A note on optimal L 2 -estimates of solutions to some strongly damped σ -evolution equations

Abstract

We consider the Cauchy problem in $R^{n}$ for the so-called σ-evolution equations with damping terms. We derive asymptotic profiles of solutions with weighted $L^{1, 1} (R^{n})$ initial data, and investigates the optimality of estimates of solutions in $L^{2}$ -sense. The obtained results will generalize and compensate those already known in (J. Math. Anal. Appl. 478 (2019) 476–498, J. Diff. Eqns 257 (2014) 2159–2177, Diff. Int. Eqns 30 (2017), 505–520).

Keywords

damping terms optimal estimates

1. Introduction

We are concerned with the Cauchy problem for wave equations in $R^{n}$ ( $n ⩾ 1$ ) with the structural damping term $\begin{array}{l} (1.1) & u_{t t} (t, x) + {(- Δ)}^{σ} u (t, x) + {(- Δ)}^{σ} u_{t} (t, x) = 0, (t, x) \in (0, \infty) \times R^{n}, \\ (1.2) & u (0, x) = u_{0} (x), u_{t} (0, x) = u_{1} (x), x \in R^{n}, \end{array}$ with $σ ⩾ 1$ , and $\begin{matrix} ({(- Δ)}^{σ} u) (x) : = F^{- 1} (| ξ |^{2 σ} (F u) (ξ)) (x), \end{matrix}$ where $F$ represents the usual Fourier transform in $L^{2} (R^{n})$ . The initial data $u_{0}$ and $u_{1}$ are also chosen from the energy space (at present) $\begin{matrix} [u_{0}, u_{1}] \in H^{σ} (R^{n}) \times L^{2} (R^{n}) . \end{matrix}$ One defines the linear operator $A : D (A) (\subset L^{2} (R^{n})) \to L^{2} (R^{n})$ as $\begin{matrix} (1.3) & D (A) : = H^{2 σ} (R^{n}), (A u) (x) : = ({(- Δ)}^{σ} u) (x) (u \in D (A)) . \end{matrix}$ Then, the operator A is non-negative, and self-adjoint in $L^{2} (R^{n})$ . Under these conditions, it is already known in [12] that the problem (1.1)–(1.2) has a unique weak solution $\begin{matrix} u \in C ([0, \infty); H^{σ} (R^{n})) \cap C^{1} ([0, \infty); L^{2} (R^{n})) . \end{matrix}$

Let us make mentioning of several related previous results by limiting the discussion basically only to the equation (1.1) with $σ ⩾ 1$ and constant coefficients.

In the case when $σ = 1$ pioneering works are introduced by Ponce [14] in 1983, and Shibata [15] in 1999, in there some $L^{p}$ – $L^{q}$ estimates are derived. After [14] and [15] it seems that few works are devoted to the study of asymptotic behavior of solutions even in the case of $σ = 1$ . Among others Ikehata–Todorova–Yordanov [12] discovers the asymptotic profile of solutions to the following abstract evolution equations: $\begin{matrix} u^{″} (t) + A u (t) + A u^{'} (t) = 0, t \in (0, \infty), \end{matrix}$ and applied the theory to the concrete exterior problems of (1.1) with $σ = 1$ , where A is a nonnegative self adjoint operator in a real Hilbert space. Furthermore, in the concrete setting Ikehata [9] and Ikehata–Onodera [11] have re-studied the problem (1.1)–(1.2) with $σ = 1$ to construct the asymptotic profile of the solution systematically, and applied them to investigate the optimality of the obtained decay/non-decay estimates of solutions. In this connection, Charão–da Luz–Ikehata [1] studies the optimality of the decay estimates of solutions by introducing a new energy method in the Fourier space combined with the Haraux–Komornik inequality (see also Ikehata–Natsume [10] for $L^{1}$ – $L^{\infty}$ estimates of solutions for the case of $σ = 1$ ). While, basing on Shibata’s $L^{p}$ – $L^{q}$ estimates D’Abbicco–Reissig [4] studies the semi-linear problems to the equation (1.1) with $σ = 1$ and the nonlinearity $| u |^{p}$ , and investigates the critical exponent $p^{*}$ of the power p. It seems that nobody has ever succeeded to determine the precise critical exponent $p^{*}$ to the equation (1.1) with $σ = 1$ . For the unique existence and smoothing property of solutions to the abstract equation: $\begin{matrix} (1.4) & u^{″} (t) + A u (t) + 2 δ A^{σ} u^{'} (t) = 0, t \in (0, \infty), δ > 0, \end{matrix}$ one can cite the paper due to Ghisi–Gobbibo–Haraux [7]. From these previous works one can see that at present nobody studies the problem (1.1)–(1.2) with $σ > 1$ , which includes the so-called plate equation with strong damping term. In this connection, concerning the asymptotic profile of solutions to the concrete damped wave equation with $σ = 0$ , and $A : = - Δ$ in $L^{2} (R^{3})$ in (1.4), one can cite the celebrated paper due to Nishihara [13], in which a diffusive aspect together with the wave effect of the solutions is precisely investigated.

Quite recently, D’Abbicco–Girardi–Liang [2] have published the paper concerning the equation (1.1)–(1.2) with $σ = 2$ , and studies $L^{1}$ – $L^{1}$ estimates of solutions in the case of $n ⩾ 5$ , and infinite time blowup property of solutions: $\begin{matrix} {‖ u (t, \cdot) ‖}_{L^{1}} ⩽ C {(1 + t)}^{\frac{n}{4}} (‖ u_{0} ‖_{L^{1}} + {(1 + t)}^{1 / 2} ‖ u_{1} ‖_{L^{1}}) (t ⩾ 0) . \end{matrix}$ Furthermore, Dao–Reissig [6] also studies the $L^{1}$ -estimates of solutions to the more generalized equation including (1.1), and applied it to the nonlinear problems with nonlinearities $| u |^{p}$ and/or $| u_{t} |^{p}$ . However, their main concern in [6] is also in the $L^{1}$ -estimates of solutions, and so even in $L^{2}$ -sense the optimality of the estimates of solutions are not investigated well to the linear models (1.1) (as a special topic, see also [3] for a plate model with memory terms). In some suitable sense, the equation (1.1) with $σ = 2$ is included as an example one in the previous abstract result due to [12], so at this second stage of the study concerning the equation (1.1) one should investigate the optimality in the estimates of the $L^{2}$ norm of solutions and/or total energy for the general case $σ > 1$ . The optimality implies the lower and upper estimates of solutions in $L^{2}$ -sense. The problem itself of this research is strongly inspired from the paper recently published by Dao–Michihisa [5], there asymptotic profiles and the critical exponents are investigated precisely to the following semi-linear equations: $\begin{matrix} (1.5) & u_{t t} + {(- Δ)}^{σ} u + u_{t} + {(- Δ)}^{σ} u_{t} = | u |^{p} . \end{matrix}$ The frictional damping is more dominant than the visco-elastic one. It is quite natural to consider the equation (1.1) first before dealing with (1.5) in order to compare the effect of the frictional term $u_{t}$ .

Our purpose of this paper is to obtain optimal decay/non-decay estimates of solutions in $L^{2}$ -sense to problem (1.1)–(1.2) with general $σ > 1$ , especially we would like to face up to the low dimensional case, which is frequently removed in the study of (1.1) with higher $σ > 1$ . A study of this paper never overlaps with any topics considered in [2].

In order to state our main results we set $\begin{matrix} (1.6) & I_{0} : = ‖ u_{0} ‖_{H^{σ}} + ‖ u_{0} ‖_{L^{1}} + ‖ u_{1} ‖_{L^{2}} + {‖ (1 + | x |) u_{1} ‖}_{L^{1}} . \end{matrix}$ Our results read as follows.

Theorem 1.1.
Let $σ ⩾ 1$ and $n > 2 σ$ . If $[u_{0}, u_{1}] \in (H^{σ} (R^{n}) \cap L^{1} (R^{n})) \times (L^{2} (R^{n}) \cap L^{1, 1} (R^{n}))$ , then there exists a constant $C > 0$ such that the unique weak solution $u (t, x)$ to problem ( 1.1 )–( 1.2 ) satisfies $\begin{matrix} C | \int_{R^{n}} u_{1} (x) d x | t^{- \frac{n - 2 σ}{4 σ}} ⩽ {‖ u (t, \cdot) ‖}_{L^{2}} ⩽ C I_{0} t^{- \frac{n - 2 σ}{4 σ}} (t ≫ 1) . \end{matrix}$
Theorem 1.2.
Let $σ ⩾ 1$ and $n = 2 σ$ . If $[u_{0}, u_{1}] \in (H^{σ} (R^{n}) \cap L^{1} (R^{n})) \times (L^{2} (R^{n}) \cap L^{1, 1} (R^{n}))$ , then there exists a constant $C > 0$ such that the unique weak solution $u (t, x)$ to problem ( 1.1 )–( 1.2 ) satisfies $\begin{matrix} C | \int_{R^{n}} u_{1} (x) d x | \sqrt{log t} ⩽ {‖ u (t, \cdot) ‖}_{L^{2}} ⩽ C I_{0} \sqrt{log t} (t ≫ 1) . \end{matrix}$
Theorem 1.3.
Let $σ ⩾ 1$ and $n < 2 σ$ . If $[u_{0}, u_{1}] \in (H^{σ} (R^{n}) \cap L^{1} (R^{n})) \times (L^{2} (R^{n}) \cap L^{1, 1} (R^{n}))$ , then there exists a constant $C > 0$ such that the unique weak solution $u (t, x)$ to problem ( 1.1 )–( 1.2 ) satisfies $\begin{matrix} C | \int_{R^{n}} u_{1} (x) d x | t^{\frac{2 σ - n}{2 σ}} ⩽ {‖ u (t, \cdot) ‖}_{L^{2}} ⩽ C I_{0} t^{\frac{2 σ - n}{2 σ}} (t ≫ 1) . \end{matrix}$
Remark 1.1.
If one chooses $σ = 1$ , the obtained results completely coincide with that of [9,11]. In this sense, the results above are a generalization to the case of $σ > 1$ . Furthermore, $n^{} : = 2 σ$ on the spatial dimension n is a kind of threshold, which divides the phenomena into two parts: one is $L^{2}$ -decay for $n > n^{}$ , the other is $L^{2}$ -blowup in infinite time for $1 ⩽ n ⩽ n^{}$ .
Remark 1.2.
One can obtain similar estimates to the higher order time and spatial derivatives of solutions, however, in that case one can get only more fast optimal “decay” estimates even in parts corresponding to Theorems 1.2 and 1.3, and the infinite time blowup property such as Theorems 1.2 and 1.3 never appears. This case is not so interesting. By this reason one should investigate only the $L^{2}$ -estimates of solutions.
Remark 1.3.
If one chooses $σ = 2$ as an example (plate equation case), one has the critical value $n^{} = 4$ , and this corresponds to the results in [2], which studies mainly the case of $n ⩾ 5$ . In the case when (for example) $σ = 8 / 3$ , then one has $n^{*} = 16 / 3$ , and the corresponding results show that the log-order blowup in infinite time never occurs for any $n ⩾ 1$ .
Remark 1.4.
In [2, Lemma 2.2] the $L^{p}$ – $L^{q}$ estimates ( $1 ⩽ p ⩽ 2 ⩽ q ⩽ \infty$ ) of the low-frequency part of solutions to (1.1)–(1.2) with $σ = 2$ are studied. The low frequency part of solutions are essential in such estimates. By applying Theorems 1.1, 1.2 and 1.3 of this paper with $σ = 2$ one can check that in the case when $p = 1$ and $q = 2$ in [2, Lemma 2.2], (26) and (28) in [2, Lemma 2.2] are exactly optimal, and by checking carefully the proof of [2, Lemma 2.2] with $p = 1$ and $q = 2$ one can see that (27) of [2, Lemma 2.2] is also optimal under the condition $\int_{R^{n}} u_{1} (x) d x \neq 0$ .

Our plan in this paper is as follows. In Section 2, we prepare several preliminary formulas based on abstract theory driven in [12] on the asymptotic profile of solutions to some abstract evolution equations, and in Section 3 we shall prove Theorems 1.1, 1.2 and 1.3 by combining such profiles with the method introduced in [9,11]. The main topic is in the manner that how we use the $L^{1, 1}$ -assumption on the initial data to catch a leading term of the decay of the solution. Section 4 will be devoted to the full proof of main theorems. Notation.
Throughout this paper, $‖ \cdot ‖_{q}$ and $‖ \cdot ‖_{H^{η}}$ stand for the usual $L^{q} (R^{n})$ -norm, and $H^{η} (R^{n})$ -norm ( $η ⩾ 0$ ), respectively. For simplicity of notations, in particular, we use $‖ \cdot ‖$ instead of $‖ \cdot ‖_{2}$ . $\begin{matrix} f \in L^{1, γ} (R^{n}) \Leftrightarrow f \in L^{1} (R^{n}), ‖ f ‖_{1, γ} : = \int_{R^{n}} {(1 + | x |)}^{γ} | f (x) | d x < + \infty, γ ⩾ 0 . \end{matrix}$ In this connection, the quantity $I_{0}$ defined in (1.6) can be rewritten in terms of these notations as follows: $\begin{matrix} I_{0} = ‖ u_{0} ‖_{H^{σ}} + ‖ u_{0} ‖_{1} + ‖ u_{1} ‖ + ‖ u_{1} ‖_{1, 1} . \end{matrix}$ Furthermore, we express the Fourier transform of the function $ϕ (x)$ by $F (ϕ) (ξ) = \hat{ϕ} (ξ)$ , and the inverse Fourier transform of $F$ is denoted by $F^{- 1}$ . Finally, we denote by $ω_{n} : = \int_{| ω | = 1} d ω$ the surface measure of the unit ball in $R^{n}$ .

2. Preliminary results

In this section, one shall study asymptotic profiles of the solutions to (1.1)–(1.2) by relying on the abstract theory constructed in [12]. In fact, by using the operator A defined in (1.3), the problem (1.1)–(1.2) can be converted into the following abstract form: $\begin{array}{l} (2.1) & u^{″} (t) + A u (t) + A u^{'} (t) = 0, t \in (0, \infty), \\ (2.2) & u (0) = u_{0} \in D (A^{1 / 2}), u^{'} (0) = u_{1} \in H . \end{array}$ Here, $H : = L^{2} (R)$ , and the square root $A^{1 / 2}$ of the operator A is defined as $\begin{matrix} D (A^{1 / 2}) = H^{σ} (R^{n}), (A^{1 / 2} u) (x) = ({(- Δ)}^{σ / 2} u) : = F^{- 1} (| ξ |^{σ} \hat{u} (ξ)) (x) (u \in H^{σ} (R^{n})) . \end{matrix}$

Under this setting, in [12, (II) with $k = 0$ in Theorem 1.3] the following asymptotic estimates are derived to problem (2.1)–(2.2): $\begin{matrix} (2.3) & ‖ u (t) - u_{diff} (t) ‖ ⩽ C (‖ e^{- t ζ A} u_{0} ‖ + ‖ e^{- t ζ A} u_{1} ‖) + C e^{- η t} (‖ u_{0} ‖_{H^{σ}} + ‖ u_{1} ‖), \end{matrix}$ where $C > 0$ , $η > 0$ and $ζ > 0$ are some constants, and $\begin{matrix} (2.4) & u_{diff} (t) : = e^{- t A / 2} (A^{- 1 / 2} sin (t A^{1 / 2}) u_{1} + cos (t A^{1 / 2}) u_{0}) . \end{matrix}$ Let us translate the obtained estimates in (2.3) into the concrete ones in $L^{2}$ -framework. The following decay estimate is quite standard.

Lemma 2.1.
Let $σ ⩾ 1$ , $n ⩾ 1$ , $ζ > 0$ , and $v \in L^{2} (R^{n}) \cap L^{1} (R^{n})$ . Then, it holds that $\begin{matrix} ‖ e^{- t ζ A} v ‖ ⩽ C {(1 + t)}^{- \frac{n}{4 σ}} ‖ v ‖_{1} + e^{- ζ t / 2} ‖ v ‖ (t ⩾ 0) . \end{matrix}$
Proof.
For making this paper self-contained as possible as one can do, let us draw its full proof. Indeed, $\begin{array}{rcl} {‖ e^{- t ζ A} v ‖}^{2} & = & \int_{R^{n}} e^{- 2 t ζ | ξ |^{2 σ}} {| \hat{v} (ξ) |}^{2} d ξ = \int_{| ξ | ⩽ 1} + \int_{| ξ | ⩾ 1} {e^{- 2 t ζ | ξ |^{2 σ}} {| \hat{v} (ξ) |}^{2} d ξ} \\ (2.5) & = & I_{low} (t) + I_{high} (t) . \end{array}$ The high frequency estimate is quite easy. $\begin{matrix} (2.6) & I_{high} (t) ⩽ e^{- t ζ} ‖ v ‖^{2} (t ⩾ 0) . \end{matrix}$ Concerning the low frequency estimate one uses the variable transformation to derive $\begin{array}{rcl} I_{low} (t) & ⩽ & C ω_{n} ‖ \hat{v} ‖_{\infty}^{2} \int_{0}^{1} e^{- 2 t ζ r^{2 σ}} r^{n - 1} d r ⩽ C ω_{n} ‖ v ‖_{1}^{2} \int_{0}^{1} e^{- 2 t ζ r^{2 σ}} r^{n - 1} d r \\ (2.7) & ⩽ & C ω_{n} {(2 ζ)}^{- \frac{n}{2 σ}} (\int_{0}^{\infty} e^{- x^{2 σ}} x^{n - 1} d x) ‖ v ‖_{1}^{2} {(1 + t)}^{- \frac{n}{2 σ}} (t ⩾ 0), \end{array}$ where $C > 0$ is a generous constant. Then (2.5)–(2.7) imply the desired estimates. □
Remark 2.1.
One can also derive the following estimate under less regularity on the data: for $v \in L^{1} (R^{n})$ and $ζ > 0$ there exists a constant $C > 0$ such that $\begin{matrix} ‖ e^{- t ζ A} v ‖ ⩽ C t^{- \frac{n}{4 σ}} ‖ v ‖_{1} (t > 0) . \end{matrix}$ However, it suffices to use Lemma 2.1 because one has chosen $v : = u_{j} \in L^{2} \cap L^{1}$ ( $j = 0, 1$ ) in our setting.

Now, it follows from (2.3) and Lemma 2.1 one can get the following estimate. $\begin{array}{rcl} ‖ u (t) - u_{diff} (t) ‖ & ⩽ & C {(1 + t)}^{- \frac{n}{4 σ}} ‖ u_{0} ‖_{1} + C e^{- α t} ‖ u_{0} ‖_{H^{σ}} \\ (2.8) & + C {(1 + t)}^{- \frac{n}{4 σ}} ‖ u_{1} ‖_{1} + C e^{- α t} ‖ u_{1} ‖ (t ⩾ 0) . \end{array}$ Here, $α > 0$ is a constant depending on η and ζ. In the next Section 3 we shall derive a more precise leading term from $u_{diff}$ in the $L^{1, 1}$ -framework, and use it in order to prove our main Theorems 1.1, 1.2 and 1.3.
3. Optimal estimates for the leading term

In this section, among others one will derive the lower and upper bounds for the possible leading terms defined by $\begin{matrix} (3.1) & K_{n} (t) : = \int_{R^{n}} e^{- t | ξ |^{2 σ}} \frac{{sin}^{2} (t | ξ |^{σ})}{| ξ |^{2 σ}} d ξ = {‖ e^{- t | ξ |^{2 σ} / 2} \frac{sin (t | ξ |^{σ})}{| ξ |^{σ}} ‖}^{2} . \end{matrix}$ Let us divide $K_{n} (t)$ into two parts by using the variable transform: $\begin{array}{l} (3.2) & K_{n} (t) = t^{- \frac{n - 2 σ}{2 σ}} I^{(n)} (t), \\ I^{(n)} (t) : = \int_{R^{n}} e^{- | η |^{2 σ}} \frac{{sin}^{2} (t^{1 / 2} | η |^{σ})}{| η |^{2 σ}} d η \\ (3.3) & = (\int_{| η | ⩽ t^{- 1 / 2 σ}} + \int_{| η | ⩾ t^{- 1 / 2 σ}}) e^{- | η |^{2 σ}} \frac{{sin}^{2} (t^{1 / 2} | η |^{σ})}{| η |^{2 σ}} d η \\ (3.4) & = : I_{low}^{(n)} (t) + I_{high}^{(n)} (t) . \end{array}$

One prepares the following fundamental remark, which will be used in the course of proof.

Remark 3.1.
Let $σ ⩾ 1$ . Then it holds that
$0 ⩽ sin (t^{1 / 2} | η |^{σ}) ⩽ t^{1 / 2} | η |^{σ}$ for $| η | ⩽ t^{- 1 / 2 σ}$ ,

there exists a large $t_{0} > 1$ such that $\frac{1}{2} ⩽ e^{- \frac{25 π^{2}}{16 t}} ⩽ e^{- \frac{1}{t}} ⩽ 2$ for $t ⩾ t_{0}$ ,

$2 sin (t^{1 / 2} | η |^{σ}) ⩾ t^{1 / 2} | η |^{σ}$ for $| η | ⩽ t^{- 1 / 2 σ}$ ,

$| sin (t^{\frac{1}{2}} | η |^{σ}) | ⩾ \frac{1}{\sqrt{2}}$ for ${(\frac{1}{4} + j) \frac{π}{\sqrt{t}}}^{\frac{1}{σ}} ⩽ | η | ⩽ {(\frac{3}{4} + j) \frac{π}{\sqrt{t}}}^{\frac{1}{σ}}$ ( $j = 1, 2, \dots$ ).

The following lemma is also useful.
Lemma 3.1.
Let $σ ⩾ 1$ , and $2 σ < n$ . Then, it holds that $\begin{matrix} lim_{t \to \infty} \int_{0}^{\infty} e^{- z^{2}} z^{\frac{n - 3 σ}{σ}} cos (2 \sqrt{t} z) d z = 0 . \end{matrix}$
Proof.
It is obvious from the Riemann–Lebesgue lemma and the fact that $\begin{matrix} \int_{0}^{1} z^{\frac{n - 3 σ}{σ}} d z = \frac{σ}{n - 2 σ} . \end{matrix}$ □

We first treat the higher dimensional case. This part is rather standard nowadays (see [9]).
Proposition 3.1.
Let $σ ⩾ 1$ , and $2 σ < n$ . Then, there exists a constant $C_{n, σ} > 0$ such that $\begin{matrix} C_{n, σ} t^{- \frac{n - 2 σ}{4 σ}} ⩽ ‖ e^{- t | ξ |^{2 σ} / 2} \frac{sin (t | ξ |^{σ})}{| ξ |^{σ}} ‖ ⩽ C_{n, σ} t^{- \frac{n - 2 σ}{4 σ}} (t ≫ 1) . \end{matrix}$
Proof.
It follows that $\begin{array}{rcl} K_{n} (t) & = & \frac{1}{2} \int_{R^{n}} e^{- t | ξ |^{2 σ}} \frac{1}{| ξ |^{2 σ}} (1 - cos (2 t | ξ |^{σ})) d ξ \\ = & \frac{ω_{n}}{2} \int_{0}^{\infty} e^{- {t r}^{2 σ}} r^{n - 2 σ - 1} d r - \frac{ω_{n}}{2} \int_{0}^{\infty} e^{- {t r}^{2 σ}} r^{n - 2 σ - 1} cos (2 {t r}^{σ}) d r \\ = & \frac{ω_{n}}{2} t^{- \frac{n - 2 σ}{2 σ}} \int_{0}^{\infty} e^{- x^{2 σ}} x^{n - 2 σ - 1} d x \\ (3.5) & - \frac{ω_{n}}{2} t^{- \frac{n - 2 σ}{2 σ}} \int_{0}^{\infty} e^{- x^{2 σ}} x^{n - 2 σ - 1} cos (2 \sqrt{t} x^{σ}) d x . \end{array}$ Since $\begin{matrix} \int_{0}^{\infty} e^{- x^{2 σ}} x^{n - 2 σ - 1} cos (2 \sqrt{t} x^{σ}) d x = \frac{1}{σ} \int_{0}^{\infty} e^{- z^{2}} z^{\frac{n - 3 σ}{σ}} cos (2 \sqrt{t} z) d z, \end{matrix}$ by using Lemma 3.1 and (3.5), one has $\begin{matrix} K_{n} (t) = A_{n, σ} t^{- \frac{n - 2 σ}{2 σ}} + o (t^{- \frac{n - 2 σ}{2 σ}}) (t \to \infty), \end{matrix}$ which implies the desired estimate. Here, $\begin{matrix} A_{n, σ} : = \frac{ω_{n}}{2} \int_{0}^{\infty} e^{- x^{2 σ}} x^{n - 2 σ - 1} d x . \end{matrix}$ □

Now, we concentrate on considering the case for $n ⩽ 2 σ$ . One follows the method introduced in [11].
Lemma 3.2.
Let $σ ⩾ 1$ , and $1 ⩽ n < 2 σ$ . Then, there exists a constant $C_{n, σ} > 0$ such that $\begin{matrix} \sqrt{K_{n} (t)} ⩽ C_{n, σ} t^{\frac{2 σ - n}{2 σ}} (t ≫ 1) . \end{matrix}$
Proof.
It follows from (1) of Remark 3.1 and the integration by parts that $\begin{array}{rcl} I_{low}^{(n)} (t) & ⩽ & t \int_{| η | ⩽ t^{- 1 / 2 σ}} e^{- | η |^{2 σ}} d η = t ω_{n} \int_{0}^{t^{- 1 / 2 σ}} e^{- r^{2 σ}} r^{n - 1} d r \\ = & t ω_{n} {\frac{1}{n} t^{- n / 2 σ} e^{- t^{- 1}} + \frac{2 σ}{n} \int_{0}^{t^{- 1 / 2 σ}} r^{2 σ + n - 1} e^{- r^{2 σ}} d r} \\ ⩽ & \frac{2 ω_{n}}{n} t^{1 - \frac{n}{2 σ}} + \frac{4 σ ω_{n}}{n} t^{1 - \frac{1}{2 σ} - \frac{2 σ + n - 1}{2 σ}} \\ (3.6) & ⩽ & C_{n, σ} t^{1 - \frac{n}{2 σ}} (t ≫ 1), \end{array}$ where one has just used (2) of Remark 3.1 and the monotone increasing property of the function $r \mapsto r^{2 σ + n - 1} e^{- r^{2 σ}}$ on $[0, t^{- 1 / 2 σ}]$ ( $t ≫ 1$ ).

On the other hand, by using (2) of Remark 3.1 and the integration by parts one can get $\begin{array}{rcl} I_{high}^{(n)} (t) & ⩽ & \int_{| η | ⩾ t^{- 1 / 2 σ}} e^{- | η |^{2 σ}} | η |^{- 2 σ} d η = ω_{n} \int_{t^{- \frac{1}{2 σ}}}^{\infty} e^{- r^{2 σ}} r^{n - 2 σ - 1} d r \\ = & - \frac{ω_{n}}{n - 2 σ} t^{- \frac{n - 2 σ}{2 σ}} e^{- t^{- 1}} + \frac{2 σ ω_{n}}{n - 2 σ} \int_{t^{- \frac{1}{2 σ}}}^{\infty} r^{n - 1} e^{- r^{2 σ}} d r \\ ⩽ & \frac{2 ω_{n}}{2 σ - n} t^{- \frac{n - 2 σ}{2 σ}} - \frac{2 σ ω_{n}}{2 σ - n} \int_{t^{- \frac{1}{2 σ}}}^{\infty} r^{n - 1} e^{- r^{2 σ}} d r \\ (3.7) & ⩽ & \frac{2 ω_{n}}{2 σ - n} t^{- \frac{n - 2 σ}{2 σ}} (t ≫ 1) . \end{array}$ (3.1), (3.2), (3.3), (3.4), (3.6) and (3.7) imply the desired upper bound. □
Lemma 3.3.
Let $σ ⩾ 1$ , and $1 ⩽ n < 2 σ$ . Then, there exists a constant $C_{n, σ} > 0$ such that $\begin{matrix} C_{n, σ} t^{\frac{2 σ - n}{2 σ}} ⩽ \sqrt{K_{n} (t)} (t ≫ 1) . \end{matrix}$
Proof.
It follows from (3) of Remark 3.1 and the integration by parts that $\begin{array}{rcl} I_{low}^{(n)} (t) & ⩾ & \frac{t}{4} \int_{| η | ⩽ t^{- 1 / 2 σ}} e^{- | η |^{2 σ}} d η = \frac{t}{4} ω_{n} \int_{0}^{t^{- 1 / 2 σ}} e^{- r^{2 σ}} r^{n - 1} d r \\ = & \frac{t}{4} ω_{n} {\frac{1}{n} t^{- n / 2 σ} e^{- t^{- 1}} + \frac{2 σ}{n} \int_{0}^{t^{- 1 / 2 σ}} r^{2 σ + n - 1} e^{- r^{2 σ}} d r} ⩾ \frac{ω_{n}}{8 n} t^{1 - \frac{n}{2 σ}} \\ (3.8) & = & C_{n, σ} t^{1 - \frac{n}{2 σ}} (t ≫ 1), \end{array}$ where one has just used (2) of Remark 3.1.

(3.1), (3.2), (3.3), (3.4) and (3.8) imply the desired lower bound. □

By summarizing Lemmas 3.2 and 3.3 one has arrived at the following proposition.
Proposition 3.2.
Let $σ ⩾ 1$ , and $1 ⩽ n < 2 σ$ . Then, there exist constants $C_{j, n, σ} > 0$ ( $j = 1, 2$ ) depending only on n and σ such that $\begin{matrix} C_{1, n, σ} t^{\frac{2 σ - n}{2 σ}} ⩽ ‖ e^{- t | ξ |^{2 σ} / 2} \frac{sin (t | ξ |^{σ})}{| ξ |^{σ}} ‖ ⩽ C_{2, n, σ} t^{\frac{2 σ - n}{2 σ}} (t ≫ 1) . \end{matrix}$

Let us consider the critical case of $n = 2 σ$ . One can get the following lemma.
Lemma 3.4.
Let $σ ⩾ 1$ , and $n = 2 σ$ . Then, there exists a constant $C_{n, σ} > 0$ such that $\begin{matrix} K_{n} (t) ⩽ C_{n, σ} log t (t ≫ 1) . \end{matrix}$
Proof.
It follows from (1) of Remark 3.1 and the integration by parts that $\begin{array}{rcl} I_{low}^{(n)} (t) & ⩽ & t ω_{n} \int_{0}^{t^{- \frac{1}{n}}} r^{n - 1} e^{- r^{n}} d r = \frac{ω_{n}}{n} e^{- t^{- 1}} + ω_{n} t \int_{0}^{t^{- \frac{1}{n}}} r^{2 n - 1} e^{- r^{n}} d r \\ ⩽ & \frac{2 ω_{n}}{n} + ω_{n} t^{1 - \frac{1}{n} - \frac{2 n - 1}{n}} e^{- t^{- 1}} \\ (3.9) & = & \frac{2 ω_{n}}{n} + 2 ω_{n} t^{- 1} ⩽ C_{n, σ} (t ≫ 1), \end{array}$ where one has just used (2) of Remark 3.1, the assumption $n = 2 σ$ , and the monotone increasing property of the function $r \mapsto r^{2 n - 1} e^{- r^{n}}$ on $[0, t^{- 1 / n}]$ ( $t ≫ 1$ ).

On the other hand, by using (2) of Remark 3.1 and the integration by parts one can get $\begin{array}{rcl} I_{high}^{(n)} (t) & ⩽ & \int_{| η | ⩾ t^{- 1 / 2 σ}} e^{- | η |^{2 σ}} | η |^{- 2 σ} d η = ω_{n} \int_{t^{- \frac{1}{2 σ}}}^{\infty} e^{- r^{2 σ}} r^{- 1} d r \\ = & - ω_{n} e^{- t^{- 1}} log (t^{- \frac{1}{2 σ}}) + 2 σ ω_{n} \int_{t^{- \frac{1}{2 σ}}}^{\infty} r^{2 σ - 1} (log r) e^{- r^{2 σ}} d r \\ ⩽ & \frac{ω_{n}}{2 σ} e^{- t^{- 1}} log t + 2 σ ω_{n} \int_{1}^{\infty} r^{2 σ - 1} log r e^{- r^{2 σ}} d r \\ (3.10) & ⩽ & \frac{ω_{n}}{σ} log t + 2 σ ω_{n} G_{0} ⩽ C_{n, σ} log t (t ≫ 1), \end{array}$ where $\begin{matrix} G_{0} : = \int_{1}^{\infty} r^{2 σ - 1} e^{- r^{2 σ}} log r d r < + \infty, \end{matrix}$ where one has just taken $t ≫ 1$ sufficiently large such that $t^{- \frac{1}{2 σ}} < 1$ , and $\begin{matrix} 2 σ ω_{n} G_{0} < (C_{n, σ} - \frac{ω_{n}}{σ}) log t \end{matrix}$ with a constant $C_{n, σ} > ω_{n} / σ$ . (3.1), (3.2), (3.3), (3.4), (3.9) and (3.10) imply the desired upper bound. □

Furthermore, one can also obtain the following result.
Lemma 3.5.
Let $σ ⩾ 1$ , and $n = 2 σ$ . Then, there exists a constant $C_{n, σ} > 0$ such that $\begin{matrix} C_{n, σ} log t ⩽ K_{n} (t) (t ≫ 1) . \end{matrix}$
Proof.
It follows from (3) of Remark 3.1 and the integration by parts that $\begin{array}{rcl} I_{low}^{(n)} (t) & ⩾ & \frac{t}{4} ω_{n} \int_{0}^{t^{- 1 / n}} e^{- r^{n}} r^{n - 1} d r \\ = & \frac{t ω_{n}}{4 n} t^{- 1} e^{- t^{- 1}} + \frac{ω_{n}}{4} t \int_{0}^{t^{- 1 / n}} r^{2 n - 1} e^{- r^{n}} d r \\ (3.11) & = & \frac{ω_{n}}{4 n} e^{- t^{- 1}} + \frac{ω_{n}}{4} t \int_{0}^{t^{- 1 / n}} r^{2 n - 1} e^{- r^{n}} d r ⩾ \frac{ω_{n}}{8 n} (t ≫ 1), \end{array}$ where one has just used (2) of Remark 3.1 and the relation $n = 2 σ$ .

On the other hand, by using (4) of Remark 3.1 and the relation $n = 2 σ$ one can get $\begin{array}{rcl} I_{high}^{(n)} (t) & = & ω_{n} \int_{t^{- 1 / 2 σ}}^{\infty} e^{- r^{2 σ}} r^{n - 1 - 2 σ} {sin}^{2} (t^{1 / 2} r^{σ}) d r \\ ⩾ & \frac{ω_{n}}{2} \sum_{j = 1}^{\infty} \int_{k_{j}}^{k_{j}^{'}} e^{- r^{2 σ}} r^{- (2 σ - n + 1)} d r = \frac{ω_{n}}{2} \sum_{j = 1}^{\infty} \int_{k_{j}}^{k_{j}^{'}} e^{- r^{2 σ}} r^{- 1} d r, \end{array}$ where $\begin{matrix} k_{j} : = {(\frac{1}{4} + j) \frac{π}{\sqrt{t}}}^{1 / σ}, k_{j}^{'} : = {(\frac{3}{4} + j) \frac{π}{\sqrt{t}}}^{1 / σ} (j = 1, 2, \dots) . \end{matrix}$ Thus, by relying on (2) of Remark 3.1 and the integration by parts it follows that $\begin{array}{rcl} I_{high}^{(n)} (t) & ⩾ & \frac{ω_{n}}{4} \int_{{(\frac{5 π}{4 \sqrt{t}})}^{1 / σ}}^{\infty} e^{- r^{2 σ}} r^{- 1} d r \\ = & - \frac{ω_{n}}{4} log {{(\frac{5 π}{4})}^{\frac{1}{σ}} t^{- \frac{1}{2 σ}}} e^{- 25 π^{2} / 16 t} + \frac{σ ω_{n}}{2} \int_{{(5 π / 4 \sqrt{t})}^{1 / σ}}^{\infty} e^{- r^{2 σ}} r^{2 σ - 1} (log r) d r \\ ⩾ & \frac{ω_{n}}{16 σ} log t - \frac{ω_{n}}{2 σ} log (\frac{5 π}{4}) - \frac{σ ω_{n}}{2} \int_{0}^{\infty} r^{2 σ - 1} | log r | e^{- r^{2 σ}} d r \\ = & \frac{ω_{n}}{16 σ} log t - \frac{ω_{n}}{2 σ} log (\frac{5 π}{4}) - \frac{σ ω_{n}}{2} G_{1} \\ (3.12) & ⩾ & C_{n, σ} log t (t ≫ 1) \end{array}$ with some generous constant $C_{n, σ} > 0$ . Here, $\begin{matrix} G_{1} : = \int_{0}^{\infty} r^{2 σ - 1} | log r | e^{- r^{2 σ}} d r < + \infty \end{matrix}$ because of the following computation: $\begin{matrix} \int_{0}^{1} r^{2 σ - 1} (- log r) d r = \frac{1}{4 σ^{2}} (σ > 0) . \end{matrix}$ (3.1), (3.2), (3.3), (3.4), (3.11) and (3.12) imply the desired lower bound. □

By summarizing Lemmas 3.4 and 3.5 one has arrived at the following proposition. Proposition 3.3.
Let $σ ⩾ 1$ , and $n = 2 σ$ . Then, there exist constants $C_{j, n, σ} > 0$ ( $j = 1, 2$ ) depending only on n and σ such that $\begin{matrix} C_{1, n, σ} \sqrt{log t} ⩽ ‖ e^{- t | ξ |^{2 σ} / 2} \frac{sin (t | ξ |^{σ})}{| ξ |^{σ}} ‖ ⩽ C_{2, n, σ} \sqrt{log t} (t ≫ 1) . \end{matrix}$

4. Proof of results

In this section, let us prove our main results by relying on the following a series of inequalities, which is based on the Plancherel theorem: $\begin{array}{rcl} C_{n} ‖ u (t, \cdot) ‖ & = & ‖ F (u (t, \cdot)) (ξ) ‖ \\ (4.1) & ⩾ & | P_{1} | ‖ e^{- t | ξ |^{2 σ} / 2} \frac{sin (t | ξ |^{σ})}{| ξ |^{σ}} ‖ - ‖ F (u (t, \cdot)) (ξ) - P_{1} e^{- t | ξ |^{2 σ} / 2} \frac{sin (t | ξ |^{σ})}{| ξ |^{σ}} ‖ \end{array}$ and $\begin{array}{rcl} C_{n} ‖ u (t, \cdot) ‖ & = & ‖ F (u (t, \cdot)) (ξ) ‖ \\ (4.2) & ⩽ & ‖ F (u (t, \cdot)) (ξ) - P_{1} e^{- t | ξ |^{2 σ} / 2} \frac{sin (t | ξ |^{σ})}{| ξ |^{σ}} ‖ + | P_{1} | ‖ e^{- t | ξ |^{2 σ} / 2} \frac{sin (t | ξ |^{σ})}{| ξ |^{σ}} ‖, \end{array}$ where the moment of the initial velocity $u_{1} (x)$ is defined by $\begin{matrix} P_{1} : = \int_{R^{n}} u_{1} (x) d x \in R . \end{matrix}$

We prepare several lemmas.

Lemma 4.1.

Let $σ ⩾ 1$ , and $u_{0} \in L^{1} (R^{n}) \cap L^{2} (R^{n})$ . Then, it holds that $\begin{matrix} ‖ e^{- t | ξ |^{2 σ} / 2} cos (t | ξ |^{σ}) {\hat{u}}_{0} ‖ ⩽ C {(1 + t)}^{- \frac{n}{4 σ}} ‖ u_{0} ‖_{1} + C e^{- \frac{t}{2}} ‖ u_{0} ‖ (t ⩾ 0) . \end{matrix}$

Proof.

Indeed, $\begin{array}{rcl} \int_{R^{n}} {| cos (t | ξ |^{σ}) {\hat{u}}_{0} (ξ) |}^{2} e^{- t | ξ |^{2 σ}} d ξ & ⩽ & (\int_{| ξ | ⩽ 1} + \int_{| ξ | ⩾ 1}) {| {\hat{u}}_{0} (ξ) |}^{2} e^{- t | ξ |^{2 σ}} d ξ \\ ⩽ & e ‖ u_{0} ‖_{1}^{2} \int_{| ξ | ⩽ 1} e^{- (1 + t) | ξ |^{2 σ}} d ξ + e^{- t} ‖ u_{0} ‖^{2} . \end{array}$ It is easy to check the following inequality. $\begin{matrix} (4.3) & \int_{| ξ | ⩽ 1} e^{- (1 + t) | ξ |^{2 σ}} d ξ ⩽ C {(1 + t)}^{- \frac{n}{2 σ}} . \end{matrix}$ So, one has the desired estimate. □

Here, let $\begin{matrix} A (ξ) : = \int_{R^{n}} (cos (x \cdot ξ)) - 1) u_{1} (x) d x, B (ξ) : = \int_{R^{n}} sin (x \cdot ξ)) u_{1} (x) d x . \end{matrix}$ Then, one can get the following decomposition of ${\hat{u}}_{1} (ξ)$ : $\begin{matrix} (4.4) & {\hat{u}}_{1} (ξ) = P_{1} + A (ξ) - i B (ξ), \end{matrix}$ where $i : = \sqrt{- 1}$ . We recall the following inequality coming from [8].

Lemma 4.2.

Let $n ⩾ 1$ , and let $u_{1} \in L^{1, 1} (R^{n})$ . Then, it holds that $\begin{array}{l} | A (ξ) | ⩽ L | ξ | ‖ u_{1} ‖_{1, 1} (ξ \in R^{n}), \\ | B (ξ) | ⩽ M | ξ | ‖ u_{1} ‖_{1, 1} (ξ \in R^{n}), \end{array}$ where $\begin{matrix} L : = sup_{θ \neq 0} \frac{| 1 - cos θ |}{| θ |}, M : = sup_{θ \neq 0} \frac{| sin θ |}{| θ |} = 1 . \end{matrix}$

Based on Lemma 4.2 above, one can prove the following lemma.

Lemma 4.3.

Let $σ ⩾ 1$ , $n ⩾ 1$ , $n ⩾ 2 σ$ and $u_{1} \in L^{1, 1} (R^{n})$ . Then, it is true that $\begin{matrix} \int_{R^{n}} e^{- t | ξ |^{2 σ}} \frac{sin (t | ξ |^{σ})}{| ξ |^{2 σ}} {| (A (ξ) - i B (ξ)) |}^{2} d ξ ⩽ C t^{- \frac{n - 2 σ + 2}{2 σ}} ‖ u_{1} ‖_{1, 1}^{2} (t > 0) . \end{matrix}$

Proof.

Indeed, it follows from Lemma 4.2 that $\begin{array}{l} \int_{R^{n}} e^{- t | ξ |^{2 σ}} \frac{{sin}^{2} (t | ξ |^{σ})}{| ξ |^{2 σ}} {| (A (ξ) - i B (ξ)) |}^{2} d ξ \\ ⩽ (L^{2} + M^{2}) ‖ u_{1} ‖_{1, 1}^{2} \int_{R^{n}} e^{- t | ξ |^{2 σ}} | ξ |^{- (2 σ - 2)} d ξ \\ = (L^{2} + M^{2}) ω_{n} ‖ u_{1} ‖_{1, 1}^{2} \int_{0}^{\infty} e^{- {t r}^{2 σ}} r^{n - 2 σ + 1} d r ⩽ C t^{- \frac{n - 2 σ + 2}{2 σ}} ‖ u_{1} ‖_{1, 1}^{2}, \end{array}$ which implies the desired estimate. □

It follows from (2.8), Lemmas 4.1 and 4.3 that (see (1.6) for $I_{0}$ ).

Lemma 4.4.

Let $σ ⩾ 1$ , $n ⩾ 1$ , $n ⩾ 2 σ$ . If $u_{0} \in H^{σ} (R^{n}) \cap L^{1} (R^{n})$ and $u_{1} \in L^{1, 1} (R^{n}) \cap L^{2} (R^{n})$ , then, it holds that $\begin{matrix} ‖ F (u (t, \cdot)) - P_{1} e^{- t | ξ |^{2 σ} / 2} \frac{sin (t | ξ |^{σ})}{| ξ |^{σ}} ‖ ⩽ C I_{0} (t^{- \frac{n}{4 σ}} + t^{- \frac{n - 2 σ + 2}{4 σ}}) (t ≫ 1) \end{matrix}$ with some constant $C > 0$ .

Proof.

Set $F (u (t, \cdot)) (ξ) - u_{diff} (t, ξ) : = ε (t, ξ)$ . Then, one has $\begin{matrix} F (u (t, \cdot)) (ξ) = e^{- t | ξ |^{2 σ} / 2} \frac{sin (t | ξ |^{σ})}{| ξ |^{σ}} {\hat{u}}_{1} (ξ) + ε (t, ξ) + e^{- t | ξ |^{2 σ} / 2} cos (t | ξ |^{σ}) {\hat{u}}_{0} (ξ) . \end{matrix}$ Now, we use the decomposition (4.4) to obtain $\begin{array}{rcl} F (u (t, \cdot)) (ξ) & = & P_{1} e^{- t | ξ |^{2 σ} / 2} \frac{sin (t | ξ |^{σ})}{| ξ |^{σ}} + e^{- t | ξ |^{2 σ} / 2} \frac{sin (t | ξ |^{σ})}{| ξ |^{σ}} (A (ξ) - i B (ξ)) \\ (4.5) & + ε (t, ξ) + e^{- t | ξ |^{2 σ} / 2} cos (t | ξ |^{σ}) {\hat{u}}_{0} (ξ) . \end{array}$ Here, one notices from (2.8) that $\begin{matrix} (4.6) & ‖ ε (t, \cdot) ‖ ⩽ C {(1 + t)}^{- \frac{n}{4 σ}} ‖ u_{0} ‖ + C e^{- α t} ‖ u_{0} ‖_{H^{σ}} + C {(1 + t)}^{- \frac{n}{4 σ}} ‖ u_{1} ‖ + C e^{- α t} ‖ u_{1} ‖ (t ⩾ 0) . \end{matrix}$ Thus, from (4.6) and Lemmas 4.1 and 4.3 one can get the desired estimate. □

Now one can prove Theorems 1.1 and 1.2.

Proof of Theorems 1.1 and 1.2.

Indeed, the results of Theorems 1.1 and 1.2 can be derived by combining Lemma 4.4, (4.1), (4.2), with Propositions 3.1 and 3.3. □

One needs one more lemmas in order to complete the proof of Theorem 1.3 for the case of $n < 2 σ$ . This case is rather delicate.

Assume $n < 2 σ$ below. Then, one has $\begin{matrix} (4.7) & \int_{R^{n}} e^{- t | ξ |^{2 σ}} \frac{{sin}^{2} (t | ξ |^{σ})}{| ξ |^{2 σ}} | ξ |^{2} d ξ = t^{- \frac{n - 2 σ + 2}{2 σ}} J^{(n)} (t), \end{matrix}$ where $\begin{array}{rcl} J^{(n)} (t) & : = & \int_{R^{n}} e^{- | η |^{2 σ}} {sin}^{2} (t^{1 / 2} | η |^{σ}) | η |^{2 - 2 σ} d η \\ (4.8) & = & (\int_{| η | ⩽ t^{- 1 / 2 σ}} + \int_{| η | ⩾ t^{- 1 / 2 σ}}) e^{- | η |^{2 σ}} {sin}^{2} (t^{1 / 2} | η |^{σ}) | η |^{2 - 2 σ} d η \\ (4.9) & = : & J_{low}^{(n)} (t) + J_{high}^{(n)} (t) . \end{array}$ Let us obtain the $(n < 2 σ)$ -version of Lemma 4.3 by estimating $J_{low}^{(n)} (t)$ and $J_{high}^{(n)} (t)$ .

Lemma 4.5.

Let $σ ⩾ 1$ , and $1 ⩽ n < 2 σ$ . Then, there exists a constant $C_{n, σ} > 0$ such that $\begin{array}{l} \int_{R^{n}} e^{- t | ξ |^{2 σ}} \frac{sin (t | ξ |^{σ})}{| ξ |^{2 σ}} {| (A (ξ) - i B (ξ)) |}^{2} d ξ \\ ⩽ \{\begin{matrix} C_{n, σ} ‖ u_{1} ‖_{1, 1}^{2} t^{\frac{2 σ - n - 2}{σ}} & (2 σ - n - 2 > 0) \\ C_{n, σ} ‖ u_{1} ‖_{1, 1}^{2} log t & (2 σ - n - 2 = 0) \\ C_{n, σ} ‖ u_{1} ‖_{1, 1}^{2} (t^{\frac{2 σ - n - 2}{2 σ}} + t^{\frac{2 σ - n - 2}{σ}}) & (2 σ - n - 2 < 0) \end{matrix} \end{array}$ for $t ≫ 1$ .

Proof.

It follows from (1) and (2) of Remark 3.1 and the integration by parts that $\begin{array}{rcl} J_{low}^{(n)} (t) & ⩽ & t \int_{| η | ⩽ t^{- 1 / 2 σ}} e^{- | η |^{2 σ}} | η |^{2} d η = t ω_{n} \int_{0}^{t^{- 1 / 2 σ}} e^{- r^{2 σ}} r^{n + 1} d r \\ = & \frac{ω_{n}}{n + 2} t^{1 - \frac{n + 2}{2 σ}} e^{- t^{- 1}} + \frac{2 σ t ω_{n}}{n + 2} \int_{0}^{t^{- 1 / 2 σ}} r^{n + 2 σ + 1} e^{- r^{2 σ}} d r \\ ⩽ & \frac{2 ω_{n}}{n + 2} t^{1 - \frac{n + 2}{2 σ}} + \frac{4 σ ω_{n}}{n + 2} t^{1 - \frac{n + 2 σ + 1}{2 σ} - \frac{1}{2 σ}} \\ (4.10) & ⩽ & C_{n, σ} t^{\frac{2 σ - (n + 2)}{2 σ}} (t ≫ 1), \end{array}$ where one has just used the monotone increasing property of the function $r \mapsto r^{n + 2 σ + 1} e^{- r^{2 σ}}$ on $[0, t^{- 1 / 2 σ}]$ ( $t ≫ 1$ ).

On the other hand, by using (1) and (2) of Remark 3.1 and the integration by parts one can get formally $\begin{array}{rcl} (4.11) & J_{high}^{(n)} (t) & ⩽ & \int_{| η | ⩾ t^{- 1 / 2 σ}} e^{- | η |^{2 σ}} | η |^{2 - 2 σ} d η = ω_{n} \int_{t^{- \frac{1}{2 σ}}}^{\infty} e^{- r^{2 σ}} r^{n - 2 σ + 1} d r \\ = & - \frac{ω_{n}}{n - 2 σ + 2} t^{- \frac{n - 2 σ + 2}{2 σ}} e^{- t^{- 1}} + \frac{2 σ ω_{n}}{n - 2 σ + 2} \int_{t^{- \frac{1}{2 σ}}}^{\infty} r^{n + 1} e^{- r^{2 σ}} d r \\ (4.12) & = & \frac{ω_{n}}{(2 σ - n) - 2} t^{\frac{2 σ - n}{2 σ} - \frac{1}{σ}} e^{- t^{- 1}} - \frac{2 σ ω_{n}}{(2 σ - n) - 2} \int_{t^{- \frac{1}{2 σ}}}^{\infty} r^{n + 1} e^{- r^{2 σ}} d r . \end{array}$ Therefore, from (4.12) if $2 σ - n > 2$ , then $\begin{matrix} (4.13) & J_{high}^{(n)} (t) ⩽ \frac{2 ω_{n}}{(2 σ - n) - 2} t^{\frac{2 σ - n}{2 σ} - \frac{1}{σ}} . \end{matrix}$ In the case when $2 σ - n = 2$ , it follows from (4.11) and the integration by parts that $\begin{array}{rcl} J_{high}^{(n)} (t) & ⩽ & ω_{n} \int_{t^{- \frac{1}{2 σ}}}^{\infty} e^{- r^{2 σ}} r^{- 1} d r \\ = & \frac{ω_{n}}{2 σ} log t + 2 σ ω_{n} \int_{t^{- \frac{1}{2 σ}}}^{1} r^{2 σ - 1} (log r) e^{- r^{2 σ}} d r + 2 σ ω_{n} \int_{1}^{\infty} r^{2 σ - 1} (log r) e^{- r^{2 σ}} d r \\ (4.14) & ⩽ & \frac{ω_{n}}{2 σ} log t + 2 σ ω_{n} \int_{1}^{\infty} r^{2 σ - 1} (log r) e^{- r^{2 σ}} d r ⩽ C_{n, σ} log t (t ≫ 1) \end{array}$ with some constant $C_{n, σ} > 0$ . Here, one takes $t > 0$ sufficiently large such that (at least) $t^{- \frac{1}{2 σ}} < 1$ .

While, in the case when $2 > 2 σ - n > 0 \Leftrightarrow 2 σ - n - 2 < 0$ , it follows from (4.12) that $\begin{array}{rcl} J_{high}^{(n)} (t) & ⩽ & \frac{ω_{n}}{(2 σ - n) - 2} t^{\frac{2 σ - n}{2 σ} - \frac{1}{σ}} e^{- t^{- 1}} + \frac{2 σ ω_{n}}{2 + n - 2 σ} \int_{t^{- \frac{1}{2 σ}}}^{\infty} r^{n + 1} e^{- r^{2 σ}} d r \\ ⩽ & - \frac{ω_{n}}{n + 2 - 2 σ} t^{\frac{2 σ - n}{2 σ} - \frac{1}{σ}} e^{- t^{- 1}} + \frac{2 σ ω_{n}}{2 + n - 2 σ} \int_{0}^{\infty} r^{n + 1} e^{- r^{2 σ}} d r \\ (4.15) & ⩽ & \frac{2 σ ω_{n}}{2 + n - 2 σ} \int_{0}^{\infty} r^{n + 1} e^{- r^{2 σ}} d r ⩽ C_{n, σ} (t ≫ 1) . \end{array}$ (4.7), (4.8), (4.9), (4.10), (4.13), (4.14), (4.15) and Lemma 4.2 imply the desired upper bound. □

Now, one can get the following important lemma from (4.5), (4.6) and Lemmas 4.1 and 4.5.

Lemma 4.6.

Let $σ ⩾ 1$ , and $1 ⩽ n < 2 σ$ . If $u_{0} \in H^{σ} (R^{n}) \cap L^{1} (R^{n})$ and $u_{1} \in L^{1, 1} (R^{n}) \cap L^{2} (R^{n})$ , then there exists a constant $C_{n, σ} > 0$ such that $\begin{matrix} ‖ F (u (t, \cdot)) - P_{1} e^{- t | ξ |^{2 σ} / 2} \frac{sin (t | ξ |^{σ})}{| ξ |^{σ}} ‖ ⩽ \{\begin{matrix} C_{n, σ} I_{0} t^{\frac{2 σ - n - 2}{2 σ}} & (2 σ - n - 2 > 0) \\ C_{n, σ} I_{0} \sqrt{log t} & (2 σ - n - 2 = 0) \\ C_{n, σ} I_{0} t^{- \frac{n + 2 - 2 σ}{4 σ}} & (2 σ - n - 2 < 0) \end{matrix} \end{matrix}$ for $t ≫ 1$ , where $I_{0}$ is defined in ( 1.6 ).

Let us finalize the proof of Theorem 1.3.

Proof of Theorem 1.3.

Indeed, the results of Theorem 1.3 can be derived by combining (4.1), (4.2), Lemma 4.6 with Proposition 3.2. □

Footnotes

Acknowledgements

The author would like to thank Professor Grozdena Todorova (the University of Tennessee, Knoxville) for her useful advice on this kind of problem. The author also would like to thank two referees for their careful reading and helpful suggestions. The work of the author was supported in part by Grant-in-Aid for Scientific Research (C) 15K04958 of JSPS.

References

R.C.

Charão,

C.R.

da Luz and

Ikehata, Sharp decay rates for wave equations with a fractional damping via new method in the Fourier space, J. Math Anal. Appl. 408(1) (2013), 247–255. doi:10.1016/j.jmaa.2013.06.016.

D’Abbicco,

Girardi and

Liang,

L^{1}

–

L^{1}

estimates for the strongly damped plate equation, J. Math. Anal. Appl. 478 (2019), 476–498. doi:10.1016/j.jmaa.2019.05.039.

D’Abbicco and

Lucente, The beam equation with nonlinear memory, Z. Angew. Math. Phys. 67 (2016), 60. doi:10.1007/s00033-016-0655-x.

D’Abbicco and

Reissig, Semilinear structural damped waves, Math. Methods Appl. Sci. 37(11) (2014), 1570–1592. doi:10.1002/mma.2913.

T.A.

Dao and

Michihisa, Study of semi-linear σ-evolution equations with frictional and visco-elastic damping, Commun. Pure Appl. Anal. 19(3) (2020), 1581–1608. doi:10.3934/cpaa.2020079.

T.A.

Dao and

Reissig,

L^{1}

estimates for oscillating integrals and their applications to semi-linear models with σ-evolution like structural damping, Discrete Contin. Dyn. Syst. A 39 (2019), 5431–5463. doi:10.3934/dcds.2019222.

Ghisi,

Gobbino and

Haraux, Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation, Trans. Amer. Math. Soc. 368(3) (2016), 2039–2079. doi:10.1090/tran/6520.

Ikehata, New decay estimates for linear damped wave equations and its application to nonlinear problem, Math. Meth. Appl. Sci. 27 (2004), 865–889. doi:10.1002/mma.476.

Ikehata, Asymptotic profiles for wave equations with strong damping, J. Diff. Eqns 257 (2014), 2159–2177. doi:10.1016/j.jde.2014.05.031.

10.

Ikehata and

Natsume, Energy decay estimates for wave equations with a fractional damping, Diff. Int. Eqns 25(9–10) (2012), 939–956.

11.

Ikehata and

Onodera, Remarks on large time behavior of the

L^{2}

-norm of solutions to strongly damped wave equations, Diff. Int. Eqns 30 (2017), 505–520.

12.

Ikehata,

Todorova and

Yordanov, Wave equations with strong damping in Hilbert spaces, J. Diff. Eqns 254 (2013), 3352–3368. doi:10.1016/j.jde.2013.01.023.

13.

Nishihara,

L^{p}

–

L^{q}

estimates to the damped wave equation in 3-dimensional space and their application, Math. Z. 244 (2003), 631–649. doi:10.1007/s00209-003-0516-0.

14.

Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal. 9(5) (1985), 399–418. doi:10.1016/0362-546X(85)90001-X.

15.

Shibata, On the rate of decay of solutions to linear viscoelastic equation, Math. Meth. Appl. Sci. 23 (2000), 203–226. doi:10.1002/(SICI)1099-1476(200002)23:3<203::AID-MMA111>3.0.CO;2-M.