text:
\section*{Acknowledgements}
The authors wish to thank Chengbo Wang for
his suggestion in the early stages of this research.
The first author was supported by Mie University Research Grant 2014 (Step-up (C)).

\bibliographystyle{amsplain}
\input{0511uvSystemArx.bbl}

\end{document}
\section{Proof of Theorem \ref{y4}}

Our aim in this section is to prove Theorem \ref{y4}.
Let us define
\begin{equation}
 F(t)=\int_{\R^n}u(t,x)\,dx,\quad
 G(t)=\int_{\R^n}v(t,x)\,dx.
\end{equation}
We adopt the strategy of deriving a system of ordinary differential inequalities
with respect to $F(t)$ and $G(t)$,
which causes blow-up of solutions.
Remark that if $(u, v)$ solves the Cauchy problem in the sense
stated above Theorem \ref{y4},
then we can see that $F''(t)=\|v(t,\cdot)\|_{L^q}^q$ and
$G''(t)=\|\pa_tu(t,\cdot)\|_{L^p}^p$ for $t \in (0, T)$.
Hence we conclude from (\ref{y2})--(\ref{y3}) that $F, G \in C^2((0,T))
\cap C^1([0, T))$. Furthermore, $F'(t)$ and $G(t)$ are non-negative.
Thus the following lemma is crucial for the proof of Theorem \ref{y4}.
\begin{lemma}\label{y8}
Let $F, G \in C^2((0, T)) \cap C^1([0, T))$. We suppose that
they satisfy
$F'(t), G(t) \geq 0$ and
\begin{align}
\label{y9}
 &F''(t) \geq A(t+1)^{-\alpha}G(t)^q,\\
\label{y10}
 &G''(t) \geq B(t+1)^{-\beta}F'(t)^p
\end{align}
for all $t \in (0,T)$,
with some positive constants $A, B$ and exponents
$p, q > 1$, $\alpha, \beta \geq 0$.
Assume also $G'(0) > 0$ and
\begin{align}
\label{y11}
 &G(t) \geq \kappa t^a
\end{align}
on some sub-interval $[t_0, T)$, where $\kappa, a > 0$ are constants.
If $\max \left\{t_0, G(0)/G'(0), 1\right\}$ $< T/2$ and
\begin{equation}
 \beta + \alpha p < p+2 + a(pq-1),
\label{y12}
\end{equation}
then
\begin{equation}
 T \leq C\kappa^{-\frac{pq-1}{p+2+a(pq-1)-(\beta + \alpha p)}},
\end{equation}
where $C$ is a constant depending only on $A, B, p, q, \alpha, \beta, a$.
\end{lemma}

\subsection{Proof of Lemma {\rm \ref{y8}}}
We follow the method of Kurokawa, Takamura and Wakasa (see
Lemma 2.1 of \cite{KTW}).
Note beforehand that we have $G'(t) > 0$ as a result of $G'(0)>0$ and (\ref{y10}).
Hence by (\ref{y9}), we have
\begin{align}
 \int_0^t F''(s)G'(s)\,ds
 &\geq A\int_0^t (s+1)^{-\alpha}G(s)^q G'(s)\,ds
\nonumber\\
 &\geq A(t+1)^{-\alpha}\int_0^t G(s)^q G'(s)\,ds.
\nonumber
\end{align}
Applying the integration by parts formula on the left-hand side,
we get
\begin{equation}
 F'(t)G'(t) \geq \frac{A}{q+1}(t+1)^{-\alpha}\left(
G(t)^{q+1}-G(0)^{q+1}\right)
\label{y13}
\end{equation}
for $t > 0$.

We turn to the inequality (\ref{y10}) next.
Multiplying both sides of it by $G'(t)^p$ and using (\ref{y13}),
we get
\begin{align}
 G'(t)^pG''(t) &\geq B(t+1)^{-\beta}\left(
\frac{A}{q+1}(t+1)^{-\alpha}\left(
G(t)^{q+1}-G(0)^{q+1}\right)
\right)^p \nonumber
\end{align}
for $t > 0$. Thus we immediately obtain
\begin{align}
 \int_0^tG'(s)^pG''(s)\,ds
&\geq \frac{A^pB}{(q+1)^p}(t+1)^{-\beta-\alpha p}\int_0^t
\left(G(s)^{q+1}-G(0)^{q+1}\right)^p\,ds.
\nonumber
\end{align}
We can directly compute the left-hand side of this inequality.
As for the right-hand side, we estimate the integral as follows:
\begin{align}
 \lefteqn{\int_0^t \left(G(s)^{q+1}-G(0)^{q+1}\right)^p\,ds}
\nonumber\\
 &= \int_0^t \left(G(s)^{q+1}-G(0)^{q+1}\right)^p
\frac{\left(G(s)^{q+1}\right)'}{(q+1)G(s)^q G'(s)}\,ds
\nonumber\\
 &\geq \frac{1}{(q+1)G(t)^q G'(t)}\int_0^t \left(G(s)^{q+1}-G(0)^{q+1}\right)^p
\left(G(s)^{q+1}\right)'\,ds.
\nonumber
\end{align}
In the last inequality, we have used the fact that $G(s)$ and $G'(s)$ are
increasing functions.
Combining the estimates above, we have
\begin{equation}
 G'(t)^{p+2}
\geq \frac{A^pB}{(q+1)^{p+1}}(t+1)^{-\beta-\alpha p}
\frac{\left(G(t)^{q+1}-G(0)^{q+1}\right)^{p+1}}{G(t)^q}
\label{y14}
\end{equation}
for $t > 0$.

Now we suppose $t \geq G(0)/G'(0)$.
Then we see $G(t) \geq 2G(0)$,
because
\begin{equation}
 G(t) = G(0) + \int_0^t G'(s)\,ds \geq G(0) + tG'(0)
\geq 2G(0).
\nonumber
\end{equation}
Hence (\ref{y14}) implies
\begin{align}
 G'(t)^{p+2} &
\geq \frac{A^pB}{(q+1)^{p+1}}(t+1)^{-\beta-\alpha p}
(1-2^{-q-1})^{p+1}G(t)^{(q+1)(p+1)-q}
\nonumber
\end{align}
for $t \geq G(0)/G'(0)$.
Since $t+1 \leq 2t$ if $t \geq 1$, we further get
\begin{equation}
 G'(t)
\geq Dt^{-\frac{\beta+\alpha p}{p+2}}
G(t)^{\frac{pq+p+1}{p+2}}
\label{y15}
\end{equation}
for $t \geq \max \left\{G(0)/G'(0), 1\right\}$,
where
\begin{equation}
 D=\left\{\frac{A^pB(1-2^{-q-1})^{p+1}}{(q+1)^{p+1}2^{\beta +
    \alpha p}}\right\}^{\frac{1}{p+2}}.
\nonumber
\end{equation}

\bigskip

In what follows, we will show that the inequality (\ref{y15})
cannot hold if $t$ is large enough.
We use (\ref{y11}) in this position.
By (\ref{y15}) and (\ref{y11}),
we have
\begin{align}
 G(t)^{-1-\delta}G'(t) &\geq Dt^{-\frac{\beta+\alpha p}{p+2}}G(t)^{\frac{pq-1}{p+2}-\delta}
\nonumber \\
 &\geq D\kappa^{\frac{pq-1}{p+2}-\delta}t^{\frac{a(pq-1)-(\beta+\alpha p)}{p+2}-a\delta}
\label{y16}
\end{align}
for $t \geq \max \{t_0, G(0)/G'(0), 1\}$,
where we choose $\delta$ so that (recall (\ref{y12}))
\[
 0 < \delta \leq \frac{pq-1}{p+2}, \quad \frac{a(pq-1)-(\beta+\alpha
 p)}{p+2}-a\delta > -1.
\]
Now we take arbitrary $\tilde{T} < T$ so that
$\max \{t_0, G(0)/G'(0), 1\} \leq \tilde{T}/2$.
Integrating (\ref{y16}) over $\left[\tilde{T}/2, \tilde{T}\right]$,
we obtain
\begin{equation}
 \frac{1}{\delta}G\left(\frac{\tilde{T}}{2}\right)^{-\delta}
\geq \frac{D\kappa^{\frac{pq-1}{p+2}-\delta}%
(1-2^{-1-\frac{a(pq-1)-(\beta+\alpha p)}{p+2}+a\delta})}{%
1+\frac{a(pq-1)-(\beta+\alpha p)}{p+2}-a\delta}\tilde{T}^{1+\frac{a(pq-1)-(\beta+\alpha p)}{p+2}-a\delta}.
\label{y17}
\end{equation}
Using (\ref{y11}) again, we see that
\begin{equation}
 \mbox{LHS of (\ref{y17})} \leq \frac{1}{\delta}\kappa^{-\delta}\left(\frac{\tilde{T}}{2}\right)^{-a\delta}.
\label{y18}
\end{equation}
Hence, it follows from (\ref{y17}) and (\ref{y18}) that
\begin{equation}
 \tilde{T} \leq C \kappa^{-\frac{pq-1}{p+2+a(pq-1)-(\beta + \alpha p)}},
\end{equation}
where $C$ is a constant depending only on $A, B, p, q, \alpha, \beta, a$.
This completes the proof of Lemma \ref{y8}.
\hfill $\square$

\subsection{Proof of Theorem {\rm \ref{y4}}}

Following Yordanov--Zhang \cite{YZ05}, we introduce the two positive functions
\begin{equation}
 \ph_1(x) = \int_{S^{n-1}}e^{x\cdot \omega}\,d\omega,\quad
 \psi_1(t,x) = \ph_1(x)e^{-t}.
\nonumber
\end{equation}
Note that $\psi_1$ satisfies $\square \psi_1 = 0$.
By a standard manner, we can deduce from (\ref{y1})--(\ref{y3})
that
\begin{align}
\nonumber
 \int_0^t \lb |v(s,\cdot)|^q, \psi(s,\cdot)\rb\,ds
&= \lb \pa_t u(s,\cdot), \psi(s,\cdot)\rb\Big|_{s=0}^{s=t}
-\lb u(s,\cdot), \pa_t\psi(s,\cdot)\rb\Big|_{s=0}^{s=t}\\
&\quad +\int_0^t \lb u(s,\cdot),(\pa_t^2\psi -\Delta \psi)(s,\cdot)\rb\,ds
\nonumber
\end{align}
for any $\psi \in C^{\infty}([0, T) \times \R^n)$ such that $\psi (s,\cdot)
\in C_0^{\infty}(\R^n)$ for each $s \in [0, T)$. Thanks to (\ref{y6}),
we may substitute $\psi = \psi_1$. As a result, we easily obtain
\begin{equation}
 \int_0^t \lb |v(s,\cdot)|^q, \psi_1(s,\cdot)\rb\,ds
= \lb \pa_t u(t,\cdot) + u(t,\cdot), \psi_1(t,\cdot)\rb
-\ep \lb f + g, \ph_1\rb.
\nonumber
\end{equation}
Moreover, we see from (\ref{y1}) that
\begin{align}
\nonumber
 \lefteqn{\frac{d}{dt}\lb \pa_t u(t,\cdot), \psi_1(t,\cdot)\rb}\\
 & = \lb |v(t,\cdot)|^q, \psi_1(t,\cdot)\rb + \lb u(t,\cdot),\Delta \psi_1(t,\cdot)\rb
-\lb \pa_t u(t,\cdot),\psi_1(t,\cdot)\rb \nonumber\\
 & = \lb |v(t,\cdot)|^q, \psi_1(t,\cdot)\rb + \lb u(t,\cdot), \psi_1(t,\cdot)\rb
-\lb \pa_t u(t,\cdot),\psi_1(t,\cdot)\rb \nonumber
\end{align}
for $t \in (0, T)$. Combining these estimates, we get
\begin{align}
e^{-2t}\frac{d}{dt}\left(e^{2t}\lb \pa_t u(t,\cdot), \psi_1(t,\cdot)\rb\right)
&= \frac{d}{dt}\lb \pa_t u(t,\cdot), \psi_1(t,\cdot)\rb
+ 2\lb \pa_t u(t,\cdot),\psi_1(t,\cdot)\rb\nonumber\\
  &\geq \ep \lb f+g, \ph_1\rb,\nonumber
\end{align}
which gives
\begin{equation}
\label{y19}
 \lb \pa_t u(t,\cdot), \psi_1(t,\cdot)\rb \geq \frac{1}{4}\ep \lb f+g, \ph_1\rb
\end{equation}
for $t \geq \log \sqrt{2}$.

On the other hand, according to Lemma 2.3 of Yordanov--Zhang \cite{YZ05},
\begin{equation}
\label{y20}
 \|\psi_1(t,\cdot) \|_{L^{p'}(|x| \leq t+1)} \leq C(t+1)^{(n-1)\left(\frac{1}{2}-\frac{1}{p}\right)}
\end{equation}
for $t \geq 0$ (see also Zhou--Han \cite{ZH14}). Hence, by (\ref{y19}) and (\ref{y20}),
we have
\begin{equation}
\nonumber
 G''(t) = \|\pa_t u(t,\cdot)\|_{L^p}^p \geq C\ep^p(t+1)^{-\frac{(n-1)(p-2)}{2}}
\end{equation}
for $t \geq \log \sqrt{2}$. Since $(n-1)(p-2)/2<1$ by assumption,
we eventually get the estimate
\begin{equation}
\nonumber
 G(t) \geq C\ep^p (t+1)^{2-\frac{(n-1)(p-2)}{2}}
\end{equation}
for $t \geq t_0$, if $t_0$ is large enough.

Meanwhile, by (\ref{y6}) and the H\"older inequality, we
immediately have
\begin{align}
 &\left(F'(t)\right)^p \leq C(t+1)^{n(p-1)}\|\pa_tu(t,\cdot)\|_{L^p}^p
= C(t+1)^{n(p-1)}G''(t),\nonumber\\
&\left(G(t)\right)^q \leq C(t+1)^{n(q-1)}\|v(t,\cdot)\|_{L^q}^q
= C(t+1)^{n(q-1)}F''(t)\nonumber
\end{align}
for $t > 0$.

Now we are ready to apply Lemma \ref{y8} to the present problem.
Set
\begin{equation}
\nonumber
 a = 2-\frac{(n-1)(p-2)}{2},\quad \alpha = n(q-1),\quad
\beta = n(p-1)
\end{equation}
and $\kappa = C\varepsilon^p$.
Noting $\beta + \alpha p = n(pq-1)$,
we see that (\ref{y12}) is equivalent to (\ref{y5}).
Hence we have (\ref{y7}) as a conclusion of Lemma \ref{y8}.
\qed

\section{Proof of Theorem \ref{z7}}

To control the iteration procedure, we need the estimates of derivatives.
Computing the first derivatives of (\ref{z1}) and (\ref{z2}), we find that
\begin{align}
 r\pa_t u(t, r) &= \ep r\pa_t u_0(t, r)+K_+|v|^q (t, r),
\nonumber\\
 \pa_r \left\{rv(t, r)\right\} &= \ep \pa_r \left\{rv_0(t, r)\right\}
+K_-|\pa_tu|^p (t, r),\nonumber
\end{align}
where
\begin{equation}
 K_{\pm}F(t, r)
= \frac{1}{2}\int_0^t \left\{(r+t-s) F(s,r+t-s) \pm (r-t+s)
F(s,r-t+s)\right\}\,ds. \nonumber
\end{equation}
As functions of $r$,
$K_+F(t,r)$ is odd and $K_-F(t,r)$ is even.

In order to prove Theorem \ref{z7}, we use the following lemmas.
Let us recall the definitions of the operator $L$ and the weights $w_1$,
$w_2$, $w_3$ for convenience:
\begin{equation}
 LF(t,r) = \frac{1}{2r}\int_0^t \int_{r-(t-s)}^{r+(t-s)}\rho
  F(s,\rho)\,d\rho ds,
\nonumber
\end{equation}
\begin{align*}
 w_1(t,r)&=\lb r\rb \lb t-r\rb^{\mu/p}, \\
 w_2(t,r)&=
 \begin{cases}
  \lb r \rb^{p-2} \lb t+r\rb^{\mu} & \left(r < t/2\right) \\
  \lb t-r\rb^{p-3+\mu}\lb t+r\rb & \left(r \geq t/2\right)
 \end{cases},\\
 w_3(t,r)&= \lb t-r\rb^{\mu}.
\end{align*}
\begin{lemma}\label{z8}
Suppose {\rm (\ref{z3})}, {\rm (\ref{z4})}.
Then we have
 \begin{align}
 \label{z9}
&\|w_2L|w|^{p}\|_{L^{\infty}_{t,r}}
 \leq  C\|w_1 w\|^{p}_{L^{\infty}_{t,r}},\\
\label{z10}
&\|\lb t-r \rb^{\mu/p}K_+|v|^{q}\|_{L^{\infty}_{t,r}}
 \leq  C\|w_2 v\|^{q}_{L^{\infty}_{t,r}},\\
\label{z11}
&\left\|\lb t \rb^{\mu/p} r^{-1}
K_+|v|^{q}\right\|_{L^{\infty}_{t,r}(r < 1)}
 \leq  C\|w_2 v\|^{q}_{L^{\infty}_{t,r}}
+C\|w_2 v\|^{q-1}_{L^{\infty}_{t,r}}
\|w_3 r\pa_rv\|_{L^{\infty}_{t,r}},\\
\label{z12}
&\|w_3K_-|w|^{p}\|_{L^{\infty}_{t,r}}
 \leq  C\|w_1 w\|^{p}_{L^{\infty}_{t,r}}.
 \end{align}
\end{lemma}

\begin{lemma}\label{z13}
Suppose {\rm (\ref{z3})}, {\rm (\ref{z4})}.
Then we have
 \begin{align}
&\|w_2L[|w|^{p}-|\bar{w}|^p]\|_{L^{\infty}_{t,r}}
 \leq  C\left(\|w_1 w\|^{p-1}_{L^{\infty}_{t,r}}
+\|w_1 \bar{w}\|^{p-1}_{L^{\infty}_{t,r}}
\right)
 \|w_1 (w-\bar{w})\|_{L^{\infty}_{t,r}},\\
\lefteqn{\|\lb t-r \rb^{\mu/p}K_+[|v|^{q}-|\bar{v}|^q]\|_{L^{\infty}_{t,r}}}
\nonumber\\
& \leq  C\left(\|w_2 v\|^{q-1}_{L^{\infty}_{t,r}}
+\|w_2 \bar{v}\|^{q-1}_{L^{\infty}_{t,r}}
\right)
\|w_2 (v-\bar{v})\|_{L^{\infty}_{t,r}},
\label{z26}\\
\lefteqn{\left\|\lb t \rb^{\mu/p}r^{-1}K_+[|v|^{q}-|\bar{v}|^q]\right\|_{L^{\infty}_{t,r}(r < 1)}}
\nonumber\\
& \leq  C\left(\|w_2 v\|^{q-1}_{L^{\infty}_{t,r}}
+\|w_2 \bar{v}\|^{q-1}_{L^{\infty}_{t,r}}
\right)
\|w_2 (v-\bar{v})\|_{L^{\infty}_{t,r}}\nonumber\\
&\quad +C\left(\|w_2 v\|^{q-2}_{L^{\infty}_{t,r}}
+\|w_2 \bar{v}\|^{q-2}_{L^{\infty}_{t,r}}
\right)\|w_3r\pa_r v\|_{L^{\infty}_{t,r}}
\|w_2 (v-\bar{v})\|_{L^{\infty}_{t,r}}
\nonumber\\
&\quad+ C\|w_2\bar{v}\|^{q-1}_{L^{\infty}_{t,r}}
\|w_3r(\pa_r v - \pa_r \bar{v})\|_{L^{\infty}_{t,r}},
\label{z27}\\
&\|w_3K_-[|w|^{p}-|\bar{w}|^p]\|_{L^{\infty}_{t,r}}
 \leq  C\left(\|w_1 w\|^{p-1}_{L^{\infty}_{t,r}}
+\|w_1 \bar{w}\|^{p-1}_{L^{\infty}_{t,r}}
\right)
 \|w_1 (w-\bar{w})\|_{L^{\infty}_{t,r}}.
 \end{align}
\end{lemma}

{\it Proof of Theorem {\rm \ref{z7}}}.\ We consider the system of integral equations
\begin{align}
\label{z14}
 w(t,r) & = \ep \pa_tu_0(t,r)+ r^{-1}K_+|v|^q (t,r), \\
 \label{z15}
 v(t,r) & = \ep v_0(t,r)+ L|w|^p (t,r)
\end{align}
in the complete metric space
\begin{align}
 X_{\ep}=&\{(w, v) \in C([0,\infty)\times \R) \times C([0,\infty)\times \R)\,:\,\nonumber\\
  &\quad w(t,-r)=u(t,r),\ v(t,r)=v(t,-r),\nonumber\\
  &\quad  r\pa_r v \in C([0,\infty)\times \R),\nonumber\\
&\quad \|(w,v)\| =\|w_1w\|_{L^{\infty}_{t,r}}
+\|w_2v\|_{L^{\infty}_{t,r}}
+\|w_3r\pa_r v\|_{L^{\infty}_{t,r}} \leq C_2\ep\}.\nonumber
\end{align}
Lemmas \ref{z8}, \ref{z13} imply that
(\ref{z14})--(\ref{z15}) can be solved by
applying the Banach fixed-point theorem for the mapping
\begin{equation}
 (w, v) \longmapsto (\ep \pa_tu_0+ r^{-1}K_+|v|^q,\
 \ep v_0+ L|w|^p),\nonumber
\end{equation}
if $C_2$ is suitably chosen and $\ep$ is sufficiently small.
Since the argument is standard, we omit the details.
Setting
\begin{equation}
 u(t,r) := \ep f(r) + \int_0^tw(s,r)\,ds, \nonumber
\end{equation}
we see that $(u, v)$ satisfies {\rm (\ref{z1})--(\ref{z2})}.
Moreover, we can easily check the regularity property by using the
integral equations (\ref{z1})--(\ref{z2}).
\qed

We will prove Lemma \ref{z8} in the following subsections.
We omit the proof of Lemma \ref{z13}, since it is similar to the proof
of Lemma \ref{z8}.
To prove Lemma \ref{z13}, we adapt the proof of
Lemma \ref{z8} by noting
\begin{align}
 &\big||v|^q-|\bar{v}|^q\big|
\leq C\left(|v|^{q-1}+|\bar{v}|^{q-1}\right)|v-\bar{v}|
\nonumber\\
\intertext{for (\ref{z26}),}
 &\lefteqn{
\big|
\left(|v(s,r^+)|^q-|\bar{v}(s,r^+)|^q\right)
-\left(|v(s,|r^-|)|^q-|\bar{v}(s,|r^-|)|^q\right)\big|}
\nonumber\\
& \leq C|r^+-|r^-||\biggl\{\int_0^1
\left(|v|^{q-2}+|\bar{v}|^{q-2}\right)|\pa_rv||v-\bar{v}|
(s,\theta r^+ + (1-\theta)|r^-|)
\,d\theta\nonumber\\
&\quad + \int_0^1|\bar{v}|^{q-1}|\pa_r(v-\bar{v})|
(s,\theta r^+ +(1-\theta)|r^-|)\,d\theta
\biggr\}
\nonumber
\end{align}
for (\ref{z27}), and so on. Here we set $r^{\pm}=r \pm (t-s)$ (see
Sections \ref{z28}--\ref{z29} below).

\subsection{Proof of {\rm (\ref{z9})}.}
Let $r > 0$. Using the obvious pointwise estimate
\begin{equation}
 |w(s,\rho)| \leq w_1^{-1}(s,\rho) \|w_1 w\|_{L^{\infty}_{t,r}},
\nonumber
\end{equation}
we immediately get
\begin{align}
 |L|w|^{p}(t,r)|
 & \leq CLw_1^{-p}(t,r)\cdot
 \|w_1 w\|^{p}_{L^{\infty}_{t,r}}.
\nonumber
\end{align}
Thus we need to show
\begin{equation}
Lw_1^{-p} \leq Cw_2(t, r)^{-1}.
\nonumber
\end{equation}
Define the two sets $D_1$, $D_2$ by
\begin{align}
 D_1&=\left\{(s, \rho) \,:\, 0<s<t,\ |r-(t-s)|<\rho<r+(t-s),\ s-\rho >
 -|t-r|\right\}, \nonumber\\
 D_2&=\left\{(s, \rho) \,:\, 0<s<t,\ |r-(t-s)|<\rho<r+(t-s),\ s-\rho \leq
 -|t-r|\right\}.\nonumber
\end{align}
Note that $D_1 = \varnothing$ if $r\geq t$.
In what follows, we will estimate $L[\chi_{D_1}w_1^{-p}]$ and $L[\chi_{D_2}w_1^{-p}]$ separately.

\bigskip

\emph{Estimate of $L[\chi_{D_1}w_1^{-p}]$}.
We assume $t > r$, since otherwise $D_1 = \varnothing$.
We further write $\chi_{D_1}=\chi_{D_{11}}+\chi_{D_{12}}$, where
\begin{align}
 D_{11}&=\left\{(s, \rho) \,:\, (s, \rho) \in D_1,\ s-\rho < \frac{s+\rho}{2} \right\}, \nonumber\\
 D_{12}&=\left\{(s, \rho) \,:\, (s, \rho) \in D_1,\ s-\rho \geq  \frac{s+\rho}{2} \right\}.\nonumber
\end{align}
Since $\lb \rho \rb \geq 4^{-1}\lb s+\rho \rb$ on $D_{11}$, we have
\begin{align}
 L[\chi_{D_{11}}w_1^{-p}]
 & \leq \frac{C}{r}\iint_{D_{11}}\lb s+\rho \rb^{1-p}\lb s-\rho \rb^{-\mu}\,d\rho ds,
\nonumber\\
\intertext{and applying the transformation $\tau = s+\rho$, $\sigma = s
 - \rho$, we have}
 & \leq \frac{C}{r} \int_{t-r}^{t+r}\int_{-(t-r)}^{t-r}\lb \tau \rb^{1-p}
 \lb \sigma \rb^{-\mu}\,d\sigma d\tau
\nonumber\\
 & \leq \frac{C}{r} \int_{t-r}^{t+r}\lb \tau \rb^{1-p}\,d\tau
 \int_0^{t-r} \lb \sigma \rb^{-\mu}\,d\sigma.
\label{z30}
\end{align}
Meanwhile, we can check that
\begin{equation}
\label{z17}
\frac{1}{r} \int_{|t-r|}^{t+r} \lb \tau \rb^{-\kappa}\,d\tau
 \leq C \lb t+r \rb^{-1} \lb t-r \rb^{1-\kappa} \quad \mbox{for}\quad \kappa >1.
\end{equation}
Indeed, if $r > t/2$ and $r > 1$, we easily see this inequality by $r \geq C\lb
t+r \rb$; otherwise, we can use $t+r-|t-r| \leq 2r$ and $|t-r| \geq C(t+r)$.
Putting (\ref{z30}) and (\ref{z17}) together, we get
\begin{equation}
\label{z18}
 L[\chi_{D_{11}}w_1^{-p}] \leq Cw_2(t,r)^{-1}.
\end{equation}

We next consider $L[\chi_{D_{12}}w_1^{-p}]$.
Making the change of variables $\tau = s+\rho$, $\sigma = s
 - \rho$ in the integral, we obtain
\begin{align}
 L[\chi_{D_{12}}w_1^{-p}]
&\leq \frac{C}{r}\iint_{D_{12}}
\lb \rho \rb^{1-p}\lb s+\rho \rb^{-\mu}\,d\rho ds
\nonumber\\
 & \leq \frac{C}{r} \int_{t-r}^{t+r}\int_{\frac{t-r}{2}}^{t-r}
 \lb \tau - \sigma \rb^{1-p}\lb \tau \rb^{-\mu}\,d\sigma d\tau
\nonumber\\
 & \leq \frac{C}{r} \int_{t-r}^{t+r} \lb \tau-t+r \rb^{2-p}
\lb \tau \rb^{-\mu}\,d\tau.
\label{z19}
\end{align}

Now suppose $r > t/2$ and $r > 1$ first. Then we see
\begin{align}
 \lefteqn{\mbox{RHS of (\ref{z19})}}\nonumber\\
 & \leq \frac{C}{r}\lb t-r \rb^{-\mu}
\int_{t-r}^{2(t-r)} \lb \tau -t+r \rb^{2-p}\,d\ta
+ \frac{C}{r}\int_{2(t-r)}^{t+r}\lb \tau \rb^{2-p-\mu}\,d\tau
\nonumber\\
 & \leq  C w_2(t,r)^{-1}.\nonumber
\end{align}
If, on the other hand, $r \leq t/2$ or $r \leq 1$,
\begin{align}
\mbox{RHS of (\ref{z19})}
 &\leq
 \frac{1}{r}\lb t-r \rb^{-\mu}\int_{t-r}^{t+r} \lb \tau-t+r \rb^{2-p}\,d\tau
\nonumber\\
 & \leq C\frac{(1+r)^{3-p}-1}{r}\lb t+r \rb^{-\mu} \nonumber\\
 & \leq Cw_2(t,r)^{-1}.\nonumber
\end{align}
Thus we have shown $L[\chi_{D_{12}}w_1^{-p}] \leq Cw_2(t, r)^{-1}$. Together with
(\ref{z18}), we obtain
\begin{equation}
  L[\chi_{D_{1}}w_1^{-p}]  \leq Cw_2(t,r)^{-1}.
\nonumber
\end{equation}

\bigskip

\emph{Estimate of $L[\chi_{D_2}w_1^{-p}]$}.\
By direct computation, we have
\begin{align}
L[\chi_{D_2}w_1^{-p}]   & =
 \frac{1}{r}\int_{|t-r|}^{t+r}\int_0^{%
 \min\{\rho-|t-r|, t+r-\rho\}}
 \lb \rho \rb^{1-p} \lb \rho-s \rb^{-\mu}\,ds d\rho
\nonumber\\
  & \leq
 \frac{C}{r}\int_{|t-r|}^{t+r} \lb \rho \rb^{2-p-\mu}\,d\rho.
\nonumber
\end{align}
Thus (\ref{z17}) gives $L[\chi_{D_2}w_1^{-p}] \leq Cw_2(t, r)^{-1}$.
\qed

\subsection{Proof of {\rm (\ref{z10})}.}\label{z28}

We set
\begin{equation}
 r^{\pm} = r \pm (t-s)
\nonumber
\end{equation}
for brevity. Then we have
\begin{align}
\label{z20}
 |K_+|v|^{q}(t,r)|
 \leq \int_0^t \left\{
\frac{r^+}{w_2(s,r^+)^{q}} + \frac{|r^-|}{w_2(s,|r^-|)^{q}}\right\}\,ds
 \cdot \|w_2 v\|_{L^{\infty}_{t,r}}^{q}.
\end{align}
In order to estimate this integral,
we note that
\begin{align}
 & \lb t-r \rb \leq \lb s+|r^-| \rb \leq \lb s+r^+ \rb \quad \mbox{for} \quad 0 \leq s \leq t,\nonumber\\
 & \lb t-r \rb = \lb s-|r^-| \rb \quad \mbox{if} \quad r^- \geq 0.
\nonumber
\end{align}
Here the first inequality is a consequence of $\big||t-r| - s\big| \leq |t-r-s|
= |r^-|$.

Recalling (\ref{z5}), we see that, for the first part of the integral in (\ref{z20}),
\begin{align}
\lb t-r \rb^{\mu/p}\int_0^t \frac{r^+}{w_2(s,r^+)^{q}}\,ds
&\leq \int_0^t \frac{r^+ \lb s+r^+ \rb^{\mu/p}}{w_2(s,r^+)^{q}}\,ds
\nonumber\\
&\leq \int_{r^+ <s/2} \lb r^+ \rb^{1-(p-2)q}
 \lb s+r^+ \rb^{\mu/p-\mu q}\,ds\nonumber\\
&\quad + \int_{r^+ \geq s/2}
\lb s+r^+ \rb^{1+\mu/p-q} \lb s-r^+ \rb^{-(p-3+\mu)q}\,ds.
\nonumber
\end{align}
As for the second part, we estimate it as
\begin{align}
\lefteqn{\lb t-r \rb^{\mu/p}\int_0^t \frac{|r^-|}{w_2(s,|r^-|)^{q}}\,ds}
\nonumber\\
&\leq \int_{|r^-|<s/2} \lb r^- \rb^{1-(p-2)q}\lb s+|r^-| \rb^{\mu/p-\mu q}\,ds\nonumber\\
&\quad + \int_{r^- \leq -s/2} \lb s+|r^-| \rb^{1+\mu/p-q}\lb s-|r^-| \rb^{-(p-3+\mu)q}\,ds
\nonumber\\
&\quad + \int_{r^- \geq s/2} \frac{\lb t-r \rb^{\mu/p}}{\lb s+|r^-| \rb^{q-1}\lb s-|r^-| \rb^{(p-3+\mu)q}}\,ds.
\label{z21}
\end{align}
Since $\mu$ has chosen so that (\ref{z6}) holds, we see
\begin{equation}
\label{z22}
 \frac{\mu}{p}+1-(p-2+\mu)q<-1.
\end{equation}
Therefore, for example,
\begin{align}
\lefteqn{\sum_{i=+,-}\int_{|r^i|<s/2}
\lb r^i \rb^{1-(p-2)q} \lb s+|r^i| \rb^{\mu/p-\mu q}\,ds}
\nonumber\\
&\leq \sum_{i=+,-}\int_{|r^i|<s/2} \lb r^i \rb^{1-(p-2)q-\mu q+\mu/p}\,ds
\leq C. \nonumber
\end{align}
All the terms other than the last term of (\ref{z21}) can be treated similarly,
and we find that they are bounded by a constant.
Finally, using (\ref{z22}) again, we obtain
\begin{align}
\lefteqn{\int_{r^- \geq s/2} \frac{\lb t-r \rb^{\mu/p}}{\lb s+|r^-| \rb^{q-1}\lb s-|r^-| \rb^{(p-3+\mu)q}}\,ds}\nonumber\\
& = \lb t-r \rb^{\mu/p-(p-3+\mu)q}\int_{s \geq \max\{0, 2(t-r)\}} \lb
 r-t+2s \rb^{1-q}\,ds\nonumber\\
& \leq C\lb t-r \rb^{\mu/p-(p-3+\mu)q+2-q} \leq C.\nonumber
\end{align}
\qed

\subsection{Proof of {\rm (\ref{z11})}.}
We assume $0<r < 1$ throughout the proof of (\ref{z11}).
To cancel out the unbounded factor $r^{-1}$, we begin with
\begin{align}
\frac{1}{r}|K_+|v|^{q}(t,r)|
 & =  \frac{1}{r}\left|\int_0^t \left\{r^+|v|^{q}(s,r^+) + r^-|v|^{q}(s,|r^-|)\right\}\,ds\right|
\nonumber \\
 & \leq \frac{1}{r}\int_0^t |r^+ + r^-||v|^{q}(s,r^+)\,ds
\nonumber\\
 &\quad + \frac{1}{r}\int_0^t |r^-|\left||v|^{q}(s,r^+)-|v|^{q}(s,|r^-|)\right|\,ds.
\label{z23}
\end{align}

Firstly, let us consider the simpler part,
the first integral of the last expression.
Since $r^++r^-=2r$, we have
\begin{align}
\frac{1}{r}\int_0^t |r^+ + r^-||v|^{q}(s,r^+)\,ds
  \leq 2\int_0^t w_2(s,r^+)^{-q}\,ds \cdot
\|w_2v\|^{q}_{L^{\infty}_{t,r}}.
\nonumber
\end{align}
Using $\lb t \rb \leq \lb s+r^+ \rb$, we see that
\begin{align}
\lb t \rb^{\mu/p}\int_0^tw_2(s,r^+)^{-q}\,ds
 & \leq \int_{r^+<s/2}\lb r^+ \rb^{-(p-2)q}\lb s+r^+ \rb^{\mu/p-\mu q}\,ds
\nonumber\\
 & \quad +
\int_{r^+ \geq s/2}\lb s-r^+ \rb^{-(p-3+\mu)}
\lb s+r^+ \rb^{\mu/p-q}\,ds\nonumber\\
 & \leq C \int_{r^+<s/2}\lb r^+ \rb^{-(p-2)q+\mu/p-\mu q}\,ds\nonumber\\
 & \quad +
 C\int_{r^+ \geq s/2}\lb s-r^+ \rb^{-(p-3+\mu)q+\mu/p - q}\,ds.\nonumber
\end{align}
Thus we obtain the desired estimate by (\ref{z22}).

We turn to the remaining part of (\ref{z23}) next.
Noting that $|r^+-|r^-|| \leq 2r$, we obtain
\begin{align}
\lefteqn{\frac{1}{r}\int_0^t |r^-|\left||v|^{q}(s,r^+)-|v|^{q}(s,|r^-|)\right|\,ds}
\nonumber\\
&\leq C\int_0^1\int_0^t |r^-|
 |v(s, r_{\theta})|^{q-1}|\pa_r v(s, r_{\theta})|\,dsd\theta,
\label{z24}
\end{align}
where we set
\begin{equation}
r_{\theta}=\theta r^+ + (1-\theta)|r^-|.
\nonumber
\end{equation}
Now we use the weighted norms as bofore, and see that
\begin{align}
 \lefteqn{\mbox{RHS of } (\ref{z24})}
\nonumber\\
&\leq C\int_0^1\int_0^t
\frac{|r^-|}{w_2(s,r_{\theta})^{q-1}}
\frac{1}{|r_\theta|
w_3(s,r_\theta)}\,dsd\theta
 \cdot \|w_2v\|^{q-1}_{L^{\infty}_{t,r}}
\|w_3r\pa_rv\|_{L^{\infty}_{t,r}}
\nonumber\\
&\leq C\int_0^1\int_0^t
\frac{1}{w_2(s,r_{\theta})^{q-1}
w_3(s,r_\theta)}\,dsd\theta
 \cdot \|w_2v\|^{q-1}_{L^{\infty}_{t,r}}
\|w_3r\pa_rv\|_{L^{\infty}_{t,r}}.
\nonumber
\end{align}
In what follows, we will show that
\begin{equation}
\label{z25}
 \int_0^1\int_0^t
\frac{1}{w_2(s,r_{\theta})^{q-1}w_3(s,r_\theta)}\,dsd\theta
\leq C \lb t \rb^{-\mu/p}.
\end{equation}
We may assume $t \geq 1$,
because otherwise (\ref{z25}) is easy (recall that we are assuming $0 < r < 1$).
Note that the region $r^- \geq 0$ is included in $t - 1 < s < t$.
Therefore,
\begin{align}
\lefteqn{\lb t \rb^{\mu/p}
\int_0^1\int_{r_- \geq 0}
\frac{1}{w_2(s,r_{\theta})^{q-1}w_3(s,r_\theta)}\,dsd\theta}
\nonumber\\
&\leq C\int_0^1\int_{r^- \geq 0}
\frac{\lb s+r_{\theta} \rb^{\mu/p}}{\lb s+r_{\theta}\rb^{\mu (q-1)}}
\,dsd\theta\nonumber\\
&\leq C\int_0^1\int_{t-1}^t
\,dsd\theta \leq C.\nonumber
\end{align}
When $r^- < 0$, we see that
the coefficients of $s$ in $r_{\theta}=(2\theta -1)r+t-s$
and $s-r_{\theta}=2s + (1-2\theta)r-t$
are independent of $\theta$. Thus we estimate the integral over
the region $r^-<0$ as
\begin{align}
\lefteqn{\lb t \rb^{\mu/p}
\int_0^1\int_{r^-<0}
\frac{1}{w_2(s,r_{\theta})^{q-1}w_3(s,r_\theta)}\,dsd\theta}
\nonumber\\
&\leq C\int_0^1\int_{r_{\theta} < s/2,\ r^-<0}
\lb r_{\theta} \rb^{-(p-2)(q-1)}\lb s+r_{\theta} \rb^{\mu/p-\mu (q-1)-\mu}
\,dsd\theta
\nonumber\\
&\quad + C\int_0^1\int_{s/2 \leq r_{\theta},\ r^-<0}
\lb s-r_{\theta} \rb^{-(p-3+\mu)(q-1)-\mu}\lb s+r_{\theta} \rb^{\mu/p-(q-1)}
\,dsd\theta\nonumber\\
&\leq C\int_0^1\int_{r_{\theta} < s/2,\ r^-<0}
\lb r_{\theta} \rb^{-(p-2)(q-1)+\mu/p-\mu (q-1)-\mu}
\,dsd\theta
\nonumber\\
&\quad + C\int_0^1\int_{s/2 \leq r_{\theta},\ r^-<0}
\lb s-r_{\theta} \rb^{-(p-3+\mu)(q-1)-\mu+\mu/p-(q-1)}
\,dsd\theta.\nonumber
\end{align}
Since
\begin{align}
& \frac{\mu}{p}-\mu (q-1)-(p-2)(q-1) - \mu
 < p-4 < -1
\nonumber
\end{align}
by (\ref{z22}),
the integrals above are bounded by a constant.
We have finished the proof of (\ref{z11}).
\qed

\subsection{Proof of {\rm (\ref{z12})}}\label{z29}
We follow the same process as before.
\begin{align}
|K_-|w|^{p}(t,r)|
 \leq C\int_0^t \left\{ \frac{r^+}{w_1(s,r^+)^{p}}
 + \frac{|r^-|}{w_1(s,|r^-|)^{p}}\right\}\,ds \cdot
\|w_1w\|^{p}_{L^{\infty}_{t,r}}.
\nonumber
\end{align}
Both of integrands $|r^{\pm}|w_1(s,|r^{\pm}|)^{-1}$ can be treated similarly.
We only show the estimate of the second part.
We note that $\lb t-r \rb \leq \lb s+|r^-| \rb$ and  get
\begin{align}
\lefteqn{\lb t-r \rb^{\mu}
\int_0^t \frac{|r^-|}{w_1(s,|r^-|)^q}\,ds}
\nonumber\\
&\leq C\int_{|r^-|<s/2} \lb r^- \rb^{1-p}\,ds
 + C\int_{r^- \leq -s/2} \lb s-|r^-| \rb^{1-p}\,ds
\nonumber\\
&\quad + C\int_{r^- \geq s/2} \lb s+|r^-| \rb^{1-p}\,ds
\leq C.
\nonumber
\end{align}
\qed

\section{Introduction}

We are interested in the study of systems of semilinear wave equations of the form
\begin{equation}
\label{b1}
 \square u = |v|^q,\quad
 \square v = |\pa_t u|^p.
\end{equation}
Here the unknown real-valued functions $u$, $v$ depend on $(t, x) \in
[0,T) \times \R^n$ for some $T \in (0, \infty]$.
Throughout this paper, we suppose $n \geq 2$.
Given some functions $f$, $g$, $\tilde{f}$, $\tilde{g}$,
we consider the Cauchy problem for (\ref{b1}) with initial data
\begin{equation}
\label{b2}
 u(0,x)=\ep f(x),\quad \pa_t u(0,x)=\ep g(x),\quad
 v(0,x)=\ep \tilde{f}(x),\quad \pa_t v(0,x)=\ep \tilde{g}(x),
\end{equation}
where $\ep > 0$ is small.
We want to determine, in the set $(1, \infty)^2$
of index-pairs $(p, q)$, the borderline
between global-in-time existence and finite time blow-up
for (\ref{b1})--(\ref{b2}) when $\ep$ is small.

The system (\ref{b1}) reminds us of some related problems.
Among them, the Cauchy problem for
\begin{equation}
\label{b10}
 \square u = |u|^q
\end{equation}
has been especially well-studied since John's pioneering work \cite{J79}. If $q > q_0(n)$ where
\[
 q_0(n) = \frac{n+1+\sqrt{n^2+10n-7}}{2(n-1)},
\]
then the Cauchy problem for (\ref{b10}) has a unique global-in-time
solution for small and smooth initial data with compact support.
If $1 < q \leq q_0(n)$ on the contrary, then there exists
$(f, g) \in C_0^{\infty}(\R^n)^2$ such that the Cauchy problem for (\ref{b10})
with initia data $(u(0), \pa_tu(0)) =
(\ep f, \ep g)$ has a solution which blows up
in finite time no matter how small $\ep > 0$ is.
This phenomenon has been verified for all $n \geq 2$ through efforts
of several authors: see, e.g.,
\cite{J79}, \cite{Gl2}, \cite{Z95},
\cite{KK98},
\cite{LS}, \cite{GLS} for global existence results,
\cite{J79}, \cite{Gl1}, \cite{S84}, \cite{Sc85}, \cite{R}, \cite{YZ06}
for blow-up results. Thus we have already understood that
$q_0(n)$ is the critical exponent which separates
global-in-time solvability and unsolvability for (\ref{b10}).

The similar phenomenon can be observed for the equation
\begin{equation}
\label{b11}
 \square v = |\pa_t v|^p.
\end{equation}
For this equation, the critical value of $p$ is
\[
 p_0(n) = \frac{n+1}{n-1}.
\]
In the case $1 < p \leq p_0(n)$, we know that solutions blow up
in finite time in general, no matter how small the initial data are
chosen. We also know that (\ref{b11}) with $p > p_0(n)$ has a unique global-in-time
solution for given small and smooth data, though the radial symmetry
assumption is needed for $n \geq 4$ so far.
See, e.g., \cite{HT}, \cite{Tz},
\cite{HWY2014} for global existence results,
\cite{A}, \cite{J81},
\cite{Z01} for blow-up results.

Recently, combined effects of the two nonlinearities above have been studied
by \cite{HZ} and \cite{HWY2014}. They considered the Cauchy problem for
\begin{equation}
\label{b12}
 \square u = |u|^q + |\pa_t u|^p
\end{equation}
(only when $n=2, 3$ in \cite{HWY2014}).
According to their results, we can conclude that
\begin{equation}
\label{b14}
 q = \frac{4}{(n-1)p-2}+1 \quad (q>q_0(n),\ p>p_0(n))
\end{equation}
is a portion of the critical curve in the $p$-$q$ plane of index-pairs $(p, q)$
(see \cite{HZ}, \cite{HWY2014} for details).
Observe that $(p_0(n), q_0(n))$ belongs to the domain
$q < 4/\{(n-1)p-2\}+1$, in which $(p, q)$
corresponds to the blow-up case.
As is pointed out in \cite{HZ}, this means that
there exists $(p, q)$ such that a solution of the Cauchy problem
(\ref{b12}) blows up in finite time for arbitrarily small initial data,
while the Cauchy problems for (\ref{b1}) and (\ref{b11}) with
small data have global solutions.
Another remarkable feature of (\ref{b12}) is that,
unlike (\ref{b1}) and (\ref{b11}),
we can establish global existence for $(p, q)$ being on the
curve (\ref{b14}) (see \cite{HWY2014}).

When it comes to systems of wave equations,
the Cauchy problem for
\begin{equation}
 \square u = |v|^q,\quad \square v = |u|^p
\label{b15}
\end{equation}
has been well-studied. It is known that
\begin{equation}
\label{b13}
 \max \left\{\frac{p+2+q^{-1}}{pq-1},\
\frac{q+2+p^{-1}}{pq-1}\right\}-\frac{n-1}{2}=0
\end{equation}
is the critical curve in the $p$-$q$ plane of index-pairs $(p, q)$.
See, e.g.,

   \cite{DGM}, \cite{DM},
   \cite{KO99}, \cite{GTZ}.

Remark that $(q_0(n), q_0(n))$ is on the critical curve. Also,
the curve (\ref{b13}) lies above the curve (\ref{b14})
when $q > q_0(n)$.

Now let us turn to the problem (\ref{b1})--(\ref{b2}).
We first show a blow-up result which is valid for all $n \geq 2$.
When discussing blow-up,
we say that $(u, v)$ satisfies (\ref{b1}) if the following two conditions
(i), (ii) hold:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item
The equalities
\begin{align}
\label{y1}
 &\int_0^T \lb |v(s,\cdot)|^q,\psi(s,\cdot)\rb\,ds
=\int_0^T \lb u(s,\cdot), (\pa_s^2 \psi -\Delta \psi)(s,\cdot)\rb\,ds,\\
 &\int_0^T \lb |\pa_s u(s,\cdot)|^p,\psi(s,\cdot)\rb\,ds
=\int_0^T \lb v(s,\cdot), (\pa_s^2 \psi -\Delta \psi)(s,\cdot)\rb\,ds
\end{align}
hold for any $\psi \in C_0^{\infty}((0,T)\times \R^n)$.
\item
$u$ and $v$ have the continuity and integrability properties
\begin{align}
\label{y2}
 & u, v, \pa_t u, \pa_t v \in C([0,T);\,L^1(\R^n)),\\
\label{y3}
 & v \in C((0,T);\,L^q(\R^n)),\quad
 \pa_t u \in C((0,T);\,L^p(\R^n)).
\end{align}
\end{enumerate}
In particular, the initial conditions (\ref{b2}) should be satisfied
in the sense of (\ref{y2}).
\begin{theorem}\label{y4}
 Let $n \geq 2$, $1<q$, $1 < p < 2n/(n-1)$ and
\begin{equation}
\label{y5}
 \left(\frac{n-1}{2}p-1\right)(pq-1) < p+2.
\end{equation}
Assume that the initial value problem {\rm (\ref{b1})--(\ref{b2})}
with non-negative initial data has a solution
$(u, v)$ in the sense described above, which is localized inside a cone:
\begin{equation}
\label{y6}
 \supp (u, v, \pa_t u, \pa_t v) \subset \{(t,\,x) \in [0, T) \times \R^n\,:\,
|x| \leq t+1\}.
\end{equation}
We also assume that $f+g$ and $\tilde{g}$ do not vanish identically.
Then there exists a positive constant $C_1$,
depending only on $n, p, q, f, g, \tilde{f}, \tilde{g}$, such that
\begin{equation}
\label{y7}
 T \leq C_1 \ep^{-\frac{p(pq-1)}{p+2-\left(\frac{n-1}{2}p-1\right)(pq-1)}}.
\end{equation}
\end{theorem}

Observe that $(p_0(n), q_0(n))$ belongs to the domain
(\ref{y5}). Thus the same remark as the one stated on (\ref{b12}) above
applies to (\ref{b1}).
We infer from Theorem \ref{y4} that a portion of the curve
\begin{equation}
\label{b5}
 \left(\frac{n-1}{2}p-1\right)(pq-1) = p+2
\end{equation}
gives a component part of the critical curve in the $p$-$q$ plane.
In order to confirm this expectation,
we should establish a global existence theorem for $(p, q)$
above the curve (\ref{b5}).
We solve the system of the integral equations
\begin{align}
\label{z1}
 u(t,r) & = \ep U_0(t,r)+ L|v|^q (t,r), \\
 \label{z2}
 v(t,r) & = \ep V_0(t,r)+ L|\pa_t u|^p (t,r),
\end{align}
where
\begin{align}
 U_0(t,r)&=\frac{1}{2r}\left\{(r+t)f(r+t)+(r-t)f(r-t)\right\}
+\frac{1}{2r}\int_{r-t}^{r+t}\rho g(\rho)\,d\rho,
\nonumber\\
 V_0(t,r)&=\frac{1}{2r}\left\{(r+t)\tilde{f}(r+t)+(r-t)\tilde{f}(r-t)\right\}
+\frac{1}{2r}\int_{r-t}^{r+t}\rho \tilde{g}(\rho)\,d\rho
\nonumber
\end{align}
and
\begin{equation}
 LF(t,r) = \frac{1}{2r}\int_0^t \int_{r-(t-s)}^{r+(t-s)}\rho
  F(s,\rho)\,d\rho ds.
\nonumber
\end{equation}
The system (\ref{z1})--(\ref{z2}) is a radially symmetric version of the
original system with $n=3$,
and we naturally suppose that $f(r)$, $g(r)$, $u(t,r)$, $\ldots$ are even functions of $r$.
Hence, the lower limits of the integrals above may be replaced by $|r-t|$ or $|r-(t-s)|$.
$LF(t,r)$ is also an even function of $r$.

Throughout the proof of global existence, we suppose that
\begin{equation}
\label{z3}
 q > 2, \quad 2 < p < 3.
\end{equation}
We want to show that if in addition
\begin{equation}
\label{z4}
 (p-1)(pq-1)>p+2
\end{equation}
is assumed, then the system (\ref{z1})--(\ref{z2}) has
a global solution for sufficiently small data.

For this purpose, we introduce a weighted norm.
We define weight functions $w_1$, $w_2$, $w_3$ by
\begin{align}
 w_1(t,r)&=\lb r\rb \lb t-r\rb^{\mu/p}, \\
 \label{z5}
 w_2(t,r)&=
 \begin{cases}
  \lb r \rb^{p-2} \lb t+r\rb^{\mu} & \left(r < t/2\right) \\
  \lb t-r\rb^{p-3+\mu}\lb t+r\rb & \left(r \geq t/2\right)
 \end{cases},\\
 w_3(t,r)&= \lb t-r\rb^{\mu}
\end{align}
for $t \geq 0$ and $r \geq 0$. Here we use the concise notation $\lb \xi\rb=\sqrt{1+|\xi|^2}$ as usual, but
we may replace it by $1+|\xi|$ below.
By (\ref{z4}), we can choose $\mu$ so that
\begin{equation}
\label{z6}
 3-p < \mu < 1,\quad
p\left\{2-(p-2)q\right\} < (pq-1)\mu
\end{equation}
are satisfied. Then we define
\begin{equation}
 \|(w, v)\|
=\|w_1w\|_{L^{\infty}_{t,r}}
+\|w_2v\|_{L^{\infty}_{t,r}}
+\|w_3r\pa_r v\|_{L^{\infty}_{t,r}}
\nonumber
\end{equation}
where $L^{\infty}_{t,r} = L^{\infty}([0, \infty) \times [0, \infty))$.

\begin{theorem}\label{z7}
Suppose {\rm (\ref{z3})}, {\rm (\ref{z4})}.
Suppose also that $f, \tilde{f} \in C_0^2(\R)$,\ $g, \tilde{g} \in C_0^1(\R)$ are even functions.
There exist positive numbers $\ep_2$, $C_2$ such that
we have a unique global solution $(u, v)$ of {\rm (\ref{z1})--(\ref{z2})}
for $0 < \ep < \ep_2$,
satisfying $\|(\pa_tu, v)\| \leq C_2\ep$ and
\begin{align}
 &u(t,-r)=u(t,r),\ v(t,-r)=v(t,r),\nonumber\\
 &(u, v) \in C([0, \infty)\times \R) \times C([0,\infty)\times \R),
\quad \pa_t u \in C([0, \infty)\times \R), \nonumber\\
  & (ru, rv) \in C^2([0,\infty)\times \R) \times C^2([0,\infty)\times \R).
\nonumber
\end{align}
\end{theorem}
Thus we may say that a portion of the curve (\ref{b5}) is truely the  borderline
between global-in-time existence and finite time blow-up
for (\ref{b1})--(\ref{b2}).
It is known for (\ref{b15})
that finite time blow-up occurs for $(p, q)$ being on the critical curve,
but this problem is poorly understood for (\ref{b1}) and
we have no results about the case where $(p, q)$ belongs to
 the critical curve at this time.
Note that the curve (\ref{b5}) lies below the curve (\ref{b14}) for $p > p_0(n)$.

If radial symmetry is not assumed,
we can use the method of \cite{H98}, \cite{HWY2014} to get
some partial results of global existence. However, the present
authors have not yet suceeded in
obtaining a global existence result right up to the critical curve.
Instead, we treat the case $(p, q) = (2, 2)$ for $n = 2, 3$
to describe the method of \cite{HWY2014}. It is of some interest
since we can solve (\ref{b1})--(\ref{b2}) up to
$T=c\ep^{-\frac{6}{10-3n}}$ in this way (recall (\ref{y7})).

We assume that
\begin{gather}
\label{b8}
 f \in L^2(\R^n),\quad x g \in L^2 (\R^n), \quad
x \tilde{f} \in \dot{H}^{1/4}(\R^n),\\
 x^a \pa_x^b \pa_x f \in L^2 (\R^n), \quad
 x^a \pa_x^b g \in L^2 (\R^n),\\
 x^a \pa_x^b \tilde{f} \in \dot{H}^{1/4}(\R^n),\quad
 x^a \pa_x^b \tilde{g} \in \dot{H}^{-3/4}(\R^n)
\label{b9}
\end{gather}
for $|a| \leq |b| \leq 2$, where
${\dot H}^s({\mathbb R}^n)$
is the standard homogeneous Sobolev space. We set
\begin{align}
\label{b16}
\Lambda_1&
:=\|f\|_{L^2}+\sum_{i=1}^n\|x_ig\|_{L^2}
+\sum_{|a|\leq |b| \leq 2}\left(
\|x^a\pa_x^b \pa_x f\|_{L^2}
+\|x^a\pa_x^b g\|_{L^2}\right)\\
\intertext{and}
\label{b17}
\Lambda_2&
:=\sum_{i=1}^n\|x_i\tilde{f}\|_{\dot{H}^{1/4}}
+\sum_{|a|\leq |b|\leq 2}
\left(
\|x^a\pa_x^b{\tilde f}\|_{\dot{H}^{1/4}}
+\|x^a\pa_x^b{\tilde g}\|_{\dot{H}^{-3/4}}
\right).
\end{align}

Let us denote $\pa_0 = \pa_t$ as usual. Following Klainerman \cite{Kl85}--\cite{Kl87},
we introduce several partial differential operators as follows:
$L_j=t\partial_j+x_j\partial_0 $ $(j=1,\dots,n)$,
$\Omega_{kl}=x_k\partial_l-x_l\partial_k$ $(1\leq k<l\leq n)$,
$L_0=t\partial_0+x_1\partial_1+\cdots+x_n\partial_n$.
Operators
$\partial_0,\dots,\partial_n$, $L_1,\dots,L_n$, $\Omega_{12},\dots,
\Omega_{1n},\Omega_{23},\dots,\Omega_{n-1n}$ and $L_0$ are denoted by
$\Gamma_0,\dots,\Gamma_\nu$ in this order, where
$\nu:=(n^2+3n+2)/2$.
For a multi-index $\alpha=(\alpha_0,\dots,\alpha_\nu)$,
$\Gamma^{\alpha_0}_0\cdots\Gamma^{\alpha_\nu}_\nu$ is denoted by
$\Gamma^\alpha$.

\begin{theorem}\label{b6}
 Let $(p, q)=(2, 2)$ and $n=2, 3$. Assume {\rm (\ref{b8})--(\ref{b9})}.
Then, there exist $\ep_3$, $C_3$ depending on $n$, $\Lambda_1$ and
$\Lambda_2$ such that the Cauchy problem {\rm (\ref{b1})--(\ref{b2})}
with $0 < \ep < \ep_3$ and $T = C_3\ep^{-\frac{6}{10-3n}}$
admits a unique solution satisfying
\begin{eqnarray}
& &
\Gamma^\alpha u\in C([0,T];H^1({\mathbb R}^n)),\quad
\partial_t\Gamma^\alpha u\in C([0,T];L^2({\mathbb R}^n)),
\nonumber\\
& &
\Gamma^\alpha v\in C([0,T];\dot{H}^{1/4}({\mathbb R}^n)),
\nonumber
\end{eqnarray}
for $|\alpha| \leq 2$ and
\begin{eqnarray*}
& &
\sup_{0<t<T}
(1+t)^{-1}\|u(t)\|_{L^2}
+
\sum_{|\alpha|\leq 2\atop 0\leq j\leq n}
\sup_{0<t<T}
\|\partial_j\Gamma^\alpha u(t)\|_{L^2}\\
& &
\hspace{2cm}
+
\sum_{|\alpha|\leq 2}
\sup_{0<t<T}
(1+t)^{-1/12}\|\Gamma^\alpha v(t)\|_{\dot{H}^{1/4}}
\leq C\ep.\nonumber
\end{eqnarray*}
\end{theorem}

In the rest of this paper, we prove Theorem \ref{y4},
Theorem \ref{z7} and Theorem \ref{b6}, in Sections 2, 3 and 4, respectively.
\section{Proof of Theorem \ref{b6}}

We start this section by explaining some additional notation used
in this section.
It is necessary to define the norm for $1\leq p,\,q<\infty$
\begin{equation}
 \|v(\cdot)\|_{p,q}
  :=
  \left(
   \int_0^\infty
    \left(
    \int_{S^{n-1}}|v(r\omega)|^qdS_\omega
  \right)^{p/q}r^{n-1}dr
  \right)^{1/p}
\nonumber
\end{equation}
with an obvious modification for $p=\infty$
\begin{equation}
\|v(\cdot)\|_{\infty,q}
:=
\sup_{r>0}
 \left(
  \int_{S^{n-1}}
  |v(r\omega)|^qdS_\omega
  \right)^{1/q}
\nonumber
\end{equation}
where $r=|x|$, $\omega\in S^{n-1}$.
These types of norms have been effectively used
for the existence theory of solutions to fully nonlinear wave equations
in \cite{LY}, \cite{LZ}.
Let $N$ be a nonnegative integer and $\Psi$ a characteristic function
of a set of ${\mathbb R}^n$.
We define the norm
\begin{equation}
\|u(t,\cdot)\|_{\Gamma,N,p,q,\Psi}
:=\sum_{|\alpha|\leq N}
\|\Psi(\cdot)\Gamma^\alpha u(t,\cdot)\|_{p,q}.
\label{a1}
\end{equation}
For $\Psi\equiv 1$ in (\ref{a1}), we omit the subscript $\Psi$.
If $p=q$, then we omit $q$.
If $N=0$, then we omit both the subscripts $\Gamma$ and $N$.
In sum, we abbreviate the notation of the norm
$\|u(t,\cdot)\|_{\Gamma,N,p,q,\Psi}$ to
\begin{eqnarray*}
& &
\|u(t,\cdot)\|_{\Gamma,N,p,q}, \mbox{~when~}\Psi\equiv 1,\\
& &
\|u(t,\cdot)\|_{\Gamma,N,p,\Psi}, \mbox{~when~}p=q,\\
& &
\|u(t,\cdot)\|_{\Gamma,N,p}, \mbox{~when~}p=q\mbox{~and~}\Psi\equiv 1,\\
& &
\|u(t,\cdot)\|_{p,q,\Psi}, \mbox{~when~}N=0,\\
& &
\|u(t,\cdot)\|_{p,q}, \mbox{~when~}N=0\mbox{~and~}\Psi\equiv 1,\\
& &
\|u(t,\cdot)\|_{p,\Psi}, \mbox{~when~}N=0\mbox{~and~}p=q,\\
& &
\|u(t,\cdot)\|_{p}, \mbox{~when~}N=0,\,p=q,\,\mbox{and~}\Psi\equiv 1.
\end{eqnarray*}

We find solutions to the Cauchy problem to the system of wave equations
\begin{equation}
\label{a12}
\partial_t^2 u-\Delta u=v^2,\quad
\partial_t^2 v-\Delta v=(\partial_t u)^2,\quad
t>0,\,\,x\in{\mathbb R}^n
\end{equation}
with initial data (\ref{b2})
by iteration.
For any $T>0$,
$\varepsilon>0$, and
$(f,g)$, $({\tilde f}, {\tilde g})$
satisfying (\ref{b8})--(\ref{b9}),
let us define the set of functions
\begin{align*}
Z:=&
Z(T,\varepsilon,f,g,{\tilde f},{\tilde g})\\
=&
\{\,(u,v)\in
C([0,T];H^1({\mathbb R}^n)
\times
{\dot H}^{1/4}({\mathbb R}^n)):\nonumber\\
&\qquad
\partial_j\Gamma^\alpha u\in C([0,T];L^2({\mathbb R}^n))\quad
(|\alpha|\leq 2,\,0\leq j\leq n),\nonumber\\
&\qquad
|D|^{1/4}\Gamma^\alpha v\in C([0,T];L^2({\mathbb R}^n))\quad
(|\alpha|\leq 2),\nonumber\\
&\qquad
u(0)=\varepsilon f,\,
\partial_t u(0)=\varepsilon g,\,
v(0)=\varepsilon\tilde f,\,
\partial_t v(0)=\varepsilon\tilde g\,\},\nonumber
\end{align*}
where $|D|:=\sqrt{-\Delta}$.
Denote by $u_0$ and $v_0$ the solutions to
$\partial_t^2 u-\Delta u=0$ with data
$(\varepsilon f,\varepsilon g)$ and
$(\varepsilon\tilde f,\varepsilon\tilde g)$
at $t=0$, respectively.
We obviously see from the discussion below that
$(u_0, v_0)\in Z$.
Therefore, the set $Z$ is nonempty.
Let us define the sequence
$\{(u_m,v_m)\}$ inductively by solving
\begin{equation}
\label{a14}
\partial_t^2 u_m-\Delta u_m=(v_{m-1})^2,\quad
\partial_t^2 v_m-\Delta v_m=(\partial_t u_{m-1})^2,\quad
t>0,\,\,x\in{\mathbb R}^n
\end{equation}
with initial data
\begin{equation}
(u_m(0),\partial_t u_m(0))
=
(\varepsilon f,\varepsilon g),
\quad
(v_m(0),\partial_t v_m(0))
=
(\varepsilon\tilde f,\varepsilon\tilde g).
\nonumber
\end{equation}
It is obvious from the discussion below that
this is a well-defined sequence in $Z$.

Using the quantity defined as
$M:=\Lambda_1+\Lambda_2+\Lambda_1^2+\Lambda_2^2
+\Lambda_1\Lambda_2^2$ (see (\ref{b16}) and (\ref{b17}) for the
definitions of $\Lambda_1$, $\Lambda_2$), we set
$Z(2C_0M\varepsilon):=\{\,(u,v)\in Z\,;\,N((u,v))\leq 2C_0M\varepsilon\,\}$,
where
\begin{align*}
N((u,v))
:=&
\sup_{0<t<T}
(1+t)^{-1}
\|u(t)\|_2
+
\sum_{|\alpha|\leq 2\atop 0\leq j\leq n}
\sup_{0<t<T}
\|\partial_j\Gamma^\alpha u(t)\|_2\\
&+
\sum_{|\alpha|\leq 2}
\sup_{0<t<T}
(1+t)^{-1/12}\||D|^{1/4}\Gamma^\alpha v(t)\|_2.\nonumber
\end{align*}
For the constant $C_0$, see (\ref{a2}) below.
The set $Z(2C_0M\varepsilon)$ is complete with respect to the metric
$d((u,v),(u',v')):=N((u-u',v-v'))$
$((u,v),(u',v')\in Z(2C_0M\varepsilon))$.

Let us first recall how to bound
$(\Gamma^\alpha u_m,\partial_t\Gamma^\alpha u_m)|_{t=0}$
and
$(\Gamma^\alpha v_m,\partial_t\Gamma^\alpha v_m)|_{t=0}$
for $|\alpha|\leq 2$; see, e.g., Section 4 of \cite{HWY2014}.
We will rewrite them in terms of
$\varepsilon f$, $\varepsilon g$, $\varepsilon\tilde f$,
and $\varepsilon\tilde g$, and obtain
for $0<\varepsilon<1$
\begin{align}
\lefteqn{\|u_m(0)\|_2
+\sum_{|\alpha|\leq 2\atop 0\leq j\leq n}
\|(\partial_j\Gamma^\alpha u_m)(0)\|_2}
\nonumber\\
&+
\sum_{|\alpha|\leq 2}
\bigl(
\|(|D|^{1/4}\Gamma^\alpha v_m)(0)\|_2
+
\|(|D|^{-3/4}\partial_t\Gamma^\alpha v_m)(0)\|_2
\bigr)
\leq
C_0M\varepsilon.
\label{a2}
\end{align}
We like to put off the proof of this bound
until the end of this section,
and see how it is useful in the iteration argument.
The crucial point in the proof of Theorem \ref{b6}
is to prove the following.
\begin{proposition}\label{a22}
For $|\alpha|\leq 2$ and $0\leq j\leq n$,
the following estimates hold$:$
\begin{eqnarray*}
& &
\sup_{0<t<T}
\|\partial_j\Gamma^\alpha u_m(t)\|_2\\
& &
\leq
CM\varepsilon
+
C(1+T)^{(10-3n)/6}
\biggl(
\sum_{|\beta|\leq 2}
\sup_{0<t<T}
\langle t\rangle^{-1/12}
\||D|^{1/4}\Gamma^\beta v_{m-1}(t)\|_2
\biggr)^2,\nonumber\\
& &
\sup_{0<t<T}
\langle t\rangle^{-1/12}
\||D|^{1/4}\Gamma^\alpha v_m(t)\|_2\\
& &
\leq
CM\varepsilon
+
C(1+T)^{(10-3n)/6}
\biggl(
\sum_{|\beta|\leq 2\atop 0\leq j\leq n}
\sup_{0<t<T}
\|\partial_j\Gamma^\beta u_{m-1}(t)\|_2
\biggr)^2.\nonumber
\end{eqnarray*}
Here, $C$ denotes a positive constant
independent of $m$ and $T$.
\end{proposition}
For the proof of this proposition, we use several inequalities.
\begin{proposition}\label{a25}
For any $2<q<\infty$, there exists a constant $C>0$ such that
the inequality
\begin{equation}
\bigl\|\,
\|v(r\omega)\|_{L^2(S^{n-1})}
\,\bigr\|
_{L^q((\lambda,\infty);r^{n-1}dr)}
\leq
C
\lambda^{-(n-1)s(q)}\||D|^{s(q)}v\|_2
\nonumber
\end{equation}
holds for all $\lambda>0$, where $s(q):=1/2-1/q$.
\end{proposition}

{\it Proof}. See Theorem 2.10 of Li and Zhou \cite{LZ}.
See also Section 4 of \cite{HWY2014} where a different proof is given.
\qed

\begin{proposition}
If $1\leq p<q<\infty$ and $1/q\geq 1/p-1/n$,
then the inequality
\begin{equation}
\label{a4}
\|v(t,\cdot)\|_{q,\chi_1}
\leq
C(1+|t|)^{-n(1/p-1/q)}
\|v(t,\cdot)\|_{\Gamma,1,p}
\end{equation}
holds. Here $\chi_1$ denotes the characteristic function of the set
$\{x\in{\mathbb R}^n:|x|<(1+|t|)/2\}$.
\end{proposition}

{\it Proof}.
See Theorem 2.9 of Li and Zhou \cite{LZ}.
\qed

\begin{proposition}\label{a23}
Suppose $n\geq 2$. $(1)$ If $1/2<s<n/2$, then
the inequality
\begin{equation}
\label{a5}
\sup_{r>0}
r^{(n/2)-s}
\|v(r\cdot)\|_{L^p(S^{n-1})}
\leq
C\||D|^s v\|_2
\end{equation}
holds,
where $p$ is defined as
\begin{equation}
\frac1p=\frac12-\frac{s-\frac12}{n-1}.
\nonumber
\end{equation}
$(2)$ If $\sigma$ satisfies $1/2<1-\sigma<n/2$,
then the solution $u$ to the inhomogeneous wave equation
$\partial_t^2 u-\Delta u=F$ in
${\mathbb R}^n\times (0,\infty)$
with data $(f,g)$ at $t=0$ satisfies
\begin{align}
\label{a6}
\||D|^\sigma u(t,\cdot)\|_2
\leq &
\||D|^\sigma f\|_2
+
\||D|^{\sigma-1}g\|_2
+
C\int_0^t\|F(\tau,\cdot)\|_{p_1,\chi_1}d\tau
\nonumber\\
&+
C\int_0^t
\langle\tau\rangle^{-(n/2)+1-\sigma}
\|F(\tau,\cdot)\|_{1,p_2,\chi_2}d\tau.
\end{align}
Here $p_1$ and $p_2$ are defined as
\begin{equation}
\frac{1}{p_1}
=\frac12+\frac{1-\sigma}{n},
\quad
\frac{1}{p_2}
=
\frac12
+\frac{\frac12-\sigma}{n-1}.
\nonumber
\end{equation}
The functions $\chi_1$ and $\chi_2$ denote
the characteristic functions of
$\{x\in{\mathbb R}^n:|x|<(1+\tau)/2\}$ and
$\{x\in{\mathbb R}^n:|x|>(1+\tau)/2\}$, respectively.
The solution $u$ also satisfies
for $j=0,\dots,n$
\begin{equation}
\label{a7}
\|\partial_j u(t,\cdot)\|_2
\leq
\||D|f\|_2+\|g\|_2
+
\int_0^t\|F(\tau,\cdot)\|_2d\tau.
\end{equation}
\end{proposition}

{\it Proof}.
The inequality (\ref{a5}) is an immediate consequence of
Trace Lemma (see \cite{Ho} for $n\geq 3$ and \cite{FW} for $n\geq 2$)
and the Sobolev embedding on the unit sphere $S^{n-1}$.
The inequality (\ref{a6}) is also a direct consequence of
the standard Sobolev embedding
$\|v\|_{L^{p^*}({\mathbb R}^n)}\leq C\|v\|_{{\dot H}^s({\mathbb R}^n)}$
$(1/p^*=1/2-s/n)$, (\ref{a5}) and the standard duality argument.
For details, see the proof of Theorem 2.11 of Li and Zhou \cite{LZ}.
The inequality (\ref{a7}) is the standard estimate.
\qed

\begin{proposition}
The following commuting relations hold$:$
\begin{eqnarray*}
& &
[\Gamma_i,\Box]=0\,\,\,\mbox{for $i=0,\dots,\nu-1$, and}\,\,\,
[L_0,\Box]=-2\Box,\\
& &
[\Gamma_j,\Gamma_k]=\sum_{l=0}^\nu C^{j,k}_l\Gamma_l,\,\,\,
j,\,k=0,\dots,\nu,\\
& &
[\Gamma_j,\partial_k]
=
\sum_{l=0}^nC^{j,k}_l\partial_l,\,\,\,j=0,\dots,\nu,\,\,k=0,\dots,n.
\end{eqnarray*}
Here $C^{j,k}_l$ denotes a constant depending on
$j$, $k$, and $l$.
\end{proposition}

{\it Proof}. We can verify these relations by direct computations.
\qed

\begin{remark}\label{a24}
In particular, we see by this proposition
$$
\|\partial_t u(t,\cdot)\|_{\Gamma,2,2}
\leq
C
\sum_{0\leq j\leq n\atop|\alpha|\leq 2}
\|\partial_j \Gamma^\alpha u(t,\cdot)\|_2.
$$
This fact will be employed in {\rm (\ref{a8})} below.
\end{remark}

{\it Proof of Proposition {\rm \ref{a22}}.}\,\,We drop the subscript $m-1$
until the last step of the proof.
We also note that, in what follows, the functions $\chi_1$ and $\chi_2$ denote
the characteristic functions of
$\{x\in{\mathbb R}^n:|x|<(1+\tau)/2\}$ and
$\{x\in{\mathbb R}^n:|x|>(1+\tau)/2\}$, respectively.

In view of (\ref{a6}) with $\sigma=1/4$ and (\ref{a7}),
our task is to bound
$\|v(\tau,\cdot)^2\|_{\Gamma,2,2}$,
$\|(\partial_t u(\tau,\cdot))^2\|_{\Gamma,2,p_1,\chi_1}$,
and
$\|(\partial_t u(\tau,\cdot))^2\|_{\Gamma,2,1,p_2,\chi_2}$,
where
\begin{equation}
\frac{1}{p_1}
=
\frac12
+
\frac{3}{4n},
\quad
\frac{1}{p_2}
=
\frac12
+
\frac{1}{4(n-1)}.
\nonumber
\end{equation}
Let us start with the estimate of
$\|v(\tau,\cdot)^2\|_{\Gamma,2,2}$.
We carry it out by
dealing with
$\|v(\tau,\cdot)^2\|_{\Gamma,2,2,\chi_1}$
and
$\|v(\tau,\cdot)^2\|_{\Gamma,2,2,\chi_2}$,
separately.

{\it Estimate of $\|v(\tau,\cdot)^2\|_{\Gamma,2,2,\chi_1}$}.
Define $p^*$ and $p_3$
as $1/p^*=1/2-1/(4n)$
and
$1/2=1/p_3+1/p^*$, respectively.
Using the H\"older inequality, (\ref{a4}),
and the Sobolev embedding
${\dot H}^{1/4}({\mathbb R}^n)\hookrightarrow L^{p^*}({\mathbb R}^n)$,
we get
\begin{eqnarray}
\lefteqn{\sum_{|\alpha|\leq |\beta|\atop |\alpha|+|\beta|\leq 2}
\|
(\Gamma^\alpha v(\tau))(\Gamma^\beta v(\tau))
\|_{2,\chi_1}
\leq
C\|v(\tau)\|_{\Gamma,1,p_3,\chi_1}
\|v(\tau)\|_{\Gamma,2,p^*}}\nonumber\\
& &
\leq
C(1+\tau)^{-n(1/p^*-1/p_3)}\|v(\tau)\|_{\Gamma,2,p^*}^2\nonumber\\
& &
\leq
C(1+\tau)^{-(n-1)/2}
\biggl(
\sum_{|\alpha|\leq 2}
\||D|^{1/4}\Gamma^\alpha v(\tau)\|_2
\bigg)^2.
\label{a10}
\end{eqnarray}
Here, we have used
$-n(1/p^*-1/p_3)=-n(2/p^*-1/2)=-(n-1)/2$.

{\it Estimate of $\|v(\tau,\cdot)^2\|_{\Gamma,2,2,\chi_2}$}.
We use the H\"older inequality,
the Sobolev embedding on $S^{n-1}$,
and Proposition \ref{a25} to get
\begin{eqnarray}
\lefteqn{\sum_{|\alpha|\leq |\beta|\atop |\alpha|+|\beta|\leq 2}
\|
(\Gamma^\alpha v(\tau))(\Gamma^\beta v(\tau))
\|_{2,\chi_2}}\nonumber\\
& &
\leq
\sum_{|\alpha|\leq 2}
\|v(\tau)\|_{4,\infty,\chi_2}
\|\Gamma^\alpha v(\tau)\|_{4,2,\chi_2}
+
\sum_{|\alpha|,|\beta|\leq 1}
\|\Gamma^\alpha v(\tau)\|_{4,\chi_2}
\|\Gamma^\beta v(\tau)\|_{4,\chi_2}\nonumber\\
& &
\leq
C\|v(\tau)\|_{\Gamma,2,4,2,\chi_2}^2
\leq
C(1+\tau)^{-(n-1)/2}
\biggl(
\sum_{|\alpha|\leq 2}
\||D|^{1/4}\Gamma^\alpha v(\tau)\|_2
\bigg)^2.
\end{eqnarray}
We have finished the estimate of $\|v(\tau)^2\|_{\Gamma,2,2}$.

{\it Estimate of $\|(\partial_t u(\tau))^2\|_{\Gamma,2,p_1,\chi_1}$}.
Recall $1/p_1=1/2+3/(4n)$.
Using the H\"older inequality and (\ref{a4}), we get
\begin{align}
\sum_{|\alpha|\leq 2}
\|\partial_t u(\tau)\Gamma^\alpha\partial_t u(\tau)\|_{p_1,\chi_1}
&\leq
\sum_{|\alpha|\leq 2}
\|\partial_t u(\tau)\|_{4n/3,\chi_1}
\|\Gamma^\alpha\partial_t u(\tau)\|_2
\nonumber\\
&\leq
C(1+\tau)^{-n(1/2-3/(4n))}
\|\partial_t u(\tau)\|_{\Gamma,2,2}^2.
\label{a20}
\end{align}
Moreover, we also obtain
\begin{eqnarray}
\lefteqn{\sum_{|\alpha|,|\beta|\leq 1}
\|(\Gamma^\alpha\partial_t u(\tau))
(\Gamma^\beta\partial_t u(\tau))\|_{p_1,\chi_1}}
\nonumber\\
& &
\leq
\|\partial_t u(\tau)\|_{\Gamma,1,2p_1,\chi_1}^2
\leq
C(1+\tau)^{-n/2+3/4}
\|\partial_t u(\tau)\|_{\Gamma,2,2}^2.
\end{eqnarray}
Here, we have used $-2n(1/2-1/(2p_1))=-n/2+3/4$.

{\it Estimate of $\|(\partial_t u(\tau))^2\|_{\Gamma,2,1,p_2,\chi_2}$ }.
Recall $1/p_2=1/2+1/(4(n-1))$.
Using the H\"older inequality and
then the Sobolev embedding on $S^{n-1}$, we get
\begin{align}
\sum_{|\alpha|\leq 2}
\|\partial_t u(\tau)\Gamma^\alpha\partial_t u(\tau)\|_{1,p_2,\chi_2}
\leq &
\sum_{|\alpha|\leq 2}
\|\partial_t u(\tau)\|_{2,4(n-1)}
\|\Gamma^\alpha\partial_t u(\tau)\|_2 \nonumber\\
\leq &
C\|\partial_t u(\tau)\|_{\Gamma,1,2}
\|\partial_t u(\tau)\|_{\Gamma,2,2}.
\label{a9}
\end{align}
We also have by the Sobolev embedding on $S^{n-1}$
\begin{eqnarray}
& &
\sum_{|\alpha|,|\beta|\leq 1}
\|
(\Gamma^\alpha\partial_t u(\tau))(\Gamma^\beta\partial_t u(\tau))
\|_{1,p_2,\chi_2}\nonumber\\
& &
\leq
\|\partial_t u(\tau)\|_{\Gamma,1,2,2p_2}^2
\leq
C\|\partial_t u(\tau)\|_{\Gamma,2,2}^2,
\end{eqnarray}
which, together with (\ref{a9}), implies for $\sigma=1/4$
\begin{equation}
\label{a11}
\langle\tau\rangle^{-(n/2)+1-\sigma}
\|(\partial_t u(\tau))^2\|_{\Gamma,2,1,p_2,\chi_2}
\leq
C\langle\tau\rangle^{-(n/2)+3/4}
\|\partial_t u(\tau)\|_{\Gamma,2,2}^2.
\end{equation}
By (\ref{a10})--(\ref{a11}), together with Remark \ref{a24},
we have shown for $|\alpha|\leq 2$,
\begin{align*}
\lefteqn{\sum_{j=0}^n\|\partial_j\Gamma^\alpha u_m(t)\|_2}
\nonumber\\
&\leq
\sum_{j=0}^n\|(\partial_j\Gamma^\alpha u_m)(0)\|_2\nonumber\\
&\quad +
C\int_0^t
\langle\tau\rangle^{-(n-1)/2+1/6}d\tau
\biggl(
\sum_{|\beta|\leq 2}
\sup_{0<t<T}
\langle t\rangle^{-1/12}
\||D|^{1/4}\Gamma^\beta v_{m-1}(t)\|_2
\biggr)^2,
\end{align*}
\begin{align}
\lefteqn{\langle t\rangle^{-1/12}\||D|^{1/4}\Gamma^\alpha v_m(t)\|_2}
\nonumber\\
&
\leq
\||D|^{1/4}((\Gamma^\alpha v_m)(0))\|_2\nonumber\\
&\quad +
C\langle t\rangle^{-1/12}
\int_0^t\langle\tau\rangle^{-n/2+3/4}d\tau
\biggl(
\sum_{|\beta|\leq 2\atop 0\leq j\leq n}
\sup_{0<t<T}
\|\partial_j\Gamma^\beta u_{m-1}(t)\|_2
\biggr)^2.
\label{a8}
\end{align}
We also note that
\begin{eqnarray*}
& &
\int_0^t
\langle\tau\rangle^{-(n-1)/2+1/6}d\tau
\leq
C(1+t)^{(10-3n)/6},\\
& &
\langle t\rangle^{-1/12}
\int_0^t\langle\tau\rangle^{-n/2+3/4}d\tau
\leq
C(1+t)^{(10-3n)/6}.
\end{eqnarray*}
Hence we have obtained
\begin{equation}
N((u_m,v_m))
\leq
C_0M\varepsilon
+
\tilde{C}(1+T)^{(10-3n)/6}
N((u_{m-1},v_{m-1}))^2
\nonumber
\end{equation}
for positive constants $C_0$, $\tilde{C}$ independent of $m$, $T$.

In the same way, we get
\begin{align*}
\lefteqn{N((u_{m+1}-u_m,v_{m+1}-v_m))}\\
&\leq
\hat{C}(1+T)^{(10-3n)/6}
\bigl(
N((u_m,v_m))+N((u_{m-1},v_{m-1}))
\bigr)\nonumber\\
&\quad
\times
N((u_m-u_{m-1},v_m-v_{m-1}))\quad(m=1,2,\dots)\nonumber
\end{align*}
for a constant $\hat{C}>0$ independent of $m$, $T$.

If we choose $T$ and $\varepsilon$ so that
\begin{equation}
\tilde{C}(1+T)^{(10-3n)/6}C_0M\varepsilon\leq 1,
\quad
\hat{C}(1+T)^{(10-3n)/6}(2C_0M\varepsilon)\leq\frac12
\nonumber
\end{equation}
may hold,
then it follows from the standard argument that
$\{(u_m,v_m)\}$ converges to the limit
in $Z(T,\varepsilon,f,g,\tilde f,\tilde g)$,
which means that
for any $\varepsilon$ with
$0<\varepsilon<\min\{1,1/(\tilde{C}C_0M),1/(4\hat{C}C_0M)\}$,
the Cauchy problem (\ref{a12}) with initial data (\ref{b2}) admits a solution $(u,v)$ in
$Z(T_\varepsilon,\varepsilon,f,g,\tilde f,\tilde g)$
satisfying
$N((u,v))\leq 2C_0M\varepsilon$.
Here, $T_\varepsilon$ is defined as
$$
1+T_\varepsilon
:=
\left(
\min\left\{\frac{1}{\tilde{C}C_0M}, \frac{1}{4\hat{C}C_0M}\right\}
\right)^{6/(10-3n)}
\varepsilon^{-6/(10-3n)}.
$$
Uniqueness of solutions in
$Z(T_\varepsilon,\varepsilon,f,g,\tilde f,\tilde g)$
follows from essentially the same argument.
We have finished the proof.
\qed

\vspace{0.5cm}

{\it Proof of $(\ref{a2})$}. We follow the argument of Section 4 of \cite{HWY2014}.
We know for $|\alpha|\leq 2$
\begin{align}
\label{a16}
&
\Gamma^\alpha u_m(0)
=
\sum_{|b|\leq 2}
\sum_{|a|\leq |b|}
C_{ab}^\alpha x^a\partial^b u_m(0),\\
&
\partial_t\Gamma^\alpha u_m(0)
=
\sum_{1\leq |b|\leq 2}
\sum_{|a|\leq |b|-1}
\tilde C_{ab}^\alpha x^a\partial^b u_m(0)
+
\sum_{|a|\leq 2 \atop |b|=2}\hat C_{ab}^\alpha x^a\partial^b\partial_t u_m(0),
\end{align}
where $x^a=x_1^{a_1}\cdots x_n^{a_n}$,
$\partial^b=\partial_t^{b_0}\cdots\partial_n^{b_n}$;
see (4.9)--(4.10) of \cite{HWY2014}.
Thus we have for $|\alpha|\leq 2$
\begin{eqnarray}
\lefteqn{\sum_{j=1}^n
\|\partial_j\Gamma^{\alpha} u_m(0)\|_2
+
\|\partial_t\Gamma^\alpha u_m (0)\|_2}
\nonumber\\
&\leq&
C\sum_{j=1}^n
\sum_{|b|\leq 2}
\sum_{|a|\leq |b|}
\|\partial_j(x^a\partial^b u_m(0))\|_2\nonumber\\
& &
+C\sum_{|b|=1,2}
\sum_{|a|\leq |b|-1}
\|x^a\partial^b u_m(0)\|_2
+
C\sum_{|a|\leq 2 \atop |b|=2}
\|x^a\partial^b\partial_t u_m(0)\|_2\nonumber\\
&\leq&
C\varepsilon\Lambda_1
+
C\|\partial_t^2 u_m(0)\|_2
+
C\sum_{j=1}^n
\|x_j\partial_t^2 u_m(0)\|_2\nonumber\\
& &
+
C\sum_{|a|\leq 2}
\bigl(
\|x^a\partial_x\partial_t^2 u_m(0)\|_2
+
\|x^a\partial_t^3 u_m(0)\|_2
\bigr)\nonumber\\
&\leq&
C\varepsilon\Lambda_1
+
C\sum_{|a|\leq 1}\|x^a\Box u_m(0)\|_2\nonumber\\
& &
+
C\sum_{|a|\leq 2}
\bigl(
\|x^a\partial_x\Box u_m(0)\|_2
+
\|x^a\partial_t\Box u_m(0)\|_2
\bigr).
\label{a17}
\end{eqnarray}
Using the 1st equation of (\ref{a14}),
we get for $|a|\leq 1$
\begin{eqnarray}
& &
\|x^a\Box u_m(0)\|_2
=
\|x^a(v_{m-1}(0))^2\|_2
=
\varepsilon^2\|x^a\tilde f^2\|_2
\nonumber\\
& &
\leq
\varepsilon^2
\|x^a\tilde f\|_{p^*}\|\tilde f\|_{4n}
\leq
C\varepsilon^2
\||D|^{1/4}(x^a\tilde f)\|_2
\||D|^{n/2-1/4}\tilde f\|_2\nonumber\\
& &
\leq
\left\{
\begin{array}{ll}
\displaystyle{C\varepsilon^2\Lambda_2
\||D|^{1/4}\tilde f\|_2^{1/2}
\||D|^{5/4}\tilde f\|_2^{1/2}},&\displaystyle{n=2},\\
\displaystyle{C\varepsilon^2\Lambda_2
\||D|^{5/4}\tilde f\|_2},&\displaystyle{n=3}
\end{array}
\right.
\nonumber\\
& &
\leq
C\varepsilon^2\Lambda_2^2,
\label{a15}
\end{eqnarray}
where $1/p^*=1/2-1/(4n)$. Moreover, we get for $j=1,\dots,n$
\begin{eqnarray*}
& &
\sum_{|a|\leq 2}
\|x^a\partial_j\Box u_m(0)\|_2
=
2\varepsilon^2
\sum_{|a|\leq 2}
\|x^a\tilde f\partial_j\tilde f\|_2\\
& &
\leq
2\varepsilon^2
\sum_{|a|\leq 1}
\|x^a\tilde f\|_{p^*}\|\partial_j\tilde f\|_{4n}
+
2\varepsilon^2
\sum_{|a|,|b|=1}
\|x^a\tilde f\|_{p^*}
\|x^b\partial_j\tilde f\|_{4n}\nonumber
\leq
C\varepsilon^2\Lambda_2^2,\nonumber
\end{eqnarray*}
where, as in (\ref{a15}), we have proceeded for $|b|=1$
\begin{equation}
\|x^b\partial_j\tilde f\|_{4n}
\leq
\left\{
\begin{array}{ll}
\displaystyle{C\||D|^{1/4}(x^b\partial_j\tilde f)\|_2^{1/2}
\||D|^{5/4}(x^b\partial_j\tilde f)\|_2^{1/2}},&\displaystyle{n=2},\\
\displaystyle{C\||D|^{5/4}(x^b\partial_j\tilde f)\|_2},&\displaystyle{n=3}
\end{array}
\right.
\nonumber
\end{equation}
and dealt with $\||D|^{5/4}(x^b\partial_j\tilde f)\|_2$ as
\begin{equation}
\||D|^{5/4}(x^b\partial_j\tilde f)\|_2
\leq
C
\biggl(
\||D|^{1/4}\partial_j\tilde f\|_2
+
\sum_{|\alpha|=2}
\||D|^{1/4}(x^b\partial_x^\alpha\tilde f)\|_2
\biggr).
\nonumber
\end{equation}
We also obtain for $|a|\leq 2$
\begin{eqnarray*}
& &
\|x^a(\partial_t\Box u_m)(0)\|_2
=
2\varepsilon^2\|x^a\tilde f\tilde g\|_2\\
& &
\leq
2\varepsilon^2
\sum_{|b|\leq 1}
\|x^b\tilde f\|_{p^*}\|\tilde g\|_{4n}
+
2\varepsilon^2
\biggl(
\sum_{|b|=1}
\|x^b\tilde f\|_{p^*}
\biggr)
\biggl(
\sum_{|b|=1}
\|x^b\tilde g\|_{4n}
\biggr)
\leq
C\varepsilon^2\Lambda_2^2,\nonumber
\end{eqnarray*}
where we have handled
$\|x^b\tilde g\|_{4n}$ $(|b|=1)$ as
\begin{equation}
\|x^b\tilde g\|_{4n}
\leq
\left\{
\begin{array}{ll}
\displaystyle{C\||D|^{1/4}(x^b\tilde g)\|_2^{1/2}
\||D|^{5/4}(x^b\tilde g)\|_2^{1/2}},&\displaystyle{n=2},\\
\displaystyle{C\||D|^{5/4}(x^b\tilde g)\|_2},&\displaystyle{n=3}
\end{array}
\right.
\nonumber
\end{equation}
and
$\||D|^{1/4}(x^b\tilde g)\|_2$,
$\||D|^{5/4}(x^b\tilde g)\|_2$ as
\begin{align*}
\||D|^{1/4}(x^b\tilde g)\|_2
&\leq
C\biggl(
\||D|^{-3/4}\tilde g\|_2
+
\sum_{|\alpha|=1}
\||D|^{-3/4}(x^b\partial_x^\alpha\tilde g)\|_2
\biggr),\\
\||D|^{5/4}(x^b\tilde g)\|_2
&\leq
C\biggl(
\||D|^{1/4}\tilde g\|_2
+
\sum_{|\alpha|=1}
\||D|^{1/4}(x^b\partial_x^\alpha\tilde g)\|_2
\biggr)\\
&\leq
C
\biggl(
\sum_{|\alpha|=1}
\||D|^{-3/4}\partial_x^\alpha\tilde g\|_2
+
\sum_{|\alpha|=2}
\||D|^{-3/4}(x^b\partial_x^\alpha\tilde g)\|_2
\biggr),\nonumber
\end{align*}
respectively.
Summing up, we have obtained
\begin{equation}
\sum_{|\alpha|\leq 2}
\sum_{j=0}^n
\|\partial_j\Gamma^\alpha u_m(0)\|_2
\leq
C\varepsilon\Lambda_1
+
C\varepsilon^2\Lambda_2^2.
\nonumber
\end{equation}
We next prove the estimate related to $v_m(0)$.
As in (\ref{a16})--(\ref{a17}),
we get for $|\alpha|\leq 2$
\begin{eqnarray}
& &
\||D|^{1/4}\Gamma^\alpha v_m(0)\|_2
+
\||D|^{-3/4}\partial_t\Gamma^\alpha v_m(0)\|_2\\
& &
\leq
C\sum_{|b|\leq 2 \atop |a|\leq |b|}
\||D|^{1/4}(x^a\partial^b v_m(0))\|_2
+
C\sum_{1\leq |b|\leq 2 \atop |a|\leq |b|-1}
\||D|^{-3/4}(x^a\partial^b v_m(0))\|_2\nonumber\\
& &
+
C\sum_{|a|\leq 2 \atop |b|=2}
\||D|^{-3/4}(x^a\partial^b\partial_t v_m(0))\|_2.
\label{a18}
\end{eqnarray}
The 1st term on the right-hand side above is handled as
\begin{eqnarray*}
& &
\sum_{|b|\leq 1 \atop |a|\leq |b|}
\||D|^{1/4}(x^a\partial^b v_m(0))\|_2
+
\sum_{|b|=2 \atop |a|\leq 2}
\||D|^{1/4}(x^a\partial^b v_m(0))\|_2\\
& &
\leq
C\varepsilon\Lambda_2
+
C\sum_{|a|\leq 2}
\||D|^{1/4}(x^a\Box v_m(0))\|_2.\nonumber
\end{eqnarray*}
The 2nd term on the right-hand side of (\ref{a18})
is treated as
\begin{eqnarray*}
& &
\sum_{|b|=1}
\||D|^{-3/4}(\partial^b v_m(0))\|_2
+
\sum_{|b|=2 \atop |a|\leq 1}
\||D|^{-3/4}(x^a\partial^b v_m(0))\|_2\\
& &
\leq
C\varepsilon\Lambda_2
+
C\sum_{|a|\leq 1}
\||D|^{-3/4}(x^a\Box v_m(0))\|_2.\nonumber
\end{eqnarray*}
We remark that we have dealt with
$\||D|^{-3/4}(x^a\partial_i\partial_j\tilde f)\|_2$
$(|a|=1)$ as
\begin{eqnarray*}
& &
\||D|^{-3/4}\partial_i(x^a\partial_j\tilde f)\|_2
+
\||D|^{-3/4}((\partial_i x^a)\partial_j\tilde f)\|_2\\
& &
\leq
\||D|^{1/4}(x^a\partial_j\tilde f)\|_2
+
\||D|^{1/4}\tilde f\|_2
\leq
2\Lambda_2.
\end{eqnarray*}
The 3rd term on the right-hand side of (\ref{a18})
is treated as, for $|a|\leq 2$,
\begin{eqnarray*}
& &
\sum_{i,j=1}^n
\varepsilon
\||D|^{-3/4}(x^a\partial_i\partial_j\tilde g)\|_2\\
& &
+
\sum_{i=1}^n
\||D|^{-3/4}(x^a\partial_i\partial_t^2 v_m(0))\|_2
+
\||D|^{-3/4}(x^a\partial_t^3 v_m(0))\|_2\nonumber\\
& &
\leq
C\varepsilon\Lambda_2
+
\sum_{i=1}^n
\||D|^{-3/4}\partial_i(x^a\partial_t^2 v_m(0))\|_2
+
\sum_{|b|\leq 1}
\||D|^{-3/4}(x^b\partial_t^2 v_m(0))\|_2\nonumber\\
& &
+
\||D|^{-3/4}(x^a\Delta\partial_t v_m(0))\|_2
+
2\||D|^{-3/4}(x^a\partial_t u_{m-1}(0)\partial_t^2 u_{m-1}(0))\|_2\nonumber\\
& &
\leq
C\varepsilon\Lambda_2
+
C\||D|^{1/4}(x^a\Box v_m(0))\|_2
+
\sum_{|b|\leq 1}
\||D|^{-3/4}(x^b\Box v_m(0))\|_2\nonumber\\
& &
+2\||D|^{-3/4}(x^a\partial_t u_{m-1}(0)\partial_t^2 u_{m-1}(0))\|_2.\nonumber
\end{eqnarray*}
Since the term $\||D|^{1/4}(x^a\Box v_m(0))\|_2$ $(|a|\leq 2)$
can be treated as
\begin{eqnarray*}
& &
\sum_{j=1}^n
\||D|^{-3/4}\partial_j(x^a\Box v_m(0))\|_2\\
& &
\leq
C\sum_{|b|\leq 1}
\||D|^{-3/4}(x^b\Box v_m(0))\|_2
+
\sum_{j=1}^n
\||D|^{-3/4}(x^a\partial_j\Box v_m(0))\|_2,\nonumber
\end{eqnarray*}
we finally arrive at, for $|\alpha|\leq 2$,
\begin{align}
\lefteqn{\||D|^{1/4}\Gamma^\alpha v_m(0)\|_2
+
\||D|^{-3/4}\partial_t\Gamma^\alpha v_m(0)\|_2}
\nonumber\\
&\leq
C\varepsilon\Lambda_2
+C\varepsilon^2
\sum_{|b|\leq 1}
\||D|^{-3/4}(x^b g^2)\|_2
+C\varepsilon^2
\sum_{|b|\leq 2 \atop 1\leq j\leq n}
\||D|^{-3/4}(x^b g\partial_j g)\|_2\nonumber\\
&\quad +
C\sum_{|b|\leq 2}
\||D|^{-3/4}(x^b\partial_t u_{m-1}(0)\partial_t^2 u_{m-1}(0))\|_2.
\label{a19}
\end{align}
The 2nd term on the right-hand side of (\ref{a19}) is treated as,
for $|b|\leq 1$,
\begin{eqnarray}
& &
\||D|^{-3/4}(x^b g^2)\|_2
\leq
C\|x^b g^2\|_{p_1}
\leq
C\|g\|_{4n/3}\|x^b g\|_2\nonumber\\
& &
\leq
C\|g\|_{H^1}\|x^b g\|_2
\leq C\Lambda_1^2,
\end{eqnarray}
where, as in (\ref{a20}) above,
$p_1$ is defined as
$1/p_1=1/2+3/(4n)$.
Similarly, the 3rd term on the right-hand side of (\ref{a19})
is treated as, for $j=1,\dots,n$,
\begin{align}
\lefteqn{\sum_{|b|\leq 1}
\||D|^{-3/4}(x^b g\partial_j g)\|_2
+\sum_{|b|=2}
\||D|^{-3/4}(x^b g\partial_j g)\|_2}
\nonumber\\
&\leq
C\sum_{|b|\leq 1}
\|g\|_{4n/3}\|x^b\partial_j g\|_2
+
C\sum_{|a|,|b|=1}
\|x^a g\|_{4n/3}\|x^b\partial_j g\|_2
\leq
C\Lambda_1^2.
\end{align}
Finally, using the 1st equation of (\ref{a14}),
we deal with the 4th term
on the right-hand side of (\ref{a19}) as, for $|b|\leq 2$,
\begin{align}
&
\||D|^{-3/4}(x^b\partial_t u_{m-1}(0)\Box u_{m-1}(0))\|_2
\nonumber\\
&
\quad +
\||D|^{-3/4}(x^b\partial_t u_{m-1}(0)\Delta u_{m-1}(0))\|_2\nonumber\\
&=
\varepsilon^3\||D|^{-3/4}(x^b g{\tilde f}^2)\|_2
+
\varepsilon^2\||D|^{-3/4}(x^b g\Delta f)\|_2.
\label{a26}
\end{align}
We treat the 1st term above as
\begin{eqnarray}
& &
\sum_{|b|\leq 1}
\||D|^{-3/4}(x^b g{\tilde f}^2)\|_2
+
\sum_{|b|=2}
\||D|^{-3/4}(x^b g{\tilde f}^2)\|_2\nonumber\\
& &
\leq
C
\sum_{|b|\leq 1}
\|x^b g\|_2
\|\tilde f\|_{8n/3}^2
+
C\sum_{|b|=1}
\|g\|_2\|x^b\tilde f\|_{8n/3}^2.
\end{eqnarray}
Using the Sobolev embedding
${\dot H}^{(4n-3)/8}_2({\mathbb R}^n)\hookrightarrow L^{8n/3}({\mathbb R}^n)$,
we can handle
$\|x^b\tilde f\|_{8n/3}$ $(|b|\leq 1)$ as
\begin{align}
\|x^b\tilde f\|_{8n/3}
&\leq
C\||D|^{(4n-3)/8}(x^b\tilde f)\|_2
\nonumber\\
&\leq
C\||D|^{1/4}(x^b\tilde f)\|_2^{(13-4n)/8}
\||D|^{5/4}(x^b\tilde f)\|_2^{(4n-5)/8}
\leq
C\Lambda_2.
\end{align}
We treat the 2nd term on the right-hand side of (\ref{a26}) as
\begin{equation}
\label{a21}
\sum_{|b|\leq 1}
\|x^b g\|_2
\|\Delta f\|_{4n/3}
+
\sum_{|a|,|b|=1}
\|x^a g\|_{4n/3}\|x^b\Delta f\|_2
\leq
C\Lambda_1^2.
\end{equation}
Summing up,
we have obtained by (\ref{a19})--(\ref{a21})
\begin{eqnarray*}
& &
\sum_{|\alpha|\leq 2}
\biggl(
\||D|^{1/4}\Gamma^\alpha v_m(0)\|_2
+
\||D|^{-3/4}\partial_t\Gamma^\alpha v_m(0)\|_2
\biggr)\\
& &
\leq
C\varepsilon\Lambda_2
+
C\varepsilon^2\Lambda_1^2
+
C\varepsilon^3\Lambda_1\Lambda_2^2.\nonumber
\end{eqnarray*}
We have finished the proof of (\ref{a2}).
\qed

redpajama_set_name: RedPajamaArXiv


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Our research focuses on connecting the dimensions of people, the environment and policy to broad thematic areas including, but not limited to:
and Wellbeing
Our research embodies inter- and transdisciplinary principles to solve today's environmental and societal challenges. We conduct research using concepts and principles from knowledge exchange and mobilization theory, social ecology, and human dimensions research. Our work spans the interface of natural and social sciences, as well as science and policy. We are committed to knowledge mobilization including co-producing knowledge, co-creating research agendas and science communication.
VIEW OUR RESEARCH THEMES
Conducting inter- and transdisciplinary research to solve environmental and conservation challenges.
Bridging the Knowledge-Action Gap for Conservation and Natural Resource Management
Overview of theme:
Mobilizing evidence for decision-making, management and conservation practices is important for implementing effective solutions to increasingly complex environmental, societal and policy problems such as climate change, population growth, and biodiversity loss. Using evidence, however, remains a significant challenge. As a result, greater research has been devoted to understanding the "science-policy" or "knowledge-action" gap. One of our major research themes is to better understand this gap between knowledge and action, and how we can bridge that gap to bring relevant and best-available knowledge into policy and practice.
The Human Dimensions of Natural Resources Management
Conservation and natural resource management have traditionally been dominated by knowledge informed by natural sciences. The natural sciences have often revealed potential solutions, but implementation of solutions require the social sciences because we deal with humans and society. The human dimensions are thus critical for ensuring that conservation and natural resource management strategies are effective. Studying people matters if we are to solve today's increasingly complex problems.
Integrated Social Ecological Research
Social ecological system (SES) was first coined in the 70s (Ratzlaff 1970). Since then, the SES concept has been widely used in both environmental and social sciences among other disciplines. In our lab, we integrate approaches from both social and ecological sciences to tackle complex environmental and conservation issues. We do this by collaborating with academics and non-academics towards a solution-oriented approach. We ensure that all of our projects are partnered with knowledge users to ensure our work is relevant, legitimate, and applicable. It is not possible for one discipline or knowledge system to solve wicked problems, and thus by working together with relevant stakeholders and rightsholders, with different knowledge systems and disciplines, we strive for more socially and ecologically robust solutions.
Join Our Lab

redpajama_set_name: RedPajamaCommonCrawl


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Jon Vroman
Jon Vroman is an international speaker voted "Campus Speaker of The Year" and best-selling author. His travels the world talking with students about what it takes to Live College Life In The Front Row™. His mission is to encourage individuals to find and develop their unique talents so that they will make the biggest impact within their campus and community. Jon lives in New Jersey with his wife and son. He's passionate about squeezing the juice out of life by creating extraordinary experiences with incredible people. In 2005, Jon founded the charitable organization Front Row Foundation, which has raised over $1,000,000 to help people braving life-threatening illnesses get front row seats to the concert or sporting event of their dream.
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Childhood Hunger, College Life, and Iridology
Dr. Ellen Cutler
Dr. Ellen Cutler has acquired great expertise in many fields of health care and is a passionate advocate of natural health and healing. She is the founder of BioSET (Bioenergetic Sensitivity and Enzyme Therapy), a system of healing that incorporates enzyme therapy with nutritional counseling, detoxification, and the identification and clearing of food and other sensitivities. Dr. Cutler has authored a number of books and articles, and has been a frequent guest speaker at numerous venues. She continues her clinical work at the BioSET Clinic in the San Francisco Bay Area. Dr. Cutler's newest book was recently released. Entitled "Clearing the Way to Health and Wellness", the book provides an overview of the BioSET system. In it, she gives a number of case studies documenting the efficacy of the BioSET system in a wide range of health problems. Dr. Cutler also explains how individuals can help themselves with proper diet, supplementation, and desensitization.
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Children's Dreams and Clearing Sensitivities
Kelly Bulkeley, Ph.D.
Kelly Bulkeley, Ph.D., is a Visiting Scholar at the Graduate Theological Union in Berkeley, California. He earned a doctorate in Religion and Psychological Studies from the University of Chicago Divinity School, an M.T.S. from Harvard Divinity School, and a B.A. from Stanford University. A former President of the International Association for the Study of Dreams and a Senior Editor of the APA journal Dreaming, he has written and edited several books on dream research. His recent works include Dreaming in the World's Religions: A Comparative History (New York University Press, 2008), American Dreamers: What Dreams Tell Us about the Political Psychology of Conservatives, Liberals, and Everyone Else (Beacon Press, 2008), and Dreaming in the Classroom: Practices, Methods, and Resources in Dream Education (SUNY Press, 2011). He is the director of the Sleep and Dream Database (SDDb), a digital archive and search engine designed to facilitate empirical dream research.
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Scott Williams is currently senior vice president of Men's Health Network (MHN), a Washington, D.C.-based national non-profit organization whose mission is to reach men and their families where they live, work, play and pray with health prevention messages and tools, screening programs, educational materials, advocacy opportunities and patient navigation. He has been featured as an expert in the New York Times, CNN, MSNBC, Washington Post, Baltimore Times, Medicare Reimbursement Weekly, Chicago Tribune, National Public Radio, St. Petersburg Times, Inside CMS, Dallas Morning News and Seattle Post-Intelligencer, among others. Scott is a peer reviewer and contributing author to the American Journal of Men's Health, a SAGE Publication. He founded the Men's Health Caucus within the American Public Health Association and also serves as the membership chair for the Community Health Planning and Policy Development Section.
Listen to Scott Williams on VoiceAmerica
Letting Go, Persevering, Preventing, and Listening
John Nichols
In 1993, John Nichols had a near-death experience from a water skiing accident that left him paralyzed from his neck to his toes. Doctors weren't sure he would ever walk again. Through six years of rehabilitation, he reached a level of recovery that less than 1 percent of all spinal cord injury patients attain. Today, John does more than just walk: He runs marathons. In fact, he's completed a marathon a year for four straight years. John is a nationally recognized disability insurance benefits consultant and serves as president of Chicago-based Disability Resource Group Inc., a national insurance agency that he founded in 1999. John also serves as the president-elect of the National Association of Insurance and Financial Advisors and previously as a national board trustee since 2007. He obtained his CLU designation in 2003 and graduated with his Masters in Science Management with an emphasis in Leadership in 2011 from The American College.
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Nicholas Corrin
Nicholas Corrin as a child was very curious, always asking questions. He still does. He wants to understand the deeper reasons for things being the way they are, not just respond to external "facts" as they may present themselves to us. For this reason, he may have tended to be a thorn in the side of the establishment, challenging and occasionally mocking received views anchored in conformism and rigid thinking. The air outside the box has always appealed to him as sweeter and more invigorating than the stale stuff inside it. His training at school, both in Europe and the US, has been long and diverse, spanning literature, medicine, history, art and foreign languages. But most of all, he have learned what he knows from direct observation of nature. In a world of "specialists", he seeks out the middle ground, the metaphorical vantage point from which the entire landscape can be surveyed. Only from here can the bigger picture be comprehended, and the various broken parts re-integrated.
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Matthew J. Bassiur
Matthew J. Bassiur is vice president and deputy, Chief Security Office (CSO) of Pfizer's Global Security division, where he oversees anti-counterfeiting operations, large-scale investigations, physical security and crisis management for North, Central and South America. Mr. Bassiur also manages the Strategic Planning and Intelligence directorates globally, and serves as deputy to Pfizer's CSO. In 2008, Mr. Bassiur received and completed a United States Fulbright Scholarship to teach intellectual property law at Renmin University in Beijing, China. Mr. Bassiur earned his J.D. from the University of Virginia, School of Law in 1998 and was inducted into the Order of the Coif.
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Sylvia Hawthorn-Deppen
Sylvia Hawthorn-Deppen is a licensed practical nurse passionate about helping others live comfortably despite illness. An active member of Chronic Pain Anonymous, she also volunteers as a speaker and peer support in behavioral health at a hospital in Camp Hill, Pa. Hawthorn-Deppen, after abusing pain pills for years to numb her pain, credits the 12-step program of Alcoholics Anonymous with reinforcing her recovery. She enjoys riding her bicycle in her spare time and dreams of cycling cross-country from Pennsylvania to Florida one day.
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Whole Foods' Benefits and Living Comfortably with Illness
T. Colin Campbell, PhD
T. Colin Campbell, PhD, for more than 40 years has been at the forefront of nutrition research. His legacy, the China Study, is the most comprehensive study of health and nutrition ever conducted. Dr. Campbell is the author of the bestselling book, The China Study, and the Jacob Gould Schurman Professor Emeritus of Nutritional Biochemistry at Cornell University. He has received more than 70 grant-years of peer-reviewed research funding and authored more than 300 research papers. The China Study was the culmination of a 20-year partnership of Cornell University, Oxford University, and the Chinese Academy of Preventive Medicine. Dr. Campbell has just released his new book Whole…Rethinking the Science of Nutrition. The China Study revealed what we should eat and provided the powerful empirical support for this answer. Whole answers the question of why we should eat this way.
Listen to T. Colin Campbell, PhD on VoiceAmerica
Elizabeth "Coach" Scala, MSN/MBA, RN
Elizabeth "Coach" Scala, MSN/MBA, RN is passionate about helping healthcare professionals, nurses in particular, to embody holistic living and embrace self-care. Through her business, Living Sublime Wellness, she writes regularly on the topic of self-care, conducts a free, monthly wellness workshop, and offers both in-person and online seminars for busy nurses. She is a Certified Health and Wellness Coach and holds a Bachelor of Arts degree in Psychology, a Bachelor of Science in Nursing, and Master's degrees in both Business and Nursing. She writes for Scrubs Magazine and is a monthly contributor to the online community, Nurses Together.com">Nurses Together.com. Originally from Carmel, NY, Elizabeth lives in Baltimore, MD with her husband and two dogs. Her websites are: http://livingsublimewellness.com/ and http://rejuvenationcollaboration.com/
Listen to Elizabeth "Coach" Scala, MSN/MBA, RN on VoiceAmerica
Caring for Self While Helping Others


Jeffrey Sachs, 20th February 2017, 19:45, Great Hall, Durham Castle.
The world is on a dangerous and unsustainable path regarding climate change, other environmental threats, and sociopolitical instability. In response, governments have adopted several high-level goals and objectives, including the Sustainable Development Goals and the commitment to keep global warming to "well below 2-degree C." Yet we are aware that our social and political systems are profoundly difficult to reorient. While we call for "rational" responses to global challenges, we too often face paralysis or even open conflict. I will discuss the challenges, and possible solutions, to create a global response commensurate with our declared objectives. My focus will be on fostering a "world consciousness" that at a global scale can effectively recognize challenges, adopt goals and targets, identify and test potential solutions, and choose collectively on actions to achieve the agreed goals. I will draw on my 16 years experience as senior UN advisor, and on analogies from the neuroscience of rationality at the level of the individual to develop concepts of collective rationality at the global scale.


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50 Cent Hanging Up His Mic After "Street King Immortal"
by S. Leslie December 21, 2016
S. Leslie Author
50 Cent is strongly suggesting that he will be retiring from rap after his last album "Street King Immortal."
The New York rap mogul is dropping a lot less music these days and you can't blame him. With the success of his acting career, things seems to be moving towards Hollywood for 50 Cent. This year he had one of the hottest shows on TV, Power. Speaking with Muscle & Fitness magazine, the hip hop titan didn't just talk about how he build and maintained his muscles, but he also opened up about his rap and acting career.
"I want to finish with [Street King Immortal]," he said. "I don't want to write another record after that. You know how some artists got to have confirmation that they're right. I have that confirmation in f**king 35, 40 million [albums] that I already sold."
50 Cent first announced the project from back in 2011 and went on to release one album since then, Animal Ambition in 2014. That last project didn't saw the same success of his previous platinum releases.
Fifty also recently hinted that he will be quitting his hugely successful TV series Power after getting snubbed by the Golden Globe. "I'm off the POWER train, champ is gonna do the show," he wrote on Instagram. "Go to him for more info, I gotta do something different.#EFFENVODKA."
He continued to rant in another post, writing "I'm on the move again, Golden globe's can s*ck a d**k. I accept my series POWER was not intended to be a signature show for the network but it is the highest rated show. I know my audience very well, I'm done promoting this out side of contractual obligations. SEASON 4 KANAN MUST DIE, I got other sh*t to do."
Whether or not he is serious, we will have to wait on season 4 of Power to see what will happen to his character Kanan.
More 50 Cent News
50 Cent Proclaim His Greatness After New Song Goes No. 1, "I'm Still 50 Cent"
50 Cent Shares Appreciation For Eminem Amid Snoop Dogg Beef
Cardi B & 50 Cent React To Trump Supporters Overrunning Capitol Buildings


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Una capitanía de puerto es una oficina encargada de hacer cumplir las normas de un refugio marítimo o puerto en particular, con el fin de garantizar la seguridad de la navegación, la seguridad portuaria y el correcto funcionamiento de las instalaciones portuarias.

Responsabilidades
Los capitanes de puerto normalmente son responsables de emitir información de seguridad local, a veces conocida como aviso a los navegantes.

Oficiales civiles y navales
Un capitán de puerto puede ser un civil o un oficial naval comisionado de cualquier rango.

Históricamente, todos los capitanes de puerto eran oficiales navales; incluso hoy en día, deben poseer conocimientos y experiencia previa en navegación a través del servicio en una marina mercante o una armada.

Los términos naval y civil se utilizan aquí para distinguir quién está empleado por una fuerza militar y quién está empleado por un puerto público o privado.

Reino Unido y Canadá
En el Reino Unido, cuando una persona ha sido nombrada para supervisar un astillero portuario, el oficial se desempeña en esta capacidad como capitán de puerto de la Reina (o del Rey) y tiene derecho a enarbolar una bandera de la Unión con un borde blanco y un disco central blanco. con las iniciales "QHM" (o "KHM" durante el reinado de un rey) debajo de una corona. Aunque la legislación no lo requiere, la mayoría de los QHM son oficiales en servicio en la Marina Real. También existen puestos equivalentes en Canadá, donde el capitán de puerto de la reina se conoce en francés como capitaine de port de Sa Majesté (literalmente, "Capitán de puerto de Su Majestad").

Estados Unidos
En los Estados Unidos, el Capitán del Puerto es un oficial de la Guardia Costera de los Estados Unidos, responsable de estos deberes en una zona predefinida que generalmente incluye múltiples puertos y vías fluviales que conducen a esos puertos, generalmente en aguas federales. Las directivas de los capitanes de puerto están sujetas a la supervisión de la Guardia Costera.

Chile

En Chile, capitán de puerto es la denominación genérica que se da a las autoridades marítimas, siendo la capitanía de Puerto es la que se da al asiento de estos funcionarios. En los puertos o radas de pequeña población o escaso comercio marítimo, que estén tan lejanos al lugar de la capitanía de que dependen, desempeñarán las funciones de capitanes de puerto, los alcaldes de mar. Los capitanes de puerto, en general, en su carácter militar, están bajo las órdenes del comandante en jefe del apostadero a cuya jurisdicción pertenecen. El capitán de puerto especialmente, tendrá a su cargo la policía del mar territorial, de los puertos, bahías canales, lagos y ríos navegables e islas, y la inmediata inspección de toda maniobra o faena que deba efectuarse en su jurisdicción. Igualmente toda operación que deba efectuarse a bordo de embarcaciones de cualquier clase, por orden de otra autoridad, se hará previo su conocimiento.

Referencias

Enlaces externos

 Asociación de Capitanes de Puerto de California
 Asociación de Capitán de Puerto y Constable de la Bahía del Estado de Nueva York
 Capitán de puerto de la reina Portsmouth, Plymouth y Clyde
 Día Nacional de Agradecimiento al Capitán de Puerto (EE. UU. )

Terminología naval
Autoridades portuarias

redpajama_set_name: RedPajamaWikipedia


Do you dream like becoming a real hero of the ring?
In this case, wrestling simulator, Wrestling Revolution 3D allows you to fully experience the adrenaline of the battle. The game opens up a realistic world of battles in the ring where the strongest wins. Choose any opponent, depending on your level, from 28 million other users and fight to win!
If you do not like to participate in battles – bet on players and pump your bookmaker career. As you can evaluate the battle on both sides, you will never get tired of the atmosphere of wrestling! Both game modes are available for free download.
A distinctive feature of the game is high-quality graphics, as well as realistic sound and visual effects. In addition, the game is downloaded absolutely free.
The game will allow to carry out any captures, receptions, and throws of opponents. You will support the audience if you are good enough to play a role in the ring. Win and unlock new fighters who can surprise you with incredible strength and endurance. Choose one of ten battle modes and go for great victories!


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Council Decision of 8 November 2007 amending Annex I to the 2005 Act of Accession (2007/857/EC)
Decisions originating from the EU
Whole Decision without Annexes
Original (As adopted by EU)
Open whole Decision
EU Official Journal Version
This is a Decision originating from the EU
See the EU version of this legislation on EUR-Lex
See an archived version from EUR-Lex in the web archive
This is a legislation item that originated from the EU
After exit day there will be three versions of this legislation to consult for different purposes. The legislation.gov.uk version is the version that applies in the UK. The EU Version currently on EUR-lex is the version that currently applies in the EU i.e you may need this if you operate a business in the EU.
The web archive version is the official version of this legislation item as it stood on exit day before being published to legislation.gov.uk and any subsequent UK changes and effects applied. The web archive also captured associated case law and other language formats from EUR-Lex.
This is the original version (as it was originally adopted).
The following indent shall be added to point 1 of Annex I to the 2005 Act of Accession:
'Convention of 14 April 2005 on the accession of the Czech Republic, the Republic of Estonia, the Republic of Cyprus, the Republic of Latvia, the Republic of Lithuania, the Republic of Hungary, the Republic of Malta, the Republic of Poland, the Republic of Slovenia and the Slovak Republic to the Convention on the law applicable to contractual obligations opened for signature in Rome on 19 June 1980, and to the First and Second Protocols on its interpretation by the Court of Justice of the European Communities (OJ C 169, 8.7.2005, p. 1).'
This Decision shall take effect on the day following its publication in the Official Journal of the European Union.
Done at Brussels, 8 November 2007.
For the Council
R. Pereira
PrintThe Whole Decision
PDF The Whole Decision
Web page The Whole Decision
PrintThe Decision without Annexes
PDF The Decision without Annexes
Web page The Decision without Annexes
Original (As adopted by EU): The original version of the legislation as it stood when it was first adopted in the EU. No changes have been applied to the text.
the original print PDF of the as adopted version that was used for the EU Official Journal
the original print PDF of the as adopted version that was used for the print copy


1. Check the provided Readme! Please read carefully if you are replacing the car and if you want everything to work correctly (dials, liveries, LOD distances)!
v1.1 is here! Pending approval!
No LODs, cuz it will make the game crash. The amount of polygons is barely reached.
@MC_WRX Cool Car, and Nice Livery !
dial is not working properly for me, also car disappears when going away for around 50m. will rate 5 stars after these are patched or explained. good mod overall.
Nice mod, but would be nice to see changeable interior colours and please fix the gaps between windows and doors. Also, the brake lights are way too dark.
@JoeChen0102 well, you better read the Readme file and after that complain about dials.
@MC_WRX This is the best Porsche Panamera mod, but why the 4S and not the Turbo Executive?
@MC_WRX Hey, very nice work ! the forza horizon 3 panamera turbo 2017 just appeared on gamemodels.ru if you'd like to do some tweaks!
Too slow, can you make it a little bit faster?