Asymptotic profile of solutions for wave equations with frictional and viscoelastic damping terms

Abstract

We consider the Cauchy problem in $R^{n}$ for some types of damped wave equations. We derive asymptotic profiles of solutions with weighted $L^{1, 1} (R^{n})$ initial data by employing a simple method introduced in [Math. Meth. Appl. Sci. 27 (2004), 865–889].

Keywords

wave equation frictional term viscoelastic term Gauss kernel asymptotic profiles Fourier analysis low frequency high frequency

1. Introduction

We are concerned with the Cauchy problem for wave equations in $R^{n}$ ( $n ⩾ 1$ ) with frictional and viscoelastic terms $\begin{array}{l} (1.1) & u_{t t} (t, x) - Δ u (t, x) + u_{t} (t, x) - Δ u_{t} (t, x) = 0, (t, x) \in (0, \infty) \times R^{n}, \\ (1.2) & u (0, x) = u_{0} (x), u_{t} (0, x) = u_{1} (x), x \in R^{n} . \end{array}$ The initial data $u_{0}$ and $u_{1}$ are also chosen from the weighted $L^{1}$ -space $\begin{matrix} [u_{0}, u_{1}] \in (H^{1} (R^{n}) \cap L^{1, 1} (R^{n})) \times (L^{2} (R^{n}) \cap L^{1, 1} (R^{n})), \end{matrix}$ where $\begin{matrix} f \in L^{1, k} (R^{n}) \Leftrightarrow f \in L^{1} (R^{n}), ‖ f ‖_{1, k} : = \int_{R^{n}} {(1 + | x |)}^{k} | f (x) | d x < + \infty, k \in N \cup {0} . \end{matrix}$ Then we can see that the problem (1.1)–(1.2) admits a unique weak solution in the class (cf. [16, Proposition 2.1]) $\begin{matrix} u \in C ([0, + \infty); H^{1} (R^{n})) \cap C^{1} ([0, + \infty); L^{2} (R^{n})) . \end{matrix}$ The purpose of this research is to find an asymptotic profile as $t \to + \infty$ of the solution $u (t, x)$ to problem (1.1)–(1.2).

As for the Cauchy problem of the usual damped wave equation: $\begin{matrix} (1.3) & u_{t t} (t, x) - Δ u (t, x) + u_{t} (t, x) = 0, (t, x) \in (0, \infty) \times R^{n}, \end{matrix}$ it is well-known due to Chill–Haraux [2], D’Abbicco–Ebert [4], Han–Milani [6], Hayashi–Kaikina–Naumkin [7], Hosono [8], Hosono–Ogawa [9], Ikehata–Nishihara [15], Kawakami–Ueda [18], Marcati–Nishihara [20], Narazaki [22], Nishihara [23], Radu–Todorova–Yordanov [28], Said-Houari [29], Takeda [31] and Wakasugi [32] that (roughly speaking) $\begin{matrix} u (t, x) \sim v (t, x) (t \to + \infty), \end{matrix}$ where $v (t, x)$ is the corresponding solution to the heat equation: $\begin{array}{l} - Δ v (t, x) + v_{t} (t, x) = 0, (t, x) \in (0, \infty) \times R^{n}, \\ v (0, x) = u_{0} (x) + u_{1} (x), x \in R^{n}, \end{array}$ and in fact, we can have a more deep result due to Karch [17] that the asymptotic profile is a constant multiple of the Gauss kernel. Furthermore, Nishihara [24] investigated the profile like $\begin{matrix} u (t, x) \sim v (t, x) + e^{- t / 2} w (t, x), \end{matrix}$ where $w (t, x)$ is the corresponding solution to the free wave equation $\begin{array}{l} w_{t t} (t, x) - Δ w (t, x) = 0, (t, x) \in (0, \infty) \times R^{n}, \\ w (0, x) = u_{0} (x), w_{t} (0, x) = u_{1} (x), x \in R^{n} . \end{array}$ On the other hand, concerning the Cauchy problem for the viscoelastic equation $\begin{matrix} (1.4) & u_{t t} (t, x) - Δ u (t, x) - Δ u_{t} (t, x) = 0, (t, x) \in (0, \infty) \times R^{n}, \end{matrix}$ recently Ikehata–Todorova–Yordanov [16] in an abstract framework, and Ikehata [12] in a concrete setting have derived its asymptotic profile like $\begin{matrix} u (t, x) \sim e^{- t A / 2} (cos (t A^{1 / 2}) u_{0} + A^{- 1 / 2} sin (t A^{1 / 2}) u_{1}) (t \to + \infty), \end{matrix}$ where $A = - Δ$ in $L^{2} (R^{n})$ . This implies the oscillation property of the solution to (1.4).

So, a natural question arises such as:

with regard to the asymptotic behavior of solutions to problem (1.1)–(1.2) having two types of damping terms, which is dominant (as $t \to + \infty$ ), $u_{t}$ or $- Δ u_{t}$ ?

Our answer is summarized as follows in terms of the weighted $L^{1, 1}$ -initial data. To begin with, we set the Gauss kernel $\begin{matrix} G (t, x) : = \frac{1}{{(\sqrt{4 π t})}^{n}} e^{- \frac{| x |^{2}}{4 t}}, \end{matrix}$ and the $L^{2}$ -norm $\begin{matrix} ‖ f ‖ : = {(\int_{R^{n}} {| f (x) |}^{2} d x)}^{1 / 2} . \end{matrix}$

Theorem 1.1.
Let $n ⩾ 1$ . If $[u_{0}, u_{1}] \in (H^{1} (R^{n}) \cap L^{1, 1} (R^{n})) \times (L^{2} (R^{n}) \cap L^{1, 1} (R^{n}))$ , then the solution $u (t, x)$ to problem ( 1.1 )–( 1.2 ) satisfies $\begin{matrix} ‖ u (t, \cdot) - (P_{00} + P_{01}) G (t, \cdot) ‖ ⩽ C t^{- \frac{n}{4} - \frac{1}{2}} (‖ u_{0} ‖_{1, 1} + ‖ u_{1} ‖_{1, 1} + ‖ u_{0} ‖ + ‖ \nabla u_{0} ‖ + ‖ u_{1} ‖) \end{matrix}$ for large $t ≫ 1$ , where $\begin{matrix} P_{00} : = \int_{R^{n}} u_{0} (x) d x, P_{01} : = \int_{R^{n}} u_{1} (x) d x . \end{matrix}$
Remark 1.1.
It should be noticed that as for the Gauss kernel, one can easily check that $‖ G (t, \cdot) ‖ \sim t^{- \frac{n}{4}}$ ( $t \to + \infty$ ), so that the result of Theorem 1.1 provides exactly leading terms in asymptotic sense.
Remark 1.2.
As a result, the effect of the frictional damping is more dominant for the asymptotic profile as $t \to \infty$ than that of the viscoelastic one, so that the oscillation property vanishes as time goes to infinity. People sometimes say that the term $- Δ u_{t}$ is strong damping, however, we can see from the result above that the effect of the strong damping is not too strong for large $t ≫ 1$ as compared with the frictional damping.

Finally, we have to mention related previous results based on the Fourier (or generalized Fourier) Analysis about the “decay” property of the solution in some norms. Concerning the Cauchy or mixed problem for (1.3) and/or (1.4), we can cite many results written by Charão–da Luz–Ikehata [1], D’Abbicco–Reissig [5], Ikehata [10,11], Ikehata–Natsume [14], da Luz–Ikehata–Charão [3], Lu–Reissig [19], Matsumura [21], Ponce [26], Racke [27], Shibata [30], Wirth [33] and the references therein.

Open question: what is the dominant profile of the solution to the equation $\begin{matrix} u_{t t} (t, x) + A u (t, x) + A^{ρ} u_{t} (t, x) + A^{θ} u_{t} (t, x) = 0, (t, x) \in (0, \infty) \times R^{n}, \end{matrix}$ where $ρ, θ \in [0, 1]$ with $ρ < θ$ , and $A : D (A) (\subset H) \to H$ is a nonnegative self-adjoint operator in a real Hilbert space.

Our strategy to prove Theorem 1.1 is as follows: we first get a special explicit representation of the Fourier transformed solution (which is introduced by Chill–Haraux [2]), and then we make a decomposition of initial data in the obtained expression of the Fourier transformed solution to proceed the low frequency estimate. The latter decomposition of initial data has its origin in [11]. In Section 2 we will prove Theorem 1.1 by dividing the proof into the low and high frequency parts of the Fourier transformed solution. In Section 3, in order to observe the possibility of the asymptotic effect of the oscillation property coming from viscoelastic damping term we will study further asymptotic expansions by imposing moment conditions on the initial data. Notation.
Throughout this paper, $‖ \cdot ‖_{q}$ stands for the usual $L^{q} (R^{n})$ -norm. For simplicity of notations, in particular, we use $‖ \cdot ‖$ instead of $‖ \cdot ‖_{2}$ . Furthermore, we denote the Fourier transform $\hat{ϕ} (ξ)$ of the function $ϕ (x)$ by $\begin{matrix} (1.5) & F (ϕ) (ξ) : = \hat{ϕ} (ξ) : = \frac{1}{{(2 π)}^{n / 2}} \int_{R^{n}} e^{- i x \cdot ξ} ϕ (x) d x, \end{matrix}$ where $i : = \sqrt{- 1}$ , and $x \cdot ξ = \sum_{i = 1}^{n} x_{i} ξ_{i}$ for $x = (x_{1}, \dots, x_{n})$ and $ξ = (ξ_{1}, \dots, ξ_{n})$ , and the inverse Fourier transform of $F$ is denoted by $F^{- 1}$ . When we estimate several functions by applying the Fourier transform sometimes we can also use the following definition in place of (1.5) $\begin{matrix} F (ϕ) (ξ) : = \int_{R^{n}} e^{- i x \cdot ξ} ϕ (x) d x \end{matrix}$ without loss of generality. We also use the notation $\begin{matrix} v_{t} = \frac{\partial v}{\partial t}, v_{t t} = \frac{\partial^{2} v}{\partial t^{2}}, Δ = \sum_{i = 1}^{n} \frac{\partial^{2}}{\partial x_{i}^{2}}, x = (x_{1}, \dots, x_{n}) . \end{matrix}$

2. Proof of Theorem 1.1

Let us prove Theorem 1.1 based on an idea due to [11, Lemma 3.1]. The essential part of the proof corresponds to the low frequency estimate of the solution.

Lemma 2.1.
There are two constants $γ > 0$ and $C > 0$ such that for all $t > 0$ one has $\begin{array}{l} \int_{| ξ | ⩽ γ} {| \hat{u} (t, ξ) - (P_{00} + P_{01}) e^{- t | ξ |^{2}} |}^{2} d ξ \\ ⩽ C t^{- \frac{n}{2} - 1} (‖ u_{0} ‖_{1, 1}^{2} + ‖ u_{1} ‖_{1, 1}^{2}) + C t^{- \frac{n}{2} - 2} (‖ u_{0} ‖_{1}^{2} + ‖ u_{1} ‖_{1}^{2}) + C e^{- t} (‖ u_{0} ‖^{2} + ‖ u_{1} ‖^{2}) . \end{array}$
Proof.
We apply the Fourier transform of both sides of (1.1)–(1.2), then in the Fourier space $R_{ξ}^{n}$ one has the reduced problem $\begin{array}{l} (2.1) & {\hat{u}}_{t t} (t, ξ) + | ξ |^{2} \hat{u} (t, ξ) + (1 + | ξ |^{2}) {\hat{u}}_{t} (t, ξ) = 0, (t, ξ) \in (0, \infty) \times R_{ξ}^{n}, \\ (2.2) & \hat{u} (0, ξ) = {\hat{u}}_{0} (ξ), {\hat{u}}_{t} (0, ξ) = {\hat{u}}_{1} (ξ), x \in R_{ξ}^{n} . \end{array}$ Let us solve (2.1)–(2.2) directly under the condition that $| ξ | ≪ 1$ . In this case we get $\begin{matrix} (2.3) & \hat{u} (t, ξ) = \frac{{\hat{u}}_{1} (ξ) - σ_{2} {\hat{u}}_{0} (ξ)}{σ_{1} - σ_{2}} e^{σ_{1} t} + \frac{{\hat{u}}_{0} (ξ) σ_{1} - {\hat{u}}_{1} (ξ)}{σ_{1} - σ_{2}} e^{σ_{2} t}, \end{matrix}$ where $σ_{j} \in (- \infty, 0]$ ( $j = 1, 2$ ) have forms $\begin{matrix} σ_{1} = \frac{- (1 + | ξ |^{2}) + \sqrt{{(1 + | ξ |^{2})}^{2} - 4 | ξ |^{2}}}{2}, σ_{2} = \frac{- (1 + | ξ |^{2}) - \sqrt{{(1 + | ξ |^{2})}^{2} - 4 | ξ |^{2}}}{2} . \end{matrix}$ The smallness of $| ξ | ≪ 1$ guarantees $σ_{j} \in R$ . Here, we notice that $\begin{array}{l} (2.4) & σ_{1}^{2} = - σ_{1} - | ξ |^{2} - σ_{1} | ξ |^{2}, \\ (2.5) & σ_{2}^{2} = - σ_{2} - | ξ |^{2} - σ_{2} | ξ |^{2} . \end{array}$ By rewriting (2.3) by using (2.4) and (2.5) one has $\begin{matrix} (2.6) & \hat{u} (t, ξ) = e^{- t | ξ |^{2}} {K_{1} (t, ξ) + K_{2} (t, ξ)}, \end{matrix}$ where $\begin{array}{l} K_{1} (t, ξ) = \frac{σ_{2} {\hat{u}}_{0} (ξ) - {\hat{u}}_{1} (ξ)}{σ_{2} - σ_{1}} e^{- σ_{1}^{2} t - σ_{1} | ξ |^{2} t}, \\ K_{2} (t, ξ) = \frac{{\hat{u}}_{0} (ξ) σ_{1} - {\hat{u}}_{1} (ξ)}{σ_{1} - σ_{2}} e^{- σ_{2}^{2} t - σ_{2} | ξ |^{2} t} . \end{array}$ It is important to know that $K_{1} (t, ξ)$ can be decomposed into the following style. This decomposition comes from an idea due to Chill–Haraux [2]. $\begin{array}{rcl} K_{1} (t, ξ) & = & {\hat{u}}_{0} (ξ) + {\hat{u}}_{1} (ξ) + \frac{σ_{1} {\hat{u}}_{0} (ξ)}{σ_{2} - σ_{1}} \\ + \frac{σ_{2} {\hat{u}}_{0} (ξ) (1 - e^{- σ_{1}^{2} t - σ_{1} | ξ |^{2} t})}{σ_{1} - σ_{2}} + \frac{{\hat{u}}_{1} (ξ) (e^{- σ_{1}^{2} t - σ_{1} | ξ |^{2} t} - (σ_{1} - σ_{2}))}{σ_{1} - σ_{2}} . \end{array}$ So one has arrived at the important equality $\begin{array}{rcl} \hat{u} (t, ξ) & = & e^{- t | ξ |^{2}} {{\hat{u}}_{0} (ξ) + {\hat{u}}_{1} (ξ) + \frac{σ_{1} {\hat{u}}_{0} (ξ)}{σ_{2} - σ_{1}} \\ + \frac{σ_{2} {\hat{u}}_{0} (ξ) (1 - e^{- σ_{1}^{2} t - σ_{1} | ξ |^{2} t})}{σ_{1} - σ_{2}} + \frac{{\hat{u}}_{1} (ξ) (e^{- σ_{1}^{2} t - σ_{1} | ξ |^{2} t} - (σ_{1} - σ_{2}))}{σ_{1} - σ_{2}}} \\ (2.7) & + e^{- t | ξ |^{2}} K_{2} (t, ξ) . \end{array}$ Now let us use the idea coming from [11, Lemma 3.1]. Then we have a decomposition $\begin{matrix} (2.8) & {\hat{u}}_{j} (ξ) = A_{j} (ξ) - i B_{j} (ξ) + P_{0 j} (j = 0, 1), \end{matrix}$ where $\begin{matrix} A_{j} (ξ) : = \int_{R^{n}} (cos (x \cdot ξ) - 1) u_{j} (x) d x, B_{j} (ξ) : = \int_{R^{n}} sin (x \cdot ξ) u_{j} (x) d x (j = 0, 1) . \end{matrix}$ Because of (2.7) and (2.8) we get the useful identity for small $| ξ | ≪ 1$ $\begin{array}{l} \hat{u} (t, ξ) - (P_{00} + P_{01}) e^{- t | ξ |^{2}} \\ (2.9) & = (A_{0} (ξ) - i B_{0} (ξ) + A_{1} (ξ) - i B_{1} (ξ)) e^{- t | ξ |^{2}} \\ + e^{- t | ξ |^{2}} {\frac{σ_{1} {\hat{u}}_{0} (ξ)}{σ_{2} - σ_{1}} + \frac{σ_{2} {\hat{u}}_{0} (ξ) (1 - e^{- σ_{1}^{2} t - σ_{1} | ξ |^{2} t})}{σ_{1} - σ_{2}} + \frac{{\hat{u}}_{1} (ξ) (e^{- σ_{1}^{2} t - σ_{1} | ξ |^{2} t} - (σ_{1} - σ_{2}))}{σ_{1} - σ_{2}}} \\ (2.10) & + e^{- t | ξ |^{2}} K_{2} (t, ξ) . \end{array}$ The essential part of the proof result is in the estimation for (2.9). Although the estimates for (2.10) can be done almost similar to [2], for the sake of completeness of the proof we write down all estimates for (2.9) and (2.10).

(I) The $L^{2}$ estimates for (2.10).

Set $\begin{matrix} D : = {(1 + | ξ |^{2})}^{2} - 4 | ξ |^{2} > 0 \end{matrix}$ for small $| ξ | ≪ 1$ . At first, let us note the inequality that $\begin{matrix} (2.11) & 0 < - σ_{1} = \frac{(1 + | ξ |^{2}) - \sqrt{D}}{2} = \frac{1}{2} \frac{4 | ξ |^{2}}{(1 + | ξ |^{2}) + \sqrt{D}} ⩽ \frac{2 | ξ |^{2}}{(1 + | ξ |^{2})} ⩽ 2 | ξ |^{2}, \end{matrix}$ and, since $\begin{matrix} σ_{1} - σ_{2} = \sqrt{D} \to 1 (| ξ | \to + 0), \end{matrix}$ one can assume that for small $| ξ | ≪ 1$ , $\begin{matrix} (2.12) & σ_{1} - σ_{2} > 1 / 2 . \end{matrix}$ So, because of (2.11) and (2.12), the first term of (2.10) can be estimated in terms of $L^{2}$ -norm as follows $\begin{array}{rcl} J_{0} (t) & : = & \int_{| ξ | ≪ 1} e^{- 2 t | ξ |^{2}} {| \frac{σ_{1} {\hat{u}}_{0} (ξ)}{σ_{2} - σ_{1}} |}^{2} d ξ \\ (2.13) & ⩽ & C ‖ {\hat{u}}_{0} ‖_{\infty}^{2} \int_{| ξ | ≪ 1} e^{- 2 t | ξ |^{2}} | ξ |^{4} d ξ ⩽ C t^{- \frac{n}{2} - 2} ‖ u_{0} ‖_{1}^{2} . \end{array}$ Furthermore, because of the mean value theorem we get $\begin{matrix} (2.14) & | 1 - e^{- (σ_{1}^{2} + σ_{1} | ξ |^{2}) t} | ⩽ t | σ_{1}^{2} + σ_{1} | ξ |^{2} |, \end{matrix}$ so that from (2.12) one has $\begin{array}{rcl} J_{1} (t) & : = & \int_{| ξ | ≪ 1} \frac{σ_{2}^{2} {(1 - e^{- (σ_{1}^{2} + σ_{1} | ξ |^{2}) t})}^{2}}{| σ_{1} - σ_{2} |^{2}} {| {\hat{u}}_{0} (ξ) |}^{2} e^{- 2 t | ξ |^{2}} d ξ \\ ⩽ & t^{2} ‖ u_{0} ‖_{1}^{2} \int_{| ξ | ≪ 1} σ_{2}^{2} {| σ_{1}^{2} + σ_{1} | ξ |^{2} |}^{2} e^{- 2 t | ξ |^{2}} d ξ . \end{array}$ Since $σ_{1}^{4} ⩽ 16 | ξ |^{8}$ (see (2.11)), and $\begin{matrix} 0 ⩽ - σ_{2} = \frac{(1 + | ξ |^{2}) + \sqrt{{(1 + | ξ |^{2})}^{2} - 4 | ξ |^{2}}}{2} ⩽ \frac{2 (1 + | ξ |^{2})}{2} = (1 + | ξ |^{2}) ⩽ 2 \end{matrix}$ for small $| ξ | ≪ 1$ , it follows from (2.11) that $\begin{matrix} \int_{| ξ | ≪ 1} σ_{2}^{2} (σ_{1}^{4} + σ_{1}^{2} | ξ |^{4}) e^{- 2 t | ξ |^{2}} d ξ ⩽ C \int_{| ξ | ≪ 1} | ξ |^{8} e^{- 2 t | ξ |^{2}} d ξ ⩽ C t^{- \frac{n}{2} - 4}, \end{matrix}$ so that one has $\begin{matrix} (2.15) & J_{1} (t) ⩽ C t^{- \frac{n}{2} - 2} ‖ u_{0} ‖_{1}^{2} . \end{matrix}$ On the other hand, since $\begin{array}{rcl} | e^{- (σ_{1}^{2} + σ_{1} | ξ |^{2}) t} - (σ_{1} - σ_{2}) | & = & | (e^{- (σ_{1}^{2} + σ_{1} | ξ |^{2}) t} - 1) + 1 - (σ_{1} - σ_{2}) | \\ = & | (e^{- (σ_{1}^{2} + σ_{1} | ξ |^{2}) t} - 1) + (1 - \sqrt{D}) | \\ ⩽ & | (e^{- (σ_{1}^{2} + σ_{1} | ξ |^{2}) t} - 1) | + | \frac{1 - {(1 + | ξ |^{2})}^{2} + 4 | ξ |^{2}}{(1 + \sqrt{D})} | \\ ⩽ & | e^{- (σ_{1}^{2} + σ_{1} | ξ |^{2}) t} - 1 | + | 2 | ξ |^{2} - | ξ |^{4} |, \end{array}$ because of (2.11) and (2.14), we see that for small $| ξ | ≪ 1$ $\begin{matrix} (2.16) & | e^{- (σ_{1}^{2} + σ_{1} | ξ |^{2}) t} - (σ_{1} - σ_{2}) | ⩽ t | σ_{1}^{2} + σ_{1} | ξ |^{2} | + 2 | ξ |^{2} ⩽ 4 t | ξ |^{4} + 2 | ξ |^{2} . \end{matrix}$ Thus, from (2.16) one can estimate as follows $\begin{array}{rcl} J_{2} (t) & : = & \int_{| ξ | ≪ 1} e^{- 2 t | ξ |^{2}} \frac{| {\hat{u}}_{1} (ξ) |^{2} | e^{- (σ_{1}^{2} + σ_{1} | ξ |^{2}) t} - (σ_{1} - σ_{2}) |^{2}}{| σ_{1} - σ_{2} |^{2}} d ξ \\ (2.17) & ⩽ & C ‖ u_{1} ‖_{1}^{2} \int_{| ξ | ≪ 1} e^{- 2 t | ξ |^{2}} (t^{2} | ξ |^{8} + | ξ |^{4}) d ξ ⩽ C ‖ u_{1} ‖_{1}^{2} t^{- \frac{n}{2} - 2} . \end{array}$

(II) The $L^{2}$ estimates for (2.9).

Let us estimate (2.9) in terms of $L^{2}$ -norm, which is the main part of our result. The original idea comes from [11]. In fact, we first get $\begin{array}{rcl} J_{3} (t) & : = & \int_{| ξ | ≪ 1} {| A_{0} (ξ) - i B_{0} (ξ) + A_{1} (ξ) - i B_{1} (ξ) |}^{2} e^{- 2 t | ξ |^{2}} d ξ \\ (2.18) & ⩽ & C \int_{| ξ | ≪ 1} {(| A_{0} (ξ) | + | B_{0} (ξ) | + | A_{1} (ξ) | + | B_{1} (ξ) |)}^{2} e^{- 2 t | ξ |^{2}} d ξ . \end{array}$ In order to estimate (2.18), we prepare two quantities $\begin{array}{l} L : = sup_{θ \neq 0} \frac{| 1 - cos θ |}{| θ |} < + \infty, \\ M : = sup_{θ \neq 0} \frac{| sin θ |}{| θ |} < + \infty . \end{array}$ Then, in case of $ξ \neq 0$ , for small $ε > 0$ one has $\begin{array}{l} \int_{| x | ⩾ ε} | u_{j} (x) | | (cos (x \cdot ξ) - 1) | d x \\ ⩽ \int_{| x | ⩾ ε} | u_{j} (x) | \frac{| (cos (x \cdot ξ) - 1) |}{| x \cdot ξ |} | x \cdot ξ | d x \\ (2.19) & ⩽ L | ξ | \int_{R^{n}} | u_{j} (x) | | x | d x (j = 0, 1) . \end{array}$ Letting $ε ↓ 0$ in (2.19), one has $\begin{matrix} | A_{j} (ξ) | ⩽ L | ξ | \int_{R^{n}} | u_{j} (x) | | x | d x ⩽ L | ξ | ‖ u_{j} ‖_{1, 1} (ξ \in R^{n}, j = 0, 1) . \end{matrix}$ Note that the estimate just above holds true also in the case when $ξ = 0$ . Similarly to the computation above one can also get $\begin{matrix} | B_{j} (ξ) | ⩽ M | ξ | \int_{R^{n}} | u_{j} (x) | | x | d x ⩽ M | ξ | ‖ u_{j} ‖_{1, 1} (ξ \in R^{n}, j = 0, 1) . \end{matrix}$ Therefore, one can deduce important estimates $\begin{array}{rcl} J_{3} (t) & ⩽ & C (‖ u_{0} ‖_{1, 1}^{2} + ‖ u_{1} ‖_{1, 1}^{2}) \int_{| ξ | ≪ 1} {(L + M)}^{2} | ξ |^{2} e^{- 2 t | ξ |^{2}} d ξ \\ (2.20) & ⩽ & C {(L + M)}^{2} (‖ u_{0} ‖_{1, 1}^{2} + ‖ u_{1} ‖_{1, 1}^{2}) t^{- \frac{n}{2} - 1} . \end{array}$

(III) The $L^{2}$ estimates for the last term of (2.10).

Let us estimate the following $J_{4} (t)$ $\begin{array}{rcl} J_{4} (t) & : = & \int_{| ξ | ≪ 1} e^{- 2 t | ξ |^{2}} {| K_{2} (t, ξ) |}^{2} d ξ \\ = & \int_{| ξ | ≪ 1} e^{- 2 t | ξ |^{2}} {| \frac{{\hat{u}}_{0} (ξ) σ_{1} - {\hat{u}}_{1} (ξ)}{σ_{1} - σ_{2}} |}^{2} e^{- 2 (σ_{1}^{2} + σ_{1} | ξ |^{2}) t} d ξ . \end{array}$ Because of (2.11) and (2.12) one can get $\begin{matrix} J_{4} (t) ⩽ C \int_{| ξ | ≪ 1} e^{- 2 t | ξ |^{2}} ({| {\hat{u}}_{0} (ξ) |}^{2} + {| {\hat{u}}_{1} (ξ) |}^{2}) e^{- 2 (σ_{1}^{2} + σ_{1} | ξ |^{2}) t} d ξ . \end{matrix}$ And also, since $\begin{matrix} σ_{2}^{2} + σ_{2} | ξ |^{2} \to 1 (| ξ | \to + 0), \end{matrix}$ we can assume that for small $| ξ | ≪ 1$ $\begin{matrix} σ_{2}^{2} + σ_{2} | ξ |^{2} > 1 / 2, \end{matrix}$ so that one gets $\begin{matrix} e^{- 2 (σ_{1}^{2} + σ_{1} | ξ |^{2}) t} ⩽ e^{- t} \end{matrix}$ provided that $| ξ | ≪ 1$ . By the Plancherel theorem one has $\begin{array}{rcl} J_{4} (t) & ⩽ & C e^{- t} \int_{| ξ | ≪ 1} e^{- 2 t | ξ |^{2}} ({| {\hat{u}}_{0} (ξ) |}^{2} + {| {\hat{u}}_{1} (ξ) |}^{2}) d ξ \\ (2.21) & ⩽ & C e^{- t} \int_{| ξ | ≪ 1} ({| {\hat{u}}_{0} (ξ) |}^{2} + {| {\hat{u}}_{1} (ξ) |}^{2}) d ξ ⩽ C e^{- t} (‖ u_{0} ‖^{2} + ‖ u_{1} ‖^{2}) . \end{array}$

Finally, because of (2.9), (2.10), (2.13), (2.15), (2.17), (2.20) and (2.21) one has arrived at the desired estimate for Lemma 2.1 $\begin{array}{l} \int_{| ξ | ⩽ γ} {| \hat{u} (t, ξ) - (P_{00} + P_{01}) e^{- t | ξ |^{2}} |}^{2} d ξ \\ ⩽ C t^{- \frac{n}{2} - 1} (‖ u_{0} ‖_{1, 1}^{2} + ‖ u_{1} ‖_{1, 1}^{2}) + C t^{- \frac{n}{2} - 2} (‖ u_{0} ‖_{1}^{2} + ‖ u_{1} ‖_{1}^{2}) + C e^{- t} (‖ u_{0} ‖^{2} + ‖ u_{1} ‖^{2}) \end{array}$ with some constants $γ > 0$ , $C > 0$ . □

In order to get the high frequency estimate of the solution $u (t, x)$ , one first considers the energy localized in the high frequency region in the Fourier space. For this purpose we shall rely on the powerful tool, which has its origin in [3, Theorem 2.1, Corollary 2.1]. To apply the result of [3, Theorem 2.1, Corollary 2.1] to problem (2.1)–(2.2) in the high frequency region, we choose $\begin{matrix} P_{1} (ξ) = 1, P_{2} (ξ) = 1 + | ξ |^{2}, P_{3} (ξ) = | ξ |^{2}, \end{matrix}$ and for fixed $γ > 0$ , we set $\begin{matrix} Ω = {ξ \in R_{ξ}^{n} | | ξ | ⩾ γ > 0}, \end{matrix}$ where the constant $γ > 0$ has already been defined in Lemma 2.1. Then the energy in the high frequency region Ω can be defined as $\begin{matrix} E_{h} (t) : = \frac{1}{2} \int_{Ω} {{| {\hat{u}}_{t} (t, ξ) |}^{2} + | ξ |^{2} {| \hat{u} (t, ξ) |}^{2}} d ξ . \end{matrix}$ Now, since $\begin{matrix} P_{1} (ξ) ⩽ P_{2} (ξ) ⩽ \frac{1}{γ} | ξ |^{2} + | ξ |^{2} = (1 + \frac{1}{γ}) | ξ |^{2} = (1 + \frac{1}{γ}) P_{3} (ξ), \end{matrix}$ it follows from Corollary 2.1 of [3] that $\begin{matrix} E_{h} (t) ⩽ C e^{- η t}, t ≫ 1 \end{matrix}$ with some constant $C > 0$ and $η > 0$ . In fact, one can get more precise estimate from the direct computations based on [3] $\begin{matrix} (2.22) & E_{h} (t) ⩽ C E_{h} (0) e^{- η t} ⩽ C (‖ u_{1} ‖^{2} + ‖ \nabla u_{0} ‖^{2}) e^{- η t}, t ≫ 1 \end{matrix}$ because of the Plancherel Theorem. As a consequence of (2.22) one can derive the useful estimate, for large $t ≫ 1$ $\begin{matrix} (2.23) & \int_{| ξ | ⩾ γ} {| \hat{u} (t, ξ) |}^{2} d ξ ⩽ \frac{1}{γ^{2}} \int_{Ω} | ξ |^{2} {| \hat{u} (t, ξ) |}^{2} d ξ ⩽ \frac{1}{γ^{2}} E_{h} (t) ⩽ C (‖ u_{1} ‖^{2} + ‖ \nabla u_{0} ‖^{2}) e^{- η t} . \end{matrix}$ Furthermore, the following lemma is important to get the high frequency estimate of the main result. Lemma 2.2.
Under the assumptions as in Theorem 1.1, it is true that for $t > 0$ $\begin{matrix} \int_{| ξ | ⩾ γ} {(P_{00} + P_{01})}^{2} e^{- 2 | ξ |^{2} t} d ξ ⩽ e^{- 2 γ^{2} t} ‖ u_{0} + u_{1} ‖^{2} + (L^{2} + M^{2}) ‖ u_{0} + u_{1} ‖_{1, 1}^{2} t^{- \frac{n}{2} - 1} (t > 0), \end{matrix}$ where $γ > 0$ is a constant defined in Lemma 2.1, and $L, M > 0$ are constants defined by (2.19 ) .
Proof.
We first consider the following Cauchy problem on the heat equation in $R^{n}$ ( $n ⩾ 1$ ) $\begin{array}{l} v_{t} (t, x) - Δ v (t, x) = 0, (t, x) \in (0, \infty) \times R^{n}, \\ v (0, x) = u_{0} (x) + u_{1} (x), x \in R^{n}, \end{array}$ where the initial data $u_{0} + u_{1}$ comes from (1.2). Under the assumptions on the initial data as in Theorem 1.1, it is known that the problem above admits a unique solution $v \in C ([0, + \infty); L^{2} (R^{n})) \cap C^{1} ((0, + \infty); H^{2} (R^{n}))$ (cf. [25]).

Now, after taking the Fourier transform of the equation above, we get the reduced ODE in the Fourier space $R_{ξ}^{n}$ $\begin{array}{l} (2.24) & {\hat{v}}_{t} (t, ξ) + | ξ |^{2} \hat{v} (t, ξ) = 0, (t, ξ) \in (0, \infty) \times R_{ξ}^{n}, \\ (2.25) & \hat{v} (0, ξ) = {\hat{u}}_{0} (ξ) + {\hat{u}}_{1} (ξ), ξ \in R_{ξ}^{n} . \end{array}$ Then we can solve (2.24)–(2.25) directly $\begin{matrix} \hat{v} (t, ξ) = ({\hat{u}}_{0} (ξ) + {\hat{u}}_{1} (ξ)) e^{- | ξ |^{2} t} . \end{matrix}$ We notice that $\begin{array}{rcl} {\hat{u}}_{0} (ξ) + {\hat{u}}_{1} (ξ) & = & \int_{R^{n}} (u_{0} (x) + u_{1} (x)) cos (x \cdot ξ) d x - i \int_{R^{n}} (u_{0} (x) + u_{1} (x)) sin (x \cdot ξ) d x \\ = & \int_{R^{n}} (u_{0} (x) + u_{1} (x)) (cos (x \cdot ξ) - 1) d x - i \int_{R^{n}} (u_{0} (x) + u_{1} (x)) sin (x \cdot ξ) d x \\ + \int_{R^{n}} (u_{0} (x) + u_{1} (x)) d x \\ = : & A (ξ) - i B (ξ) + (P_{00} + P_{01}), \end{array}$ so that as in (2.8) one has $\begin{matrix} \hat{v} (t, ξ) - (P_{00} + P_{01}) e^{- | ξ |^{2} t} = A (ξ) e^{- | ξ |^{2} t} - i B (ξ) e^{- | ξ |^{2} t} . \end{matrix}$ This implies $\begin{matrix} (2.26) & (P_{00} + P_{01}) e^{- | ξ |^{2} t} = \hat{v} (t, ξ) - A (ξ) e^{- | ξ |^{2} t} + i B (ξ) e^{- | ξ |^{2} t} . \end{matrix}$ As in the same computations around (2.19) one can get $\begin{matrix} (2.27) & | A (ξ) | ⩽ L | ξ | \int_{R^{n}} | u_{0} (x) + u_{1} (x) | | x | d x ⩽ L | ξ | ‖ u_{0} + u_{1} ‖_{1, 1} (ξ \in R^{n}) . \end{matrix}$ Similarly to (2.27), one also has $\begin{matrix} (2.28) & | B (ξ) | ⩽ M | ξ | \int_{R^{n}} | u_{0} (x) + u_{1} (x) | | x | d x ⩽ M | ξ | ‖ u_{0} + u_{1} ‖_{1, 1} (ξ \in R^{n}) . \end{matrix}$ Therefore, from (2.27) one has $\begin{array}{rcl} \int_{| ξ | ⩾ γ} {| A (ξ) |}^{2} e^{- 2 | ξ |^{2} t} d ξ & ⩽ & L^{2} ‖ u_{0} + u_{1} ‖_{1, 1}^{2} \int_{| ξ | ⩾ γ} | ξ |^{2} e^{- 2 | ξ |^{2} t} d ξ \\ (2.29) & ⩽ & L^{2} ‖ u_{0} + u_{1} ‖_{1, 1}^{2} t^{- \frac{n}{2} - 1} (t > 0), \end{array}$ and similarly, from (2.28) one gets $\begin{matrix} (2.30) & \int_{| ξ | ⩾ γ} {| B (ξ) |}^{2} e^{- 2 | ξ |^{2} t} d ξ ⩽ M^{2} ‖ u_{0} + u_{1} ‖_{1, 1}^{2} t^{- \frac{n}{2} - 1} (t > 0) . \end{matrix}$ On the other hand, since $\hat{v} (t, ξ) = ({\hat{u}}_{0} (ξ) + {\hat{u}}_{1} (ξ)) e^{- | ξ |^{2} t}$ , one can estimate $\begin{matrix} (2.31) & \int_{| ξ | ⩾ γ} {| \hat{v} (t, ξ) |}^{2} d ξ ⩽ \int_{| ξ | ⩾ γ} e^{- 2 | ξ |^{2} t} {| {\hat{u}}_{0} (ξ) + {\hat{u}}_{1} (ξ) |}^{2} d ξ ⩽ e^{- 2 γ^{2} t} ‖ u_{0} + u_{1} ‖^{2} . \end{matrix}$ Under these preparations one can prove the statement of Lemma 2.2 as follows. Because of (2.26), (2.29), (2.30) and (2.31) one can get the desired estimate $\begin{array}{l} \int_{| ξ | ⩾ γ} {(P_{00} + P_{01})}^{2} e^{- 2 | ξ |^{2} t} d ξ \\ ⩽ \int_{| ξ | ⩾ γ} {| \hat{v} (t, ξ) |}^{2} d ξ + \int_{| ξ | ⩾ γ} {| A (ξ) |}^{2} e^{- 2 | ξ |^{2} t} d ξ + \int_{| ξ | ⩾ γ} {| B (ξ) |}^{2} e^{- 2 | ξ |^{2} t} d ξ \\ ⩽ e^{- 2 γ^{2} t} ‖ u_{0} + u_{1} ‖^{2} + (L^{2} + M^{2}) ‖ u_{0} + u_{1} ‖_{1, 1}^{2} t^{- \frac{n}{2} - 1} . \end{array}$ □
Proof of Theorem 1.1.
Under the preparation from Lemmas 2.1 and 2.2, and (2.23) we can prove Theorem 1.1. We first recall the relation $\begin{matrix} G (t, x) = F^{- 1} (e^{- t | ξ |^{2}}) (x) . \end{matrix}$ Then, we can consider the following decomposition based on the Plancherel theorem $\begin{array}{l} {‖ u (t, \cdot) - (P_{00} + P_{01}) G (t, \cdot) ‖}^{2} \\ ⩽ (\int_{| ξ | ⩽ γ} + \int_{γ ⩽ | ξ |}) {| \hat{u} (t, ξ) - (P_{00} + P_{01}) e^{- t | ξ |^{2}} |}^{2} d ξ \\ = : R_{l} (t) + R_{h} (t) . \end{array}$ We can rely on Lemma 2.1 to get $\begin{matrix} (2.32) & R_{l} (t) ⩽ C t^{- \frac{n}{2} - 1} (‖ u_{0} ‖_{1, 1}^{2} + ‖ u_{1} ‖_{1, 1}^{2}) + C t^{- \frac{n}{2} - 2} (‖ u_{0} ‖_{1}^{2} + ‖ u_{1} ‖_{1}^{2}) + C e^{- t} (‖ u_{0} ‖^{2} + ‖ u_{1} ‖^{2}) . \end{matrix}$ On the other hand, it follows from Lemma 2.2 and (2.23) that $\begin{array}{rcl} R_{h} (t) & ⩽ & C \int_{γ ⩽ | ξ |} {| \hat{u} (t, ξ) |}^{2} d ξ + C \int_{γ ⩽ | ξ |} {(P_{00} + P_{01})}^{2} e^{- 2 t | ξ |^{2}} d ξ \\ ⩽ & C (‖ u_{1} ‖^{2} + ‖ \nabla u_{0} ‖^{2}) e^{- η t} + C e^{- 2 γ^{2} t} ‖ u_{0} + u_{1} ‖^{2} \\ (2.33) & + C (L^{2} + M^{2}) ‖ u_{0} + u_{1} ‖_{1, 1}^{2} t^{- \frac{n}{2} - 1} (t ≫ 1) \end{array}$ with some constant $C > 0$ . The desired estimate of Theorem 1.1 can be deduced by (2.32) and (2.33). □

3. Further trials

By reconsidering the computation in Section 2, one can get more improved asymptotic profiles of solutions. For this purpose, we fully use a special form of equations. It should be mentioned that the computation in Section 2 can be also applied to a more general shape of equations with constant coefficients $c > 0$ , $δ > 0$ and $γ > 0$ , $\begin{matrix} u_{t t} - c Δ u + δ u_{t} - γ Δ u_{t} = 0 . \end{matrix}$

Now let us reconsider the following one dimensional damped wave equation, $\begin{array}{l} (3.1) & u_{t t} (t, x) - u_{x x} (t, x) + u_{t} (t, x) - u_{x x t} (t, x) = 0, (t, x) \in (0, \infty) \times R, \\ (3.2) & u (0, x) = u_{0} (x), u_{t} (0, x) = u_{1} (x), x \in R . \end{array}$ The restriction to the one dimensional case just comes from a technical problem.

By applying the Fourier transform, we restudy the following ordinary differential equations with parameters ξ, $\begin{array}{l} (3.3) & {\hat{u}}_{t t} (t, ξ) + | ξ |^{2} \hat{u} (t, ξ) + (1 + | ξ |^{2}) {\hat{u}}_{t} (t, ξ) = 0, (t, ξ) \in (0, \infty) \times R_{ξ}, \\ (3.4) & \hat{u} (0, ξ) = {\hat{u}}_{0} (ξ), {\hat{u}}_{t} (0, ξ) = {\hat{u}}_{1} (ξ), x \in R_{ξ} . \end{array}$ Since $\begin{matrix} \sqrt{{(1 + | ξ |^{2})}^{2} - 4 | ξ |^{2}} = | 1 - | ξ |^{2} |, \end{matrix}$ in the case when $| ξ | ≪ 1$ one has (see around (2.11) and (2.12)) $\begin{matrix} σ_{1} = - | ξ |^{2}, σ_{2} = - 1, \end{matrix}$ so that one can get explicit solutions $\hat{u} (t, ξ)$ to problem (3.3)–(3.4) for small parameters $| ξ | ≪ 1$ as follows (see (2.3)). $\begin{array}{rcl} \hat{u} (t, ξ) & = & \frac{{\hat{u}}_{1} (ξ) + {\hat{u}}_{0} (ξ)}{1 - | ξ |^{2}} e^{- t | ξ |^{2}} + \frac{{\hat{u}}_{1} (ξ) + | ξ |^{2} {\hat{u}}_{0} (ξ)}{| ξ |^{2} - 1} e^{- t} \\ = : & I_{1} (t, ξ) + I_{2} (t, ξ) . \end{array}$ Concerning $I_{2} (t, ξ)$ one can get the estimate, $\begin{array}{rcl} \int_{| ξ | ≪ 1} {| I_{2} (t, ξ) |}^{2} d ξ & ⩽ & C \int_{| ξ | ≪ 1} \frac{| {\hat{u}}_{1} (ξ) |^{2} + | ξ |^{4} | {\hat{u}}_{0} (ξ) |^{2}}{| | ξ |^{2} - 1 |^{2}} e^{- 2 t} d ξ \\ ⩽ & 2 C \int_{| ξ | ≪ 1} ({| {\hat{u}}_{1} (ξ) |}^{2} + | ξ |^{4} {| {\hat{u}}_{0} (ξ) |}^{2}) d ξ \cdot e^{- 2 t} \\ (3.5) & ⩽ & C (‖ u_{1} ‖^{2} + ‖ u_{0} ‖^{2}) e^{- 2 t}, \end{array}$ where one has used the fact that $| | ξ |^{2} - 1 |^{2} > 1 / 2$ for small $| ξ | ≪ 1$ . Thus, by using the mean value theorem one can get $\begin{matrix} (3.6) & \hat{u} (t, ξ) = ({\hat{u}}_{1} (ξ) + {\hat{u}}_{0} (ξ)) e^{- t | ξ |^{2}} + \frac{| ξ |^{2} ({\hat{u}}_{1} (ξ) + {\hat{u}}_{0} (ξ))}{{(1 - θ^{2} | ξ |^{2})}^{2}} e^{- t | ξ |^{2}} + I_{2} (t, ξ) (| ξ | ≪ 1) \end{matrix}$ with some $θ \in (0, 1)$ .

Now, one uses an idea which was already used in [13] to obtain a decomposition of the Fourier transformed initial data, $\begin{matrix} (3.7) & {\hat{u}}_{j} (ξ) = C_{j} (ξ) - i D_{j} (ξ) + Q_{j} (ξ) - i R_{j} (ξ) + P_{0 j} (j = 0, 1, ξ \in R), \end{matrix}$ where $\begin{array}{l} C_{j} (ξ) : = \int_{R} (cos (x ξ) - 1 + \frac{{(x ξ)}^{2}}{2}) u_{j} (x) d x, \\ D_{j} (ξ) : = \int_{R} (sin (x ξ) - (x ξ)) u_{j} (x) d x, \\ Q_{j} (ξ) : = - \frac{ξ^{2}}{2} \int_{R} x^{2} u_{j} (x) d x, \\ R_{j} (ξ) : = ξ \int_{R} x u_{j} (x) d x . \end{array}$ By substituting those decompositions above to the first term of the right hand side of (3.6) one can get the following equality for $| ξ | ≪ 1$ , $\begin{array}{l} \hat{u} (t, ξ) - (P_{01} - i R_{1} (ξ) + P_{00} - i R_{0} (ξ)) e^{- t | ξ |^{2}} \\ = (Q_{1} (ξ) + Q_{0} (ξ)) e^{- t | ξ |^{2}} + (C_{1} (ξ) - i D_{1} (ξ) + C_{0} (ξ) - i D_{0} (ξ)) e^{- t | ξ |^{2}} \\ + \frac{| ξ |^{2} ({\hat{u}}_{1} (ξ) + {\hat{u}}_{0} (ξ))}{{(1 - θ^{2} | ξ |^{2})}^{2}} e^{- t | ξ |^{2}} + I_{2} (t, ξ) \\ (3.8) & = : E (t, ξ) . \end{array}$ Let us check that $E (t, ξ)$ is the error term. To begin with, one has $\begin{array}{l} \int_{| ξ | ≪ 1} {| \frac{| ξ |^{2} ({\hat{u}}_{1} (ξ) + {\hat{u}}_{0} (ξ))}{{(1 - θ^{2} | ξ |^{2})}^{2}} |}^{2} e^{- 2 t | ξ |^{2}} d ξ \\ (3.9) & ⩽ C \int_{| ξ | ≪ 1} | ξ |^{4} ({| {\hat{u}}_{1} (ξ) |}^{2} + {| {\hat{u}}_{0} (ξ) |}^{2}) e^{- 2 t | ξ |^{2}} d ξ ⩽ C (‖ u_{1} ‖_{1}^{2} + ‖ u_{0} ‖_{1}^{2}) t^{- \frac{n}{2} - 2} \end{array}$ for some constant $C > 0$ . In order to estimate the second term of (3.8) one prepares the following lemma (see [13, Lemma 2.2]).

Lemma 3.1.

It is true that for all $ξ \in R$ , $\begin{array}{l} | C_{j} (ξ) | ⩽ L_{*} | ξ |^{2} ‖ u_{j} ‖_{1, 2} (j = 0, 1), \\ | D_{j} (ξ) | ⩽ M_{*} | ξ |^{2} ‖ u_{j} ‖_{1, 2} (j = 0, 1) \end{array}$ with some positive constants $L > 0$ and $M > 0$ defined by $\begin{matrix} L_{*} : = sup_{θ \neq 0} \frac{| cos θ - 1 + θ^{2} / 2 |}{θ^{2}} < + \infty, M_{*} : = sup_{θ \neq 0} \frac{| sin θ - θ |}{θ^{2}} < + \infty . \end{matrix}$

Let us observe the rate of decay for the second term of the right hand side of (3.8). Indeed, because of Lemma 3.1 one can estimate as follows, $\begin{array}{l} \int_{| ξ | ≪ 1} {| (C_{1} (ξ) - i D_{1} (ξ) + C_{0} (ξ) - i D_{0} (ξ)) |}^{2} e^{- 2 t | ξ |^{2}} d ξ \\ ⩽ C (L_{*}^{2} + M_{*}^{2}) (‖ u_{1} ‖_{1, 2}^{2} + ‖ u_{0} ‖_{1, 2}^{2}) \int_{| ξ | ≪ 1} | ξ |^{4} e^{- 2 t | ξ |^{2}} d ξ \\ (3.10) & ⩽ C (L_{*}^{2} + M_{*}^{2}) (‖ u_{1} ‖_{1, 2}^{2} + ‖ u_{0} ‖_{1, 2}^{2}) t^{- \frac{n}{2} - 2} (t > 0) \end{array}$ with some constant $C > 0$ . Furthermore, the first term of the right hand side of (3.8) can be estimated as follows, $\begin{array}{l} \int_{| ξ | ≪ 1} {| Q_{1} (ξ) + Q_{0} (ξ) |}^{2} e^{- 2 t | ξ |^{2}} d ξ \\ ⩽ C {{(\int_{R} x^{2} u_{1} (x) d x)}^{2} + {(\int_{R} x^{2} u_{0} (x) d x)}^{2}} \int_{| ξ | ≪ 1} | ξ |^{4} e^{- 2 t | ξ |^{2}} d ξ \\ (3.11) & ⩽ C {{(\int_{R} x^{2} u_{1} (x) d x)}^{2} + {(\int_{R} x^{2} u_{0} (x) d x)}^{2}} t^{- \frac{n}{2} - 2} \end{array}$ with some constant $C > 0$ . Thus, it follows from (3.5), (3.8), (3.9), (3.10) and (3.11) that one has the following crucial lemma in the low frequency region.

Lemma 3.2.

Let $n = 1$ . Then, there exists a small $δ > 0$ such that it is true that $\begin{array}{l} \int_{| ξ | ⩽ δ} {| \hat{u} (t, ξ) - (P_{01} - i R_{1} (ξ) + P_{00} - i R_{0} (ξ)) e^{- t | ξ |^{2}} |}^{2} d ξ \\ ⩽ C t^{- \frac{n}{2} - 2} (‖ u_{1} ‖_{1}^{2} + ‖ u_{0} ‖_{1}^{2}) + C t^{- \frac{n}{2} - 2} (‖ u_{1} ‖_{1, 2}^{2} + ‖ u_{0} ‖_{1, 2}^{2}) + C e^{- 2 t} (‖ u_{1} ‖^{2} + ‖ u_{0} ‖^{2}) \\ + C {{(\int_{R} x^{2} u_{1} (x) d x)}^{2} + {(\int_{R} x^{2} u_{0} (x) d x)}^{2}} t^{- \frac{n}{2} - 2} . \end{array}$

On the other hand, estimates in the high frequency region $| ξ | ≫ 1$ are always exponential decay. In the following we shall draw its outline of proof.

For this ends, let us consider the initial value problem of the heat equation as in the proof of Lemma 2.2, $\begin{array}{l} v_{t} (t, x) - Δ v (t, x) = 0, (t, x) \in (0, \infty) \times R^{n}, \\ v (0, x) = u_{0} (x) + u_{1} (x), x \in R^{n} . \end{array}$ By applying the Fourier transform one gets $\begin{array}{l} (3.12) & {\hat{v}}_{t} (t, ξ) + | ξ |^{2} \hat{v} (t, ξ) = 0, (t, ξ) \in (0, \infty) \times R_{ξ}^{n}, \\ (3.13) & \hat{v} (0, ξ) = {\hat{u}}_{0} (ξ) + {\hat{u}}_{1} (ξ), ξ \in R_{ξ}^{n}, \end{array}$ so that $\begin{matrix} \hat{v} (t, ξ) = ({\hat{u}}_{0} (ξ) + {\hat{u}}_{1} (ξ)) e^{- t | ξ |^{2}} . \end{matrix}$

By (2.23) we first obtain $\begin{matrix} (3.14) & \int_{| ξ | ⩾ δ} {| \hat{u} (t, ξ) |}^{2} d ξ ⩽ C e^{- η t} (‖ u_{1} ‖^{2} + ‖ \nabla u_{0} ‖^{2}) (t ≫ 1), \end{matrix}$ where $δ > 0$ is a constant defined in Lemma 3.2 and $u = u (t, x)$ is the solution to problem (3.1)–(3.2).

Next, as in (2.26) one can get a decomposition of the solution $\hat{v}$ as follows (see (3.7)), $\begin{array}{l} (P_{01} - i R_{1} (ξ) + P_{00} - i R_{0} (ξ)) e^{- t | ξ |^{2}} \\ (3.15) & = \hat{v} (t, ξ) - (Q_{0} (ξ) + Q_{1} (ξ)) e^{- t | ξ |^{2}} - (C_{0} (ξ) - i D_{0} (ξ) + C_{1} (ξ) - i D_{1} (ξ)) e^{- t | ξ |^{2}} . \end{array}$ So, one can proceed the estimate with the help of (2.31), (3.10) and (3.11) as follows, $\begin{array}{l} \int_{| ξ | ⩾ δ} {| P_{01} - i R_{1} (ξ) + P_{00} - i R_{0} (ξ) |}^{2} e^{- 2 t | ξ |^{2}} d ξ \\ ⩽ C \int_{| ξ | ⩾ δ} {| \hat{v} (t, ξ) |}^{2} d ξ + C \int_{| ξ | ⩾ δ} {| Q_{0} (ξ) + Q_{1} (ξ) |}^{2} e^{- 2 t | ξ |^{2}} d ξ \\ + C \int_{| ξ | ⩾ δ} {| C_{0} (ξ) - i D_{0} (ξ) + C_{1} (ξ) - i D_{1} (ξ) |}^{2} e^{- 2 t | ξ |^{2}} d ξ \\ ⩽ C e^{- 2 δ^{2} t} ‖ u_{0} + u_{1} ‖^{2} + C {{(\int_{R} x^{2} u_{1} (x) d x)}^{2} + {(\int_{R} x^{2} u_{0} (x) d x)}^{2}} \int_{| ξ | ⩾ δ} | ξ |^{4} e^{- 2 t | ξ |^{2}} d ξ \\ + \int_{| ξ | ⩾ δ} ({| C_{0} (ξ) |}^{2} + {| D_{0} (ξ) |}^{2} + {| C_{1} (ξ) |}^{2} + {| D_{1} (ξ) |}^{2}) e^{- 2 t | ξ |^{2}} d ξ \\ ⩽ C e^{- 2 δ^{2} t} ‖ u_{0} + u_{1} ‖^{2} + C {{(\int_{R} x^{2} u_{1} (x) d x)}^{2} + {(\int_{R} x^{2} u_{0} (x) d x)}^{2}} e^{- δ t^{2}} t^{- \frac{n}{2} - 2} \\ + C (L_{*}^{2} + M_{*}^{2}) (‖ u_{1} ‖_{1, 2}^{2} + ‖ u_{0} ‖_{1, 2}^{2}) \int_{| ξ | ⩾ δ} | ξ |^{4} e^{- 2 t | ξ |^{2}} d ξ \\ ⩽ C e^{- 2 δ^{2} t} ‖ u_{0} + u_{1} ‖^{2} + C {{(\int_{R} x^{2} u_{1} (x) d x)}^{2} + {(\int_{R} x^{2} u_{0} (x) d x)}^{2}} e^{- δ t^{2}} t^{- \frac{n}{2} - 2} \\ (3.16) & + C (L_{*}^{2} + M_{*}^{2}) (‖ u_{1} ‖_{1, 2}^{2} + ‖ u_{0} ‖_{1, 2}^{2}) e^{- δ t^{2}} t^{- \frac{n}{2} - 2}, \end{array}$ where we have used the fact that $\begin{array}{rcl} \int_{| ξ | ⩾ δ} | ξ |^{4} e^{- 2 t | ξ |^{2}} d ξ & ⩽ & \int_{| ξ | ⩾ δ} | ξ |^{4} e^{- t | ξ |^{2}} e^{- t | ξ |^{2}} d ξ \\ ⩽ & e^{- t δ^{2}} \int_{| ξ | ⩾ δ} | ξ |^{4} e^{- t | ξ |^{2}} d ξ ⩽ C e^{- t δ^{2}} t^{- \frac{n}{2} - 2} (⩽ C e^{- \frac{δ^{2}}{2} t}) . \end{array}$

Therefore, from (3.14), (3.16) and Lemma 3.2 one has the following result in terms of moment conditions on the initial data with the help of the Plancherel theorem.

Theorem 3.1.

Let $n = 1$ . If $[u_{0}, u_{1}] \in (H^{1} (R) \cap L^{1, 2} (R)) \times (L^{2} (R) \cap L^{1, 2} (R))$ , then the solution $u (t, x)$ to problem ( 3.1 )–( 3.2 ) satisfies $\begin{array}{l} {‖ u (t, \cdot) - (\int_{R} u_{1} (x) d x + \int_{R} u_{0} (x) d x) G (t, \cdot) + (\int_{R} x u_{1} (x) d x + \int_{R} x u_{0} (x) d x) G_{x} (t, \cdot) ‖}^{2} \\ ⩽ C t^{- \frac{n}{2} - 2} (‖ u_{1} ‖_{1}^{2} + ‖ u_{0} ‖_{1}^{2}) + C (‖ u_{1} ‖_{1, 2}^{2} + ‖ u_{0} ‖_{1, 2}^{2}) t^{- \frac{n}{2} - 2} \\ + C (‖ u_{1} ‖^{2} + ‖ u_{0} ‖^{2}) e^{- 2 t} + C {{(\int_{R} x^{2} u_{1} (x) d x)}^{2} + {(\int_{R} x^{2} u_{0} (x) d x)}^{2}} t^{- \frac{n}{2} - 2} \\ + C e^{- 2 t δ^{2}} ‖ u_{1} + u_{0} ‖^{2} + C e^{- η t} (‖ u_{1} ‖^{2} + ‖ \nabla u_{0} ‖^{2}) \end{array}$ with some constant $δ > 0$ , $η > 0$ and $C > 0$ (as $t \to + \infty$ ), where $\begin{matrix} G_{x} (t, x) : = \frac{\partial}{\partial x} G (t, x) . \end{matrix}$

Remark 3.1.

As a result of Theorem 3.1, from the above trial in the one dimensional case even if we impose further regularity on the initial data in terms of moment conditions, one has only got the stronger effect of the frictional damping, which implies no asymptotic oscillation property of solutions. In this sense, the influence from the viscoelastic damping term is very weak.

Remark 3.2.

One can easily check that $‖ G_{x} (t, \cdot) ‖ \sim t^{- \frac{n}{4} - \frac{1}{2}}$ , so that the result of Theorem 3.1 has exactly caught leading terms in asymptotic sense.

Remark 3.3.

If we consider further decomposition of the Fourier transformed initial data ${\hat{u}}_{j}$ ( $j = 0, 1$ ), under additional moment conditions on the initial data one can obtain more precise asymptotic expansions of the solution $u (t, x)$ by a linear combination of the higher order spatial derivatives of the Gauss kernel. To check it is left to the readers’ exercise (cf. [10]).

Footnotes

Acknowledgements

The work of the first author (R. Ikehata) was supported in part by Grant-in-Aid for Scientific Research (C)15K04958 of JSPS. The first author would like to thank Dr. H. Takeda (Fukuoka Institute of Technology) for his useful advice. The authors would like to thank the Referees for their helpful comments, which help us to improve the first draft of this manuscript.

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