) with $Q : = M - 1$ , $- I_{1, 1}^{ϵ} (t)$ thus has the expansion $\begin{matrix} (2.5) & - I_{1, 1}^{ϵ} (t) = - \frac{1}{4} t^{- 1} + \sum_{n = 0}^{M - 2} \frac{1}{2} B_{1, n} t^{- n - 2} + O (t^{- M - 1}) . \end{matrix}$

Turning to $I_{1} (t)$ , we begin by expanding $\sqrt{1 - r^{2} / 4}$ in a Taylor series about $r = 0$ . Since $0 < ϵ < 2$ , for $0 ⩽ r ⩽ ϵ$ : $\begin{matrix} (2.6) & \sqrt{1 - r^{2} / 4} = \sum_{k = 0}^{L - 1} β_{k} r^{2 k} + O (r^{2 L}) (L \in N_{0}), \end{matrix}$ where $β_{k}$ is as given in (2.4) for $k \in N_{0}$ . Substituting (2.6) into the definition of $I_{1} (t)$ and estimating the integral with the $O (r^{2 L})$ -term, we obtain $\begin{array}{l} I_{1} (t) & = \sum_{k = 0}^{L - 1} β_{k} \int_{0}^{ϵ} r^{2 k} e^{- t r^{2}} sin (t r \sqrt{4 - r^{2}}) d r + O (t^{- L - 1 / 2}) \\ (2.7) & = - \sum_{k = 0}^{L - 1} β_{k} ℑ (G_{1, 2 k}^{ϵ} (t)) + O (t^{- L - 1 / 2}) . \end{array}$ We let $L : = M + 1$ and apply Lemma 2.1(b) to (2.7) with $Q = Q_{k} : = max {0, L - 2 k}$ to obtain $\begin{array}{l} I_{1} (t) & = - \sum_{2 k + n ⩽ M} β_{k} ℑ (B_{2 k, n}) t^{- 2 k - n - 1} + O (t^{- M - 1}) \\ = - β_{0} ℑ (B_{0, 0}) t^{- 1} - \sum_{1 ⩽ 2 k + n ⩽ M - 1} β_{k} ℑ (B_{2 k, n}) t^{- 2 k - n - 1} + O (t^{- M - 1}) \\ (2.8) & = \frac{1}{2} t^{- 1} - \sum_{1 ⩽ 2 k + n ⩽ M - 1} β_{k} ℑ (B_{2 k, n}) t^{- 2 k - n - 1} + O (t^{- M - 1}) . \end{array}$

Combining (2.5) and (2.8), $\begin{matrix} \frac{d}{d t} I_{1, - 1}^{ϵ} (t) = \frac{1}{4} t^{- 1} + \sum_{n = 0}^{M - 2} \frac{1}{2} B_{1, n} t^{- n - 2} - \sum_{1 ⩽ 2 k + n ⩽ M - 1} β_{k} ℑ (B_{2 k, n}) t^{- 2 k - n - 1} + O (t^{- M - 1}) . \end{matrix}$ This implies that for some constant C $\begin{array}{rcl} I_{1, - 1}^{ϵ} (t) & = & \frac{1}{4} log (t) + C - \sum_{n = 0}^{M - 2} \frac{1}{2 (n + 1)} B_{1, n} t^{- n - 1} \\ + \sum_{1 ⩽ 2 k + n ⩽ M - 1} \frac{β_{k}}{2 k + n} ℑ (B_{2 k, n}) t^{- 2 k - n} + O (t^{- M}) . \end{array}$ This fact is proven by modifying the proof given in Chapter 1 of [2].

To determine the constant C we consider the asymptotics of the following integral for $0 < ϵ < 2$ (to be proved separately): $\begin{matrix} (2.9) & J_{1, - 1}^{ϵ} (t) : = \int_{0}^{ϵ} r^{- 1} e^{- t r^{2}} {sin}^{2} (t r) d r = \frac{1}{4} (log (4 t) + γ) + O (t^{- 1}) . \end{matrix}$ We now use the fact that $| {sin}^{2} (t r) - {sin}^{2} (t r \sqrt{1 - r^{2} / 4}) | ⩽ t r | 1 - \sqrt{1 - r^{2} / 4} | ⩽ \frac{t r^{3}}{4}$ for $0 ⩽ r ⩽ 2$ . Thus $\begin{matrix} (2.10) & | J_{1, - 1}^{ϵ} (t) - I_{1, - 1}^{ϵ} (t) | ⩽ \frac{t}{4} \int_{0}^{ϵ} r^{2} e^{- t r^{2}} d r ⩽ Γ (\frac{3}{2}) t^{- 1 / 2} \to 0 . \end{matrix}$ Furthermore $\begin{array}{rcl} J_{1, - 1}^{ϵ} (t) - I_{1, - 1}^{ϵ} (t) & = & (\frac{1}{4} log (t) + \frac{γ}{4} + \frac{log (2)}{2}) - (\frac{1}{4} log (t) + C) + O (t^{- 1}) \\ = & \frac{γ}{4} + \frac{log (2)}{2} - C + O (t^{- 1}) \to \frac{γ}{4} + \frac{log (2)}{2} - C . \end{array}$ This result together with (2.10) implies $C = γ / 4 + log (2) / 2$ , which proves the lemma. □

We now must prove the claim about $J_{1, - 1}^{ϵ} (t)$ from Lemma 2.4. Lemma 2.5.
Let $0 < ϵ < 2$ and $P \in N_{0}$ . Then with $J_{1, - 1}^{ϵ} (t)$ as defined in ( 2.9 ), $\begin{matrix} J_{1, - 1}^{ϵ} (t) = \frac{1}{4} (log (4 t) + γ) - \sum_{p = 1}^{P} \frac{1}{4 p} {(\frac{1}{2})}_{p} t^{- p} + O (t^{- P - 1}) . \end{matrix}$
Proof.
Let $0 < ϵ < 2$ , $P \in N_{0}$ , and $t > 0$ . Then by the mean value theorem and Lebesgue’s dominated convergence theorem, $\frac{d}{d t} J_{1, - 1}^{ϵ} (t) = J_{1} (t) - ℑ (F_{0}^{ϵ} (t))$ , where $F_{0}^{ϵ} (t)$ is as given in Lemma 2.1 and $\begin{matrix} J_{1} (t) = - \int_{0}^{ϵ} r e^{- t r^{2}} {sin}^{2} (t r) d r . \end{matrix}$ We use the fact that ${sin}^{2} (x) = \frac{1}{2} - \frac{1}{2} cos (2 x)$ to write $\begin{matrix} (2.11) & J_{1} (t) = - \frac{1}{2} \int_{0}^{ϵ} r e^{- t r^{2}} d r + \frac{1}{2} ℜ (F_{1}^{ϵ} (t)) . \end{matrix}$ We observe the integral in the first term of (2.11) satisfies $\begin{matrix} \int_{0}^{ϵ} r e^{- t r^{2}} d r = \int_{0}^{\infty} r e^{- t r^{2}} d r + O (e^{- t ϵ^{2} / 2}) . \end{matrix}$ From this fact and (2.1) we conclude that $\begin{matrix} \frac{d}{d t} J_{1, - 1}^{ϵ} (t) = \frac{1}{4} t^{- 1} + \sum_{p = 0}^{P - 1} \frac{1}{2} A_{1, p} t^{- p - 2} + O (t^{- P - 2}) - \sum_{s = 1}^{S - 1} ℑ (A_{0, s}) t^{- s - 1} + O (t^{- S - 1}), \end{matrix}$ where $P \in N_{0}$ is as given and $S : = P + 1$ . Simplifying we obtain $\begin{matrix} \frac{d}{d t} J_{1, - 1}^{ϵ} (t) = \frac{1}{4} t^{- 1} + \sum_{p = 1}^{P} \frac{1}{4} {(\frac{1}{2})}_{p} t^{- p - 1} + O (t^{- P - 2}) . \end{matrix}$ Therefore there is some constant C such that $\begin{matrix} (2.12) & J_{1, - 1}^{ϵ} (t) = \frac{1}{4} log (t) + C - \sum_{p = 1}^{P} \frac{1}{4 p} {(\frac{1}{2})}_{p} t^{- p} + O (t^{- P - 1}) . \end{matrix}$

Now we must determine the constant C. To do so, we first note that from the integral definition in (2.9), $J_{1, - 1}^{ϵ} (t) = {\tilde{J}}_{1, - 1} (t) - \int_{ϵ}^{\infty} r^{- 1} e^{- t r^{2}} {sin}^{2} (t r) d r$ , where for $t > 0$ $\begin{matrix} {\tilde{J}}_{1, - 1} (t) = \int_{0}^{\infty} r^{- 1} e^{- t r^{2}} {sin}^{2} (t r) d r . \end{matrix}$ It is easily verified that $\int_{ϵ}^{\infty} r^{- 1} e^{- t r^{2}} {sin}^{2} (t r) d r = O (e^{- t ϵ^{2} / 2}) = o (1)$ , so we instead study ${\tilde{J}}_{1, - 1} (t)$ .

With the substitution $s = \sqrt{t} r$ we have ${\tilde{J}}_{1, - 1} (t) = \int_{0}^{\infty} s^{- 1} e^{- s^{2}} {sin}^{2} (\sqrt{t} s) d s$ . Integrating by parts we obtain $\begin{matrix} {\tilde{J}}_{1, - 1} (t) = J_{2} (t) + J_{3} (t) + J_{4} (t), \end{matrix}$ where $\begin{array}{l} J_{2} (t) = \int_{0}^{\infty} s log (s) e^{- s^{2}} d s, \\ J_{3} (t) = - \int_{0}^{\infty} s log (s) e^{- s^{2}} cos (2 \sqrt{t} s) d s, \\ J_{4} (t) = - \sqrt{t} \int_{0}^{\infty} log (s) e^{- s^{2}} sin (2 \sqrt{t} s) d s . \end{array}$

With the substitution $u = s^{2}$ , $\begin{matrix} J_{2} (t) = \frac{1}{4} \int_{0}^{\infty} log (u) e^{- u} d u . \end{matrix}$ The integral $\begin{matrix} (2.13) & \int_{0}^{\infty} log (u) e^{- a u} d u = - \frac{γ + log (a)}{a} (ℜ (a) > 0), \end{matrix}$ is a Laplace transform identity found in [3, §4.6(1)]. Therefore $J_{2} (t) = - \frac{γ}{4}$ .

We now let $f_{1} (s) = - s log (s) e^{- s^{2}} \cdot χ_{(0, \infty)} (s)$ , where $χ_{(0, \infty)} (s)$ is the characteristic function of the set $(0, \infty)$ . Then $f_{1} \in L^{1} (R)$ , and for all $t > 0$ $\begin{matrix} J_{3} (t) = ℜ (\int_{- \infty}^{\infty} f_{1} (s) e^{- 2 i \sqrt{t} s} d s) = \sqrt{2 π} ℜ ({\hat{f}}_{1} (2 \sqrt{t})) . \end{matrix}$ By the Riemann–Lebesgue lemma found in Chapter 2 of [5], ${lim}_{t \to \infty} J_{3} (t) = 0$ .

Lastly for $t > 0$ we define $\begin{matrix} J_{5} (t) : = - \sqrt{t} \int_{0}^{\infty} log (s) e^{- s} sin (2 \sqrt{t} s) d s . \end{matrix}$ Then $\begin{matrix} (2.14) & J_{4} (t) - J_{5} (t) = \sqrt{t} \int_{0}^{\infty} f_{2} (s) sin (2 \sqrt{t} s) d s, \end{matrix}$ where $f_{2} (s) = log (s) (e^{- s} - e^{- s^{2}})$ . We integrate (2.14) by parts and use the fact that ${lim}_{t \to 0 +} f_{2} (t) = 0$ to obtain $\begin{matrix} J_{4} (t) - J_{5} (t) = \frac{1}{2} \int_{0}^{\infty} f_{2}^{'} (s) cos (2 \sqrt{t} s) d s = \frac{1}{2} ℜ (\int_{- \infty}^{\infty} f_{3} (s) e^{- 2 i \sqrt{t} s} d s), \end{matrix}$ where $f_{3} : = f_{2}^{'} \cdot χ_{(0, \infty)} \in L^{1} (R)$ . Again by the Riemann–Lebesgue lemma, $\begin{matrix} lim_{t \to \infty} (J_{4} (t) - J_{5} (t)) = lim_{t \to \infty} \sqrt{\frac{π}{2}} ℜ ({\hat{f}}_{3} (2 \sqrt{t})) = 0 . \end{matrix}$

We use the fact that $sin (2 \sqrt{t} s) = ℑ (exp (2 i \sqrt{t} s))$ and (2.13) to obtain $\begin{matrix} J_{5} (t) = \sqrt{t} ℑ (\frac{γ + log (1 - 2 i \sqrt{t})}{1 - 2 i \sqrt{t}}) = \frac{1}{4} (log (4 t) + γ) + o (1) . \end{matrix}$

Combining the results of the analysis of ${\tilde{J}}_{1, - 1} (t)$ , we have $\begin{matrix} J_{1, - 1}^{ϵ} (t) = {\tilde{J}}_{1, - 1} (t) + o (1) = \frac{1}{4} (log (4 t) + γ) + o (1) . \end{matrix}$ From (2.12) we have $\begin{matrix} J_{1, - 1}^{ϵ} (t) = \frac{1}{4} log (t) + C + o (1) . \end{matrix}$ Thus $C = γ / 4 + log (2) / 2$ and the proof is complete. □

2.1.2. Intermediate computations

Ikehata [8] showed that for the fixed $0 < δ < 1$ , $\int_{| ξ | > δ} | \hat{u} (t, ξ) |^{2} d ξ$ is exponentially small as $t \to \infty$ . So we instead find the expansion of $‖ \hat{u} (t, \cdot) ‖_{2, δ}^{2} = X_{1} (t) + Y_{1} (t) + Z_{1} (t)$ . Additionally, the overlying assumption is that $K \in N$ with the initial data of (1.1) $u_{0} \in H^{1} (R^{2}) \cap L^{1, 2 K} (R^{2})$ and $u_{1} \in L^{2} (R^{2}) \cap L^{1, 2 K} (R^{2})$ .

Expansion of $X_{1} (t)$ . Since $\begin{matrix} X_{1} (t) = \int_{| ξ | ⩽ δ} {| {\hat{u}}_{1} (ξ) + \frac{| ξ |^{2}}{2} {\hat{u}}_{0} (ξ) |}^{2} {(h_{1} (t, ξ))}^{2} d ξ, \end{matrix}$ we may substitute (1.4) and (1.8) to obtain a sum of integrals indexed by the multi-indices σ plus an integral of a $O (| ξ |^{2 K})$ -term. We estimate the integral with the $O (| ξ |^{2 K})$ -term once, as all others may be estimated similarly: $\begin{array}{l} \int_{| ξ | ⩽ δ} | ξ |^{2 K} e^{- t | ξ |^{2}} \frac{{sin}^{2} (t | ξ | \sqrt{1 - | ξ |^{2} / 4})}{| ξ |^{2} (1 - | ξ |^{2} / 4)} d ξ \\ ⩽ \frac{1}{1 - δ^{2} / 4} \int_{| ξ | ⩽ δ} | ξ |^{2 K - 2} e^{- t | ξ |^{2}} d ξ \\ = O (\int_{0}^{δ} r^{2 K - 1} e^{- t r^{2}} d r) = O (t^{- K}) . \end{array}$

Since any remaining integrals with an odd entry in the multi-index evaluate to zero, we omit them and then switch to polar coordinates to obtain $\begin{matrix} (2.15) & X_{1} (t) = \sum_{| σ | ⩽ K - 1} d_{2 σ} J_{σ} \int_{0}^{δ} r^{2 | σ | - 1} e^{- t r^{2}} \frac{{sin}^{2} (t r \sqrt{1 - r^{2} / 4})}{1 - r^{2} / 4} d r + O (t^{- K}), \end{matrix}$ where $\begin{matrix} J_{σ} = \frac{2 \cdot Γ (σ_{1} + \frac{1}{2}) \cdot Γ (σ_{2} + \frac{1}{2})}{Γ (| σ | + 1)} . \end{matrix}$ We substitute the expansion $\begin{matrix} (2.16) & \frac{1}{1 - r^{2} / 4} = \sum_{k = 0}^{L - 1} \frac{r^{2 k}}{4^{k}} + O (r^{2 L}) for 0 ⩽ r ⩽ δ < 2 \end{matrix}$ into (2.15) with $L : = K - | σ | \in N$ . We obtain $\begin{matrix} (2.17) & X_{1} (t) = \sum_{| σ | + k ⩽ K - 1} \frac{d_{2 σ} J_{σ}}{4^{k}} I_{1, 2 | σ | + 2 k - 1} (t) + O (t^{- K}) . \end{matrix}$ We apply Lemma 2.4 with $M : = K$ to the term in the sum with $σ = 0$ , $k = 0$ , and Lemma 2.2(b) to all other terms. Thus with the identity $d_{0} J_{0} = 2 π | P_{1} |^{2}$ , we find $\begin{matrix} (2.18) & X_{1} (t) = \frac{π | P_{1} |^{2}}{2} (log (4 t) + γ) + \sum_{j = 1}^{K - 1}_{2} X_{1} (j) t^{- j} + O (t^{- K}), \end{matrix}$ where $\begin{array}{rcl} _{2} X_{1} (j) & = & - \frac{π | P_{1} |^{2}}{j} B_{1, j - 1} + \sum_{2 k + n = j} \frac{2 π | P_{1} |^{2} β_{k}}{j} ℑ (B_{2 k, n}) \\ + \sum_{| σ | + k = j} \frac{d_{2 σ} J_{σ}}{4^{k + 1}} Γ (j) - \sum_{\begin{array}{c} 2 | σ | + 2 k + n = j \\ | σ | + k ⩾ 1 \end{array}} \frac{d_{2 σ} J_{σ}}{2^{2 k + 1}} B_{2 | σ | + 2 k - 1, n} . \end{array}$

Expansion of $Y_{1} (t)$ . We refer back to the expansion of $Y_{1} (t)$ obtained in [1]. We see that all steps are valid here with $N = 2$ . We arrive at $\begin{matrix} (2.19) & Y_{1} (t) = \sum_{j = 1}^{K - 1}_{2} Y_{1} (j) t^{- j} + O (t^{- K}), \end{matrix}$ where $\begin{matrix} _{2} Y_{1} (j) = \sum_{| σ | = j - 1} \frac{f_{2 σ} J_{σ}}{4} Γ (j) + \sum_{2 | σ | + n = j - 2} \frac{f_{2 σ} J_{σ}}{2} B_{2 | σ | + 1, n} . \end{matrix}$

Expansion of $Z_{1} (t)$ . All steps are valid now with $N = 2$ and we deduce $\begin{matrix} (2.20) & Z_{1} (t) = \sum_{j = 1}^{K - 1}_{2} Z_{1} (j) t^{- j} + O (t^{- K}), \end{matrix}$ where $\begin{matrix} _{2} Z_{1} (j) = - \sum_{2 | σ | + 2 k + n = j - 1} ℜ (l_{2 σ}) J_{σ} α_{k} ℑ (B_{2 | σ | + 2 k, n}) \end{matrix}$ with $\begin{matrix} (2.21) & α_{k} = \frac{(2 k - 1)!!}{8^{k} \cdot k!} (k \in N_{0}) . \end{matrix}$

2.1.3. Asymptotic expansion of $‖ u (t, \cdot) ‖_{2}^{2}$

We combine results (2.18), (2.19), and (2.20) to obtain the asymptotic expansion $\begin{matrix} {‖ u (t, \cdot) ‖}_{2}^{2} = \frac{π | P_{1} |^{2}}{2} (log (4 t) + γ) + \sum_{j = 1}^{K - 1}_{2} W_{j} t^{- j} + O (t^{- K}), \end{matrix}$ where $\begin{matrix} _{2} W_{j} =_{2} X_{1} (j) +_{2} Y_{1} (j) +_{2} Z_{1} (j) . \end{matrix}$ For example, if $K ⩾ 2$ then $\begin{array}{rcl} _{2} W_{1} & = & - π | P_{1} |^{2} B_{1, 0} + 2 π | P_{1} |^{2} β_{0} ℑ (B_{0, 1}) + \sum_{| σ | = 1} \frac{d_{2 σ} J_{σ}}{4} Γ (1) + \frac{π | P_{1} |^{2}}{2^{3}} Γ (1) \\ + \frac{f_{0} J_{0}}{4} Γ (1) - ℜ (l_{0}) J_{0} α_{0} ℑ (B_{0, 0}) . \end{array}$ We then simplify to obtain the expressions in Remark 2.

2.2. The space dimension $N = 1$ case

2.2.1. Auxiliary results

Lemma 2.6.
For all $M \in N$ , $\tilde{I} (t) : = \int_{0}^{\infty} r^{- 1} e^{- t r^{2}} sin (2 t r) d r$ satisfies $\tilde{I} (t) = \frac{π}{2} + O (t^{- M})$ .
Proof.
It follows from [3, §2.4 (21)] that $\tilde{I} (t) = \frac{π}{2} erf (\sqrt{t})$ , where $erf (t)$ denotes the error function. Therefore, using the known asymptotic behavior of the error function [13, Chapter 8] $\begin{matrix} \tilde{I} (t) = \frac{π}{2} + O (t^{- 1 / 2} e^{- t}) = \frac{π}{2} + O (t^{- M}), \end{matrix}$ proving the lemma. □
Lemma 2.7.
For $0 < ϵ < 2$ and $t > 0$ , we define $\begin{array}{rcl} I_{3, - 1}^{ϵ} (t) & : = & \int_{0}^{ϵ} r^{- 1} e^{- t r^{2}} sin (t r \sqrt{1 - r^{2} / 4}) cos (t r \sqrt{1 - r^{2} / 4}) d r \\ = & \frac{1}{2} \int_{0}^{ϵ} r^{- 1} e^{- t r^{2}} sin (t r \sqrt{4 - r^{2}}) d r . \end{array}$ If $M \in N$ , then $I_{3, - 1}^{ϵ} (t) = \frac{π}{4} + O (t^{- M})$ .
Proof.
Let $0 < ϵ < 2$ , $t > 0$ , and $M \in N$ . We use the mean value theorem and Lebesgue’s dominated convergence theorem to justify differentiation within the integral. Hence $\begin{matrix} \frac{d}{d t} I_{3, - 1}^{ϵ} (t) = - I_{3, 1}^{ϵ} (t) + I_{1} (t), \end{matrix}$ where $I_{3, 1}^{ϵ} (t)$ is as given in Lemma 2.3 and $\begin{matrix} I_{1} (t) = \frac{1}{2} \int_{0}^{ϵ} \sqrt{4 - r^{2}} e^{- t r^{2}} cos (t r \sqrt{4 - r^{2}}) d r . \end{matrix}$ By Lemma 2.3(b) with $Q : = M - 1$ , $\begin{matrix} (2.22) & - I_{3, 1}^{ϵ} (t) = O (t^{- M - 1}) . \end{matrix}$ To determine the asymptotics of $I_{1} (t)$ , we substitute (2.6) with $L : = M + 1$ . Hence $\begin{matrix} (2.23) & I_{1} (t) = \sum_{k = 0}^{M} β_{k} ℜ (G_{1, 2 k}^{ϵ} (t)) + O (t^{- M - 3 / 2}) . \end{matrix}$ We apply Lemma 2.1(b) to (2.23) with $Q = Q_{k} : = max {0, M - 2 k + 1}$ . We also observe that since each $2 k$ ( $k \in {0, \dots, M}$ ) is even, every $B_{2 k, n} \in i R$ . Thus $\begin{matrix} (2.24) & I_{1} (t) = O (t^{- M - 3 / 2}) = O (t^{- M - 1}) . \end{matrix}$

We combine the results (2.22) and (2.24) to obtain $\frac{d}{d t} I_{3, - 1}^{ϵ} (t) = O (t^{- M - 1})$ . Hence $I_{3, - 1}^{ϵ} (t) = C + O (t^{- M})$ . We must determine the constant C.

From the fact $M \in N$ , Lemma 2.6 implies $\tilde{I} (t) = \frac{π}{2} + O (t^{- M})$ . Since $| sin (2 t r) - sin (t r \sqrt{4 - r^{2}}) | ⩽ 2 t r | 1 - \sqrt{1 - r^{2} / 4} | ⩽ t r^{3} / 2$ for $0 ⩽ r ⩽ 2$ , we obtain $\begin{array}{rcl} | \frac{1}{2} \tilde{I} (t) - I_{3, - 1}^{ϵ} (t) | & ⩽ & \frac{t}{4} \int_{0}^{ϵ} r^{2} e^{- t r^{2}} d r + \frac{1}{2} \int_{ϵ}^{\infty} r^{- 1} e^{- t r^{2}} d r \\ (2.25) & = & O (t^{- 1 / 2}) + O (e^{- t ϵ^{2} / 2}) \to 0 . \end{array}$ Further, for the given $M \in N$ , $\begin{matrix} (2.26) & \frac{1}{2} \tilde{I} (t) - I_{3, - 1}^{ϵ} (t) = \frac{π}{4} - C + O (t^{- M}) \to \frac{π}{4} - C . \end{matrix}$ Considering (2.25) and (2.26) together implies $C = π / 4$ , completing the proof. □
Lemma 2.8.
For $0 < ϵ < 2$ and $t > 0$ , we define $\begin{matrix} I_{1, - 2}^{ϵ} (t) : = \int_{0}^{ϵ} r^{- 2} e^{- t r^{2}} {sin}^{2} (t r \sqrt{1 - r^{2} / 4}) d r . \end{matrix}$ If $M \in N$ , then $I_{1, - 2}^{ϵ} (t) = \frac{π}{2} t - \frac{\sqrt{π}}{2} t^{1 / 2} + O (t^{- M})$ .
Proof.
Let $0 < ϵ < 2$ , $t > 0$ , and $M \in N$ . Using integration by parts, $I_{1, - 2}^{ϵ} (t) = I_{1} (t) - 2 t I_{1, 0}^{ϵ} (t) + I_{2} (t)$ , where $\begin{array}{l} (2.27) & I_{1} (t) = - ϵ^{- 1} e^{- t ϵ^{2}} {sin}^{2} (t ϵ \sqrt{1 - ϵ^{2} / 4}) = O (e^{- t ϵ^{2}}) = O (t^{- M}), \\ (2.28) & I_{2} (t) = t \int_{0}^{ϵ} r^{- 1} \frac{d}{d r} [r \sqrt{1 - r^{2} / 4}] e^{- t r^{2}} sin (2 t r \sqrt{1 - r^{2} / 4}) d r . \end{array}$ We apply Lemma 2.2(a) with $Q : = M$ to $- 2 t I_{1, 0}^{ϵ} (t)$ . Therefore, $\begin{matrix} (2.29) & - 2 t I_{1, 0}^{ϵ} (t) = - \frac{\sqrt{π}}{2} t^{1 / 2} + O (t^{- M}) . \end{matrix}$ To analyze (2.28), we substitute $\begin{matrix} (2.30) & \frac{d}{d r} [r \sqrt{1 - r^{2} / 4}] = \sum_{k = 0}^{M} (2 k + 1) β_{k} r^{2 k} + O (r^{2 M + 2}), \end{matrix}$ where each $β_{k}$ is as given in (2.4). Then $\begin{array}{rcl} t^{- 1} I_{2} (t) & = & 2 I_{3, - 1}^{ϵ} (t) - \sum_{k = 1}^{M} (2 k + 1) β_{k} ℑ (G_{1, 2 k - 1}^{ϵ}) \\ (2.31) & + \int_{0}^{ϵ} O (r^{2 M + 1}) e^{- t r^{2}} sin (2 t r \sqrt{1 - r^{2} / 4}) d r . \end{array}$ By Lemma 2.7, the first term of (2.31) is $\frac{π}{2} + O (t^{- M - 1})$ . We see the second term of (2.31) is $O (t^{- M - 1})$ by applying Lemma 2.1(b) with $Q = Q_{k} = max {0, M - 2 k + 1}$ for each $k \in {1, \dots, M}$ . Finally we observe $\begin{matrix} \int_{0}^{ϵ} r^{2 M + 1} e^{- t r^{2}} d r ⩽ \int_{0}^{\infty} r^{2 M + 1} e^{- t r^{2}} d r = O (t^{- M - 1}) . \end{matrix}$ Thus, $I_{2} (t) = t (\frac{π}{2} + O (t^{- M - 1})) = \frac{π}{2} t + O (t^{- M})$ . Combining this result for $I_{2} (t)$ with (2.27) and (2.29) completes the proof. □
2.2.2. Intermediate computations

Once again we compute $X_{1} (t) + Y_{1} (t) + Z_{1} (t)$ with the assumptions that $K \in N$ and the initial data of (1.1) satisfy $u_{0} \in H^{1} (R) \cap L^{1, 2 K} (R)$ and $u_{1} \in L^{2} (R) \cap L^{1, 2 K} (R)$ .

Expansion of $X_{1} (t)$ . The work from the $N = 2$ case in Section 2.1.2 carries over to this case. Since the space dimension is $N = 1$ , the “multi-indices” are simply non-negative integers. We have $\begin{matrix} (2.32) & X_{1} (t) = \sum_{σ = 0}^{K - 1} 2 d_{2 σ} \int_{0}^{δ} r^{2 σ - 2} e^{- t r^{2}} \frac{{sin}^{2} (t r \sqrt{1 - r^{2} / 4})}{1 - r^{2} / 4} d r + O (t^{- K + 1 / 2}) . \end{matrix}$ We substitute (2.16) into (2.32) with $L : = K - σ$ and find $\begin{matrix} (2.33) & X_{1} (t) = \sum_{σ + k ⩽ K - 1} \frac{d_{2 σ}}{2^{2 k - 1}} I_{1, 2 σ + 2 k - 2}^{δ} (t) + O (t^{- K + 1 / 2}) . \end{matrix}$ We treat the term in the sum with $σ = k = 0$ by using Lemma 2.8 with $M : = K$ , and all other terms by using Lemma 2.2(a) with $Q = Q_{k} : = max {0, K - 2 k + 1}$ . Then we obtain $\begin{matrix} (2.34) & X_{1} (t) = | P_{1} |^{2} (π t - \sqrt{π t}) + \sum_{j = 1}^{K - 1}_{1} X_{1} (j) t^{- j + 1 / 2} + O (t^{- K + 1 / 2}), \end{matrix}$ where $\begin{matrix} _{1} X_{1} (j) = \sum_{σ + k = j} \frac{d_{2 σ}}{2^{2 k + 1}} Γ (j - \frac{1}{2}) . \end{matrix}$

Expansion of $Y_{1} (t)$ . The work done in [1] carries over to the case $N = 1$ as well. We obtain $\begin{matrix} (2.35) & Y_{1} (t) = \sum_{j = 1}^{K - 1}_{1} Y_{1} (j) t^{- j + 1 / 2} + O (t^{- K + 1 / 2}), \end{matrix}$ where $\begin{matrix} _{1} Y_{1} (j) = \frac{f_{2 j - 2}}{2} Γ (j - \frac{1}{2}) . \end{matrix}$

Expansion of $Z_{1} (t)$ . Given that $\begin{matrix} Z_{1} (t) = 2 ℜ (\int_{- δ}^{δ} ({\hat{u}}_{1} (r) + \frac{| r |^{2}}{2} {\hat{u}}_{0} (r)) {\bar{\hat{u}}}_{0} (r) h_{1} (t, r) h_{2} (t, r) d r), \end{matrix}$ we substitute (1.6), (1.8), and (1.9). The integral with the $O (| r |^{2 K})$ -term is $O (t^{- K + 1 / 2})$ . Thus $\begin{matrix} (2.36) & Z_{1} (t) = \sum_{σ = 0}^{K - 1} 2 ℜ (l_{2 σ}) \int_{0}^{δ} r^{2 σ - 1} e^{- t r^{2}} \frac{sin (2 t r \sqrt{1 - r^{2} / 4})}{\sqrt{1 - r^{2} / 4}} d r + O (t^{- K + 1 / 2}) . \end{matrix}$ With $α_{k}$ as given in (2.21), $\begin{matrix} (2.37) & \frac{1}{\sqrt{1 - r^{2} / 4}} = \sum_{k = 0}^{L - 1} α_{k} r^{2 k} + O (r^{2 L}) . \end{matrix}$ So to find the expansion of $Z_{1} (t)$ , we substitute (2.37) into (2.36) with $L : = K$ . Then $\begin{matrix} (2.38) & Z_{1} (t) = \sum_{σ + k ⩽ K - 1} 4 ℜ (l_{2 σ}) α_{k} I_{3, 2 σ + 2 k - 1}^{δ} (t) + O (t^{- K + 1 / 2}) . \end{matrix}$ We use Lemma 2.7 with $M : = K$ to treat the term of the sum with $σ = k = 0$ in (2.38), and Lemma 2.3(b) with $Q = Q_{k} : = max {0, K - 2 k}$ to treat all other terms. Thus $\begin{matrix} (2.39) & Z_{1} (t) = π ℜ (P_{1} {\bar{P}}_{0}) + O (t^{- K + 1 / 2}) . \end{matrix}$

2.2.3. Asymptotic expansion of $‖ u (t, \cdot) ‖_{2}^{2}$

Combining (2.34), (2.35), and (2.39), we obtain the asymptotic expansion $\begin{matrix} {‖ u (t, \cdot) ‖}_{2}^{2} = | P_{1} |^{2} (π t - \sqrt{π t}) + π ℜ (P_{1} {\bar{P}}_{0}) + \sum_{j = 1}^{K - 1}_{1} W_{j} t^{- j + 1 / 2} + O (t^{- K + 1 / 2}), \end{matrix}$ where $\begin{matrix} _{1} W_{j} =_{1} X_{1} (j) +_{1} Y_{1} (j) . \end{matrix}$ For example, if $K ⩾ 2$ then $\begin{matrix} _{1} W_{1} = \frac{d_{0}}{2^{3}} Γ (\frac{1}{2}) + \frac{d_{2}}{2} Γ (\frac{1}{2}) + \frac{f_{0}}{2} Γ (\frac{1}{2}) . \end{matrix}$ Simplifying gives the expressions from Remark 2.

3. Proof of Theorem 1.2

We start by recalling some results from [1].

Lemma 3.1.

For $0 < ϵ < 2$ , $m \in N_{0}$ , and $t > 0$ define $\begin{matrix} G_{2, m}^{ϵ} (t) : = \int_{0}^{ϵ} r^{m} exp (- t r^{2} - i r t \sqrt{1 - r^{2} / 4} + i r t) d r . \end{matrix}$ If $Q \in N_{0}$ , then $\begin{matrix} G_{2, m}^{ϵ} (t) = \sum_{n = 0}^{2 Q - 1} C_{m, n} t^{- m / 2 - n / 2 - 1 / 2} + O (t^{- m / 2 - Q - 1 / 2}), \end{matrix}$ where $\begin{matrix} C_{m, n} \in \{\begin{matrix} R & n \in N_{0} even \\ i R & n \in N odd . \end{matrix} \end{matrix}$

For $0 < ϵ < 2$ , $m \in N_{0}$ , and $t > 0$ define $\begin{matrix} G_{3, m}^{ϵ} (t) : = \int_{0}^{ϵ} r^{m} exp (- t r^{2} - i r t \sqrt{1 - r^{2} / 4} - i r t) d r . \end{matrix}$ If $Q \in N_{0}$ , then $\begin{matrix} G_{3, m}^{ϵ} (t) = \sum_{n = 0}^{Q - 1} E_{m, n} t^{- m - n - 1} + O (t^{- m - Q - 1}), \end{matrix}$ where $\begin{matrix} E_{m, n} \in \{\begin{matrix} R & m \in N odd \\ i R & m \in N_{0} even . \end{matrix} \end{matrix}$

Formulas for $C_{m, n}$ and $E_{m, n}$ for small n are given in [1, (6.6), (3.7)]. Lemma 3.2.
Let $m \in N_{0}$ , $0 < ϵ < 2$ , and $t > 0$ . Define $\begin{array}{l} H_{1, m}^{ϵ} (t) : = \int_{0}^{ϵ} r^{m} e^{- t r^{2}} sin (t r \sqrt{1 - r^{2} / 4}) sin (t r) d r, \\ H_{2, m}^{ϵ} (t) : = \int_{0}^{ϵ} r^{m} e^{- t r^{2}} cos (t r \sqrt{1 - r^{2} / 4}) cos (t r) d r . \end{array}$
If $m \in N_{0}$ is even, $Q \in N_{0}$ and $p = 1, 2$ , then $\begin{matrix} H_{p, m}^{ϵ} (t) = \sum_{n = 0}^{Q - 1} \frac{1}{2} C_{m, 2 n} t^{- m / 2 - n - 1 / 2} + O (t^{- m / 2 - Q - 1 / 2}) . \end{matrix}$

If $m \in N_{0}$ is odd, $Q, R \in N_{0}$ and $p = 1, 2$ , then $\begin{array}{rcl} H_{p, m}^{ϵ} (t) & = & \sum_{n = 0}^{Q - 1} \frac{1}{2} C_{m, 2 n} t^{- m / 2 - n - 1 / 2} + O (t^{- m / 2 - Q - 1 / 2}) \\ + {(- 1)}^{p} \sum_{n = 0}^{R - 1} \frac{1}{2} E_{m, n} t^{- m - n - 1} + O (t^{- m - R - 1}) . \end{array}$

Lemma 3.3.
Let $m \in N_{0}$ , $0 < ϵ < 2$ , and $t > 0$ . Define $\begin{array}{l} H_{3, m}^{ϵ} (t) : = \int_{0}^{ϵ} r^{m} e^{- t r^{2}} sin (t r \sqrt{1 - r^{2} / 4}) cos (t r) d r, \\ H_{4, m}^{ϵ} (t) : = \int_{0}^{ϵ} r^{m} e^{- t r^{2}} cos (t r \sqrt{1 - r^{2} / 4}) sin (t r) d r . \end{array}$
If $m \in N_{0}$ is even, $Q, R \in N_{0}$ and $p = 3, 4$ then $\begin{array}{rcl} H_{p, m}^{ϵ} (t) & = & - \sum_{n = 0}^{R - 1} \frac{1}{2} ℑ (E_{m, n}) t^{- m - n - 1} + O (t^{- m - R - 1}) \\ + {(- 1)}^{p} \sum_{n = 0}^{Q - 1} \frac{1}{2} ℑ (C_{m, 2 n + 1}) t^{- m / 2 - n - 1} + O (t^{- m / 2 - Q - 1 / 2}) . \end{array}$

If $m \in N_{0}$ is odd, $Q \in N_{0}$ and $p = 3, 4$ , then $\begin{matrix} H_{p, m}^{ϵ} (t) = {(- 1)}^{p} \sum_{n = 0}^{Q - 1} \frac{1}{2} ℑ (C_{m, 2 n + 1}) t^{- m / 2 - n - 1} + O (t^{- m / 2 - Q - 1 / 2}) . \end{matrix}$

3.1. The dimension $N = 2$ case

3.1.1. Auxiliary result

Lemma 3.4.
For $0 < ϵ < 2$ and $t > 0$ , define $\begin{matrix} H_{1, - 1}^{ϵ} (t) : = \int_{0}^{ϵ} r^{- 1} e^{- t r^{2}} sin (t r \sqrt{1 - r^{2} / 4}) sin (t r) d r . \end{matrix}$ If $M \in N$ , then $\begin{matrix} H_{1, - 1}^{ϵ} (t) = \frac{1}{4} (log (4 t) + γ) + \sum_{j = 1}^{M - 1} S_{j} t^{- j} + O (t^{- M}), \end{matrix}$ where $\begin{array}{rcl} S_{j} & = & \frac{1}{2 j} C_{1, 2 j} - \frac{1}{2 j} E_{1, j - 1} + \frac{1}{2 j} ℑ (E_{0, j}) + \frac{1}{2 j} ℑ (C_{0, 2 j + 1}) \\ + \frac{1}{2 j} \sum_{2 k + n = j} β_{k} ℑ (E_{2 k, n}) - \frac{1}{2 j} \sum_{k + n = j} β_{k} ℑ (C_{2 k, 2 n + 1}) . \end{array}$
Proof.
Let $0 < ϵ < 2$ , $t > 0$ , and $M \in N$ . Using the mean value theorem and Lebesgue’s dominated convergence theorem, it is a routine argument to show that $\frac{d}{d t} H_{1, - 1}^{ϵ} (t) = - H_{1, 1}^{ϵ} (t) + H_{1} (t) + H_{3, 0}^{ϵ} (t)$ , where $\begin{matrix} H_{1} (t) = \int_{0}^{ϵ} \sqrt{1 - r^{2} / 4} e^{- t r^{2}} cos (t r \sqrt{1 - r^{2} / 4}) sin (t r) d r, \end{matrix}$ and $H_{1, 1}^{ϵ} (t)$ and $H_{3, 0}^{ϵ} (t)$ are as given in Lemmas 3.2 and 3.3, respectively.

By Lemma 3.2(b) with $Q : = M$ and $R : = M - 1$ , and the fact that $C_{1, 0} = \frac{1}{2}$ $\begin{matrix} (3.1) & - H_{1, 1}^{ϵ} (t) = - \frac{1}{4} t^{- 1} - \frac{1}{2} \sum_{n = 1}^{M - 1} C_{1, 2 n} t^{- n - 1} + \frac{1}{2} \sum_{n = 0}^{M - 2} E_{1, n} t^{- n - 2} + O (t^{- M - 1}) . \end{matrix}$

Similarly, by Lemma 3.3(a) with $R : = M$ and $Q : = M + 1$ , and the facts that $E_{0, 0} = - \frac{i}{2}$ and $C_{0, 1} = \frac{i}{16}$ $\begin{matrix} (3.2) & H_{3, 0}^{ϵ} (t) = \frac{7}{32} t^{- 1} - \frac{1}{2} \sum_{n = 1}^{M - 1} (ℑ (E_{0, n}) + ℑ (C_{0, 2 n + 1})) t^{- n - 1} + O (t^{- M - 1}) . \end{matrix}$

To determine the asymptotic expansion of $H_{1} (t)$ , we expand $\sqrt{1 - r^{2} / 4}$ in a Taylor polynomial with remainder about $r = 0$ as in (2.6). We then estimate the resulting integral with the $O (r^{2 L})$ -term and find it to be $O (t^{- L - 1 / 2})$ , where $L \in N$ . Hence $\begin{matrix} (3.3) & H_{1} (t) = \sum_{k = 0}^{L - 1} β_{k} H_{4, 2 k}^{ϵ} (t) + O (t^{- L - 1 / 2}) . \end{matrix}$ Let $L : = M + 1$ and substitute the result of Lemma 3.3(a) into (3.3) with $R = R_{k} : = max {0, L - 2 k}$ and $Q = Q_{k} : = L - k$ so that both O-terms are instead $O (t^{- M - 1})$ . Thus $\begin{matrix} H_{1} (t) = - \sum_{2 k + n ⩽ M} \frac{β_{k}}{2} ℑ (E_{2 k, n}) t^{- 2 k - n - 1} + \sum_{k + n ⩽ M} \frac{β_{k}}{2} ℑ (C_{2 k, 2 n + 1}) t^{- k - n - 1} + O (t^{- M - 1}) . \end{matrix}$ We use the fact that $β_{0} = 1$ , $E_{0, 0} = - i / 2$ , and $C_{0, 1} = i / 16$ to obtain $\begin{array}{rcl} H_{1} (t) & = & \frac{9}{32} t^{- 1} - \sum_{1 ⩽ 2 k + n ⩽ M - 1} \frac{β_{k}}{2} ℑ (E_{2 k, n}) t^{- 2 k - n - 1} \\ (3.4) & + \sum_{1 ⩽ k + n ⩽ M - 1} \frac{β_{k}}{2} ℑ (C_{2 k, 2 n + 1}) t^{- k - n - 1} + O (t^{- M - 1}) . \end{array}$

We now combine results (3.1), (3.2), and (3.4) with the fact that $\frac{d}{d t} H_{1, - 1}^{ϵ} (t) = - H_{1, 1}^{ϵ} (t) + H_{1} (t) + H_{3, 0}^{ϵ} (t)$ to obtain $\begin{matrix} H_{1, - 1}^{ϵ} (t) = \frac{1}{4} log (t) + C + \sum_{j = 1}^{M - 1} S_{j} t^{- j} + O (t^{- M}), \end{matrix}$ where C is a constant. To determine the constant C we proceed as in the proof of Lemma 2.4 and compare $H_{1, - 1}^{ϵ} (t)$ to $J_{1, - 1}^{ϵ} (t)$ (see (2.9)). We note $| {sin}^{2} (t r) - sin (t r \sqrt{1 - r^{2} / 4}) sin (t r) | ⩽ t r | 1 - \sqrt{1 - r^{2} / 4} | ⩽ t r^{3} / 4$ for $0 ⩽ r ⩽ 2$ and $t > 0$ . Therefore $\begin{matrix} (3.5) & | J_{1, - 1}^{ϵ} (t) - H_{1, - 1}^{ϵ} (t) | ⩽ \frac{t}{4} \int_{0}^{ϵ} r^{2} e^{- t r^{2}} d r = \frac{1}{8} Γ (\frac{3}{2}) t^{- 1 / 2} \to 0 . \end{matrix}$ Furthermore $\begin{array}{rcl} J_{1, - 1}^{ϵ} (t) - H_{1, - 1}^{ϵ} (t) & = & \frac{1}{4} (log (4 t) + γ + O (t^{- 1})) - (\frac{1}{4} log (t) + C + O (t^{- 1})) \\ = & \frac{γ}{4} + \frac{log (2)}{2} - C + O (t^{- 1}) \to \frac{γ}{4} + \frac{log (2)}{2} - C . \end{array}$ This result together with (3.5) implies $C = γ / 4 + log (2) / 2$ , which proves the lemma. □
3.1.2. Intermediate computations

We may now set about proving the dimension $N = 2$ case of Theorem 1.2. The proof will rely on the previous results for $X_{1} (t)$ , $Y_{1} (t)$ , and $Z_{1} (t)$ , as well as the results of the following several sections. As usual, $K \in N$ and the initial data of (1.1) satisfy $u_{0} \in H^{1} (R^{2}) \cap L^{1, 2 K} (R^{2})$ and $u_{1} \in L^{2} (R^{2}) \cap L^{1, 2 K} (R^{2})$ .

Expansion of $X_{2} (t)$ . Since $\begin{matrix} X_{2} (t) = | P_{1} |^{2} \int_{| ξ | ⩽ δ} e^{- t | ξ |^{2}} \frac{{sin}^{2} (t | ξ |)}{| ξ |^{2}} d ξ, \end{matrix}$ we see that, upon switching to polar coordinates, $X_{2} (t) = 2 π | P_{1} |^{2} J_{1, - 1}^{δ} (t)$ , where $J_{1, - 1}^{δ} (t)$ is as given in (2.9). We apply Lemma 2.5 with $P = K - 1$ to obtain the asymptotic expansion $\begin{matrix} (3.6) & X_{2} (t) = \frac{π | P_{1} |^{2}}{2} (log (4 t) + γ) + \sum_{j = 1}^{K - 1}_{2} X_{2} (j) t^{- j} + O (t^{- K}), \end{matrix}$ where $\begin{matrix} _{2} X_{2} (j) = - \frac{π | P_{1} |^{2}}{2 j} {(\frac{1}{2})}_{j} . \end{matrix}$

Expansion of $X_{3} (t)$ . Into $\begin{matrix} X_{3} (t) = - 2 ℜ ({\bar{P}}_{1} \int_{| ξ | ⩽ δ} e^{- \frac{t | ξ |^{2}}{2}} ({\hat{u}}_{1} (ξ) + \frac{| ξ |^{2}}{2} {\hat{u}}_{0} (ξ)) \frac{sin (t | ξ |)}{| ξ |} h_{1} (t, ξ) d ξ) \end{matrix}$ we substitute (1.3) and (1.8). The resulting integral with the $O (| ξ |^{2 K})$ -term is $O (t^{- K})$ . We switch to polar coordinates and obtain $\begin{matrix} (3.7) & X_{3} (t) = - \sum_{| σ | ⩽ K - 1} 2 ℜ ({\bar{P}}_{1} c_{2 σ}) J_{σ} \int_{0}^{δ} r^{2 | σ | - 1} e^{- t r^{2}} \frac{sin (t r \sqrt{1 - r^{2} / 4}) sin (t r)}{\sqrt{1 - r^{2} / 4}} d r + O (t^{- K}) . \end{matrix}$ We substitute (2.37) into (3.7) with $L : = L_{σ} = K - | σ |$ . Then $\begin{matrix} (3.8) & X_{3} (t) = - \sum_{k + | σ | ⩽ K - 1} 2 ℜ ({\bar{P}}_{1} c_{2 σ}) J_{σ} α_{k} H_{1, 2 | σ | + 2 k - 1}^{δ} (t) + O (t^{- K}) . \end{matrix}$ For the term in the sum of (3.8) with $σ = 0$ , $k = 0$ , we use Lemma 3.4 with $M : = K$ , and for all other terms we use Lemma 3.2(b) with $Q = Q_{σ, k} : = K - | σ | - k$ and $R = R_{σ} - k : = max {0, K - 2 | σ | - 2 k}$ . We obtain $\begin{matrix} (3.9) & X_{3} (t) = - π | P_{1} |^{2} (log (4 t) + γ) + \sum_{j = 1}^{K - 1}_{2} X_{3} (j) t^{- j} + O (t^{- K}), \end{matrix}$ where $\begin{array}{rcl} _{2} X_{3} (j) & = & - 4 π | P_{1} |^{2} S_{j} - \sum_{\begin{array}{c} | σ | + k + n = j \\ | σ | + k ⩾ 1 \end{array}} ℜ ({\bar{P}}_{1} c_{2 σ}) J_{σ} α_{k} C_{2 | σ | + 2 k - 1, 2 n} \\ + \sum_{\begin{array}{c} 2 | σ | + 2 k + n = j \\ | σ | + k ⩾ 1 \end{array}} ℜ ({\bar{P}}_{1} c_{2 σ}) J_{σ} α_{k} E_{2 | σ | + 2 k - 1, n} . \end{array}$

Asymptotic expansion of $X (t)$ . We combine results (2.18), (3.6), and (3.9) with the fact that $X (t) = X_{1} (t) + X_{2} (t) + X_{3} (t)$ to obtain $\begin{matrix} X (t) = \sum_{j = 1}^{K - 1}_{2} V_{j}^{X} t^{- j} + O (t^{- K}), \end{matrix}$ where $\begin{matrix} _{2} V_{j}^{X} =_{2} X_{1} (j) +_{2} X_{2} (j) +_{2} X_{3} (j) . \end{matrix}$ For example, when $K ⩾ 2$ we have $\begin{matrix} _{2} V_{1}^{X} = \frac{π | P_{1} |^{2}}{64} + \frac{π}{4} \sum_{j = 1}^{2} d_{2 e_{j}} - \frac{π}{2} \sum_{j = 1}^{2} ℜ ({\bar{P}}_{1} c_{2 e_{j}}) . \end{matrix}$

Expansions of $Y_{2} (t)$ and $Y_{3} (t)$ . The analysis from Sections 3.2.4 and 3.2.5 of [1] fully carries over here with $N = 2$ . Hence $\begin{array}{l} (3.10) & Y_{2} (t) = \sum_{j = 1}^{K - 1}_{2} Y_{2} (j) t^{- j} + O (t^{- K}), \\ (3.11) & Y_{3} (t) = \sum_{j = 1}^{K - 1}_{2} Y_{3} (j) t^{- j} + O (t^{- K}), \end{array}$ where $\begin{matrix} _{2} Y_{2} (j) = \{\begin{matrix} \frac{1}{2} π | P_{0} |^{2} & j = 1 \\ π | P_{0} |^{2} A_{1, j - 2} & j ⩾ 2 \end{matrix} \end{matrix}$ and $\begin{matrix} _{2} Y_{3} (j) = - \sum_{| σ | + n = j - 1} ℜ ({\bar{P}}_{0} b_{2 σ}) J_{σ} C_{2 | σ | + 1, 2 n} - \sum_{2 | σ | + n = j - 2} ℜ ({\bar{P}}_{0} b_{2 σ}) J_{σ} E_{2 | σ | + 1, n} . \end{matrix}$

Asymptotic expansion of $Y (t)$ . We combine results (2.19), (3.10), and (3.11) with the fact that $Y (t) = Y_{1} (t) + Y_{2} (t) + Y_{3} (t)$ to obtain the expansion $\begin{matrix} Y (t) = \sum_{j = 1}^{K - 1}_{2} V_{j}^{Y} t^{- j} + O (t^{- K}), \end{matrix}$ where $\begin{matrix} _{2} V_{j}^{Y} =_{2} Y_{1} (j) +_{2} Y_{2} (j) +_{2} Y_{3} (j) . \end{matrix}$ For example, when $K ⩾ 2$ we have $V_{1}^{Y} = 0$ .

Expansions of $Z_{2} (t)$ , $Z_{3} (t)$ , and $Z_{4} (t)$ . The analysis from Sections 3.2.7, 3.2.8, and 3.2.9 of [1] also carries over with $N = 2$ . We have $\begin{array}{l} (3.12) & Z_{2} (t) = \sum_{j = 1}^{K - 1}_{2} Z_{2} (j) t^{- j} + O (t^{- K}), \\ (3.13) & Z_{3} (t) = \sum_{j = 1}^{K - 1}_{2} Z_{3} (j) t^{- j} + O (t^{- K}), \\ (3.14) & Z_{4} (t) = \sum_{j = 1}^{K - 1}_{2} Z_{4} (j) t^{- j} + O (t^{- K}), \end{array}$ where $\begin{array}{l} _{2} Z_{2} (j) = \sum_{2 | σ | + 2 k + n = j - 1} ℜ ({\bar{P}}_{0} c_{2 σ}) J_{σ} α_{k} ℑ (E_{2 | σ | + 2 k, n}) \\ + \sum_{| σ | + k + n = j - 1} ℜ ({\bar{P}}_{0} c_{2 σ}) J_{σ} α_{k} ℑ (C_{2 | σ | + 2 k, 2 n + 1}), \\ _{2} Z_{3} (j) = \sum_{2 | σ | + n = j - 1} ℜ (P_{1} {\bar{b}}_{2 σ}) J_{σ} ℑ (E_{2 | σ |, n}) \\ - \sum_{| σ | + n = j - 1} ℜ (P_{1} {\bar{b}}_{2 σ}) J_{σ} ℑ (C_{2 | σ |, 2 n + 1}), \\ _{2} Z_{4} (j) = - 2 π ℜ (P_{1} {\bar{P}}_{0}) ℑ (A_{0, j - 1}) . \end{array}$

Asymptotic expansion of $Z (t)$ . We combine the results (2.20), (3.12), (3.13), and (3.14) with the fact that $Z (t) = Z_{1} (t) + Z_{2} (t) + Z_{3} (t) + Z_{4} (t)$ to obtain the expansion $\begin{matrix} Z (t) = \sum_{j = 1}^{K - 1}_{2} V_{j}^{Z} t^{- j} + O (t^{- K}), \end{matrix}$ where $\begin{matrix} _{2} V_{j}^{Z} =_{2} Z_{1} (j) +_{2} Z_{2} (j) +_{2} Z_{3} (j) +_{2} Z_{4} (j) . \end{matrix}$ We note that $_{2} V_{1}^{Z} = 0$ when $K ⩾ 2$ .

3.1.3. Asymptotic expansion of $‖ \hat{u} (t, \cdot) - ν (t, \cdot) ‖_{2}^{2}$

We recall from Section 1.3 that for the fixed $0 < δ < 1$ there exists $η > 0$ such that $‖ \hat{u} (t, \cdot) - ν (t, \cdot) ‖_{2}^{2} = ‖ \hat{u} (t, \cdot) - ν (t, \cdot) ‖_{2, δ}^{2} + O (e^{- η t}) = X (t) + Y (t) + Z (t) + O (e^{- η t})$ . We combine the results of the analysis for the case $N = 2$ to obtain the full asymptotic expansion of $‖ \hat{u} (t, \cdot) - ν (t, \cdot) ‖_{2}^{2}$ . Further, we suppress the $O (e^{- η t})$ -term since it is dominated by the larger $O (t^{- K})$ -term.

Let $K ⩾ 1$ be an integer as given in the statement of Theorem 1.2. Then $\begin{matrix} {‖ \hat{u} (t, \cdot) - ν (t, \cdot) ‖}_{2}^{2} = \sum_{j = 1}^{K - 1}_{2} V_{j} t^{- j} + O (t^{- K}), \end{matrix}$ where $_{2} V_{j} =_{2} V_{j}^{X} +_{2} V_{j}^{Y} +_{2} V_{j}^{Z}$ . The proof of Theorem 1.2 for dimension $N = 2$ is complete.

3.1.4. Asymptotic expansion of $‖ ν (t, \cdot) ‖_{2}^{2}$

Let us again recall that for the fixed $0 < δ < 1$ , $‖ ν (t, \cdot) ‖_{2}^{2} = ‖ ν (t, \cdot) ‖_{2, δ}^{2} + O (e^{- t δ^{2} / 2}) = X_{2} (t) + Y_{2} (t) + Z_{4} (t) + O (e^{- t δ^{2} / 2})$ . We then combine results (3.6), (3.10), and (3.14) to obtain the asymptotic expansion of Corollary 1.3(a) with $\begin{matrix} _{2} U_{j} =_{2} X_{2} (j) +_{2} Y_{2} (j) +_{2} Z_{4} (j) . \end{matrix}$

3.2. The space dimension $N = 1$ case

3.2.1. Auxiliary results

Lemma 3.5.
For $0 < ϵ < \infty$ and $t > 0$ , we define $\begin{matrix} (3.15) & J_{1, - 2}^{ϵ} (t) : = \int_{0}^{ϵ} r^{- 2} e^{- t r^{2}} {sin}^{2} (t r) d r . \end{matrix}$ If $M \in N$ , then $J_{1, - 2}^{ϵ} (t) = \frac{π}{2} t - \frac{\sqrt{π}}{2} t^{1 / 2} + O (t^{- M})$ .
Proof.
Let $0 < ϵ < \infty$ and $M \in N$ . We first note that $\begin{matrix} J_{1, - 2}^{ϵ} (t) = (\int_{0}^{\infty} - \int_{ϵ}^{\infty}) r^{- 2} e^{- t r^{2}} {sin}^{2} (t r) d r . \end{matrix}$ The latter integral is $O (e^{- t ϵ^{2} / 2})$ . The former integral we denote by ${\tilde{J}}_{1, - 2} (t)$ . Integrating by parts and using known integrals [3, §1.4(11), §2.4(21)], we find $\begin{matrix} {\tilde{J}}_{1, - 2} (t) = \frac{\sqrt{π}}{2} t^{1 / 2} e^{- t} - \frac{\sqrt{π}}{2} t^{1 / 2} + \frac{π}{2} t erf (\sqrt{t}) . \end{matrix}$ This gives the desired statement. □

We omit the proof of the following lemma because it is very similar to the proof of Lemma 3.4. Lemma 3.6.
For $0 < ϵ < 2$ and $t > 0$ , define $\begin{array}{l} H_{3, - 1}^{ϵ} (t) : = \int_{0}^{ϵ} r^{- 1} e^{- t r^{2}} sin (t r \sqrt{1 - r^{2} / 4}) cos (t r) d r, \\ H_{4, - 1}^{ϵ} (t) : = \int_{0}^{ϵ} r^{- 1} e^{- t r^{2}} cos (t r \sqrt{1 - r^{2} / 4}) sin (t r) d r . \end{array}$ If $M \in N$ and $p = 3, 4$ then $\begin{array}{rcl} H_{p, - 1}^{ϵ} (t) & = & \frac{π}{4} + {(- 1)}^{p} \sum_{n = 0}^{M - 2} \frac{1}{2 n + 1} ℑ (C_{1, 2 n + 1}) t^{- n - 1 / 2} \\ + {(- 1)}^{p} \sum_{\begin{array}{c} k + n ⩽ M - 1 \\ k ⩾ 1 \end{array}} \frac{β_{k}}{2 k + 2 n - 1} C_{2 k, 2 n} t^{- k - n + 1 / 2} + O (t^{- M + 1 / 2}) . \end{array}$
Lemma 3.7.
For $0 < ϵ < 2$ and $t > 0$ , define $\begin{matrix} H_{1, - 2}^{ϵ} (t) : = \int_{0}^{ϵ} r^{- 2} e^{- t r^{2}} sin (t r \sqrt{1 - r^{2} / 4}) sin (t r) d r . \end{matrix}$ If $M \in N$ then $\begin{array}{rcl} H_{1, - 2}^{ϵ} (t) & = & \frac{π}{2} t - \frac{\sqrt{π}}{2} t^{1 / 2} - \sum_{n = 1}^{M - 1} C_{0, 2 n} t^{- n + 1 / 2} \\ + \sum_{\begin{array}{c} k + n ⩽ M - 1 \\ k ⩾ 1 \end{array}} \frac{(2 k + 1) β_{k}}{2} ℑ (C_{2 k - 1, 2 n + 1}) t^{- k - n + 1 / 2} + O (t^{- M + 1 / 2}) . \end{array}$
Proof.
Let $0 < ϵ < 2$ and $M \in N$ . Using integration by parts, $H_{1, - 2}^{ϵ} (t) = H_{1} (t) - 2 t H_{1, 0}^{ϵ} (t) + H_{2} (t) + t H_{3, - 1}^{ϵ} (t)$ , where $H_{1, 0}^{ϵ} (t)$ is as given in Lemma 3.2, $H_{3, - 1}^{ϵ} (t)$ is as given in Lemma 3.6, and $\begin{array}{l} H_{1} (t) = - ϵ^{- 1} e^{- t ϵ^{2}} sin (t ϵ \sqrt{1 - ϵ^{2} / 4}) sin (t ϵ) = O (t^{- M + 1 / 2}), \\ H_{2} (t) = t \int_{0}^{ϵ} r^{- 1} \frac{d}{d r} [r \sqrt{1 - r^{2} / 4}] e^{- t r^{2}} cos (t r \sqrt{1 - r^{2} / 4}) sin (t r) d r . \end{array}$ Using (2.30), we find $\begin{matrix} H_{2} (t) = t H_{4, - 1}^{ϵ} (t) + t \sum_{k = 1}^{M} (2 k + 1) β_{k} H_{4, 2 k - 1}^{ϵ} (t) + O (t^{- M + 1 / 2}) . \end{matrix}$ Therefore, $\begin{matrix} (3.16) & \begin{matrix} H_{1, - 2}^{ϵ} (t) = & t (H_{3, - 1}^{ϵ} (t) + H_{4, - 1}^{ϵ} (t)) - 2 t H_{1, 0}^{ϵ} (t) \\ + t \sum_{k = 1}^{M} (2 k + 1) β_{k} H_{4, 2 k - 1}^{ϵ} (t) + O (t^{- M + 1 / 2}) . \end{matrix} \end{matrix}$ Lemma 3.6 gives $\begin{matrix} (3.17) & t (H_{3, - 1}^{ϵ} (t) + H_{4, - 1}^{ϵ} (t)) = \frac{π}{2} t + O (t^{- M + 1 / 2}) . \end{matrix}$ From Lemma 3.2(a) and $C_{0, 0} = \frac{1}{2} \sqrt{π}$ , we have $\begin{matrix} (3.18) & - 2 t H_{1, 0}^{ϵ} (t) = - \frac{\sqrt{π}}{2} t^{1 / 2} - \sum_{n = 1}^{M - 1} C_{0, 2 n} t^{- n + 1 / 2} + O (t^{- M + 1 / 2}) . \end{matrix}$ Applying Lemma 3.3 with $Q = Q_{k} : = M - k + 1$ , we obtain $\begin{matrix} (3.19) & \begin{matrix} t \sum_{k = 1}^{M} (2 k + 1) β_{k} H_{4, 2 k - 1}^{ϵ} (t) \\ = \sum_{\begin{array}{c} k + n ⩽ M - 1 \\ k ⩾ 1 \end{array}} \frac{(2 k + 1) β_{k}}{2} ℑ (C_{2 k - 1, 2 n + 1}) t^{- k - n + 1 / 2} + O (t^{- M + 1 / 2}) . \end{matrix} \end{matrix}$ Using (3.18), (3.17), and (3.19) in (3.16) completes the proof. □

3.2.2. Intermediate computations

We are now in a position to prove the dimension $N = 1$ case of Theorem 1.2. We assume that $K \in N$ and the initial data of (1.1) satisfy $u_{0} \in H^{1} (R) \cap L^{1, 2 K} (R)$ and $u_{1} \in L^{2} (R) \cap L^{1, 2 K} (R)$ .

Expansion of $X_{2} (t)$ . Observe that $\begin{matrix} X_{2} (t) = 2 | P_{1} |^{2} \int_{0}^{δ} r^{- 2} e^{- t r^{2}} {sin}^{2} (t r) d r = 2 | P_{1} |^{2} J_{1, - 2}^{δ} (t) \end{matrix}$ with $J_{1, - 2}^{δ} (t)$ as given in (3.15). By Lemma 3.5 with $M : = K$ , $\begin{matrix} (3.20) & X_{2} (t) = π | P_{1} |^{2} t - \sqrt{π} | P_{1} |^{2} t^{1 / 2} + O (t^{- K + 1 / 2}) . \end{matrix}$

Expansion of $X_{3} (t)$ . Using a similar argument for the expansion of $X_{3} (t)$ in the $N = 2$ case we find that $\begin{matrix} (3.21) & X_{3} (t) = - \sum_{σ = 0}^{K - 1} 4 ℜ ({\bar{P}}_{1} c_{2 σ}) \int_{0}^{δ} r^{2 σ - 2} e^{- t r^{2}} \frac{sin (t r \sqrt{1 - r^{2} / 4}) sin (t r)}{\sqrt{1 - r^{2} / 4}} d r . \end{matrix}$ We substitute (2.37) into (3.21) with $L = L_{σ} : = K - σ$ . Hence $\begin{matrix} (3.22) & X_{3} (t) = - \sum_{σ + k ⩽ K - 1} 4 ℜ ({\bar{P}}_{1} c_{2 σ}) α_{k} H_{1, 2 σ + 2 k - 2}^{δ} (t) + O (t^{- K + 1 / 2}) . \end{matrix}$ To the term in (3.22) with $σ = k = 0$ we apply Lemma 3.7 with $M : = K$ , and to all other terms Lemma 3.2(a) with $Q = Q_{σ, k} : = K - σ - k$ . Then we obtain the asymptotic expansion $\begin{matrix} (3.23) & X_{3} (t) = - 2 π | P_{1} |^{2} t + 2 \sqrt{π} | P_{1} |^{2} t^{1 / 2} + \sum_{j = 1}^{K - 1}_{1} X_{3} (j) t^{- j + 1 / 2} + O (t^{- K + 1 / 2}), \end{matrix}$ where $\begin{array}{rcl} _{1} X_{3} (j) & = & 4 | P_{1} |^{2} C_{0, 2 j} - 2 | P_{1} |^{2} \sum_{\begin{array}{c} k + n = j \\ k ⩾ 1 \end{array}} (2 k + 1) β_{k} ℑ (C_{2 k - 1, 2 n + 1}) \\ - \sum_{\begin{array}{c} σ + k + n = j \\ σ + k ⩾ 1 \end{array}} 2 ℜ ({\bar{P}}_{1} c_{2 σ}) α_{k} C_{2 σ + 2 k - 2, 2 n} . \end{array}$

Asymptotic expansion of $X (t)$ . We combine results (2.34), (3.20), and (3.23) for $X_{1} (t)$ , $X_{2} (t)$ , and $X_{3} (t)$ , respectively, with the fact that $X (t) = X_{1} (t) + X_{2} (t) + X_{3} (t)$ to obtain the asymptotic expansion $\begin{matrix} X (t) = \sum_{j = 1}^{K - 1}_{1} V_{j}^{X} t^{- j + 1 / 2} + O (t^{- K + 1 / 2}), \end{matrix}$ where $\begin{matrix} _{1} V_{j}^{X} =_{1} X_{1} (j) +_{1} X_{3} (j) . \end{matrix}$ For example, if $K ⩾ 2$ then $\begin{matrix} _{1} V_{1}^{X} = \frac{3 \sqrt{π} | P_{1} |^{2}}{512} + \frac{\sqrt{π} d_{2}}{2} - \sqrt{π} ℜ ({\bar{P}}_{1} c_{2}) . \end{matrix}$

Expansions of $Y_{2} (t)$ and $Y_{3} (t)$ . The analysis of $Y_{2} (t)$ and $Y_{3} (t)$ from Sections 3.2.4 and 3.2.5 of [1] applies to the $N = 1$ case as well. Hence $\begin{array}{l} (3.24) & Y_{2} (t) = \sum_{j = 1}^{K - 1}_{1} Y_{2} (j) t^{- j + 1 / 2} + O (t^{- K + 1 / 2}), \\ (3.25) & Y_{3} (t) = \sum_{j = 1}^{K - 1}_{1} Y_{3} (j) t^{- j + 1 / 2} + O (t^{- K + 1 / 2}), \end{array}$ where $\begin{array}{l} _{1} Y_{2} (j) = \{\begin{matrix} \frac{\sqrt{π} | P_{0} |^{2}}{2} & j = 1 \\ 0 & otherwise, \end{matrix} \\ _{1} Y_{3} (j) = - \sum_{σ + n = j - 1} 2 ℜ ({\bar{P}}_{0} b_{2 σ}) C_{2 σ, 2 n} . \end{array}$

Asymptotic expansion of $Y (t)$ . Since $Y (t) = Y_{1} (t) + Y_{2} (t) + Y_{3} (t)$ we combine results (2.35), (3.24), and (3.25) to obtain the asymptotic expansion $\begin{matrix} Y (t) = \sum_{j = 1}^{K - 1}_{1} V_{j}^{Y} t^{- j + 1 / 2} + O (t^{- K + 1 / 2}), \end{matrix}$ where $\begin{matrix} _{1} V_{j}^{Y} =_{1} Y_{1} (j) +_{1} Y_{2} (j) +_{1} Y_{3} (j) . \end{matrix}$ We note that $_{1} V_{1}^{Y} = 0$ for $K ⩾ 2$ .

Expansion of $Z_{2} (t)$ . $\begin{matrix} Z_{2} (t) = - 2 ℜ ({\bar{P}}_{0} \int_{- δ}^{δ} ({\hat{u}}_{1} (r) + \frac{r^{2}}{2} {\hat{u}}_{0} (r)) e^{- \frac{t r^{2}}{2}} h_{1} (t, r) cos (t r) d r), \end{matrix}$ into which we substitute (1.3) and (1.8). With our overlying assumption that $K \in N$ , the resulting integral with the $O (r^{2 K})$ -term can be shown to be $O (t^{- K + 1 / 2})$ . Hence $\begin{matrix} (3.26) & Z_{2} (t) = - \sum_{σ = 0}^{K - 1} 4 ℜ ({\bar{P}}_{0} c_{2 σ}) \int_{0}^{δ} r^{2 σ - 1} e^{- t r^{2}} \frac{sin (t r \sqrt{1 - r^{2} / 4}) cos (t r)}{\sqrt{1 - r^{2} / 4}} d r . \end{matrix}$ We substitute (2.37) into (3.26) with $L = L_{σ} : = K - σ$ . Hence $\begin{matrix} (3.27) & Z_{2} (t) = - \sum_{σ + k ⩽ K - 1} 4 ℜ ({\bar{P}}_{0} c_{2 σ}) α_{k} H_{3, 2 σ + 2 k - 1}^{δ} (t) + O (t^{- K + 1 / 2}) . \end{matrix}$ We treat the term $σ = k = 0$ in the sum of (3.27) using Lemma 3.6 with $M : = K$ , and all other terms by Lemma 3.3(b) with $Q = Q_{σ, k} : = K - σ - k$ . Hence $\begin{matrix} (3.28) & Z_{2} (t) = - π ℜ (P_{1} {\bar{P}}_{0}) + \sum_{j = 1}^{K - 1}_{1} Z_{2} (j) t^{- j + 1 / 2} + O (t^{- K + 1 / 2}), \end{matrix}$ where $\begin{array}{rcl} _{1} Z_{2} (j) & = & \frac{4 ℜ (P_{1} {\bar{P}}_{0})}{2 j - 1} ℑ (C_{1, 2 j - 1}) + \sum_{\begin{array}{c} k + n = j \\ k ⩾ 1 \end{array}} \frac{4 ℜ (P_{1} {\bar{P}}_{0}) β_{k}}{2 j - 1} C_{2 k, 2 n} \\ + \sum_{\begin{array}{c} σ + k + n = j - 1 \\ σ + k ⩾ 1 \end{array}} 2 ℜ ({\bar{P}}_{0} c_{2 σ}) α_{k} ℑ (C_{2 σ + 2 k - 1, 2 n + 1}) . \end{array}$

Expansion of $Z_{3} (t)$ . We analyze $Z_{3} (t)$ similarly to $Z_{2} (t)$ and we find $\begin{matrix} (3.29) & Z_{3} (t) = - \sum_{σ = 0}^{K - 1} 4 ℜ (P_{1} {\bar{b}}_{2 σ}) H_{4, 2 σ - 1}^{δ} (t) + O (t^{- K + 1 / 2}) . \end{matrix}$ We use Lemma 3.6 and Lemma 3.3(b) and obtain $\begin{matrix} (3.30) & Z_{3} (t) = - π ℜ (P_{1} {\bar{P}}_{0}) + \sum_{j = 1}^{K - 1}_{1} Z_{3} t^{- j + 1 / 2} + O (t^{- K + 1 / 2}), \end{matrix}$ where $\begin{array}{rcl} _{1} Z_{3} (j) & = & - \frac{4 ℜ (P_{1} {\bar{P}}_{0})}{2 j - 1} ℑ (C_{1, 2 j - 1}) - \sum_{\begin{array}{c} k + n = j \\ k ⩾ 1 \end{array}} \frac{4 ℜ (P_{1} {\bar{P}}_{0}) β_{k}}{2 j - 1} C_{2 k, 2 n} \\ - \sum_{\begin{array}{c} σ + n = j - 1 \\ σ ⩾ 1 \end{array}} 2 ℜ (P_{1} {\bar{b}}_{2 σ}) ℑ (C_{2 σ - 1, 2 n + 1}) . \end{array}$

Expansion of $Z_{4} (t)$ . From the definition of $Z_{4} (t)$ , we see that $\begin{matrix} Z_{4} (t) = 2 ℜ (P_{1} {\bar{P}}_{0}) \tilde{I} (t) + O (t^{- 1 / 2} e^{- t δ^{2} / 2}), \end{matrix}$ where $\tilde{I} (t)$ is as given in Lemma 2.6. Therefore $\begin{matrix} (3.31) & Z_{4} (t) = π ℜ (P_{1} {\bar{P}}_{0}) + O (t^{- K + 1 / 2}) . \end{matrix}$

Asymptotic expansion of $Z (t)$ . We combine results (2.39), (3.28), (3.30), and (3.31) for $Z_{1} (t)$ , $Z_{2} (t)$ , $Z_{3} (t)$ , and $Z_{4} (t)$ , respectively, with the fact that $Z (t) = Z_{1} (t) + Z_{2} (t) + Z_{3} (t) + Z_{4} (t)$ to obtain the asymptotic expansion $\begin{matrix} Z (t) = \sum_{j = 0}^{K - 1}_{1} V_{j}^{Z} t^{- j + 1 / 2} + O (t^{- K + 1 / 2}), \end{matrix}$ where $\begin{matrix} _{1} V_{j}^{Z} =_{1} Z_{2} (j) +_{1} Z_{3} (j) . \end{matrix}$ We note that $_{1} V_{1}^{Z} = 0$ if $K ⩾ 2$ .

3.2.3. Asymptotic expansion of $‖ \hat{u} (t, \cdot) - ν (t, \cdot) ‖_{2}^{2}$

Again from Section 1.3, for the fixed $0 < δ < 1$ there exists $η > 0$ such that $‖ \hat{u} (t, \cdot) - ν (t, \cdot) ‖_{2}^{2} = ‖ \hat{u} (t, \cdot) - ν (t, \cdot) ‖_{2, δ}^{2} + O (e^{- η t}) = X (t) + Y (t) + Z (t) + O (e^{- η t})$ . We now combine the results we have for dimension $N = 1$ to obtain the full asymptotic expansion of $‖ \hat{u} (t, \cdot) - ν (t, \cdot) ‖_{2}^{2}$ . The $O (e^{- η t})$ -term will hereafter be suppressed since it is dominated by the larger $O (t^{- K + 1 / 2})$ -term.

Let $K ⩾ 1$ be an integer as given in the statement of Theorem 1.2. Then $\begin{matrix} {‖ \hat{u} (t, \cdot) - ν (t, \cdot) ‖}_{2}^{2} = \sum_{j = 1}^{K - 1}_{1} V_{j} t^{- j + 1 / 2} + O (t^{- K + 1 / 2}), \end{matrix}$ where $_{1} V_{j} =_{1} V_{j}^{X} +_{1} V_{j}^{Y} +_{1} V_{j}^{Z}$ . Since cases $N = 1, 2$ are complete, here ends the proof of Theorem 1.2.

3.2.4. Asymptotic expansion of $‖ ν (t, \cdot) ‖_{2}^{2}$

Let us again recall that for the fixed $0 < δ < 1$ , $‖ ν (t, \cdot) ‖_{2}^{2} = ‖ ν (t, \cdot) ‖_{2, δ}^{2} + O (e^{- t δ^{2} / 2}) = X_{2} (t) + Y_{2} (t) + Z_{4} (t) + O (e^{- t δ^{2} / 2})$ . We then combine results (3.20), (3.24), and (3.31) to obtain the asymptotic expansion of Corollary 1.3(b) with $_{1} U_{j} =_{1} Y_{2} (j)$ .

This completes the proof of Corollary 1.3. Furthermore, we have completed the proofs of all three main results.

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Asymptotic expansion of the L 2 -norm of a solution of the strongly damped wave equation in space dimension 1 and 2

Abstract

Keywords

1. Introduction

1.3. Main results

2.1.1. Auxiliary results

2.1.3. Asymptotic expansion of ‖ u ( t , · ) ‖ 2 2

2.2. The space dimension N = 1 case

2.2.1. Auxiliary results

2.2.3. Asymptotic expansion of ‖ u ( t , · ) ‖ 2 2

3. Proof of Theorem 1.2

3.1.1. Auxiliary result

3.1.3. Asymptotic expansion of ‖ u ˆ ( t , · ) − ν ( t , · ) ‖ 2 2

3.1.4. Asymptotic expansion of ‖ ν ( t , · ) ‖ 2 2

3.2. The space dimension N = 1 case

3.2.1. Auxiliary results

3.2.3. Asymptotic expansion of ‖ u ˆ ( t , · ) − ν ( t , · ) ‖ 2 2

3.2.4. Asymptotic expansion of ‖ ν ( t , · ) ‖ 2 2

References

2.1.3. Asymptotic expansion of $‖ u (t, \cdot) ‖_{2}^{2}$

2.2. The space dimension $N = 1$ case

2.2.3. Asymptotic expansion of $‖ u (t, \cdot) ‖_{2}^{2}$

3.1.3. Asymptotic expansion of $‖ \hat{u} (t, \cdot) - ν (t, \cdot) ‖_{2}^{2}$

3.1.4. Asymptotic expansion of $‖ ν (t, \cdot) ‖_{2}^{2}$

3.2. The space dimension $N = 1$ case

3.2.3. Asymptotic expansion of $‖ \hat{u} (t, \cdot) - ν (t, \cdot) ‖_{2}^{2}$

3.2.4. Asymptotic expansion of $‖ ν (t, \cdot) ‖_{2}^{2}$