Estimates for Linear Wave Equation with Scale-Invariant Damping

Abstract

This study is devoted to deriving precise pointwise estimates for the linear Cauchy problem of the wave equation with scale-invariant damping. The proof relies on representing the solution via Fourier integral operators and employing techniques from microlocal analysis. As an application, we establish weighted Strichartz estimates for the linear system.

1. Introduction

We are interested in the properties of the solutions to the Cauchy problem

$${\partial }_t^{2}u-\Delta u+\frac{\mu }{t}{\partial }_{t}u=0,u(t_0,·)=u_0,{\partial }_t\begin{array}{c}\hfill u\end{array}(t_0,·)=u_1,$$

(1)

where $\mu >0$, $t>t_0>0$, $x\in {\mathbb{R}}^n$ and $u_0,u_1\in C_c^{\infty }\left({\mathbb{R}}^n\right)$, $n\ge \begin{array}{c}\hfill 4\end{array}$. Since Equation (1) is invariant under the scaling

$$\tilde{u}(t,x)=u(\sigma x,\sigma t),$$

where $\sigma >0$, the damping term is called scale-invariant.

The operator ${\partial }_t^2-\Delta +\frac{\mu }{t}{\partial }_t$ arises naturally in several physical models. In particular, it appears in three-dimensional compressible Euler equations with time-dependent damping and is closely related to the global existence and stability of smooth supersonic polytropic gas flows in a three-dimensional infinitely long divergent nozzle. We first consider three-dimensional compressible isentropic and irrotational Euler equations for $t>t_0>0$

$$\left\{\begin{array}{c}{\partial }_t\rho +\mathrm{div}\left(\rho \begin{array}{c}\hfill U\end{array}\right)=0,\hfill \\ {\partial }_t\left(\rho U\right)+\mathrm{div}(\rho \begin{array}{c}\hfill U\end{array}\otimes \begin{array}{c}\hfill U\end{array}+p{\mathrm{I}}_3)=-\frac{\mu }{t}\rho \begin{array}{c}\hfill U\end{array},\hfill \\ \rho (t_0,x)=\overline{\rho }+\epsilon {\rho }_0\left(x\right),\begin{array}{c}\hfill U\end{array}(t_0,x)=\epsilon {\begin{array}{c}\hfill U\end{array}}_0\left(x\right),\hfill \end{array}\right.$$

(2)

where $x=(x_1,x_2,x_3)$, and $\rho$, $\begin{array}{c}\hfill U\end{array}=({\begin{array}{c}\hfill U\end{array}}_1,{\begin{array}{c}\hfill U\end{array}}_2,{\begin{array}{c}\hfill U\end{array}}_3)$, and p denote the density, velocity, and pressure, respectively. Here, ${\mathrm{I}}_3$ denotes a $3\times 3$ identity matrix, while $\mu >0$ and $\overline{\rho }>0$ are constants. We assume that $({\rho }_0,U_0)\not\equiv 0$, $\rho (t_0,x)>0$, and that $\epsilon >0$ is sufficiently small. In addition, the equation of state is $p\left(\rho \right)={\rho }^{\gamma }$, where $\gamma >1$ is a constant. In this case, by introducing a scalar potential function $\phi =\phi (t,x)$ with $\begin{array}{c}\hfill U\end{array}={\nabla }_x\phi$, we substitute $\psi =\frac{\phi }{t}$ into the second equation of system (2) to obtain

$$\square \psi +\frac{\mu +2}{t}{\partial }_t\psi =Q(\psi ,\partial \psi ,{\partial }^2\psi ),$$

Thus, the linear wave operator

$$\square +\frac{\mu +2}{t}{\partial }_t$$

with scale-invariant damping arises naturally from this physical model. In the present paper, we study the more general operator

$$\square +\frac{\mu }{t}{\partial }_t,\mu >0$$

which includes the physically relevant case described above.

Additionally, we discuss the global existence and stability of smooth supersonic polytropic gas flow in a 3D infinitely long divergent nozzle. Consider the following steady compressible isentropic Euler equations

$$\left\{\begin{array}{c}{\sum }_{j=1}^3{\partial }_j\left(\rho v_j\right)=0,\hfill \\ {\sum }_{j=1}^3{\partial }_j\left(\rho v_i{v}_j\right)+{\partial }_{i}p=0,i=1,2,3,\hfill \end{array}\right.$$

where $\rho ,v=(v_1,v_2,v_3)$ and p represent the density, velocity and pressure of polytropic gases, respectively. In addition, the state equation is $p={\rho }^{\gamma }$ for $\gamma >1$. Introducing a potential function $\mathsf{\Phi }\left(x\right)$ such that $v={\nabla }_x\mathsf{\Phi }$, we find that $\mathsf{\Phi }$ satisfies a quasilinear wave equation whose linearization contains a scale-invariant damping term.

The behavior of solutions to (1) is closely related to the long-time behavior of solutions to the following semilinear Cauchy problem:

$$\left\{\begin{array}{c}{u}_{tt}-\Delta u+\frac{\mu }{t}{u}_t={\left|u\right|}^p,t>t_0>0,x\in {\mathbb{R}}^n,\hfill \\ u(t_0,x)=u_0\left(x\right),u_t(t_0,x)=u_1\left(x\right),x\in {\mathbb{R}}^n.\hfill \end{array}\right.$$

(3)

The solution’s behavior transitions between wave-like and heat-like regimes depending on the size of the positive constant $\mu$. More precisely, small values of $\mu$ correspond to wave-like behavior, whereas large values of $\mu$ lead to heat-like behavior. Motivated by this observation, it is conjectured in [1,2] that there exists a threshold constant

$${\mu }_c\left(n\right)=\frac{n^2+n+2}{n+2},$$

such that the small-data Cauchy problem (3) admits a critical power

$$p_{\mathrm{crit}}=p_{\mathrm{crit}}(n,\mu )=\left\{\begin{array}{cc}{p}_S(n+\mu ),\hfill & \mathrm{if}0\le \mu \le {\mu }_c\left(n\right),\hfill \\ p_F\left(n\right),\hfill & \mathrm{if}\mu >{\mu }_c\left(n\right),\hfill \end{array}\right.$$

where “critical” means that solutions blow up in finite time when $1<p<p_{\mathrm{crit}}(n,\mu )$, whereas they exist globally in time when $p>p_{\mathrm{crit}}(n,\mu )$ with sufficiently small initial data. Here, $p_S\left(n\right)$ denotes the positive root of the quadratic equation

$$(n-1)p^2-(n+1)p-2=0,$$

and is the critical exponent for the small-data Cauchy problem of the classical semilinear wave equation

$$u_{tt}-\Delta u={\left|u\right|}^p,$$

while $p_F\left(n\right)=1+\frac{2}{n}$ denotes the critical exponent for the small-data Cauchy problem of the semilinear heat equation

$$u_t-\Delta u={\left|u\right|}^p.$$

For a general $\mu$, the finite-time blow-up part of the above conjecture has been established; see [3,4]. As for global existence, the case of $n=1$ was considered by D’Abbicco [5], who proved the existence of global solutions for small initial data when $\mu >0$ and $p>p_{\mathrm{crit}}(n,\mu )$. For $n\ge 2$, D’Abbicco [6] also obtained the global existence result for $p>p_F\left(n\right)$, provided that $n=2$ and $\mu \ge 3$, or $n\ge 3$ and $\mu \ge n+2$.

Recently, the global existence problem for (3) in the $0<\mu \le {\mu }_c\left(n\right)$ regime has attracted considerable attention. It is conjectured that the small-data solution would exist globally provided that $p>p_c(n,\mu )=p_S(n+\mu )$, where $p_S(n+\mu )$ is the positive root of

$$(n+\mu -1)p^2-(n+\mu +1)p-2=0.$$

He et al. [7] and Li et al. [8] considered this problem and obtained analogous results for $n=2$ and $0<\mu <{\mu }_c\left(2\right)$. In a recent preprint [9], He et al. established the global existence of solutions for sufficiently small initial data when $n\ge 3$, $\mu \in (0,1)\cup (1,2)$ and $p_S(n+\mu )<p\le p_{\mathrm{conf}}(n,\mu )$, where

$$p_{\mathrm{conf}}(n,\mu )=\frac{n+\mu +3}{n+\mu -1}$$

is the conformally invariant exponent associated with (3). The case $n\ge 4$ and $\mu =1$ was handled by He et al. [10] for $p_S(n+1)<p\le p_{\mathrm{conf}}(n,1)$. For $n\ge 4$ and $\mu =2$, global existence has been obtained only in the radially symmetric case; see D’Abbicco et al. [11] and Palmieri [12]. Motivated by these results, we restrict our attention to the case of $n\ge 4$ and establish the pointwise decay estimate for the linear homogeneous equation

$$\left\{\begin{array}{c}{\partial }_t^{2}v-\Delta v+\frac{\mu }{t}{\partial }_{t}v=0,(t,x)\in (2,\infty )\times {\mathbb{R}}^n,\hfill \\ \begin{array}{c}\hfill v\end{array}(2,x)=v_0\left(x\right),{\partial }_{t}v(2,x)=v_1\left(x\right),\hfill \end{array}\right.$$

(4)

for the following range of $\mu$:

$$2\le \mu \le {\mu }_c\left(n\right).$$

(5)

Without loss of generality, we assume throughout the remainder of this article that $v_0,v_1$ are smooth functions with $\mathrm{supp}(v_0,v_1)\subseteq \left\{\left|x\right|\le 1\right\}$. By representing the solution v in terms of Fourier integral operators and applying techniques from microlocal analysis, we obtain the following pointwise decay estimate, which is the main result in this paper.

Theorem 1.

Let $n\ge 4$ and $2\le \mu \le {\mu }_c\left(n\right)$. Then, for $0<\delta \ll 1$ the solution v to (4) satisfies

$$\left|v(t,x)\right|\le C(v_0,v_1){(1+t)}^{-\frac{n+\mu -1}{2}}{(1+|t-\left|x\right|\left|\right)}^{-\frac{n-\mu +1}{2}+\delta }.$$

(6)

Remark 1.

We recall that Wirth, in seminal work [13], considered the following variant of (1)

$$u_{tt}-\Delta u+\frac{\mu }{1+t}{u}_t=0,u(0,x)=u_0\left(x\right),u_t(0,x)=u_1\left(x\right).$$

By defining $\mathcal{S}\left(t\right):(u_0,{\langle D\rangle }^{-1}{u}_1)\mapsto u(t,·)$, Wirth [13] (Theorem 3.5) obtained $L^p-L^q$ estimates for $\mathcal{S}\left(t\right)$, where the decay depended solely on time. Indeed, for the case of $\mu >1$, Ref. [13] (Theorem 3.5) shows that, for $p\in (1,2]$, $pq=p+q$ and $r=n(\frac{1}{p}-\frac{1}{q})$,

$${\parallel \mathcal{S}\left(t\right)\parallel }_{W^{r,p}\left({\mathbb{R}}^n\right)\to L^q\left({\mathbb{R}}^n\right)}\le {(1+t)}^{\max \{-\frac{n-1}{2}(\frac{1}{p}-\frac{1}{q})-\frac{\mu }{2},-n(\frac{1}{p}-\frac{1}{q})\}}.$$

(7)

Since $p>1$ implies $q<\infty$, one cannot derive $L^{\infty }$ or pointwise estimates for the solution u from (7).

In [14,15], Wirth considered the Cauchy problem for the following damped wave equation:

$$u_{tt}-\Delta u+b\left(t\right)u_t=0,u(0,x)=u_0\left(x\right),u_t(0,x)=u_1\left(x\right)$$

with a general time-dependent coefficient $b\left(t\right)$. The admissible choices of $b\left(t\right)$ include $b\left(t\right)=\frac{\alpha }{{(1+t)}^{\beta }}$, where $\alpha >0$ and $\beta \ne 1$. Wirth obtained $L^p-L^q$ estimates similar to [13] (Theorem 3.5) with $p\in [1,2]$, where the decay rate still depends only on t; also see Reissig et al. [16] for related results.

In contrast to the results in [13,14,15,16], we establish precise pointwise decay estimates for solutions to the linear wave equation with scale-invariant damping in this article. Our result in Theorem 1 shows explicitly how the decay rate depends on both the time variable t and the spatial variable x. Specifically, we demonstrate that the solution exhibits enhanced decay away from the light cone, a finding that aligns perfectly with the classical theory of the wave equation.

Based on the pointwise decay estimate (6), we then establish the weighted Strichartz estimate for the solution v to (4) with

$$2\le \mu \le {\mu }_0\left(n\right),$$

where the endpoint ${\mu }_0\left(n\right)$ is defined by

$${\mu }_0\left(n\right)=\sqrt{n^2+4}-1.$$

(8)

Direct computation yields $2<{\mu }_0\left(n\right)<{\mu }_c\left(n\right)$ for all $n\ge 4$. Our second main result is stated in the following theorem.

Theorem 2.

Let $n\ge 4$ and $2\le \mu \le {\mu }_0\left(n\right)$. Assume that v solves (4). Then, we have the following:

$${∥(t^2-{\left|x\right|}^2{)}^{\gamma }{t}^{\frac{\mu }{q}}v∥}_{L^q([2,+\infty )\times {\mathbb{R}}^n)}\le C(v_0,v_1),$$

(9)

where $2\le q\le \frac{2(n+\mu +1)}{n+\mu -1}$ and $\gamma <\frac{n+\mu -1}{2}-\frac{n+\mu }{q}$.

The estimate in (9) is an important tool in proving the global existence of small-data solutions to the semilinear Cauchy problem (3). In fact, we have $p_c(n,\mu )<2$ for $n\ge 4$. Since the nonlinearity $F\left(u\right)={\left|u\right|}^p$ has only limited regularity when p is close to $p_c(n,\mu )$, in particular $F\left(u\right)\notin C^2$ in the relevant range, the vector field method cannot be directly applied to establish global existence for small-data solutions to (3). Therefore, suitable $L^p-L^q$ space–time estimates are necessary. More specifically, the upper bound of q satisfies

$$\frac{2(n+\mu +1)}{n+\mu -1}=\frac{n+\mu +3}{n+\mu -1}+1=p_{\mathrm{conf}}(n,\mu )+1.$$

Thus, estimate (9) can be applied to prove the global existence of small-data solutions to (3) when $p_S(n+\mu )<p\le p_{\mathrm{conf}}(n,\mu )$.

Finally, we mention two recent studies related to decay and weighted space–time estimates for damped wave equations. Gao et al. [17] investigated the long-time dynamic behavior of a system of coupled wave equations in parallel with nonlinear localized damping; they obtained a decay estimate of the energy. Ammari et al. [18] studied the stabilization of a damped wave equation using an internal feedback mechanism incorporating time delay and established a global optimal decay estimate of the solution.

This paper is organized as follows: In Section 2, we describe the analytic method for the pointwise estimate. In Section 3, we establish the weighted Strichartz estimate.

Notations

Throughout this paper, C denotes a generic positive constant that may vary from line to line. The notation $A\lesssim B$ signifies that $A\le CB$ for some absolute constant $C>0$.

We let $\delta$ denote a positive constant that can be chosen arbitrarily small.

Additionally, we fix a cutoff function $\rho \in C_c^{\infty }\left({\mathbb{R}}^n\right)$ satisfying

$$\mathrm{supp}\rho \subseteq \left\{\xi \mid \frac{1}{2}\le \left|\xi \right|\le 2\right\},\sum _{j=-\infty }^{\infty }\rho \left(2^{-j}\xi \right)\equiv 1\mathrm{for}\xi \in {\mathbb{R}}^n\begin{array}{c}\hfill \setminus \left\{0\right\}\end{array},$$

(10)

and set ${\rho }_j\left(\xi \right)$ as $\rho \left(2^{-j}\xi \right)$.

2. Pointwise Decay Estimate

We divide the proof into two cases according to the value of the damping parameter $\mu$. More specifically, our key methodological contribution is a bifurcated approach to proving pointwise estimates, dictated by the damping parameter $\mu$. When $\mu =2$, the Liouville transformation reduces the equation to the free wave equation, allowing us to apply classical pointwise decay estimates. When $\mu >2$, we represent the solution in terms of Fourier integral operators and employ a dyadic decomposition in frequency space to derive the desired estimates.

2.1. $\mu =2$

When $\mu =2$, applying the Liouville transformation $w(t,x)=tv(t,x)$ reduces (4) to

$$\left\{\begin{array}{c}{w}_{tt}-\Delta w=0,t⩾2,x\in {\mathbb{R}}^n\hfill \\ w(2,x)=2v_0\left(x\right),w_t(2,x)=v_0\left(x\right)+2v_1\left(x\right).\hfill \end{array}\right.$$

(11)

Note that the initial data of (11) belong to $C_c^{\infty }\left({\mathbb{R}}^n\right)$. Thus, classical pointwise decay estimates for the linear wave equation (see, for example, [19] (6.2.10)) yield

$$\left|w\right|\le C(v_0,v_1){(1+t)}^{-\frac{n-1}{2}}{(1+|t-\left|x\right|\left|\right)}^{-\frac{n-1}{2}}.$$

(12)

Substituting $w=tv$ into (12) gives:

$$\left|v\right|\le C(v_0,v_1){(1+t)}^{-\frac{n+1}{2}}{(1+|t-\left|x\right|\left|\right)}^{-\frac{n-1}{2}}.$$

(13)

This implies (6) for $\mu =2$.

2.2. $2<\mu \le {\mu }_c\left(n\right)$

The analysis in the $2<\mu \le {\mu }_c\left(n\right)$ regime is more intricate. For technical reasons, we treat the finite-time and large-time regimes separately.

2.2.1. Decay Estimate in Finite Time

We first consider Equation (4) for $\begin{array}{c}\hfill 2\le \end{array}t\le T_0$ with a fixed $T_0>2$. Note that we have the representation

$$v(t,x)={\mathsf{\Phi }}_0(t,2,D)v_0+{\mathsf{\Phi }}_1(t,2,D)v_1,$$

where

$${\mathsf{\Phi }}_0(t,2,\xi )=\frac{i\pi }{4}\left|\xi \right|\frac{t^{\nu }}{2^{\nu -1}}\left[H_{\nu -1}^{-}\left(2\left|\xi \right|\right)H_{\nu }^{+}\left(t\left|\xi \right|\right)-H_{\nu -1}^{+}\left(2\left|\xi \right|\right)H_{\nu }^{-}\left(t\left|\xi \right|\right)\right],$$

(14)

and

$${\mathsf{\Phi }}_1(t,2,\xi )=-\frac{i\pi }{4}\frac{t^{\nu }}{2^{\nu -1}}\left[H_{\nu }^{-}\left(2\right|\xi \left|\right)H_{\nu }^{+}\left(t\right|\xi \left|\right)-H_{\nu }^{+}\left(2\right|\xi \left|\right)H_{\nu }^{-}\left(t\right|\xi \left|\right)\right].$$

(15)

Here, $H_{\nu }^{\pm }\left(z\right)$ are the Hankel functions of order $\nu =\frac{1-\mu }{2}$. The representation Formulas (14) and (15) follow from [13] (Theorem 2.1). For the reader’s convenience, we provide the proof of (14) and (15) in Lemma A1 of Appendix A.

Invoking [13] (Proposition 3.1), we find a constant $K>0$ such that

$$H_{\nu }^{\pm }\left(z\right)=e^{\pm iz}{a}_{\pm }\left(z\right),a_{\pm }\left(z\right)\in S^{-\frac{1}{2}}\left([K,+\infty )\right),$$

(16)

where $S^{-\frac{1}{2}}$ represents the class of classical symbols of order $-\frac{1}{2}$. Conversely, for small arguments $0<z\le c<1$, the behavior is characterized by

$$|H_{\nu }^{\pm }\left(z\right)|\lesssim z^{-\left|\nu \right|},\mathrm{if}\nu \ne 0.$$

(17)

From (16) and (17), we have, for ${\mathsf{\Phi }}_1$ in (15),

$$|{\mathsf{\Phi }}_1(t,\begin{array}{c}\hfill 2\end{array},\xi )|\lesssim \left\{\begin{array}{ccc}\begin{array}{c}\hfill t^{\nu }{\left(2\right|\xi \left|\right)}^{\nu }{\left(t\right|\xi \left|\right)}^{\nu }\lesssim {\left(t\right|\xi \left|\right)}^{1-\mu }\end{array},\hfill & \hfill & \mathrm{if}\left|\xi \right|\le \frac{1}{t},\hfill \\ \begin{array}{c}\hfill t^{\nu }{\left(2\right|\xi \left|\right)}^{-\frac{1}{2}}{\left(t\right|\xi \left|\right)}^{\nu }\lesssim t^{1-\mu }{\left|\xi \right|}^{-\frac{\mu }{2}}\end{array},\hfill & \hfill & \mathrm{if}\frac{1}{t}<\left|\xi \right|\le \frac{1}{2},\hfill \\ \begin{array}{c}\hfill t^{\nu }{\left(2\right|\xi \left|\right)}^{-\frac{1}{2}}{\left(t\right|\xi \left|\right)}^{-\frac{1}{2}}\lesssim t^{-\frac{\mu }{2}}{\left|\xi \right|}^{-1}\end{array},\hfill & \hfill & \mathrm{if}\left|\xi \right|>\frac{1}{2},\hfill \end{array}\right.$$

which further yields for $N>n$

$$\begin{array}{cc}\hfill |{\mathsf{\Phi }}_1(t,\begin{array}{c}\hfill 2\end{array},D)v_1|=& \left|{\int }_{{\mathbb{R}}^n}{e}^{it\xi }{\mathsf{\Phi }}_1(t,\begin{array}{c}\hfill 2\end{array},\xi ){\widehat{v}}_1\left(\xi \right)\mathrm{d}\xi \right|\hfill \\ \hfill \lesssim & \begin{array}{c}\hfill {\int }_{\left|\xi \right|\le \frac{1}{t}}{\left(t\right|\xi \left|\right)}^{1-\mu }|\widehat{v_1}\left(\xi \right)|\mathrm{d}\xi +{\int }_{\frac{1}{t}<\left|\xi \right|\le \frac{1}{2}}{t}^{1-\mu }{\left|\xi \right|}^{-\frac{\mu }{2}}\left|\widehat{v_1}\left(\xi \right)\right|\mathrm{d}\xi \end{array}\hfill \\ \hfill & \begin{array}{c}\hfill +{\int }_{\left|\xi \right|>\frac{1}{2}}\frac{t^{-\frac{\mu }{2}}{\left|\xi \right|}^{-1}}{{\langle \xi \rangle }^N}\widehat{{\langle D\rangle }^N{v}_1}\left(\xi \right)\mathrm{d}\xi \end{array}\hfill \\ \hfill \lesssim & \parallel v_1{\parallel }_{W^{N,1}\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

Since (14) and (15) imply that the properties of ${\mathsf{\Phi }}_0$ and ${\mathsf{\Phi }}_1$ are similar, the same argument yields a similar estimate for ${\mathsf{\Phi }}_0(t,\begin{array}{c}\hfill 2\end{array},D)v_0$. Hence,

$$\left|v\right|\le |{\mathsf{\Phi }}_0(t,\begin{array}{c}\hfill 2\end{array},D)v_0|+|{\mathsf{\Phi }}_1(t,\begin{array}{c}\hfill 2\end{array},D)v_1|\lesssim C(v_0,v_1).$$

Furthermore, for any fixed $T_0>2$ and $2\le t\le T_0$, ${(1+t)}^{-1}$ is bounded below by ${(1+T_0)}^{-1}$. Similarly, since $|t-|x\left|\right|\le t\le T_0$, the quantity ${(1+|t-\left|x\right|\left|\right)}^{-1}$ also admits the positive lower bound ${(1+T_0)}^{-1}$. Consequently, we obtain the following estimate

$$\left|v\right|\le C(v_0,v_1,T_0){(1+t)}^{-\frac{n+\mu -1}{2}}{(1+|t-\left|x\right|\left|\right)}^{-\frac{n-\mu +1}{2}},2\le t\le T_0.$$

(18)

2.2.2. Decay Estimate in Long Time

We now consider the case of $t>T_0\ge 2$. In view of the structural similarity between ${\mathsf{\Phi }}_0$ and ${\mathsf{\Phi }}_1$ in (14) and (15), it suffices to analyze ${\mathsf{\Phi }}_1(t,\begin{array}{c}\hfill 2\end{array},D)v_1$. Moreover, since (16) and (17) reveal that $H_{\nu }^{+}$ and $H_{\nu }^{-}$ have analogous asymptotic behavior, it is enough to consider

$$\tilde{v}:=t^{\nu }{H}_{\nu }^{-}\left(2\right|D\left|\right)H_{\nu }^{+}\left(t\right|D\left|\right)v_1.$$

From Littlewood–Paley decomposition, we express

$$\tilde{v}(t,x):=\sum _{j=-\infty }^{+\infty }{\tilde{v}}_j(t,x):=\sum _{j=-\infty }^{+\infty }{t}^{\nu }{\rho }_j\left(D\right)H_{\nu }^{-}\left(2\right|D\left|\right)H_{\nu }^{+}\left(t\right|D\left|\right)v_1.$$

Then, we split the frequency space into three parts:

1.

Low frequency: $\left|\xi \right|\le \frac{1}{t}$;

2.

Medium frequency: $\frac{1}{t}<\left|\xi \right|\begin{array}{c}\hfill \le \end{array}\frac{1}{2}$;

3.

High frequency: $\left|\xi \right|\begin{array}{c}\hfill >\end{array}\frac{1}{2}$.

Recall that for $2\le \mu \le {\mu }_c\left(n\right)=\frac{n^2+n+2}{n+2}$, we have

$$\nu =\frac{1-\mu }{2}\in [-\frac{n^2}{2(n+2)},-\frac{1}{2}].$$

We now turn to the analysis of the three cases described previously.

1.

Low frequency:

$$\begin{array}{cc}\hfill |{\begin{array}{c}\hfill \tilde{v}\end{array}}_j(t,x)|& \le C(n,\nu )\left|{\int }_{{\mathbb{R}}^n}{e}^{ix·\xi }{\rho }_j\left(\xi \right){\left(2\right|\xi \left|\right)}^{-\left|\nu \right|}{\left(t\right|\xi \left|\right)}^{-\left|\nu \right|}{\widehat{v}}_1\left(\xi \right)d\xi \right|\hfill \\ \hfill & \le C(n,\nu )2^{(n+2\nu )j}{t}^{2\nu }{\parallel v_1\parallel }_{L^1\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & =C(n,\nu )2^{(n+2v-\frac{n}{2}-\nu )j}{2}^{(\frac{n}{2}+\nu )j}{t}^{2\nu }{\parallel v_1\parallel }_{L^1\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le C(n,\nu )2^{\frac{n}{n+2}j}{t}^{\nu -\frac{n}{2}}{\parallel v_1\parallel }_{L^1\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le C(n,\mu )2^{\frac{n}{n+2}j}{(1+t)}^{-\frac{n+\mu -1}{2}}{\parallel v_1\parallel }_{\begin{array}{c}\hfill W^{\frac{n}{2},1}\end{array}\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

(19)

The last inequality in (19) follows from

$$t{2}^j\le 1\Rightarrow 2^j\le t^{-1},$$

and

$$n+2\nu -\frac{n}{2}-\nu =\frac{n}{2}+\nu \ge \frac{n}{n+2}>0;$$

thus, $2^{(n+2v-\frac{n}{2}-\nu )j}\le 2^{\frac{n}{n+2}j}$ and $2^{(\frac{n}{2}+v)j}\le t^{-(\frac{n}{2}+v)}$.

2.

Medium frequency:

$$\begin{array}{cc}\hfill |{\begin{array}{c}\hfill \tilde{v}\end{array}}_j(t,x)|& \le C(n,\nu )\left|{\int }_{{\mathbb{R}}^n}{e}^{ix·\xi }{t}^{\nu }{\rho }_j\left(\xi \right)H_{\nu }^{-}\left(2\right|\xi \left|\right)e^{it\left|\xi \right|}{a}_{+}\left(t\right|\xi \left|\right){\widehat{v}}_1\left(\xi \right)d\xi \right|\hfill \\ \hfill & \le C(n,\nu )t^{\nu }{2}^{(n+\nu )j}{t}^{-\frac{n}{2}}{2}^{-\frac{n}{2}j}{\parallel v_1\parallel }_{L^1\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le C(n,\nu )2^{\frac{n}{n+2}j}{t}^{-\frac{n}{2}+\nu }{\parallel v_1\parallel }_{L^1\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le C(n,\mu )2^{\frac{n}{n+2}j}{(1+t)}^{-\frac{n+\mu -1}{2}}{\parallel v_1\parallel }_{\begin{array}{c}\hfill W^{\frac{n}{2},1}\end{array}\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

(20)

Here, the second inequality comes from the stationary phase method:

$$\left|{\int }_{{\mathbb{R}}^n}{e}^{i[x·\xi +t|\xi \left|\right]}\rho \left(2^{-j}\xi \right)d\xi \right|\le 2^{nj}{(1+t{2}^j)}^{-\frac{n-1}{2}}.$$

3.

High frequency: In this case, $H_{\nu }^{\pm }\left(z\right)=e^{\pm iz}{a}_{\pm }\left(z\right)$ with $a_{\pm }\in S^{-\frac{1}{2}}$, so

$$a_{-}\left(2\right|\xi \left|\right)a_{+}\left(t\right|\xi \left|\right)=O(t^{-\frac{1}{2}}{\left|\xi \right|}^{-1});$$

therefore, we can apply the stationary phase method again to get:

$$\begin{array}{cc}\hfill |{\begin{array}{c}\hfill \tilde{v}\end{array}}_j(t,x)|& \le C(n,\nu )\left|{\int }_{{\mathbb{R}}^n}{e}^{ix·\xi }{t}^{\nu }{\rho }_j\left(\xi \right)e^{i(t-2)\left|\xi \right|}{a}_{-}\left(2\right|\xi \left|\right)a_{+}\left(t\right|\xi \left|\right){\widehat{v}}_1\left(\xi \right)d\xi \right|\hfill \\ \hfill & \le C(n,\nu )\left|{\int }_{{\mathbb{R}}^n}{e}^{ix·\xi }{t}^{\nu }{\rho }_j\left(\xi \right)e^{i(t-2)\left|\xi \right|}{a}_{-}\left(2\right|\xi \left|\right)a_{+}{\left(t\right|\xi \left|\right)\left|\xi \right|}^{-\frac{n}{2}}{\widehat{{\left|D\right|}^{\frac{n}{2}}v}}_1\left(\xi \right)d\xi \right|\hfill \\ \hfill & \le C(n,\nu )t^{\nu }{2}^{(n-\frac{1}{2})j}{\left(t{2}^j\right)}^{-\frac{1}{2}}{\left(t{2}^j\right)}^{-\frac{n-1}{2}}{2}^{-\frac{n}{2}j}{\parallel v_1\parallel }_{{\dot{W}}^{\frac{n}{2},1}\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le C(n,\nu )2^{-\frac{j}{2}}{t}^{\nu -\frac{n}{2}}{\parallel v_1\parallel }_{{\dot{W}}^{\frac{n}{2},1}\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le C(n,\mu )2^{-\frac{j}{2}}{(1+t)}^{-\frac{n+\mu -1}{2}}{\parallel v_1\parallel }_{\begin{array}{c}\hfill W^{\frac{n}{2},1}\end{array}\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

(21)

The first inequality in (21) follows from the stationary phase method and $t>T_0\gg 1$:

$$\begin{array}{cc}\hfill & \left|{\int }_{{\mathbb{R}}^n}{e}^{i[x·\xi +(t-2\left)\right|\xi \left|\right]}{\rho }_j\left(\xi \right)a_{-}\left(2\right|\xi \left|\right)a_{+}{\left(t\right|\xi \left|\right)\left|\xi \right|}^{-\frac{n}{2}}d\xi \right|\hfill \\ \hfill & \lesssim 2^{(n-\frac{1}{2})j}{\left(t{2}^j\right)}^{-\frac{1}{2}}{2}^{-\frac{n}{2}j}{(1+(t-2)2^j)}^{-\frac{n-1}{2}}\hfill \\ \hfill & \lesssim 2^{(n-\frac{1}{2})j}{\left(t{2}^j\right)}^{-\frac{1}{2}}{2}^{-\frac{n}{2}j}{\left(t{2}^j\right)}^{-\frac{n-1}{2}}.\hfill \end{array}$$

Summing (19)–(21) over j and noting that $C(n,\mu )$ in (19)–(21) are uniformly bounded constants that only depend on n and $\mu$, we get:

$$|\tilde{v}(t,x)|\le C(v_1,T_0){(1+t)}^{\nu -\frac{n}{2}}=C(v_1,T_0){(1+t)}^{-\frac{n+\mu -1}{2}},$$

and hence

$$\left|v(t,x)\right|\le C(v_0,v_1,T_0){(1+t)}^{\nu -\frac{n}{2}}=C(v_0,v_1,T_0){(1+t)}^{-\frac{n+\mu -1}{2}}.$$

(22)

Comparing (22) with (13), it remains to recover the additional decay factor involving $1+|t-|x\left|\right|$ when $\mu >2$.

If $|t-|x\left|\right|\le 10$, then ${(1+|t-\left|x\right|\left|\right)}^{-1}\ge \frac{1}{11}$, and (22) yields:

$$\left|v(t,x)\right|\le C(v_0,v_1,T_0){(1+t)}^{-\frac{n+\mu -1}{2}}{(1+|t-\left|x\right|\left|\right)}^{-\frac{n-\mu +1}{2}}.$$

Thus, we only need to consider the case $|t-|x\left|\right|>10$. Since the analysis of this case is more involved, we divide the proof into three parts according to the frequency scale.

1.

Low frequency

From the finite speed of propagation and the support condition $\mathrm{supp}{v}_1\subset \left\{\left|x\right|\le 1\right\}$, we have

$$\left|x\right|\le t-1,\mathrm{if}(t,x)\in \mathrm{supp}v,\mathrm{and}t\ge 2;$$

hence,

$$1\le t-\left|x\right|\le t,\mathrm{and}|t-|x\left|\right|2^j\le t{2}^j\le 1,$$

thus

$$\begin{array}{cc}\hfill |{\tilde{v}}_j(t,x)|& \le C(n,\nu )2^{(\frac{n}{2}+\nu )j}{t}^{-\frac{n}{2}+\nu }{\parallel v_1\parallel }_{L^1\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le C(n,\nu )2^{(\frac{n}{2}+\nu )j}{t}^{-\frac{n}{2}+\nu }(2^j{|t-|x\left|\right|)}^{-\frac{n}{2}-\nu +\delta }{\parallel v_1\parallel }_{L^1\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le C(n,\nu )2^{\delta j}{(1+t)}^{-\frac{n}{2}+\nu }{(1+|t-\left|x\right|\left|\right)}^{-\frac{n}{2}-\nu +\delta }{\parallel v_1\parallel }_{L^1\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & =C(n,\mu )2^{\delta j}{(1+t)}^{-\frac{n+\mu -1}{2}}{(1+|t-\left|x\right|\left|\right)}^{-\frac{n-\mu +1}{2}+\delta }{\parallel v_1\parallel }_{L^1\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

2.

Medium frequency

For any $N>0$, Ref. [20] (3.29) yields

$$\begin{array}{cc}\hfill {\tilde{v}}_j(t,x)|& \le C(n,\nu )t^{\nu }{2}^{(n+\nu )j}{\left(2^{j}t\right)}^{-\frac{1}{2}}{(1+2^{j}t)}^{-\frac{n-1}{2}}{\int }_{{\mathbb{R}}^n}(1+2^j{\left|\right|x-y|-t|)}^{-N}\left|v_1\left(y\right)\right|dy\hfill \\ \hfill & \le C(n,\nu )2^{(\frac{n}{2}+\nu )j}{t}^{-\frac{n}{2}+\nu }{\int }_{{\mathbb{R}}^n}(1+2^j{\left|\right|x-y|-t|)}^{-N}\left|v_1\left(y\right)\right|dy.\hfill \end{array}$$

Since $t\ge T_0\ge 2$ and $\left|y\right|\le 1$, we have

$$\left|\right|x-y|-t|\ge \frac{1}{2}|t-\left|x\right||.$$

Thus, choosing $N=\frac{n}{2}+\nu -\delta$, we obtain

$$\begin{array}{cc}\hfill {\tilde{v}}_j(t,x)|& \le C(n,\nu )2^{(\frac{n}{2}+\nu )j}{t}^{-\frac{n}{2}+\nu }(2^j{|t-|x\left|\right|)}^{-\frac{n}{2}-\nu +\delta }{\parallel v_1\parallel }_{L^1\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le C(n,\mu )2^{\delta j}{(1+t)}^{-\frac{n+\mu -1}{2}}{(1+|t-\left|x\right|\left|\right)}^{-\frac{n-\mu +1}{2}+\delta }{\parallel v_1\parallel }_{L^1\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

(23)

3.

High frequency

Since

$$\left|\right|x-y|-(t-2)|\ge \frac{1}{2}|t-\left|x\right||$$

for $t\ge T_0\gg 1$ and $\left|y\right|\le 1$, the high-frequency case can be treated in the same way as the medium-frequency case. More specifically, Ref. [20] (3.29) implies that

$$\begin{array}{cc}\hfill & |{\tilde{v}}_j(t,x)|\hfill \\ \hfill & \le C(n,\nu )t^{\nu }{2}^{(n-\frac{1}{2})j}{\left(2^{j}t\right)}^{-\frac{1}{2}}{(1+2^{j}t)}^{-\frac{n-1}{2}}{\int }_{{\mathbb{R}}^n}(1+2^j\left|\right|x-y|-(t-2){\left|\right)}^{-N}\left|v_1\left(y\right)\right|dy\hfill \\ \hfill & \le C(n,\nu )2^{\frac{n-1}{2}j}{t}^{-\frac{n}{2}+\nu }{\int }_{{\mathbb{R}}^n}(1+2^j\left|\right|x|-{t\left|\right)}^{-N}\left|v_1\left(y\right)\right|dy.\hfill \end{array}$$

Therefore we obtain, by choosing $N=\frac{n-1}{2}+\delta$,

$$\begin{array}{cc}\hfill |{\tilde{v}}_j(t,x)|& \le C(n,\nu )2^{\frac{n-1}{2}j}{t}^{-\frac{n}{2}+\nu }(2^j{|t-|x\left|\right|)}^{-\frac{n-1}{2}-\delta }{\parallel v_1\parallel }_{L^1\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le C(n,\mu )2^{-\delta j}{(1+t)}^{-\frac{n+\mu -1}{2}}{(1+|t-\left|x\right|\left|\right)}^{-\frac{n-1}{2}-\delta }{\parallel v_1\parallel }_{L^1\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le C(n,\mu )2^{-\delta j}{(1+t)}^{-\frac{n+\mu -1}{2}}{(1+|t-\left|x\right|\left|\right)}^{-\frac{n-\mu +1}{2}-\delta }{\parallel v_1\parallel }_{L^1\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

The last inequality follows from $\frac{n-\begin{array}{c}\hfill 1\end{array}}{2}\ge \frac{n-\mu +1}{2}$ when $\mu \ge 2$.

Combining the above estimates and noting that the constants $C(n,\mu )$ are uniformly bounded and depend only on n and $\mu$, we obtain the pointwise decay estimate for $2<\mu \le {\mu }_c\left(n\right)$ and $0<\delta \ll 1$:

$$|\tilde{v}| \le C(v_1,T_0){(1+t)}^{-\frac{n+\mu -1}{2}}{(1+|t-\left|x\right|\left|\right)}^{-\frac{n-\mu +1}{2}+\delta },t\ge T_0,$$

and hence

$$\left|v\right|\le C(v_0,v_1,T_0){(1+t)}^{-\frac{n+\mu -1}{2}}{(1+|t-\left|x\right|\left|\right)}^{-\frac{n-\mu +1}{2}+\delta },t\ge T_0.$$

(24)

Fixing $T_0$, we obtain from (18) and (24) that

$$\left|v\right|\le C(v_0,v_1){(1+t)}^{-\frac{n+\mu -1}{2}}{(1+|t-\left|x\right|\left|\right)}^{-\frac{n-\mu +1}{2}+\delta },2<\mu \le {\mu }_c\left(n\right).$$

(25)

Combining (13) and (25), Theorem 1 has been established.

3. Weighted Strichartz Estimate

Next, we establish the weighted Strichartz estimate. From the finite speed of propagation, if $y\in \mathrm{supp}(v_0,v_1)$, and x lies in the domain of influence of y at time t, then

$$|x-y|\le t-2,$$

which results in

$$\left|x\right|\le \left|y\right|+t-2.$$

(26)

Since we have $\mathrm{supp}(v_0,v_1)\subseteq \{x\mid |x|\le 1\}$, (26) gives

$$\left|x\right|\le t-1$$

(27)

for $(t,x)\in \mathrm{supp}v$.

Using the support condition (27) and polar coordinates, we obtain

$$\begin{array}{cc}\hfill & {∥{(t^2-{\left|x\right|}^2)}^{\gamma }{t}^{\frac{\mu }{q}}v∥}_{L^q([2,+\infty )\times {\mathbb{R}}^n)}^q\hfill \\ \hfill & \lesssim {\int }_2^{+\infty }{t}^{\mu }dt{\int }_{\left|x\right|\le t-1}{(t^2-{\left|x\right|}^2)}^{q\gamma }{(1+t)}^{-q\frac{n+\mu -1}{2}}{(1+|t-\left|x\right|\left|\right)}^{(-\frac{n-\mu +1}{2}+\delta )q}dx\hfill \\ \hfill & \lesssim {\int }_2^{+\infty }{t}^{\mu }dt{\int }_0^{t-1}{(t^2-r^2)}^{q\gamma }{(1+t)}^{-q\frac{n+\mu -1}{2}}{(1+t-r)}^{(-\frac{n-\mu +1}{2}+\delta )q}{r}^{n-1}dr\hfill \\ \hfill & \lesssim {\int }_2^{+\infty }{t}^{q\gamma +\mu -q\frac{n+\mu -1}{2}}dt{\int }_0^{t-1}{(1+t-r)}^{q(\gamma -\frac{n-\mu +1}{2}+\delta )}{r}^{n-1}dr.\hfill \end{array}$$

Next, we prove that when $q\le \frac{2(n+\mu +1)}{n+\mu -1}$, $\gamma <\frac{n+\mu -1}{2}-\frac{n+\mu }{q}$,

$${\int }_0^{t-1}{(1+t-r)}^{q(\gamma -\frac{n-\mu +1}{2}+\delta )}{r}^{n-1}dr\lesssim t^{n-1}$$

Let $s=t-r$, then

$$\begin{array}{cc}\hfill {\int }_0^{t-1}{(1+t-r)}^{q(\gamma -\frac{n-\mu +1}{2}+\delta )}{r}^{n-1}dr& ={\int }_1^t{(1+s)}^{q(\gamma -\frac{n-\mu +1}{2}+\delta )}{(t-s)}^{n-1}ds\hfill \\ & \le t^{n-1}{\int }_1^{+\infty }{(1+s)}^{q(\gamma -\frac{n-\mu +1}{2}+\delta )}ds.\hfill \end{array}$$

Since $q\le \frac{2(n+\mu +1)}{n+\mu -1}\le 1+\frac{n}{\mu -1}$, when $n\ge 4$ and $2\le \mu \le \begin{array}{c}\hfill \sqrt{n^2+4}-1\end{array}$, we have

$$\frac{n+\mu -1}{2}-\frac{n+\mu }{q}\le \frac{n-\mu +1}{2}-\frac{1}{q}.$$

Hence,

$$q(\gamma -\frac{n-\mu +1}{2})<-1.$$

Choose $\delta >0$ sufficiently small such that

$$q(\gamma -\frac{n-\mu +1}{2}+\delta )<-1,$$

then we arrive at

$$\begin{array}{cc}\hfill {\int }_0^{t-1}{(1+t-r)}^{q(\gamma -\frac{n-\mu +1}{2}+\delta )}{r}^{n-1}dr& \le {(t-1)}^{n-1}{\int }_0^{t-1}{(1+t-r)}^{q(\gamma -\frac{n-\mu +1}{2}+\delta )}dr\hfill \\ & \le t^{n-1}.\hfill \end{array}$$

Therefore

$${∥(t^2-{\left|x\right|}^2{)}^{\gamma }{t}^{\frac{\mu }{q}}v∥}_{L^q([2,+\infty )\times {\mathbb{R}}^n)}^q\lesssim {\int }_2^{+\infty }{t}^{q(\gamma -\frac{n+\mu -1}{2})+\mu +n-1}dt.$$

(28)

Furthermore, note that

$$\gamma <\frac{n+\mu -1}{2}-\frac{n+\mu }{q}\Rightarrow q(\gamma -\frac{n+\mu -1}{2})+\mu +n-1<-1;$$

consequently, the integral in (28) converges. Thus, we establish the following weighted Strichartz estimate:

$${∥(t^2-{\left|x\right|}^2{)}^{\gamma }{t}^{\frac{\mu }{q}}v∥}_{L^q([2,+\infty )\times {\mathbb{R}}^n)}\le C(v_0,v_1),$$

which proves Theorem 2.