\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 211, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2015/211\hfil Critical exponent ]
{Critical exponent for a damped wave system with fractional integral}

\author[ M. Wu, S. Li, L. Lu \hfil EJDE-2015/211\hfilneg]
{Mijing Wu, Shengjia Li, Liqing Lu}

\address{Mijing Wu \newline
School of Mathematical Sciences,
Shanxi University,
Taiyuan, Shanxi 030006, China}
\email{mjwu@sxu.edu.cn}

\address{Shengjia Li \newline
School of Mathematical Sciences,
Shanxi University,
Taiyuan, Shanxi 030006, China}
\email{sjli@sxu.edu.cn}

\address{Liqing Lu (corresponding author) \newline
School of Mathematical Sciences,
Shanxi University,
Taiyuan, Shanxi 030006, China}
\email{lulq@sxu.edu.cn}

\thanks{Submitted July 24, 2015. Published August 12, 2015.}
\subjclass[2010]{35B33}
\keywords{Damped wave equation; fractional integral;
critical exponent; \hfill\break\indent global solution}

\begin{abstract}
 We shall present the critical exponent
 $$
 F(p, q,\alpha):=\max\big\{\alpha+\frac{(\alpha+1)(p+1)}{pq-1},
 \alpha+\frac{(\alpha+1)(q+1)}{pq-1}\big\}-\frac{1}{2}
 $$ for the Cauchy problem
 \begin{gather*}
 u_{tt}-u_{xx}+u_t=J_{0|t}^{\alpha}(|v|^{p}), \quad
 (t,x)\in\mathbb{R}^{+}\times\mathbb{R},\\
 v_{tt}-v_{xx}+v_t=J_{0|t}^{\alpha}(|u|^{q}), \quad
 (t, x)\in\mathbb{R}^{+}\times\mathbb{R},\\
 (u(0,x), u_t(0,x))=(u_0(x),u_1(x)), \quad x\in \mathbb{R},\\
 (v(0,x), v_t(0,x))=(v_0(x),v_1(x)), \quad x\in \mathbb{R},\\
 \end{gather*}
 where $p,q\geq 1, pq>1$ and $0<\alpha<1/2$; that is, the small data global
 existence of solutions can be derived to the problem above if $F(p, q, \alpha)<0$.
 Furthermore, in the case of $F(p, q, \alpha)\geq0$ the non-existence of global
 solution can be obtained with the initial data having positive average value.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

 \section{Introduction and statement of main results}

 In this article, we consider the Cauchy problem of damped
wave system
 \begin{equation}\label{1.1}
\begin{gathered}
u_{tt}-u_{xx}+u_t=J_{0|t}^{\alpha}(|v|^{p}), \quad
(t,x)\in\mathbb{R}^{+}\times\mathbb{R},\\
v_{tt}-v_{xx}+v_t=J_{0|t}^{\alpha}(|u|^{q}), \quad
(t, x)\in\mathbb{R}^{+}\times\mathbb{R},\\
(u(0,x), u_t(0,x))=(u_0(x),u_1(x)), \quad
 x\in \mathbb{R},\\
(v(0,x), v_t(0,x))=(v_0(x),v_1(x)), \quad x\in \mathbb{R},
\end{gathered}
\end{equation}
 where $p,q\geq 1, pq>1$ and $0<\alpha<1/2$. The
 initial values satisfy
\begin{gather}\label{1.2}
\operatorname{supp}\{u_i,v_i\}\subset\{|x|\leq K\}, \quad K>0,\; i=0,1, \\
\label{1.3} (u_0,u_1,v_0,v_1)\in H^{1}(\mathbb{R})\times
L^{2}(\mathbb{R})\times H^{1}(\mathbb{R})\times L^{2}(\mathbb{R}).
\end{gather}

The notation $J_{0|t}^{\alpha}$ stands for the Riemann-Liouville fractional
integral \cite{l2}
\[
J_{0|t}^{\alpha}f(t):=\begin{cases}
\frac{1}{\Gamma(\alpha)}\int_0^t(t-s)^{\alpha-1}f(s)ds, & \alpha>0,\\
f(t), & \alpha=0,
\end{cases}
\]
for $f\in L^p(0,T)(1\leq p\leq \infty)$
and $\Gamma(\cdot)$ is the Euler gamma function.

In recent decades, nonlinear hyperbolic equations and systems have been studied
extensively (see, for example,
\cite{i1,i2,l2,m1,m2,y1,z1} and the rich references therein).
Todorova and Yordanov \cite{t1}
considered the semilinear wave equation
\begin{equation}\label{1.4}
\begin{gathered}
u_{tt}-\Delta u+u_{t}=|u|^{p}, \quad
(t,x)\in(0,\infty)\times\mathbb{R}^n,\\
u(x, 0) =u_0(x),\quad u_t(x,0)=u_1(x) , \quad
x\in \mathbb{R}^n,
\end{gathered}
\end{equation}
and proved that the critical exponent of \eqref{1.4} is $p_c(n)=1+2/n$.
More precisely, if $p>p_{c}(n)$ there exists a unique global solution
of \eqref{1.4} for sufficiently small initial data, while, if $1<p<p_c(n)$
any solution with positive initial data must blow up in finite time.
Later Zhang \cite{z2} proved that the exponent $1+2/n$ belongs to
the blow up region. Fino \cite{f1} considered the damped wave equation with
nonlinear memory term
\begin{equation}\label{1.5}
u_{tt}-\Delta u+u_t=J_{0|t}^{\alpha}(|u|^p)(t).
\end{equation}
The existence of global solutions and the asymptotic behavior
of small data solutions to \eqref{1.5} as $t\to\infty$
were established when $1\leq n\leq 3$.
If $p>1+2(1+\alpha)/(n-2\alpha)$, the blow up
result was also derived under some positive data in any dimensional space.
Comparing the results of \cite{t1} with that of \cite{f1},
we derive that the fractional integral $J_{0|t}^{\alpha}$ has an influence
on the solution. The problems with the fractional integral term are interesting.

The problem \eqref{1.1} with $\alpha=0$ can be considered as the following weakly
coupled system
\begin{equation}\label{1.6}
\begin{gathered}
u_{tt}-\Delta u+u_t=|v|^{p}, \quad (t, x)\in (0, \infty)\times\mathbb{R}^n, \\
v_{tt}-\Delta v+v_t=|u|^{q}, \quad (t, x)\in (0, \infty)\times\mathbb{R}^n,\\
(u(0,x), u_t(0,x))=(u_0(x),u_1(x)), \quad x\in \mathbb{R}^n,\\
(v(0,x), v_t(0,x))=(v_0(x),v_1(x)), \quad x\in \mathbb{R}^n.
\end{gathered}
\end{equation}
Sun and Wang \cite{s1} considered \eqref{1.6} and obtained the critical
exponent
\begin{equation*}
F(p, q,n):=\max\big\{\frac{p+1}{pq-1}, \frac{q+1}{pq-1}\big\}-\frac{n}{2}.
\end{equation*}
The authors proved that if $F(p, q, n)<0$, there exists a unique
global solution of \eqref{1.6} with suitably small initial data for $n=1$ or $n=3$,
and if $F(p, q, n)\geq 0$, any solution of \eqref{1.6} with initial data having
positive integral values does not exist globally for any $n\geq 1$. Based on some
conditions on nonlinear term, the asymptotic behavior of solutions of \eqref{1.6} was
considered in \cite{n1}. Recently, Kenji and Yuta \cite{n2} showed that the number
$F(p, q, n)$ is the critical exponent of \eqref{1.6} for any dimensional space.

Motivated by the work of \cite{f1} and \cite{s1}, we aim at determining the
critical exponent of \eqref{1.1}. The global result is proved by the weighted energy
method (see \cite{t1}). For the non-existence of a global solution, we shall
use the test function method (see \cite{f1}).
Our basic definition of the solution to problem \eqref{1.1} is the following.

\begin{definition} \label{def1} \rm
 Let $T>0$. We say that a pair of functions $(u,v)$ in
 $L^q((0, T), \\ L_{\rm loc}^q(\mathbb{R}))\times L^p((0, T), L_{\rm loc}^p(\mathbb{R}))$
is a weak solution of the Cauchy problem \eqref{1.1} with the initial data
$(u_i, v_i)\in[L_{\rm loc}(\mathbb{R})]^2$ if $(u,v)$ satisfies
\begin{equation}\label{1.7}
\begin{split}
 &\int_{0}^{T}\int_{\mathbb{R}^n}u(\varphi_{tt}-\varphi_{xx} +\varphi_t)\,dx\,dt\\
&=\int_{0}^{T}\int_{\mathbb{R}^n}(J_{0|t}^{\alpha}|v|^{p})\varphi \,dx\,dt
 +\int_{\mathbb{R}^n}u_1(x)\varphi(0,x)dx
+\int_{\mathbb{R}^n}u_0(x)(\varphi(0,x)-\varphi_t(0,x))dx,
\\
 &\int_{0}^{T}\int_{\mathbb{R}^n}v(\varphi_{tt}-\varphi_{xx} +\varphi_t)\,dx\,dt\\
&=\int_{0}^{T}\int_{\mathbb{R}^n}(J_{0|t}^{\alpha}|u|^q)\varphi \,dx\,dt
 +\int_{\mathbb{R}^n}v_1(x)\varphi(0,x)dx
 +\int_{\mathbb{R}^n}v_0(x)(\varphi(0,x)-\varphi_t(0,x))dx,
\end{split}
\end{equation}
for all compactly supported test functions
$\varphi\in C^2([0,T]\times\mathbb{R})$ with $\varphi(T,\cdot)=0$ and
$\varphi_t(T,\cdot)=0$.
If $T=\infty$, we say that $(u,v)$ is a global weak solution
of \eqref{1.1}.
\end{definition}

We remark that the above definition of a weak solution is a very weak
form which will be used in the proof of non-existence of a global solution.
However, to prove the global result we need a much stronger form.
We have the following local existence result.

\begin{proposition}\label{local}
Let $T>0$. Under assumptions \eqref{1.2} and \eqref{1.3},
there exists a unique solution $(u, v)\in X(T)\times X(T)$ for \eqref{1.1}
satisfying
$$
\operatorname{supp} \{u,v\}\subset B(t+K)=\{(t, x):
|x|\leq t+K\},\; K>0
$$
where $X(T)=C([0,T); H^{1}(\mathbb{R}))\cap C^{1}([0,T);
L^{2}(\mathbb{R}))$.
\end{proposition}

Using \cite[Proposition 2.3]{i3} and
\cite[Proposition 1]{f1}, the local solvability and uniqueness
of \eqref{1.1} can be established
by a standard estimation and compactness theory.

Denote $\|\cdot\|_{r}$ and $\|\cdot\|_{H^m}$ the norms of $L^{r}(\mathbb{R})$ and
$H^{m}(\mathbb{R})$ respectively. Throughout this article, we use $C$ to stand
for a generic positive constant which may be different from line to line.
Set
$$
F(p, q, \alpha):=\max\big\{\alpha+\frac{(\alpha+1)(p+1)}{pq-1},
\alpha+\frac{(\alpha+1)(q+1)}{pq-1}\big\}-\frac{1}{2}.
$$
Based on Proposition \ref{local}, our main results read as follows

\begin{theorem}\label{global}
Assume that \eqref{1.2} and \eqref{1.3} hold. If $F(p, q, \alpha)<0$,
then there is a small constant $\varepsilon$ such that under the conditions
\begin{equation}\label{1.8}
\begin{gathered}
 I_{0, u}=\|u_0\|_{H^1}+\|u_0\|_{1}+\|u_1\|_{2}+\|u_1\|_{1}<\varepsilon,\\
 I_{0, v}=\|v_0\|_{H^1}+\|v_0\|_{1}+\|v_1\|_{2}+\|v_1\|_{1}<\varepsilon,
 \end{gathered}
 \end{equation}
problem \eqref{1.1} admits a unique global solution 
$$
(u, v)\in[C((0, \infty); H^{1}(\mathbb{R}))\cap C^{1}((0, \infty); 
L^{2}(\mathbb{R}))]^2.
$$ 
Moreover,
\begin{equation}\label{1.9} 
\begin{gathered}
\|Du(t)\|_{2}\leq(1+t)^{-\frac{(\alpha+1)(p+1)}{(pq-1)}-\frac{1}{4}}, \quad
t\to\infty,\\
\|Dv(t)\|_{2}\leq(1+t)^{-\frac{(\alpha+1)(q+1)}{(pq-1)}-\frac{1}{4}}, \quad
t\to\infty, 
\end{gathered}
\end{equation} 
where $Du=(u_{t}, u_{x})$.
\end{theorem}

\begin{theorem} \label{blow up} 
Assumed that \eqref{1.2}, \eqref{1.3} hold and
\begin{equation}\label{1.10} 
\int_{\mathbb{R}}u_{i}dx>0,\quad
\int_{\mathbb{R}}v_{i}dx>0,\quad i=0, 1. 
\end{equation} 
If $F(p, q, \alpha)\geq0$, then the weak solution $(u, v)$ of \eqref{1.1} 
does not exist globally.
\end{theorem}

\begin{remark} \label{rmk1.1} \rm
 $F(p, q, \alpha)$ is the critical exponent of \eqref{1.1}.
\end{remark}

\begin{remark} \label{rmk1.2} \rm
 If $\alpha=0$, $F(p, q, \alpha)$ is consistent with the critical
exponent of \eqref{1.6} for $n=1$. 
\end{remark}

The remainder of this paper is organized as follows. 
In Section 2, some preliminaries are collected. We will prove our global 
result (Theorem \ref{global}) in Section 3. 
Section 4 is devoted to proving the blow up result (Theorem \ref{blow up}).

\section{Preliminaries} 

We shall start this section with some basic
definitions and properties on the Riemann-Liouville fractional calculus. We refer to
\cite{k1}-\cite{f2} for more details.

Let $AC[0,T]$ denote the space of all absolutely continuous functions on $[0,T]$.
Then, if $f\in AC[0,T]$, the left-sided and the right-sided Riemann-Liouville
fractional derivatives of the function $f$ of order $\alpha\in (0,1)$ are defined by
$$ 
D_{0|t}^{\alpha}f(t):=\partial_tJ_{0|t}^{1-\alpha}f(t), \quad 
D_{t|T}^{\alpha}f(t):=-\frac{1}{\Gamma(1-\alpha)}\partial_t
\int_t^T(s-t)^{-\alpha}f(s)ds.
$$ 
Set 
$$
AC^{n+1}[0,T]:=\{f: [0,T]\to \mathbb{R}\\text{ and }
\partial_{t}^{n}f\in AC[0,T]\}.
$$ 
Then for all $f\in AC^{n+1}[0,T]$, the following
propositions are obtained in \cite{k1,o1,p1}, respectively.

\begin{proposition}[\cite{k1}] \label{frac1}
Let $0<\alpha<1$ and $p\geq 1$. If $f\in L^p(0,T)$, 
$$ 
(D_{0|t}^{\alpha}J_{0|t}^{\alpha}f)(t)=f(t),\quad 
(-1)^n\partial_t^nD_{t|T}^{\alpha}f=D_{t|T}^{n+\alpha}f, 
$$ 
for almost everywhere on $[0, T]$. 
\end{proposition}

\begin{proposition}[\cite{p1}] \label{frac2}
 Let $0<\alpha<1$. For every
$f, g\in C([0,T])$ such that $(D_{0|t}^{\alpha}f)(t), (D_{t|T}^{\alpha}g)(t)$ exist
and are continuous, the formula of integration by parts is 
\[
\int_0^T (D_{0|t}^{\alpha}f)(t)g(t)dt=\int_0^T f(t)(D_{t|T}^{\alpha}g)(t)dt,\quad
t\in[0,T]. 
\]
\end{proposition}

\begin{proposition}[\cite{o1}] \label{frac3}
 Set $\varphi_2(t):=(1-t/T)_{+}^{\eta}$. Then $\varphi_2(t)$ satisfies 
\begin{gather*}
D_{t|T}^{\alpha}\varphi_2(t)=CT^{-\eta}(T-t)_{+}^{\eta-\alpha},\quad
D_{t|T}^{\alpha+1}\varphi_2(t)=CT^{-\eta}(T-t)_{+}^{\eta-\alpha-1},\\
D_{t|T}^{\alpha+2}\varphi_2(t)=CT^{-\eta}(T-t)_{+}^{\eta-\alpha-2},
\end{gather*}
and 
\begin{gather*}
 D_{t|T}^{\alpha}\varphi_2(T)=0,\quad
D_{t|T}^{\alpha}\varphi_2(0)=CT^{-\alpha},\\
D_{t|T}^{\alpha+1}\varphi_2(T)=0,\quad
D_{t|T}^{\alpha+1}\varphi_2(0)=CT^{-\alpha-1}.
\end{gather*}
\end{proposition}

Consider the  linear damped wave equation 
\begin{equation}\label{2.1}
\begin{gathered} 
U_{tt}-U_{xx}+U_t=0, \quad (t, x)\in (0,\infty)\times\mathbb{R},\\ 
(U(0, x), U_t(0, x))=(U_0(x),U_1(x)), \quad x\in \mathbb{R}.
\end{gathered}
\end{equation} 
When $U_0=0$, the unique solution $U(t, x)$ to \eqref{2.1} can be denoted 
by $S(t)U_1$. Then the Duhamel's principle
implies the solution to \eqref{1.4} solves the  integral system
\begin{equation}\label{2.2} 
\begin{split} u(t,x)&=S(t)(u_0+u_1)+\partial_t(S(t)u_0)
 +\int_0^{t}S(t-\tau)J_{0|\tau}^{\alpha}(|v|^p)d\tau\\
&=u_{L}+\int_0^{t}S(t-\tau)J_{0|\tau}^{\alpha}(|v|^p)d\tau,\\
v(t,x)&=S(t)(v_0+v_1)+\partial_t(S(t)v_0)
 +\int_0^{t}S(t-\tau)J_{0|\tau}^{\alpha}(|u|^q)d\tau\\
&=v_L+\int_0^{t}S(t-\tau)J_{0|\tau}^{\alpha}(|u|^q)d\tau.
\end{split}
\end{equation} 
The following lemmas will be used in the proof of
Theorem \ref{global}. 

\begin{lemma}[{\cite[Proposition2.5]{t1}}] \label{linear estimate} 
Let $m\in [1, 2]$. Then 
\begin{equation}\label{2.3}
\|\partial_t^{k}\nabla_{x}^{\nu}S(t)f\|_2
\leq C(1+t)^{n/4-n/(2m)-|\nu|/2-k}(\|f\|_{m}+\|f\|_{H^{k+|\nu|-1}}),
 \end{equation} 
for each $f\in L^m(\mathbb{R}^n)\bigcap H^{k+|\nu|-1}\mathbb({R}^n)$. 
\end{lemma}

\begin{lemma}[{\cite[Proposition2.4]{t1}}] \label{weighted inequality}
 Let $\theta(r)=n(1/2-1/r)$ and $0\leq \theta(r)\leq 1, 0<\delta\leq 1$. 
If $u\in H^{1}(\mathbb{R}^n)$ with $\operatorname{supp} u\subset B(t+K)$, then
\begin{equation}\label{2.4} 
\|e^{\delta\psi(t, \cdot)}u\|_{r}\leq C(1+t)^{(1-\theta(r))/2}\|e^{\psi(t, \cdot)}
\nabla u\|_2^{\delta}\|\nabla u\|_2^{1-\delta}, 
\end{equation} 
where $\psi(t,x)=(t+K-\sqrt{(t+K)^2-|x|^2})/2$.
\end{lemma} 

\begin{lemma}[\cite{c1}] \label{int}
 Suppose that $0\leq\theta<1,a\geq 0$ and $b\geq 0$. Then there exists a 
constant $C>0$ depending only on $a,b$ and $\theta$ such that for all $t>0$, 
\begin{align*}
&\int_{0}^{t}(t-\tau)^{-\theta}(1+t-\tau)^{-a}(1+\tau)^{-b}d\tau\\
&\leq \begin{cases} 
C(1+t)^{-\min\{a+\theta,b\}},  & \max\{a+\theta,b\}>1,\\ 
C(1+t)^{-\min\{a+\theta,b\}}\ln(2+t),  & \max\{a+\theta,b\}=1,\\ 
C(1+t)^{1-\theta-a-b},  & \max\{a+\theta,b\}<1. 
\end{cases}
\end{align*} 
\end{lemma}

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

 Let $T_{\rm max}$ be the maximal existence time of the local solution of
$(u, v)$ to the problem \eqref{1.1}. Denote
 \begin{equation}\label{3.1}
 M(t)=\sup _{0\leq \tau<
 t}((1+\tau)^{k}\|Du(\tau)\|_{2}+(1+\tau)^{j}\|Dv(\tau)\|_{2}), \quad
 \forall t\in[0,  T_{\rm max}),
 \end{equation}
where $k, j$ will be determined later. We will prove the estimate
 \begin{equation}\label{3.2}
 M(t)\leq C(\varepsilon+M(t)^p+M(t)^q),\quad \forall t\in[0, T_{\rm max}),
 \end{equation}
with $C$ is independent of $\varepsilon$. Taking $\varepsilon$ and $C_1$
sufficiently small such that 
\begin{equation*}
 C\varepsilon<C_1, \quad 2^{p-1}CM(t)^{p-1}+2^{q-1}CM(t)^{q-1}<1, 
\end{equation*} 
then as the argument in \cite[Proposition 2.1]{i1}, we find from \eqref{3.2} 
that 
\[
M(t)\leq 2C_1,\quad \forall t\in[0, T_{\rm max}). 
\] 
We have that 
\[
\|Du(t)\|_{2}\leq C(1+t)^{-k}, \quad \|Dv(t)\|_{2}\leq C(1+t)^{-j},\quad
\forall t\in[0, T_{\rm max}), 
\]
which imply $T_{\rm max}=\infty$, the solution of \eqref{1.1}
exists globally in time.

Now, we prove \eqref{3.2}. From \eqref{2.2}, we deduce that
\begin{equation}\label{3.3}
\|Du(t)\|_{2}\leq\|Du_L(t)\|_{2}+\int_{0}^{t}\|DS(t-\tau)
J_{0|\tau}^{\alpha}(|v|^p)(\tau)\|_{2}d\tau.
\end{equation} 
Applying Lemma \ref{linear estimate} with $m=1$ and $n=1$, we see
that 
\begin{equation}\label{3.4} 
\|Du_L(t)\|_{2}\leq CI_{0,u}(1+t)^{-3/4},
\end{equation} 
and 
\begin{equation}\label{3.5} 
\begin{split}
 &\int_{0}^{t}\|DS(t-\tau)J_{0|\tau}^{\alpha}(|v|^p)(\tau)\|_{2}d\tau\\
 &\leq  C\int_{0}^{t}(1+t-\tau)^{-3/4}(\|J_{0|\tau}^{\alpha}(|v|^p)(\tau)\|_{1}
 +\|J_{0|\tau}^{\alpha}(|v|^p)(\tau)\|_{2})d\tau\\
 &\leq  C\int_{0}^{t}(1+t-\tau)^{-3/4}
 \int_{0}^{\tau}(\tau-s)^{-(1-\alpha)}(\|v(s)\|_{p}^p+\|v(s)\|_{2p}^p)dsd\tau.
 \end{split}
\end{equation} 
Next, we transform the $L^{p}$ norm into a weighted $L^{2p}$ norm.
Making use of the Cauchy inequality and the fact 
$\psi(t, x)\geq |x|^2/4(t+K)$ for
$x\in B(\tau+K)$, we have 
\begin{equation}\label{3.6} 
\begin{split} 
\|v(\tau,\cdot)\|_{p}^{p}
&=\int_{B(\tau+K)}|v(\tau, x)|^{p}dx\\
&\leq\Big(\int_{B(\tau+K)}e^{-2p\delta\psi(\tau,x)}dx\Big)^{1/2}
 \Big(\int_{B(\tau+K)}e^{2p\delta\psi(\tau, x)}|v(\tau, x)|^{2p}dx\Big)^{1/2}\\
&\leq\Big(\int_{B(\tau+K)}e^{-p\delta|x|^2/2(\tau+K)}dx\Big)^{1/2}
 \|e^{\delta\psi(\tau,  \cdot)}v(\tau)\|_{2p}^{p}\\
&\leq C_{K, \delta}(\tau+K)^{1/4}\|e^{\delta\psi(\tau, \cdot)}v(\tau)\|_{2p}^{p},\\
\end{split} 
\end{equation} 
where $\delta>0$. Obviously, 
\begin{equation}\label{3.7}
\|v(\tau, \cdot)\|_{2p}^{p}\leq (\tau+K)^{1/4}\|e^{\delta\psi(\tau,
\cdot)}v(\tau)\|_{2p}^{p}. 
\end{equation} 
From \eqref{3.5}-\eqref{3.7}, we obtain
\begin{equation}\label{3.8} 
\begin{split}
& \int_{0}^{t}\|DS(t-\tau)J_{0|\tau}^{\alpha}(|v|^p)(\tau)\|_{2}d\tau\\
&\leq C\sup_{[0,t)}\left[(1+\tau)^{\beta_1}\|e^{\delta\psi(\tau,
 \cdot)}v(\tau)\|_{2p}\right]^{p}\\
&\quad \times  \int_{0}^{t}(1+t-\tau)^{-3/4}
 \int_{0}^{\tau}(\tau-s)^{-(1-\alpha)}(1+s)^{-(p\beta_1-1/4)}dsd\tau.
\end{split} 
\end{equation} 
Taking $\beta_1=(\alpha+1)(q+1)/(pq-1)-1/4p$ such that
$1/4<p\beta_1<5/4$ and applying Lemma \ref{int}, we have
\begin{equation}\label{3.9}
\begin{aligned}
&\int_{0}^{t}\|DS(t-\tau)J_{0|\tau}^{\alpha}(|v|^p)(\tau)\|_{2}d\tau \\
&\leq  C(1+t)^{-k}\sup_{[0,t)}\big[(1+\tau)^{\beta_1}\|e^{\delta\psi(\tau,
\cdot)}v(\tau)\|_{2p}\big]^{p},
\end{aligned}
\end{equation} 
where $k=(\alpha+1)(p+1)/(pq-1)+1/4$. To estimate the weighted
$L^{2p}$ norm, we use Lemma \ref{weighted inequality} with $r=2p$ 
and $n=1$,
\begin{equation}\label{3.10} 
\begin{split} 
\|e^{\delta\psi(\tau, \cdot)}v(\tau)\|_{2p}
&\leq C(1+\tau)^{(1-\theta(2p))/2}\|v_{x}\|_{2}^{1-\delta}\|e^{\psi(\tau,
\cdot)}v_{x}\|_{2}^{\delta}\\
&\leq  C(1+\tau)^{(1-\theta(2p))/2}\|Dv\|_{2}.
\end{split} 
\end{equation} 
From \eqref{3.9}, \eqref{3.10} and \eqref{3.1}, we derive
that 
\begin{equation}\label{3.11}
\int_{0}^{t}\|DS(t-\tau)J_{0|\tau}^{\alpha}(|v|^p)(\tau)\|_{2}d\tau
\leq  C(1+t)^{-k}\sup_{[0,t)}
 \big[(1+\tau)^{\beta_1+(1-\theta(2p))/2-j}M(t)\big]^{p}.
\end{equation} 
Multiplying \eqref{3.3} by $(1+t)^{k}$ and from \eqref{3.4},
\eqref{3.11}, we obtain that 
\begin{equation}\label{3.12} 
(1+t)^k\|Du(t)\|_{2}\leq CI_{0,u}(1+t)^{k-3/4}+C\sup_{[0,t)}
\left[(1+\tau)^{\beta_1+(1-\theta(2p))/2-j}M(\tau)\right]^{p}.
\end{equation} 
Similarly, we can deduce that 
\begin{equation}\label{3.13}
(1+t)^j\|Dv(t)\|_{2}\leq
CI_{0,v}(1+t)^{j-3/4}+C\sup_{[0,t)}
\big[(1+\tau)^{\beta_2+(1-\theta(2q))/2-k}M(\tau)\big]^{q},
\end{equation} 
where we choose $\beta_2=(\alpha+1)(p+1)/(pq-1)-1/4q$ such that
$1/4<q\beta_2<5/4$ and get
 $j=(\alpha+1)(q+1)/(pq-1)+1/4$. It can be easily
checked that 
\begin{equation}\label{3.14} 
\begin{gathered}
\beta_1+(1-\theta(2p))/2-j=0,\\ 
\beta_2+(1-\theta(2q))/2-k=0, 
\end{gathered}.
\end{equation} 
and 
\[
 k-3/4<F(p, q, \alpha)-\alpha,\quad j-3/4<F(p, q,\alpha)-\alpha. 
\]
 As $q\beta_2<5/4$ and $p\beta_1<5/4$ imply $F(p, q,\alpha)<0$, we have 
\begin{equation}\label{3.15} 
k-3/4<-\alpha,\quad j-3/4<-\alpha.
\end{equation} 
Combining \eqref{3.12}-\eqref{3.15}, we have \eqref{3.2}. Theorem
\ref{global} is proved. 


\section{Proof of Theorem \ref{blow up}} 

In this section we prove the theorem by contraction.
 In the following, we assume that $(u,v)$ is a global weak solution of \eqref{1.1}.

Set $\varphi_1(x):=\phi^l(|x|/R), l\gg1$ with the cut-off function $\phi(r)$
satisfying 
\begin{gather} 
\phi(r)=\begin{cases}
 1,  & 0\leq r\leq 1,\\
 0,  & r\geq 2,
 \end{cases} \label{4.1}\\
0\leq \phi(r)\leq 1,\quad  |\phi'(r)|\leq C/r,\quad
|\phi''(r)|\leq C/r,\label{4.2}
\end{gather} 
and $\varphi_2(t):=(1-t/T)_{+}^{\eta}$, with $\eta\gg1$. The supports
of $\varphi_1$ and $(\varphi_1)_{xx}$ are denoted as $B_{2R}$ and 
$B_{2R}\setminus B_{R}$ respectively, where 
\[ 
B_{2R}=\{x\in\mathbb{R}:|x|\leq 2R\},\quad
B_{2R}\setminus B_{R}=\{x\in\mathbb{R}:R\leq|x|\leq 2R\}.
\]

Denote \begin{equation}\label{4.3}
\varphi(t,x):=\varphi_1(x)(D_{t|T}^{\alpha}\varphi_2)(t). 
\end{equation}
From \eqref{4.1}-\eqref{4.3} and the Proposition \ref{frac1}-\ref{frac3}, 
we obtain
\begin{equation}\label{4.4} 
\begin{gathered}
\begin{aligned}
&\int_{0}^{T}\int_{B_{2R}}|v(t,x)|^p\varphi (x,t)\,dx\,dt\\
&+T^{-\alpha}\int_{B_{2R}}(u_1(x)+u_0(x))\varphi_1(x)dx
 +T^{-\alpha-1}\int_{B_{2R}}u_0(x)\varphi_1(x)dx\\
&=\int_{0}^{T}\int_{B_{2R}}u\varphi_1(D_{t|T}^{\alpha+2}\varphi_2(t)
 +D_{t|T}^{\alpha+1}\varphi_2(t))\,dx\,dt \\
&\quad -\int_{0}^{T}\int_{B_{2R}\setminus
 B_{R}}u(\varphi_1)_{xx}(D_{t|T}^{\alpha}\varphi_2)(t) \,dx\,dt,
\end{aligned}\\
\begin{aligned}
&\int_{0}^{T}\int_{B_{2R}}|u(t,x)|^q\varphi(x,t) \,dx\,dt
 +T^{-\alpha}\int_{B_{2R}}(v_1(x)+v_0(x))\varphi_1(x)dx\\
&\quad +T^{-\alpha-1}\int_{B_{2R}}v_0(x)\varphi_1(x)dx\\
&=\int_{0}^{T}\int_{B_{2R}}v\varphi_1(x)(D_{t|T}^{\alpha+2}\varphi_2(t)
 +D_{t|T}^{\alpha+1}\varphi_2(t))\,dx\,dt \\
&\quad -\int_{0}^{T}\int_{B_{2R}\setminus  B_{R}}v(\varphi_1)_{xx}
(D_{t|T}^{\alpha}\varphi_2)(t) \,dx\,dt.
\end{aligned}
\end{gathered}
\end{equation} 
Set 
\begin{gather}\label{4.5}
J_{p}=\int_{0}^{t}\int_{B_{2R}}|v(t,x)|^{p}\varphi(t,x)\,dx\,dt ,\\
\label{4.6}
 J_{q}=\int_{0}^{t}\int_{B_{2R}}|u(t,x)|^{q}\varphi(t,x)\,dx\,dt.
\end{gather} 
From \eqref{1.10} and \eqref{4.4}, we have 
\begin{equation}\label{4.7}
\begin{split} 
J_{p}
&\leq C\int_{0}^{T}\int_{B_{2R}}|u|\varphi_1
 (D_{t|T}^{2+\alpha}\varphi_2(t)+D_{t|T}^{1+\alpha}\varphi_2(t))\,dx\,dt\\
&\quad +C\int_{0}^{T}\int_{B_{2R}\setminus
B_{R}}|u(\varphi_1)_{xx}|(D_{t|T}^{\alpha}\varphi_2)(t)\,dx\,dt=I_1+I_2.
\end{split}
\end{equation} 
Applying Holder's inequality with exponents $q$ and $q/(q-1)$, we can
achieve that 
\begin{equation}\label{4.8} 
\begin{split} 
I_1&\leq C\Big(\int_{0}^{T}\int_{B_{2R}}|u(t,x)|^q\varphi(t,x)\,dx\,dt\Big)^{1/q}\\
&\quad\times  \Big(\int_{0}^{T}\int_{B_{2R}}\varphi_1\varphi_2^{-\frac{1}{q-1}}
 (D_{t|T}^{2+\alpha}\varphi_2+D_{t|T}^{1+\alpha}\varphi_2)^{q'}\,dx\,dt
 \Big)^{1/q'}\\
&\leq C(T^{-(2+\alpha)+1/q'}+T^{-(1+\alpha)+1/q'})R^{1/q'}J_q^{1/q},
\end{split} 
\end{equation}
 with $q'=q/(q-1)$. In the same way, $I_2$ can be
estimated by 
\begin{equation}\label{4.9} 
\begin{split} 
I_2&\leq C\Big(\int_{0}^{T}\int_{B_{2R}}|u(t,x)|^q\varphi(t,x)\,dx\,dt\Big)^{1/q}\\
 &\times\Big(\int_{0}^{T}\int_{B_{2R}\setminus
 B_{R}}\varphi_1^{1-2q'/l}\varphi_2^{-\frac{1}{q-1}}
 (|\Delta \varphi_1|^{q'}+|\nabla
 \varphi_1|^{2q'})(D_{t|T}^{2+\alpha}\varphi_2)^{q'}\,dx\,dt\Big)^{1/q'}\\
 &\leq CT^{-\alpha+1/q'}R^{-2+1/q'}J_q^{1/q}.
\end{split} 
\end{equation} 
From \eqref{4.8} and \eqref{4.9}, we deduce that
\begin{equation}\label{4.10}
 J_{p}\leq CS(q', T, R)J_q^{1/q}, 
\end{equation}
where 
\begin{equation}\label{4.11} 
S(q',T, R)=T^{-(2+\alpha)+1/q'}R^{1/q'}+T^{-(1+\alpha)+1/q'}R^{1/q'}
+T^{-\alpha+1/q'}R^{-2+1/q'}.
\end{equation} 
Similarly, we can prove that 
\begin{equation}\label{4.12} 
J_{q}\leq CS(p', T, R)J_p^{\frac{1}{p}}, 
\end{equation} 
where 
\begin{equation}\label{4.13}
S(p',T,R)=T^{-(2+\alpha)+1/p'}R^{1/p'}+T^{-(1+\alpha)+1/p'}R^{1/p'}
+T^{-\alpha+1/p'}R^{-2+1/p'}.
\end{equation} 
This yields 
\begin{equation}\label{4.14} 
J_{p}\leq CS(q', T, R)S(p',T, R)^{1/q}J_{p}^{\frac{1}{pq}}. 
\end{equation} 
Taking $R=\sqrt{T}$ in \eqref{4.10}-\eqref{4.14}, and by Young's inequality, 
we have
\begin{equation}\label{4.15} 
J_{p}\leq \frac{1}{2}J_{p}+CT^{1/2-\alpha-(1+p)(1+\alpha)/(pq-1)}. 
\end{equation}
 Next, we divide into two cases to discuss the estimate of \eqref{4.15}.
\smallskip

\noindent\textbf{Case i.} 
$F(p, q, \alpha)>0$. this implies the exponent of $T$ in \eqref{4.15} is
negative. Letting $T\to \infty$ in \eqref{4.15}, we derive that
\begin{equation}\label{4.16} 
\int_{0}^{\infty}\int_{-\infty}^{+\infty}|u(t,x)|^{p}\,dx\,dt=0, 
\end{equation} 
which implies $u(t, x)=0$ for all $t$ and
$x\in\mathbb{R}$  a.e.. This is a contradiction to \eqref{1.10}.
\smallskip

\noindent\textbf{Case ii}.
$F(p, q, \alpha)=0$, we have 
\begin{equation}\label{4.17}
\lim_{T\to\infty}J_p=\int_{0}^{\infty}\int_{\mathbb{R}}|u(x,t)|^{p}\,dx\,dt
\leq D. 
\end{equation} 
It follows from \eqref{4.12} that for any $\epsilon>0$ there
exists $T_1$, such that 
\begin{equation}\label{4.18} 
J_{q}\leq C\epsilon^{1/p} T^{-(1+\alpha)+3(p-1)/2p},\quad T>T_1, 
\end{equation} 
where $C$ is independent of $\epsilon$. Combining \eqref{4.10} 
and \eqref{4.18}, we get that 
\begin{equation}
J_{p}\leq C\epsilon^{1/(pq)}, 
\end{equation} 
and the constant $C$ is also independent of $\epsilon$. 
The arbitrary of $\epsilon$ yields a contradiction with
\eqref{1.10}. This completes the proof of Theorem \ref{blow up}. 

\subsection*{Acknowledgments}
This work is supported by the National
Science Foundation of China (Nos. 61174082, 61473180, 11401351).

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\end{document}