Blow-up and lifespan estimate to a nonlinear wave equation in Schwarzschild spacetime

Nonlinear wave equations in Schwarzschild spacetime attract more and more attention, since it is natural to generalize the classical results for nonlinear wave equations in flat Minkowski spactime to the black hole spacetime. Schwarzschild metric is the first analytic solution to the vacuum Einstein equation, which was derived by Schwarzschild in 1915. And according to Birkhoff s theorem it is also a unique spherical symmetry solution of the vacuum Einstein equation. The explicit expression of the metric is

$\displaystyle g_{S}=F(r)dt^{2}-F(r)^{-1}dr^{2}-r^{2}d\omega^{2},$

(1.1)

where $F(r)=1-\frac{2M}{r}$, and $M$ is the Newtonian mass. Noting that asymptotically $r\rightarrow\infty$ or $M\rightarrow 0$ the Schwarzschild metric reduces to the Minkowski metric

$$ds^{2}=dt^{2}-dr^{2}-r^{2}d\omega^{2}.$$

The study of classical Cauchy problem to semilinear wave equations in Minkowski spacetime

$$\left\{\begin{aligned} &u_{tt}-\Delta u=|u|^{p},\quad\text{in $[0,T)\times\mathbf{R}^{n}$},\\ &u(x,0)=\varepsilon f(x),\quad u_{t}(x,0)=\varepsilon g(x),\quad x\in\mathbf{R}^{n}\end{aligned}\right.$$

(1.2)

has a long time history, and it has now been determined there exists a critical power $p_{c}(n)>1,n\geq 2$, which solves the quadratic equation

$$(n-1)p^{2}-(n+1)p-2=0.$$

Such kind problem is known as Strauss conjecture: for $1<p\leq p_{c}(n)$, the solution will blow up in a finite time, while for $p>p_{c}(n)$ the solution exists globally in time, see [5, 6, 7, 10, 14, 26, 27, 30, 35] and references therein. It is easy to see that $p_{c}(3)=1+\sqrt{2}$. If there is no global solution, it is then also interesting to estimate the lifespan($T(\varepsilon)$) with respect to the small parameter $\varepsilon$. We are now clear that there exist two positive constants $c$ and $C$ such that the lifespan satisfies for $n\geq 2$ and $\max(1,2/(n-1))<p<p_{c}(n)$

$\displaystyle c\varepsilon^{\frac{-2p(p-1)}{\gamma(n,p)}}\leq T(\varepsilon)\leq C\varepsilon^{\frac{-2p(p-1)}{\gamma(n,p)}},$

(1.3)

where $\gamma(n,p)=2+(n+1)p-(n-1)p^{2}>0$. For $(n,p)=(2,2)$,

$$\left\{\begin{array}[]{ll}\exists\lim\limits_{\varepsilon\rightarrow 0^{+}}a(\varepsilon)^{-1}T(\varepsilon)>0,&\mathrm{if}~{}\int_{\mathbf{R}^{2}}g(x)dx\neq 0,\\ \exists\lim\limits_{\varepsilon\rightarrow 0^{+}}\varepsilon T(\varepsilon)>0,&\mathrm{if}~{}\int_{\mathbf{R}^{2}}g(x)dx=0,\end{array}\right.$$

(1.4)

where $a(\varepsilon)$ denotes a number satisfying

$$a^{2}\varepsilon^{2}\log(1+a)=1.$$

For $1<p<2$ and $n=2$,

$$\left\{\begin{array}[]{ll}c\varepsilon^{-\frac{p-1}{3-p}}\leq T(\varepsilon)\leq C\varepsilon^{-\frac{p-1}{3-p}},&\mathrm{if}~{}\int_{\mathbf{R}^{2}}g(x)dx\neq 0,\\ c\varepsilon^{-\frac{2p(p-1)}{\gamma(2,p)}}\leq T(\varepsilon)\leq C\varepsilon^{-\frac{2p(p-1)}{\gamma(2,p)}},&\mathrm{if}~{}\int_{\mathbf{R}^{2}}g(x)dx=0.\end{array}\right.$$

(1.5)

For the critical case $(p=p_{c}(n),n\geq 2)$,

$\displaystyle\exp(c\varepsilon^{-p(p-1)})\leq T(\varepsilon)\leq\exp(C\varepsilon^{-p(p-1)}).$

(1.6)

See [9, 14, 16, 18, 19, 28, 29, 33, 34] and and the introduction in [12].

It is natural to consider the corresponding Cauchy problem in Schwarzschild spacetime

$$\left\{\begin{aligned} &\Box_{g_{S}}u=|u|^{p},\quad(t,x)\in\mathbf{R}_{t}^{+}\times\Sigma,\\ &u(x,0)=\varepsilon f(x),\quad u_{t}(x,0)=\varepsilon g(x),\quad x\in\Sigma,\end{aligned}\right.$$

(1.7)

where $g_{S}$ denotes the Schwarzschild metric presented in (1.1) and $\mathbf{R}_{t}^{+}\times\Sigma$ is called the exterior of the black hole:

$\displaystyle\mathbf{R}_{t}^{+}\times\Sigma=\mathbf{R}_{t}^{+}\times(2M,\infty)\times\mathbb{S}^{2}.$

The D’Alembert operator associated with the Schwarzschild metric $g$ can be written explicitly as

$\displaystyle\Box_{g_{S}}=\frac{1}{F(r)}\Big{(}\partial_{t}^{2}-\frac{F}{r^{2}}\partial_{r}(r^{2}F)\partial_{r}-\frac{F}{r^{2}}\Delta_{\mathbb{S}^{2}}\Big{)},$

(1.8)

where $\Delta_{\mathbb{S}^{2}}$ denotes the standard Laplace-Beltrami operator on $\mathbb{S}^{2}$. At first glance, a divergent coefficient(when $r$ approaches to $2M$) appears in the operator, which may cause essential difficulty near the event horizon($r=2M$). The study of power type nonlinear wave equations in Schwarzschild(-like) and Kerr black hole spacetime was initiated by Nicolas [24, 25], in which the global existence results were established for

$$\Box_{g}u+m^{2}u+\lambda|u|^{2}u=0,~{}~{}~{}\lambda>0$$

with large initial data, and $g$ denotes the Schwarzschild-like or Kerr metric. After that, more and more attention is paid to the small data Cauchy problem. Catania and Georgiev [2] first studied the blow-up phenomenon for (1.7). They considered radial solution in the Regge-Wheeler coordinate

$$r^{*}=r+2M\ln(r-2M),$$

(1.9)

and proved that the solution will blow up for $1<p<1+\sqrt{2}$, by choosing special initial data

$$f(r^{*})=\varepsilon\chi_{0}(r^{*}-r_{0}^{*}(\varepsilon)),~{}~{}g(r^{*})=\varepsilon\chi_{1}(r^{*}-r_{0}^{*}(\varepsilon)),$$

where $\chi_{j}\in C_{0}^{\infty}(\mathbf{R})(j=1,2)$ satisfy

		$\displaystyle\chi_{j}(r^{*})\geq 0,r^{*}\in\mathbf{R},$
		$\displaystyle\chi_{j}(r^{*})=1,r^{*}\in[-R/2,R/2],$
		$\displaystyle supp~{}\chi_{j}\subset[-R,R]$

for some positive constant. And

$$r_{0}^{*}(\varepsilon)=\varepsilon^{-\theta}$$

for some positive constant $\theta$ depending on $p$. Noting that $r^{*}$ varies from $-\infty$ to $+\infty$ as $r$ varies from $2M$ to $+\infty$, it means that the chosen data above have compact support far away from the event horizon, and the distance depends on the small parameter, see the picture below.

Dafermos and Rodnianski [3] first studied the global existence for the corresponding small data problem, in which they showed the radial solution for $p>4$ in Schwarzschild or more generally a Reissner-Nordström spacetime exists globally in time. Blue and Sterbenz [1] generalized the global existence of general solution for (1.7) to $p>3$. Marzuola et al [22] proved the global-in-time Strichartz estimates for wave equations in Schwarzschild spacetime and furthermore established global existence for (1.7) with $p=5$. Lindblad et al [20] then established global existence result for $p>1+\sqrt{2}$, which also holds for Kerr black hole with small angular momentum if the small data have compact support. This result takes a giant step forward for the question posed by Dafermos and Rodnianski [3], and it was generalized by Metcalfe-Wang [23] to the slowly rotating Kerr spacetime with non-compact supported small data. Wang [31] established the following lower bound of the lifespan estimate

$$T(\varepsilon)\geq\left\{\begin{aligned} &C\varepsilon^{-\frac{p(p-1)}{1+2p-p^{2}}},\quad&2\leq p<1+\sqrt{2},\\ &\exp\left(C\varepsilon^{-2\sqrt{2}}\right),&p=1+\sqrt{2},\\ \end{aligned}\right.$$

(1.10)

which also holds for the corresponding problem posed on Kerr spacetime with small angular momentum($a\ll M$), see [4]. The lower bound for $2\leq p<1+\sqrt{2}$ above coincides with that in (1.3) by setting $n=3$, while the existence time for the “critical” power $p=1+\sqrt{2}$ seems to have a long gap, comparing to the one in (1.6). Very recently, Lin et al. [17] established blow-up result for $p\in[\frac{3}{2},2]$, without assuming that the supports of the initial data should be far away from the black hole. We also refer the reader to [32] for more introduction to the Cauchy problem (1.7).

One should also notice that in [21] Luk first studied the semilinear wave equations with derivative nonlinearity in slowly rotating Kerr spacetimes$(a\ll M)$, and showed global existence for small data solution assuming the nonlinear term satisfies the null condition. Without null condition assumption, the authors [15] proved that the semilinear problem in Schwarzschild spacetime has no global solution.

The main target of this paper is to show that $p=1+\sqrt{2}$ is indeed the critical power to (1.7), thus, to show there is no global solution for $2\leq p\leq 1+\sqrt{2}$, without any assumption on the distance between the event horizon and the support of the initial data. What is more, we will prove the sharpness of the lifespan estimate for $2\leq p<1+\sqrt{2}$, while for the critical power $p=1+\sqrt{2}$, a possibly sharp lifespan estimate from above will be established, since there is a gap comparing to the second lower bound in (1.10). If we try to get blow-up result for small initial data problem, it always means that the nonlinear term will dominate the linear effect, this is why we may succeed in showing blow-up for relative small power($1<p\leq p_{c}(n)$) in the flat Minkowski spacetime. However, for our concerned problem (1.7), it is easy to see from the wave operator (1.8) with Schwarzschild metric that a factor asymptotically approaching to $0$ as coming close to the event horizon($r=2M$) will appear. This factor will make competition with the nonlinear term and it may prevent finite time blow-up in some sense. This fact is also the reason why the assumption that the support of the initial data should be far away from the event horizon is needed for small data problem in [2]. We in this paper divide the radial variable into two parts: $r^{*}<\widetilde{C}_{0}$ and $r^{*}\geq\widetilde{C}_{0}$ for some positive constant $\widetilde{C}_{0}$, which corresponds to $2M<r<2M+C_{0}$ and $r\geq 2M+C_{0}$ for some other positive constant $C_{0}$ respectively. For the latter part, the wave will not touch the event horizon then all the estimates are almost the same as that of the corresponding flat Minkowski case, hence the nonlinear term will have polynomial growth at most. We call this area polynomial zone, which is labelled as $Z_{pol}$ in the picture below. For the former part $2M<r<2M+C_{0}$($r^{*}<\widetilde{C}_{0}$), the upper bound of the nonlinear term will increase exponentially with respect to $t$, which seems impossible to obtain a lifespan estimate of polynomial type. This exponential zone is labelled by $Z_{exp}$ in the picture below. However, if we use the test function

$$\psi_{\lambda}(t,r)=e^{-\lambda t}\phi_{\lambda},$$

where $\phi_{\lambda}$ solves for $\lambda>0$

$\displaystyle\frac{1}{r^{2}}\partial_{r}\big{(}r(r-2M)\partial_{r}\phi_{\lambda}\big{)}=\frac{\lambda^{2}}{1-\frac{2M}{r}}\phi_{\lambda},$

(1.11)

we then can gain some exponential decay in $Z_{exp}$ by choosing cut-off functions delicately, and hence the exponential increase mentioned above can be cancelled out. Inspired by this observation, we find that the effect for blow-up caused by $Z_{pol}$ area will dominate $Z_{exp}$ zone. We then use a family of solutions to the elliptic equation (1.11) to construct a powerful test function $b_{q}$(see Lemma 6.1) for the critical power $p=1+\sqrt{2}$, and furthermore to establish blow-up result and lifespan estimate for the solution involved in $Z_{pol}$ zone.

We state our main results as

We organize the paper as follows. In Section 2, we list the ingredients used to construct the appropriate test function. In Section 3, the upper bound of the spacetime integral of the nonlinear term far away from the event horizon is established, by using the space and time cut-off functions, while in Section 4 the upper bound of the spacetime integral of the nonlinear term in the whole exterior of the black hole is obtained, by using only a time cut-off function. In Section 5, we demonstrate the proof for Theorem 1.1, which corresponds to the upper bound of lifespan estimate for subcritical powers. The main idea is to obtain the optimal lower bounds for the space time integrals of the nonlinear term, then the result follows by combining the upper bounds established in Section 3 and 4. In Section 6, we show the proof for Theorem 1.4, which corresponds to the blow-up and upper bound of lifespan estimate for the critical power $p=1+\sqrt{2}$. The key ingredient is that we use a family of test functions with parameter $\lambda$ to construct a new test function with better decay.

For blow-up result, the normal way is to find some appropriate test functions, and then to construct some functional including the solution which approaches to $\infty$ at some time. And hence the upper bound of lifespan estimate follows. Besides, we choose some smooth cut-off functions delicately

$$\eta(t)=\left\{\begin{array}[]{ll}1&0\leq t\leq\frac{1}{3},\\ 0&t\geq\frac{2}{3},\\ \end{array}\right.\eta_{T}(t)=\eta\left(\frac{t}{T}\right),$$

$$\alpha(r^{*})=\left\{\begin{array}[]{ll}0&-\infty\leq r^{*}\leq\frac{1}{8},\\ 1&r^{*}\geq\frac{1}{4},\\ \end{array}\right.\alpha_{T}(r^{*})=\alpha\left(\frac{r^{*}}{T}\right)$$

and

$$\chi(\theta)=\left\{\begin{array}[]{ll}1&0\leq\theta\leq\frac{3}{4},\\ 0&\theta\geq\frac{5}{6},\\ \end{array}\right.\chi_{T}(\theta)=\chi\left(\frac{\theta}{T}\right),$$

where

$$T\in[T_{0},T(\varepsilon)),~{}~{}~{}T_{0}=\max\{8(4M+e),12(R+R_{3}),16R\},$$

(2.1)

with $R,R_{3}$ the positive constants appearing in (1.14) and (5.1) respectively. The next lemma concerns to our key test function.

Proof. In Lemma 5.4 in [2], it has been proved that the equation

$$\partial_{r^{*}r^{*}}\varphi_{\lambda}-\frac{2M\left(1-\frac{2M}{r}\right)}{r^{3}}\varphi_{\lambda}=\lambda^{2}\varphi_{\lambda}$$

admits a family of nonnegative solutions $\varphi_{\lambda}(r^{*})$ satisfying

$$\varphi_{\lambda}(r^{*})\thicksim e^{\lambda r^{*}}.$$

(2.5)

Set

$$\phi_{\lambda}(r)=\frac{\varphi_{\lambda}}{r},$$

then direct computation implies that $\phi_{\lambda}(r)$ solves (2.2), and the asymptotic behavior (2.3) follows from (2.5).

For (2.4), equation (2.2) yields

	$\displaystyle\partial_{r}\phi_{\lambda}$	$\displaystyle=\frac{\lambda^{2}}{r(r-2M)}\int_{2M}^{r}\frac{\tau^{3}}{\tau-2M}\phi_{\lambda}d\tau$		(2.6)
		$\displaystyle=\frac{\lambda^{2}}{r(r-2M)}\int_{-\infty}^{r^{*}}\tau^{2}\phi_{\lambda}ds.$

We next divide the estimate into two parts. For $r^{*}\leq 0$, we have $\tau,r\thicksim 1$, then plugging (2.3) into (2.6) yields

	$\displaystyle|\partial_{r}\phi_{\lambda}|$	$\displaystyle\lesssim\frac{\lambda^{2}}{r-2M}\int_{-\infty}^{r^{*}}e^{\lambda s}ds$		(2.7)
		$\displaystyle\lesssim\frac{\lambda}{r-2M}e^{\lambda r^{*}}.$

For $r^{*}>0$, we have $r\thicksim 1+r^{*}$ and then

	$\displaystyle|\partial_{r}\phi_{\lambda}|$	$\displaystyle\lesssim\frac{\lambda^{2}}{r(r-2M)}\left(\int_{-\infty}^{0}e^{\lambda s}ds+\int_{0}^{r^{*}}se^{\lambda s}ds\right)$		(2.8)
		$\displaystyle\lesssim\frac{\lambda^{2}}{r(r-2M)}\left(\frac{1}{\lambda}+\frac{1}{\lambda}r^{*}e^{\lambda r^{*}}\right)$
		$\displaystyle\lesssim\frac{\lambda}{r-2M}e^{\lambda r^{*}},$

hence (2.4) follows from (2.7) and (2.8).

Multiplying the equation in (1.7) with $\eta_{T}^{2p^{\prime}}(t)\alpha_{T}^{2p^{\prime}}(r^{*})r^{2}$, and then integrating with respect to $\omega,r,t$, we get

		$\displaystyle\int_{0}^{T}\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}\left[\frac{r^{2}}{1-\frac{2M}{r}}\partial_{t}^{2}u-\partial_{r}\big{(}r(r-2M)\partial_{r}u\big{)}-\Delta_{\mathbb{S}^{2}}u\right]\eta_{T}^{2p^{\prime}}\alpha_{T}^{2p^{\prime}}d\omega drdt$		(3.1)
	$\displaystyle=$	$\displaystyle\int_{0}^{T}\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}|u|^{p}\eta_{T}^{2p^{\prime}}\alpha_{T}^{2p^{\prime}}r^{2}d\omega drdt$
	$\displaystyle\triangleq$	$\displaystyle F_{0}(T).$

Due to the assumption (1.14), the support of the solution satisfies

$$suppu~{}\subset\Big{\{}r^{*}\Big{|}|r^{*}|\leq t+R\Big{\}}\times\mathbb{S}^{2}.$$

(3.2)

Hence by integration by part we get from (3.1)

	$\displaystyle F_{0}(T)\lesssim$	$\displaystyle\int_{0}^{T}\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}\Big{|}\frac{r^{2}}{1-\frac{2M}{r}}u\partial_{t}^{2}\eta_{T}^{2p^{\prime}}\alpha_{T}^{2p^{\prime}}\Big{|}d\omega drdt$		(3.3)
		$\displaystyle+\int_{0}^{T}\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}\big{|}u\eta_{T}^{2p^{\prime}}(2r-2M)\partial_{r}\alpha_{T}^{2p^{\prime}}\big{|}d\omega drdt$
		$\displaystyle+\int_{0}^{T}\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}\big{|}u\eta_{T}^{2p^{\prime}}(r^{2}-2Mr)\partial_{r}^{2}\alpha_{T}^{2p^{\prime}}\big{|}d\omega drdt$
	$\displaystyle\triangleq$	$\displaystyle I_{1}+I_{2}+I_{3}.$

Since

$$|\partial_{t}^{2}\eta_{T}^{2p^{\prime}}|\lesssim T^{-2}\eta_{T}^{2p^{\prime}-2},$$

and if $r^{*}\geq\frac{T}{8}\geq 4M+e$, then

$$r\geq 2M+e,~{}~{}r^{*}\thicksim r,$$

and hence

$$\frac{r}{r-2M}\thicksim C,$$

we may estimate $I_{1}$ by Hölder inequality

		$\displaystyle\int_{0}^{T}\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}\Big{|}\frac{r^{2}}{1-\frac{2M}{r}}u\partial_{t}^{2}\eta_{T}^{2p^{\prime}}\alpha_{T}^{2p^{\prime}}\Big{|}d\omega drdt$		(3.4)
	$\displaystyle\lesssim$	$\displaystyle T^{-2}F_{0}^{\frac{1}{p}}(T)\left(\int_{\frac{T}{3}}^{\frac{2T}{3}}\int_{r^{*}\geq\frac{T}{8}}\int_{\mathbb{S}^{2}}r^{2}\big{(}\frac{r}{r-2M}\big{)}^{p^{\prime}}d\omega drdt\right)^{\frac{1}{p^{\prime}}}$
	$\displaystyle\lesssim$	$\displaystyle T^{-2}F_{0}^{\frac{1}{p}}(T)\left(\int_{\frac{T}{3}}^{\frac{2T}{3}}\int_{4M+e}^{t+R}\int_{\mathbb{S}^{2}}r^{2}\big{(}\frac{r}{r-2M}\big{)}^{p^{\prime}}\frac{r-2M}{r}d\omega dr^{*}dt\right)^{\frac{1}{p^{\prime}}}$
	$\displaystyle\lesssim$	$\displaystyle T^{-2}F_{0}^{\frac{1}{p}}(T)\left(\int_{\frac{T}{3}}^{\frac{2T}{3}}\int_{4M+e}^{t+R}(r^{*})^{2}dr^{*}dt\right)^{\frac{1}{p^{\prime}}}$
	$\displaystyle\lesssim$	$\displaystyle T^{\frac{4}{p^{\prime}}-2}F_{0}^{\frac{1}{p}}(T).$

For $r^{*}\geq 4M+e$, it has

$$r-M\thicksim r\thicksim r^{*},$$

then $I_{2}$ can be estimated as

		$\displaystyle\int_{0}^{T}\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}\big{|}u\eta_{T}^{2p^{\prime}}(2r-2M)\partial_{r}\alpha_{T}^{2p^{\prime}}\big{|}d\omega drdt$		(3.5)
	$\displaystyle\lesssim$	$\displaystyle T^{-1}F_{0}^{\frac{1}{p}}(T)\left(\int_{0}^{T}\int_{r^{*}\geq\frac{T}{8}}\int_{\mathbb{S}^{2}}r^{-\frac{2}{p-1}}r^{p^{\prime}}d\omega drdt\right)^{\frac{1}{p^{\prime}}}$
	$\displaystyle\lesssim$	$\displaystyle T^{-1}F_{0}^{\frac{1}{p}}(T)\left(\int_{0}^{T}\int_{4M+e}^{t+R}(r^{*})^{\frac{p-2}{p-1}}d\omega dr^{*}dt\right)^{\frac{1}{p^{\prime}}}$
	$\displaystyle\lesssim$	$\displaystyle T^{\frac{4}{p^{\prime}}-2}F_{0}^{\frac{1}{p}}(T),$

and we can get the similar estimate for $I_{3}$

$\displaystyle\int_{0}^{T}\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}\big{|}u\eta_{T}^{2p^{\prime}}(r^{2}-2Mr)\partial_{r}^{2}\alpha_{T}^{2p^{\prime}}\big{|}d\omega drdt\lesssim T^{\frac{4}{p^{\prime}}-2}F_{0}^{\frac{1}{p}}(T).$

(3.6)

Since it also holds

$$r-2M\thicksim r\thicksim r^{*},~{}~{}for~{}r^{*}>4M+e,$$

we then get the upper bound for $F_{0}(T)$ by combining (3.3), (3.4), (3.5) and (3.6)

$\displaystyle F_{0}(T)\lesssim T^{4-2p^{\prime}}.$

(3.7)

Comparing to the estimate of $F_{0}(T)$, we have no space cut-off for $F_{1}(T)$, so we have to complete the estimate by dividing the radial space variable into two cases: $r^{*}\geq 4M+e$ and $r^{*}\leq 4M+e$. This time we multiply the equation in (1.7) with $\eta_{T}^{2p^{\prime}}r^{2}$ to get

		$\displaystyle\int_{0}^{T}\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}\left[\frac{r^{2}}{1-\frac{2M}{r}}\partial_{t}^{2}u-\partial_{r}\big{(}r(r-2M)\partial_{r}u\big{)}-\Delta_{\mathbb{S}^{2}}u\right]\eta_{T}^{2p^{\prime}}d\omega drdt$		(4.1)
	$\displaystyle=$	$\displaystyle\int_{0}^{T}\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}|u|^{p}\eta_{T}^{2p^{\prime}}r^{2}d\omega drdt$
	$\displaystyle\triangleq$	$\displaystyle F_{1}(T).$

Again by integration by parts and Hölder inequality, we have

	$\displaystyle F_{1}(T)\leq$	$\displaystyle\int_{0}^{T}\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}\frac{r^{2}}{1-\frac{2M}{r}}u\partial_{t}^{2}\eta_{T}^{2p^{\prime}}d\omega drdt$		(4.2)
	$\displaystyle\lesssim$	$\displaystyle T^{-2}F_{1}^{\frac{1}{p}}(T)\left(\int_{\frac{T}{3}}^{\frac{2T}{3}}\int_{|r^{*}|\leq t+R}\int_{\mathbb{S}^{2}}r^{2}\left(\frac{r}{r-2M}\right)^{p^{\prime}}d\omega drdt\right)^{\frac{1}{p^{\prime}}}$
	$\displaystyle\lesssim$	$\displaystyle T^{-2}F_{1}^{\frac{1}{p}}(T)\Bigg{[}\int_{\frac{T}{3}}^{\frac{2T}{3}}\int_{4M+e}^{t+R}r^{2}\left(\frac{r}{r-2M}\right)^{p^{\prime}}\frac{r-2M}{r}dr^{*}dt$
		$\displaystyle+\int_{\frac{T}{3}}^{\frac{2T}{3}}\int_{-t-R}^{4M+e}r^{2}\left(\frac{r}{r-2M}\right)^{p^{\prime}}\frac{r-2M}{r}dr^{*}dt\Bigg{]}^{\frac{1}{p^{\prime}}}$
	$\displaystyle\triangleq$	$\displaystyle T^{-2}F_{1}^{\frac{1}{p}}(T)(I_{4}+I_{5})^{\frac{1}{p^{\prime}}}.$

We may control $I_{4}$ in the similar way as that of $I_{1}$, thus

$\displaystyle I_{4}\lesssim T^{4}.$

(4.3)

But for $I_{5}$, since

$$r^{*}\leq 4M+e\Leftrightarrow 2M\leq r\leq 2M+e,$$

and hence

$\displaystyle r-2M\sim e^{\frac{r^{*}-C}{2M}}\thicksim e^{\frac{r^{*}}{2M}},$

(4.4)

we then have

	$\displaystyle I_{5}=$	$\displaystyle\int_{\frac{T}{3}}^{\frac{2T}{3}}\int_{-t-R}^{4M+e}r^{2}\left(\frac{r}{r-2M}\right)^{p^{\prime}}\frac{r-2M}{r}dr^{*}dt$		(4.5)
	$\displaystyle\lesssim$	$\displaystyle\int_{\frac{T}{3}}^{\frac{2T}{3}}\int_{-t-R}^{4M+e}e^{-\frac{1}{p-1}\frac{r^{*}}{2M}}dr^{*}dt$
	$\displaystyle\lesssim$	$\displaystyle\int_{\frac{T}{3}}^{\frac{2T}{3}}e^{\frac{t}{2M(p-1)}}dt$
	$\displaystyle\lesssim$	$\displaystyle e^{\frac{T}{3M(p-1)}}.$

By combining (4.2), (4.3) and (4.5) we come to

	$\displaystyle F_{1}\lesssim$	$\displaystyle T^{4-2p^{\prime}}+T^{-2p^{\prime}}e^{\frac{T}{3M(p-1)}}$		(4.6)
	$\displaystyle\lesssim$	$\displaystyle T^{-2p^{\prime}}e^{\frac{T}{3M(p-1)}}.$

In order to prove Theorem 1.1, we use

$$\Phi_{\lambda}(t,r)=\eta_{T}^{2p^{\prime}}(t)\chi_{T}^{2p^{\prime}}(t-r^{*}+R_{3})e^{-\lambda t}\phi_{\lambda}(r)r^{2}$$

(5.1)

as the test function, where $R_{3}$ is a positive constant independent of $\varepsilon$. Also, we can choose $R_{3}$ and $T$(large enough) such that

		$\displaystyle supp~{}\{\chi_{T}(R_{3}-r^{*})=1\}\cap supp~{}(f,g)=supp~{}(f,g),$		(5.2)
		$\displaystyle supp~{}\{\partial_{t}\chi_{T}(R_{3}-r^{*})\}\cap supp~{}(f,g)=\varnothing.$

Multiplying the equation in (1.7) with $\Phi_{\lambda}(t,r)$ and making integration by parts, one has

		$\displaystyle\int_{0}^{T}\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}\partial_{t}\left(\frac{r^{2}}{1-\frac{2M}{r}}\partial_{t}u\eta_{T}^{2p^{\prime}}\chi_{T}^{2p^{\prime}}e^{-\lambda t}\phi_{\lambda}\right)d\omega drdt$		(5.3)
		$\displaystyle-\int_{0}^{T}\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}\partial_{t}\left[\frac{r^{2}}{1-\frac{2M}{r}}u(\eta_{T}^{2p^{\prime}}\chi_{T}^{2p^{\prime}}e^{-\lambda t}\phi_{\lambda})_{t}\right]d\omega drdt$
		$\displaystyle+\int_{0}^{T}\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}\frac{r^{2}}{1-\frac{2M}{r}}u\partial_{t}^{2}(\eta_{T}^{2p^{\prime}}\chi_{T}^{2p^{\prime}}e^{-\lambda t}\phi_{\lambda})d\omega drdt$
		$\displaystyle-\int_{0}^{T}\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}u\partial_{r}\left[r(r-2M)\partial_{r}(\eta_{T}^{2p^{\prime}}\chi_{T}^{2p^{\prime}}e^{-\lambda t}\phi_{\lambda})\right]d\omega drdt$
	$\displaystyle=$	$\displaystyle\int_{0}^{T}\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}|u|^{p}\eta_{T}^{2p^{\prime}}\chi_{T}^{2p^{\prime}}e^{-\lambda t}\phi_{\lambda}r^{2}d\omega drdt,$

which implies according to the support assumption (5.2) that

		$\displaystyle C_{1}\varepsilon$		(5.4)
	$\displaystyle\leq$	$\displaystyle\int_{0}^{T}\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}\frac{r^{2}}{1-\frac{2M}{r}}u\Big{(}\partial_{t}^{2}(\eta_{T}^{2p^{\prime}}\chi_{T}^{2p^{\prime}})e^{-\lambda t}\phi_{\lambda}-2\lambda\partial_{t}\eta_{T}^{2p^{\prime}}\chi_{T}^{2p^{\prime}}e^{-\lambda t}\phi_{\lambda}\Big{)}d\omega drdt$
		$\displaystyle-2\lambda\int_{0}^{T}\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}\frac{r^{2}}{1-\frac{2M}{r}}u\eta_{T}^{2p^{\prime}}\partial_{t}\chi_{T}^{2p^{\prime}}e^{-\lambda t}\phi_{\lambda}d\omega drdt$
		$\displaystyle-\int_{0}^{T}\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}u\Big{(}2r(r-2M)\eta_{T}^{2p^{\prime}}\partial_{r}\chi_{T}^{2p^{\prime}}e^{-\lambda t}\partial_{r}\phi_{\lambda}$
		$\displaystyle+r(r-2M)\eta_{T}^{2p^{\prime}}\partial_{rr}\chi_{T}^{2p^{\prime}}e^{-\lambda t}\phi_{\lambda}+2(r-M)\eta_{T}^{2p^{\prime}}\partial_{r}\chi_{T}^{2p^{\prime}}e^{-\lambda t}\phi_{\lambda}\Big{)}d\omega drdt$
	$\displaystyle\triangleq$	$\displaystyle I_{6}+I_{7}+I_{8}+I_{9}+I_{10}+I_{11},$

where

$$C_{1}=\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}\frac{r^{2}}{1-\frac{2M}{r}}\big{(}\lambda f(x)+g(x)\big{)}\phi_{\lambda}(r)d\omega drdt.$$

For the terms $I_{6}-I_{11}$, the key point is that each of them contains at least one derivative over one of the cut-off functions, thanks to the equation (2.2) in Lemma 2.1. We estimate $I_{7}$ first.

	$\displaystyle I_{7}\triangleq$	$\displaystyle-2\lambda\int_{0}^{T}\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}\frac{r^{2}}{1-\frac{2M}{r}}u\partial_{t}\eta_{T}^{2p^{\prime}}\chi_{T}^{2p^{\prime}}e^{-\lambda t}\phi_{\lambda}d\omega drdt$		(5.5)
	$\displaystyle\lesssim$	$\displaystyle T^{-1}\Big{[}\int_{0}^{T}\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}\frac{r^{2}}{1-\frac{2M}{r}}\big{|}u\eta_{T}^{2p^{\prime}-1}\eta_{T}{{}^{\prime}}\chi_{T}^{2p^{\prime}}\alpha_{T}^{2p^{\prime}}\big{|}e^{-\lambda t}\phi_{\lambda}d\omega drdt$
		$\displaystyle+\int_{0}^{T}\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}\frac{r^{2}}{1-\frac{2M}{r}}\big{|}u\eta_{T}^{2p^{\prime}-1}\eta_{T}{{}^{\prime}}\chi_{T}^{2p^{\prime}}(1-\alpha_{T}^{2p^{\prime}})\big{|}e^{-\lambda t}\phi_{\lambda}d\omega drdt\Big{]}$
	$\displaystyle\triangleq$	$\displaystyle T^{-1}({I_{7}}_{1}+{I_{7}}_{2}),$

where $I_{71}$ can be estimated by $F_{0}(T)$

	$\displaystyle I_{71}\triangleq$	$\displaystyle\int_{0}^{T}\int_{2M}^{\infty}\int_{\mathbb{S}^{2}}\frac{r^{2}}{1-\frac{2M}{r}}\big{|}u\eta_{T}^{2p^{\prime}-1}\eta_{T}{{}^{\prime}}\chi_{T}^{2p^{\prime}}\alpha_{T}^{2p^{\prime}}\big{|}e^{-\lambda t}\phi_{\lambda}d\omega drdt$		(5.6)
	$\displaystyle\lesssim$	$\displaystyle F_{0}(T)^{\frac{1}{p}}\left(\int_{\frac{T}{3}}^{\frac{2T}{3}}\int_{4M+e}^{t+R}r^{2}\left(1-\frac{2M}{r}\right)^{1-p^{\prime}}r^{-p^{\prime}}e^{-\lambda p^{\prime}(t-r^{*})}dr^{*}dt\right)^{\frac{1}{p^{\prime}}}$
	$\displaystyle\lesssim$	$\displaystyle F_{0}(T)^{\frac{1}{p}}\left(\int_{\frac{T}{3}}^{\frac{2T}{3}}\int_{4M+e}^{t+R}(r^{*})^{2-p^{\prime}}e^{-\lambda p^{\prime}(t-r^{*})}dr^{*}dt\right)^{\frac{1}{p^{\prime}}}$
	$\displaystyle\lesssim$	$\displaystyle F_{0}(T)^{\frac{1}{p}}(T^{3-p^{\prime}})^{\frac{1}{p^{\prime}}},$

where an elementary integral estimate (5.7) below has been used.