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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 149, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/149\hfil Existence and nonexistence of solutions]
{Existence and nonexistence of solutions of asymptotically linear
Klein-Gordon equation}

\author[P. C. Carri\~ao, R. Lehrer, A. Vicente \hfil EJDE-2016/149\hfilneg]
{Paulo C. Carri\~ao, Raquel Lehrer, Andr\'e Vicente}

\address{Paulo C. Carri\~ao \newline
Universidade Federal de Minas Gerais - ICEX - Matem\'atica\\
CEP: 31270-901, Belo Horizonte, MG, Brazil}
\email{pauloceca@gmail.com}


\address{Raquel Lehrer \newline
Universidade Estadual do Oeste do Paran\'a - CCET,
Rua Universit\'aria, 2069, Jd. Universit\'ario\\
CEP: 85819-110, Cascavel, PR, Brazil}
\email{rlehrer@gmail.com}

\address{Andr\'e Vicente \newline
Universidade Estadual do Oeste do Paran\'a - CCET,
Rua Universit\'aria, 2069, Jd. Universit\'ario\\
CEP: 85819-110, Cascavel, PR, Brazil}
\email{andre.vicente@unioeste.br}

\thanks{Submitted December 15, 2015. Published June 17, 2016.}
\subjclass[2010]{35L15, 35L70, 35B40, 35B44, 35A01, 35A15}
\keywords{Klein-Gordon equation; blow up of solution; asymptotically linear; 
\hfill\break\indent Pohozaev manifold}

\begin{abstract}
 In this article we study a nonlinear Klein-Gordon equation when the
 nonlinear term  asymptotically linear at infinity.
 We used the Pohozaev manifold to separate a subspace of $H^1(\mathbb{R}^N)$
 on a global existence region and on a blow up region.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks


\section{Introduction}

We consider the  Cauchy problem for the nonlinear Klein-Gordon equation
\begin{equation}\label{1.1}
	\begin{gathered}
 u_{tt}-\Delta u +\lambda u = a(x)f(u) \quad \text{in }
 \mathbb{R}^N\times (0,T),\\
 u(x,0)=u_0(x),\quad u_t(x,0)=u_1(x)	\quad
\text{in }\mathbb{R}^N,
	\end{gathered}
\end{equation}
where $N\geq 3$; ${\Delta =\sum_{i=1}^n\frac{\partial^2}{\partial
x_i^2}}$ is the Laplace operator; $\lambda$ is a positive constant;
 and $a:\mathbb{R}\to \mathbb{R}^+$, $f:\mathbb{R}\to\mathbb{R}^+$, 
$u_0,u_1:\mathbb{R}^N\to\mathbb{R}$ are given functions.

The motivation for studying problem \eqref{1.1} was the paper of  Kaitai and 
 Quande \cite{Kaitai-Quande}, where they studied the case $a(x)\equiv 1$ 
and $f(u)=u^2+u^3$. In that case the equation \eqref{1.1}$_1$ is associated 
with the study of crystals dislocation. The authors proved a result of global 
existence and finite time blow up to the problem when $N\leq 3$.

It is know that the evolution problem \eqref{1.1} has a strong connection 
with the elliptic problem:
\begin{equation}\label{1.2}
 -\Delta u +\lambda u = a(x)f(u) \quad \text{in }\mathbb{R}^N.
\end{equation}
In fact, defining the functional $I:H^1(\mathbb{R}^N)\to\mathbb{R}$ by
$$
	I(u)=\frac{1}{2}\int_{\mathbb{R}^N} |\nabla u|^2\,dx
+ \frac{\lambda}{2} \int_{\mathbb{R}^N} u^2\,dx - \int_{\mathbb{R}^N} a(x)F(u)dx,
$$
where $F(\xi)=\int_0^{\xi}f(s)ds$, the critical values of $I$ are the weak 
solutions of \eqref{1.2}. When the known Ambrosetti-Rabinowitz condition  
(see \cite{Ambrosetti-Rabinowitz})
$$
	0<\theta F(\xi)<\xi f(\xi),\quad
\text{for all }\xi \in\mathbb{R}\backslash\{0\},
$$
for some $\theta>2$ holds, it is possible to control the projection of 
$u\in H^1(\mathbb{R}^N)$ over the Nehari manifold
$$
	\mathcal{N}=\{u\in H^1(\mathbb{R}^N)\backslash\{0\};  I'(u)u=0\},
$$
where $I'(u)u$ is the Gateaux derivative of $u$ applied on $u$, and to 
prove that the level
\begin{equation}\label{1.3}
	d=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}I(\gamma(t)),
\end{equation}
where
$$
	\Gamma=\{\gamma\in C([0,1];H^1(\mathbb{R}^N));\:\gamma(0)=0 
\text{ and }\gamma(1)=e\},
$$
here $e\in H^1(\mathbb{R}^N); \|e\|>r>0$ and
 $\inf_{\|u\|=r}I(u)>I(0)>I(e)$), is reached over $\mathcal{N}$, i e,
$$
	d=\inf_{u\in\mathcal{N}}I(u).
$$
The number $d$ is a critical value of $I$ and is called of Mountain pass level. 
Details can be found in Willem \cite{Willem}.

Therefore, it is possible defining the energy of \eqref{1.1} by
\begin{equation}\label{1.4}
	E(t)=\frac{1}{2}\int_{\mathbb{R}^N}|u_t|^2dx+ I(u),
\end{equation}
with appropriate assumptions on $f$, 
to prove an existence/non\-existence result when the energy at $t=0$, 
i e, $E(0)$ is bellow of the mountain pass level. In many papers the 
Nehari manifold has an important role because it allows to ``separate" 
one $H^1(\mathbb{R}^N)$ subset on an existence region and on a nonexistence 
region, see 
\cite{Alves-Cavalcanti,Alves-Cavalcanti-Domingos Cavalcanti-Rammaha-Toundykov,
Gan-Guo-Zhang,Levine-Todorova,Todorova} and references therein. 
See also a more recent work of Wang \cite{Wang}.

On the other hand, when we do not have the Ambrosetti-Rabinowitz condition 
the work to get blow up results involving \eqref{1.1} can be hard. 
It holds when $f$ is, for example, defined by $f(u)=\frac{u^3}{1+u^2}$. 
In the elliptic context, this function is a prototype of a class of 
nonlinearity so called {\it asymptotic linear at infinity} which was recently 
solved, in a more general context, by Lehrer and Maia \cite{Lehrer-Maia}. 
The presence of $a(x)$ let the problem {\it nonautonomous} and it gives 
some technical difficulties. Quickly speaking, the authors showed that 
the mountain pass level is not attained. To solve the problem they 
worked with an alternative to the use of the Nehari manifold, namely, 
it was used the Pohozaev manifold:
$$
\mathcal{P}= \{u\in H^1(\mathbb{R}^N)\backslash\{0\};J(u)=0\},
$$
where $J: H^1(\mathbb{R}^N)\to\mathbb{R}$ is given by
\begin{align*}
	J(u)&=\frac{N-2}{2}\int_{\mathbb{R}^N} |\nabla u|^2\,dx
+ \frac{N\lambda}{2} \int_{\mathbb{R}^N} u^2\,dx
 -N \int_{\mathbb{R}^N} a(x)F(u)\,dx\\
&\quad  - \int_{\mathbb{R}^N} \nabla a(x)\cdot x F(u)dx.
\end{align*}
The key of their paper was to show that the functional $I$ has a 
critical value, above the mountain pass level, in a subset of $\mathcal{P}$.

Let $a:\mathbb{R}^N\to\mathbb{R}$ be a radial function satisfying the 
following assumptions
\begin{itemize}
\item[(A1)] $a \in C^2(\mathbb{R}^N,\mathbb{R}^{+})$, with 
$\inf_{x\in \mathbb{R}^N}a(x)> 0$;

\item[(A2)] $\lim_{|x|\to \infty}a(x) = a_{\infty} > \lambda$;

\item[(A3)] $\nabla a(x)\cdot x \geq 0$,  for all  
$ x \in \mathbb{R}^N$, with the strict inequality holding on a subset 
of  positive Lebesgue measure of $\mathbb{R}^N$;

\item[(A4)] $a(x) + \frac{\nabla a(x)\cdot x}{N} < a_{\infty}$, for all 
$ x \in \mathbb{R}^N$;

\item[(A5)] $\nabla a(x)\cdot x + \frac{x\cdot H(x)\cdot x}{N}\geq 0$,
for all $ x\in \mathbb{R}^N$, where $H$ represents the Hessian matrix of
the function $a$.
\end{itemize}

On the nonlinearity $f$ we assume:
\begin{itemize}
\item[(A6)] $f\in C(\mathbb{R}^{+},\mathbb{R}^{+})$, 
$\lim_{s\to 0}\frac{f(s)}{s} = 0$;

\item[(A7)] $\lim_{s \to \infty} \frac{f(s)}{s} = 1$;

\item[(A8)] if $F(s)=\int_{0}^{s}f(t)dt$
and $Q(s) = \frac{1}{2}f(s)s - F(s)$, then there exists a constant 
$D\geq 1$ such that
\begin{gather*}
0 < Q(s)\leq D \;Q(t), \quad \text{for all }   0 < s \leq t,\\
\lim_{s\to \infty}Q(s) = + \infty.
\end{gather*}
\end{itemize}

We observe that the assumptions (A1)--(A8) are the same of 
\cite{Lehrer-Maia} and the condition (A8) was introduced by 
Jeanjean and Tanaka \cite{Jeanjean-Tanaka}. 
Note that (A8) is more general than the usual assumption that
 $f(s)/s$ begin an increasing function of $s>0$. 
In particular, if $f$ is differentiable, then $f(s)/s$ is increasing if 
and only if (A8) holds with $D=1$. We extend $f$ to $\mathbb{R}^-$ by 
$f(s)=0$ if $s<0$.

Conditions (A6) and (A7) imply that, given $\varepsilon > 0$ and 
$2 \leq p \leq p^*:=\frac{2N}{N-2}$, there exists a positive constant 
$C= C(\varepsilon, p) $ such that for all $s$ in $\mathbb{R}$
\begin{equation}
|F(s)|\leq \frac{\varepsilon}{2}|s|^2 + C |s|^p \;.\label{1.5}
\end{equation}

Therefore \eqref{1.1} is asymptotic linear at infinity and nonautonomous. 
As we do not have the Ambrosetti-Rabinowitz condition then the Nehari 
manifold is not appropriate, therefore we used the Pohozaev manifold to 
find an existence and a nonexistence region. Precisely, defining
\begin{gather*}
	W_1=\{u\in H^1_{\rm rad}(\mathbb{R}^N)\backslash \{0\};
I(u)<c\text{ and }J(u)>0\}\cup \{0\},\\
W_2=\{u\in H^1_{\rm rad}(\mathbb{R}^N)\backslash \{0\};
 I(u)<c\text{ and }J(u)<0\},
\end{gather*}
where $c$ is the mountain pass level and will be defined posteriorly, 
we proved that when the initial data is taken in $W_1$, the problem 
\eqref{1.1} has a global solution which there exist for all $t\geq 0$.
 Moreover, if the initial data is in $W_2$, then the solution blow up 
(in finite or infinite time).

We also would like to cite the classical paper of Shatah \cite{Shatah}, 
where $a\equiv 1$, $f$ satisfies the Berestycki-Lions assumptions and 
$N\geq 3$. In this case the author also used the Pohozaev manifold. 
The work of Shatah was extended to $N=2$ by Jeanjean and 
Le Coz \cite{Jeanjean-LeCoz}.

The goal of our paper is to prove an existence/nonexistence result to 
\eqref{1.1} when $a$ and $f$ satisfy (A1)-(A5) and (A6)-(A8), respectively. 
This work extends, in a sence, the results of \cite{Kaitai-Quande,Shatah} 
to an other class of nonlinearities. Our paper is organized as follows: 
in Section 2 we give the notations, the preliminaries and we stablish a 
linear existence theorem, which is analogous to  
Serrin, Todorova and Vitillaro \cite[Theorem 3]{Serrin-Todorova-Vitillaro}. 
In Section 3 we prove the main result.

\section{Preliminaries}


The norms in $L^2(\mathbb{R}^N)$ and $H^1(\mathbb{R}^N)$ are
denoted, respectively, by
$$
  \|u\|_2 = \Big(\int_{\mathbb{R}^N} |u(x)|^2\,dx \Big)^{1/2},\quad
  \|u\|_{\lambda}=\Big(\lambda\int_{\mathbb{R}^N} |u(x)|^2\,dx
+\int_{\mathbb{R}^N}\nabla u\cdot \nabla u \,dx\Big)^{1/2},
$$
here $\nabla =(\frac{\partial}{\partial
x_1},\ldots,\frac{\partial}{\partial x_n})$ is the gradient operator
in spatial variable and $\cdot$ is the usual scalar product in
$\mathbb{R}^N$.

To prove our blow up result we need two technical lemmas which are
 connected with the autonomous elliptic problem:
\begin{equation}\label{2.1}
 -\Delta u +\lambda u = a_{\infty}f(u) \quad \text{in }\mathbb{R}^N,
\end{equation}
where the constant $a_{\infty}$ was given in (A2). Associated to \eqref{2.1} 
we have the Pohozaev manifold
$$
	\mathcal{P}_{\infty} = \{ u\in H^1(\mathbb{R}^N)\backslash \{ 0\} ;
 J_{\infty}(u) = 0\},
$$
where
$$
	J_{\infty}(u) = \frac{(N-2)}{2}\int_{\mathbb{R}^N} |\nabla u|^2dx 
+ \frac{\lambda N}{2}\int_{\mathbb{R}^N}u^2\,dx
- N\int_{\mathbb{R}^N} a_{\infty}F(u)dx.
$$
	
Define the functional $I_\infty$ associated with \eqref{2.1} by
$$
	I_\infty(u)=\frac{1}{2}\int_{\mathbb{R}^N} |\nabla u|^2 + \lambda u^2 
- \int_{\mathbb{R}^N} a_\infty F(u)dx,
$$
and  the mountain pass level
$$
	c_\infty = \min_{\gamma\in \Gamma_\infty}\max_{0\leq t\leq 1}I_{\infty}
(\gamma(t)),
$$
where the set of paths is given by
$$
	\Gamma_\infty=\{\gamma\in C([0,1],H^1(\mathbb{R}^N))
| \gamma(0)=0, I_{\infty}(\gamma(1))<0\}.
$$
We have the following result which show that, in the autonomous case, 
the mountain pass level is the minimum of $I_{\infty}$ over $\mathcal{P}_{\infty}$.

\begin{lemma}\label{Lemma2.1}
	Let $\varphi_{\infty}$ be the ground state solution of the autonomous 
elliptic problem. Then
	$$
I_{\infty}(\varphi_{\infty})=c_{\infty}
=\min_{v\in \mathcal{P}_{\infty}}I_{\infty}(v)>0.
$$
\end{lemma}

For a proof of the above lemma, see Jeanjean and Tanaka 
\cite[Lemma 3.1]{Jeanjean-Tanaka1}.

Let $I$, $E$, $J$, $W_1$ and $W_2$ as in the preview section and $c$ 
the mountain pass level, associated to \eqref{1.1}, defined in \eqref{1.3}, namely,
$$
	c=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}I(\gamma(t)).
$$
The following lemma establishes that the mountain pass level of the 
nonautonomous problem is attained in the Pohozaev manifold and 
it is the same level of the autonomous problem.

\begin{lemma}\label{Lemma2.2}
	It holds that
	$$
 c=c_{\infty}=\inf_{u\in\mathcal{P}}I(u).
	$$
\end{lemma}

For a proof of the above lemma, see \cite[lemmas 3.13 and 4.2]{Lehrer-Maia}.
We will need one more characterization to the level $c$ which is given
 by the next lemma.

\begin{lemma}\label{Lemma2.3}
	We have
	$$
c=\min\{T_{\infty}(v);\:J_{\infty}(v)\leq 0\},
$$
where $T_{\infty}(v)=\frac{1}{N}\int_{\mathbb{R}^N}|\nabla v|^2\,dx$.
\end{lemma}

\begin{proof}
 We observe that, for all $u\in H^1(\mathbb{R}^N)$,
$$
	I_{\infty} (u) = T_{\infty} (u) +\frac{1}{N}J_{\infty} (u).
$$
Let $v\in H^1(\mathbb{R}^N)$ be such that $J_{\infty}(v)\leq 0$. 
If $J_{\infty}(v)= 0$, then $v\in \mathcal{P}_{\infty}$ and 
$I_{\infty} (u) = T_{\infty} (u)$. Therefore
\begin{equation}\label{2.2}
	\inf_{J_{\infty}(v)=0} T_{\infty} (v)
=\inf_{J_{\infty}(v)=0} I_{\infty} (v)
=\inf_{v\in \mathcal{P}_{\infty}} I_{\infty} (v)=c,
\end{equation}
where in the last equality we used Lemmas \ref{Lemma2.1} and 
\ref{Lemma2.2}. If $J_{\infty}(v)<0$, then for $\beta>0$ define 
$v_{\beta}(x)=v(\frac{x}{\beta})$. Thus
\begin{align*}
J_{\infty}(v_{\beta})
&= \beta^{N-2}\Big[ \frac{N-2}{2}
\int_{\mathbb{R}^N}|\nabla v(x)|^2dx\\
&\quad -N\beta^2\Big(\int_{\mathbb{R}^N}a_{\infty}F(v(x))dx
+\frac{\lambda N}{2}\int_{\mathbb{R}^N}|v(x)|^2dx\Big)\Big].
\end{align*}
Note that for $\beta=1$ we have $J_{\infty}(v_{1})=J_{\infty}(v)<0$ 
and for $\beta>0$ sufficiently small $J_{\infty}(v_{\beta})>0$. 
Therefore, there exists $\beta_0\in(0,1)$ such that $J_{\infty}(v_{\beta_0})=0$. 
Thus, we infer that
\begin{equation} \label{2.3}
\begin{aligned}
I_{\infty}(v_{\beta_0})
&=T_{\infty}(v_{\beta_0})
= \frac{\beta_0^{N-2}}{N}\int_{\mathbb{R}^N}|\nabla v(x)|^2dx\\
&< \frac{1}{N}\int_{\mathbb{R}^N}|\nabla v(x)|^2dx = T_{\infty}(v).
\end{aligned}
\end{equation}
Taking into account \eqref{2.2} and \eqref{2.3} we  complete the proof.
\end{proof}

To prove the existence of solution we will need a theorem that gives 
us the existence of solution to a linear problem. The next lemma
establishes this result and its proof is analogous to 
Serrin, Todorova and Vitillaro \cite[Theorem 3]{Serrin-Todorova-Vitillaro}.

\begin{lemma}\label{Lemma2.4}
	Let $u_0\in H^1(\mathbb{R}^N)$, $u_1\in L^2(\mathbb{R}^N)$ and 
$g\in L^2(\mathbb{R}^N\times (0,T))$  be given functions. 
Then, for all $T>0$ there exists a unique   weak solution, 
$u:\mathbb{R}^N\times (0,T)\to\mathbb{R}$, of the linear problem
\begin{equation}
\begin{gathered}
 u_{tt}-\Delta u +\lambda u = g(x,t) \quad \text{in }\mathbb{R}^N\times (0,T),\\
 u(x,0)=u_0(x),\:\: u_t(x,0)=u_1(x)	\quad	\text{in }\mathbb{R}^N,
	\end{gathered}	
\end{equation}
in the class
$$
u\in C([0,T];H^1(\mathbb{R}^N)),\quad u_t\in C([0,T];L^2(\mathbb{R}^N)).
$$	
Moreover, the energy identity satisfies
\begin{equation}\label{2.5}
	E_l(t)-E_l(s)=\int_s^t\int_{\mathbb{R}^N}g(x,t)u_t(x,t)\,dxdt,
\end{equation}
for all $0\leq s\leq t$, where
$$
E_l(t)=\frac{1}{2}\left[\int_{\mathbb{R}^N}|u_t(x,t)|^2dx
+\int_{\mathbb{R}^N}|\nabla u(x,t)|^2dx
+\lambda\int_{\mathbb{R}^N}|u(x,t)|^2dx\right].
$$
\end{lemma}

\section{Main results}

\begin{theorem}[Local Existence]\label{Theorem3.1}
	Suppose that assumptions {\rm (A1)-(A8) } hold. Then for each set 
of initial conditions $(u_{0},u_{1})\in H^1(\mathbb{R}^N)\times L^2(\mathbb{R}^N)$ 
there exists a $T>0$ such that the problem \eqref{1.1} has a unique weak 
solution, $u:\mathbb{R}^N\times (0,T)\to\mathbb{R}$, in the class
	$$
 u\in C([0,T]; H^1(\mathbb{R}^N))\cap C^1([0,T]; L^2(\mathbb{R}^N)).
	$$
\end{theorem}

\begin{proof} 
Define de set
$$
	X_{T}=\{u\in C([0,T];H^1(\mathbb{R}^N))\cap C^1([0,T];L^2(\mathbb{R}^N))\}
$$
endowed with the norm
$$
	\|u\|^2=\|u\|_{L^{\infty}(0,T;H^1(\mathbb{R}^N))}^2+\|u_t\|_{L^{\infty}(0,T;L^2(\mathbb{R}^N))}^2
$$
and, for each $\alpha>0$, we consider the set
$$
	X_{T,\alpha}=\{u\in X_T; \|u\|\leq \alpha,\:u(0)=u_0\text{ and }u_t(0)=u_1\}.
$$
Define $ g:	X_{T,\alpha} \to L^2 (\mathbb{R}^N\times (0,T))$ by
$g(v)=f(v)$.

For each $v\in X_{T,\alpha}$ let $u$ be the solution of
\begin{equation}\label{3.1}
	\begin{gathered}
 u_{tt}-\Delta u +\lambda u = g(v) \quad \text{in }\mathbb{R}^N\times (0,T),\\
 u(x,0)=u_0(x),\:\: u_t(x,0)=u_1(x)	\quad	\text{in }\mathbb{R}^N
	\end{gathered}	
\end{equation}
given by Lemma \ref{Lemma2.4}. Now, we  show that for $\alpha$ large enough 
and $T$ small enough $u\in X_{T,\alpha}$. The energy identity \eqref{2.5} gives us
\begin{equation}\label{3.2}
	E_l(t)=E_l(0) +\int_0^t\int_{\mathbb{R}^N}f(v)u_t\,dxd\xi.
\end{equation}
As $|f(s)|\leq Cs$, for all $s$, then
$$
	\big|\int_0^t\int_{\mathbb{R}^N}f(v)u_t\,dx\,dt\big|
\leq C\int_0^t\int_{\mathbb{R}^N}|v||u_t|\,dxd\xi
\leq C\int_0^t\|v(\xi)\|_2\|u_t(\xi)\|_2\,d\xi,
$$
but $v\in X_{T,\alpha}$, therefore
\begin{equation}\label{3.3}
	\big|\int_0^t\int_{\mathbb{R}^N}f(v)u_t\,dxdt\big|
\leq \sqrt{2}C\alpha\int_0^tE_l^{1/2}(\xi)\,d\xi.
\end{equation}
Putting \eqref{3.3} into \eqref{3.2} we obtain
\begin{equation}\label{3.4}
	\frac{E_l(t)}{2}\leq \frac{E_l(0)}{2} 
+\frac{C\alpha}{\sqrt{2}}\int_0^tE_l^{1/2}(\xi)\,d\xi.
\end{equation}
Applying Lemma A.5 in Brezis \cite[page 157]{Brezis}, we conclude that
\begin{equation}\label{3.5}
	E_l^{1/2}(t)\leq E_l^{1/2}(0)+\frac{C\alpha}{\sqrt{2}}T, \quad\text{for all }t\in[0,T].
\end{equation}
Choosing $\alpha > \sqrt{2}E_l(0)^{1/2}$ and, posteriorly, 
$T<\frac{\alpha -\sqrt{2}E_l^{1/2}(0)}{C\alpha}$ from \eqref{3.5} 
and using the definition of $E_l$ we obtain
$$
	\Big(\|u_t(t)\|_2^2+\|\nabla u(t)\|_2^2+\lambda\|u(t)\|_2^2\Big)^{1/2}\leq \alpha,
$$
i e, $u\in X_{T,\alpha}$. This allows us to define the application
$\Phi:	X_{T,\alpha} \to X_{T,\alpha}$ by
$\Phi(v)=u$.

We will prove that for $T>0$ sufficiently small $\Phi$ is a contraction. 
In fact, consider $v_1,v_2\in X_{T,\alpha}$ and denote $u_1=\Phi(v_1)$ 
and $u_2=\Phi(v_2)$. Define also $z=u_1-u_2$ the unique solution of
\begin{equation}\label{3.6}
\begin{gathered}
 z_{tt}-\Delta z +\lambda z = g(u_1)-g(u_2) \quad \text{in }\mathbb{R}^N\times (0,T),\\
 u(x,0)=0,\quad u_t(x,0)=0	\quad	\text{in }\mathbb{R}^N.
	\end{gathered}	
\end{equation}
The energy identity \eqref{2.5} gives
\begin{equation}\label{3.7}
	\frac{1}{2}\Big(\|z_t(t)\|_2^2+\|z(t)\|_{\lambda}^2\Big)
= \int_0^t\int_{\mathbb{R}^N}(f(v_1)-f(v_2))z_t\,dx\,d\xi.
\end{equation}
From the assumptions over $f$ and H\"{o}lder inequality we obtain
$$
	\int_{\mathbb{R}^N}(f(v_1)-f(v_2))z_t\,dx
\leq C\int_{\mathbb{R}^N}|v_1-v_2||z_t|dx
\leq C \|v_1(t)-v_2(t)\|_2\|z_t(t)\|_2.
$$
Combining this with \eqref{3.7} we obtain
\begin{equation}\label{3.8}
	\frac{1}{2}\Big(\|z_t(t)\|_2^2+\|z(t)\|_{\lambda}^2\Big)
\leq C\|v_1-v_2\|\Big(\|z_t(t)\|_2^2+\|z(t)\|_{\lambda}^2\Big)^{1/2}.
\end{equation}
The inequality \eqref{3.8} and  Brezis \cite[Lemma A.5]{Brezis} gives
$$
	\Big(\|z_t(t)\|_2^2+\|z(t)\|_{\lambda}^2\Big)^{1/2}\leq C\alpha T\|v_1-v_2\|,
$$
and since $z=u_1-u_2=\Phi(v_1)-\Phi(v_2)$ it follows that
$$
	\|\Phi(v_1)-\Phi(v_2)\|\leq C\alpha T\|v_1-v_2\|.
$$
Taking $T>0$ sufficiently small we conclude that $\Phi$ is a contraction. 
Therefore, $\Phi$ has a unique fixed point which is the solution of \eqref{1.1}.
\end{proof}

Let $u$ be the local solution of \eqref{1.1} given by Theorem \ref{Theorem3.1}. 
From the linear energy identity \eqref{2.5} we have the  identity
\begin{equation}\label{3.9}
	E(t)=\frac{1}{2}\|u_t(t)\|_2^2+I(u(t))=E(0),
\end{equation}
for all $t$ in the interval of existence of solution $u$, where $E$ was 
defined in \eqref{1.4}.

We  say that a subset $V$ of $H^1(\mathbb{R}^N)$ is an {\it invariant region} 
for the solution of \eqref{1.1} when if the initial data $u_0\in V$, 
then the solution of \eqref{1.1} is in $V$.

Define a subset of $H^1(\mathbb{R}^N)$:
$$
	W=\{u\in H_{\rm rad}^1(\mathbb{R}^N); I(u)<c\}.
$$

\begin{lemma}\label{Lemma3.2}
	Under the assumption $E(0)<c$, we have that $W=W_1\cup W_2$ and $W_1$, 
$W_2$ are invariant regions for the solutions of \eqref{1.1}.
\end{lemma}

\begin{proof} 
We note that if $J(u)=0$, then $u\in\mathcal{P}$. Thus 
$I(u)\geq \inf_{u\in\mathcal{P}}I(u)=c$. From this, for all $u\in W$ 
we can conclude that $J(u)\neq 0$, i.e., $u\in W_1$ or $u\in W_2$, 
thus $W=W_1\cup W_2$.

Now, let $u_0\in W_1$ and $u$ be the solution of \eqref{1.1} associated 
to $u_0$. From \eqref{3.9} and the assumption $E(0)< c$ we obtain
$$
	I(u(t))\leq \frac{1}{2}\|u_t(t)\|_2^2+I(u(t))=E(0)<c,
$$
therefore $u(t)\in W$ for all $t$ in the interval of existence of solution. 
We affirm that $u(t)\in W_1$ for all $t$ in the interval of existence 
of the solution. If it is not hold, then there exists $t_0>0$ such that 
$u(t_0)\notin W_1$, therefore, as $W=W_1\cup W_2$ we have $u(t_0)\in W_2$. 
From the definition of $W_1$ and $W_2$ there exists $t^*\in (0,t_0)$ 
such that $J(u(t^*))=0$. If $u(t^*)\neq 0$, then $u(t^*)\in\mathcal{P}$ 
and this implies
$$
	I(u(t^*))\geq \inf_{u\in\mathcal{P}}I(u)=c,
$$
therefore $u(t^*)\notin W$, which is a contradiction. 
If $u(t^*)=0$ then
\begin{equation}\label{3.10}
\lim_{t\to t^{*+}}(\|\nabla u(t)\|_2^2+\lambda\|u(t)\|_2^2)=0.
\end{equation}
and
\begin{equation}\label{3.11}
	J(u(t))<0, \text{ for all } t^*<t\leq t_0
\end{equation}
From the definition of $J$ and \eqref{3.11} we have
\begin{align*}
&\frac{N-2}{2}\int_{\mathbb{R}^N}|\nabla u|^2dx
 +\frac{N\lambda}{2}\int_{\mathbb{R}^N}| u|^2dx \\
&< N\int_{\mathbb{R}^N}a(x)F (u)dx +\int_{\mathbb{R}^N}\nabla a(x)\cdot xF (u)dx,
\end{align*}
for all $t^*<t\leq t_0$. This inequality, (A4) and \eqref{1.5} imply 
that given $\varepsilon>0$ and $2\leq p\leq 2^*$ such that
$$
	\|u(t)\|_ {\lambda}^2 \leq \frac{\varepsilon}{2}
\int_{\mathbb{R}^N}|u|^2dx+c(\varepsilon,p)\int_{\mathbb{R}^N}|u|^pdx.
$$
Thus
$$
	\|u(t)\|_ {\lambda}^2 \leq C\|u(t)\|_ {L^{p}(\mathbb{R}^N)}^p\leq C\|u(t)\|_ {\lambda}^p,
$$
consequently
$$
	\frac{1}{C}\leq \|u(t)\|_ {\lambda}^{p-2},
$$
but this and \eqref{3.10} give us a contradiction. 
Therefore $u(t)\in W_1$, for all $t$ in the interval of existence of solution.
 The proof for $W_2$ is analogous.
\end{proof}

\begin{theorem}[Global solution]
	Suppose that {\rm (A1)--(A8)} are satisfied and that 
$(u_{0},u_{1})\in W_1\times L^2(\mathbb{R}^N)$ and $E(0)<c$. 
Then the local solution given by Theorem \ref{Theorem3.1} can be extended 
for all $t>0$.
\end{theorem}

\begin{proof} 
It is sufficient to estimate the $H^1(\mathbb{R}^N)$ norm. We observe that
\begin{equation}\label{3.12}
	\|\nabla u(t)\|_2^2+J(u)+\int_{\mathbb{R}^N}\nabla a(x)\cdot x F(u)dx=NI(u(t)).
\end{equation}
As $u_0\in W_1$, then for all $t$, $u(t)\in W_1$ and $J(u(t))>0$. 
Therefore, from \eqref{3.12} we have
\begin{equation}\label{3.13}
	\|\nabla u(t)\|_2^2<NI(u(t)) -\int_{\mathbb{R}^N}\nabla a(x)\cdot x F(u)dx<Nc,
\end{equation}
for all $t\geq 0$. By Sobolev, Gagliardo, Nirenberg inequality there exist
 $C>0$ such that
\begin{equation}\label{3.14}
	\|u(t)\|_{2^*}<C,
\end{equation}
for all $t\geq 0$. Using \eqref{1.5} for all $\varepsilon>0$ we have
$$
\int_{\mathbb{R}^N}a(x) F(u)dx 
\leq \frac{\|a\|_{\infty}\varepsilon}{2} \|u(t)\|_2^2
+C(\varepsilon) \|u(t)\|_{2^*}^{2^*}.
$$
This allows us to conclude that
\begin{equation}\label{3.15}
\begin{aligned}
	I(u(t))
&=\frac{1}{2}\int_{\mathbb{R}^N}|\nabla u|^2dx
 + \frac{\lambda}{2}\int_{\mathbb{R}^N}| u|^2dx 
 -\frac{\|a\|_{\infty}\varepsilon}{2} \|u(t)\|_2^2
 -C(\varepsilon) \|u(t)\|_{2^*}^{2^*} \\
&\leq\frac{1}{2}\int_{\mathbb{R}^N}|\nabla u|^2dx
 + \frac{\lambda}{2}\int_{\mathbb{R}^N}| u|^2dx 
 -\int_{\mathbb{R}^N}a(x)F(u)dx.
\end{aligned}
\end{equation}
Using the assumption we obtain
\begin{equation}\label{3.16}
	I(u(t))<E(t)=E(0)<c.
\end{equation}
From \eqref{3.12} and \eqref{3.16} we have
$$
	\frac{1}{2}\int_{\mathbb{R}^N}|\nabla u|^2dx
+ \frac{\lambda}{2}\int_{\mathbb{R}^N}| u|^2dx 
-\frac{\|a\|_{\infty}\varepsilon}{2} \|u(t)\|_2^2-C(\varepsilon) \|u(t)\|_{2^*}^{2^*}
	<c
$$
This inequality, \eqref{3.13}, \eqref{3.14} allow us to infer that
$$
	\Big(\frac{\lambda}{2}-\frac{\|a\|_{\infty}\varepsilon}{2}\Big) 
\|u(t)\|_2^2\leq C.
$$
Choosing $\varepsilon>0$ such that $\varepsilon < \frac{\lambda}{\|a\|_{\infty}}$, 
we conclude that $\|u(t)\|_2^2$ is bounded, this and \eqref{3.13} give the result.
\end{proof}

The next step is to prove the blow up result, for which we need some
 auxiliary results.

\begin{lemma}\label{Lemma3.4}
	Let $(u_{0},u_{1})\in W_2\times L^2(\mathbb{R}^N)$ be such that 
$E(0)<c$ and $u(t)$ the associated solution of \eqref{1.1} defined in $[0,T)$. 
Then there exists $\delta>0$ such that $J(u(t))<-\delta$, for all $t\in[0,T)$.
\end{lemma}

\begin{proof} 
Since $u_0\in W_2$ then $u(t)\in W_2$, for all $t\in (0,T)$, this implies
$$
	J_{\infty}(u(t))\leq J(u(t))<0,\: \text{for all }t\in(0,T).
$$
On the other hand, it is easy to see that
$$
	\frac{1}{N}\int_{\mathbb{R}^N}|\nabla u(t)|^2dx
= I(u(t))-\frac{J(u(t))}{N}-\frac{1}{N}\int_{\mathbb{R}^N}\nabla a(x)
\cdot x F(u(t))\,dx.
$$
As $I(u(t))\leq E(t)=E(0)$ and $\nabla a(x)\cdot x F(u(t))\geq 0$ we have
\begin{equation}\label{3.17}
	\frac{1}{N}\int_{\mathbb{R}^N}|\nabla u|^2dx
\leq  E(0)-\frac{J(u(t))}{N}= c - \nu-\frac{J(u(t))}{N},
\end{equation}
where $\nu := c - E(0)$ is a positive constant, since $E(0)<c$. 
Suppose that does not exist $\delta$ satisfying the lemma. 
Then there exists a sequence $(t_k)_{k\in\mathbb{N}}\subset (0,T)$ 
such that $J(u(t_k))\to 0$, when $k\to\infty$. Therefore, for $k$ 
large enough, we have
\begin{equation}\label{3.18}
	-\frac{\nu N}{2}<J(u(t_k))\leq 0.
\end{equation}
From \eqref{3.17} and \eqref{3.18} we obtain
$$
	T_{\infty}(u(t_k))= \frac{1}{N}\int_{\mathbb{R}^N}|\nabla u(t_k)|^2dx 
\leq c -\frac{\nu}{2},
$$
for $k$ large enough, but this contradicts the Lemma \ref{Lemma2.3}.
\end{proof}

For each $\varepsilon>0$, define $\Phi_{\varepsilon}:\mathbb{R}\to\mathbb{R}$ by
$$
	\Phi_{\varepsilon}(r)=
 \begin{cases}
 	N	&	\text{if }		0\leq r\leq \exp(\frac{1}{\varepsilon}),\\
 	2N-N\varepsilon ln (r)	&	\text{if }	
 \exp(\frac{1}{\varepsilon})<r\leq \exp(\frac{2}{\varepsilon}),\\
 	0	&	\text{if }		r>\exp(\frac{2}{\varepsilon})
 \end{cases}
$$
and $\Psi_{\varepsilon}:\mathbb{R}\to\mathbb{R}$ by
$$
	\Psi_{\varepsilon}(r)=\frac{1}{r^{N-1}}\int_0^rs^{N-1}\Phi_{\varepsilon}(s)ds.
$$
The next lemma summarizes some properties of $\Phi_{\varepsilon}$ 
and $\Psi_{\varepsilon}$.

\begin{lemma}\label{Lemma3.5}
	For each $\varepsilon>0$, we have
	\begin{gather}
 \Phi_{\varepsilon}(r)=N,\quad \Psi_{\varepsilon}(r)=r,\quad
 0\leq r\leq \exp(\frac{1}{\varepsilon});\\
 \Psi_{\varepsilon}'(r)+\frac{N-1}{r}\Psi_{\varepsilon}(r)
=\Phi_{\varepsilon}(r),\quad\text{for all }r\geq 0;\\
\label{3.21}
 \|\Psi_{\varepsilon}'-\frac{1}{r}\Psi_{\varepsilon}\|_{L^{\infty}}<\varepsilon;\\
 |\Phi_{\varepsilon}(r)|\leq k,\quad\Psi_{\varepsilon}'(r)\leq 1,
\quad \text{for all }r\geq 0; \\
 \Big(\frac{r^{N-1}}{N-1}\Psi_{\varepsilon}(r)\Big)'
=\frac{r^{N-1}}{N-1}\Phi_{\varepsilon}(r),\quad \text{for all }r\geq 0.
	\end{gather}
\end{lemma}

For a proof of the above lemma see Ohta and Todorova \cite{Ohta-Todorova} 
and Jeanjean and Le Coz \cite{Jeanjean-LeCoz}.

\begin{lemma}\label{Lemma3.6}
	Let $(u_{0},u_{1})\in W_2\times L^2(\mathbb{R}^N)$ be such that 
$E(0)<c$ and $u(t)$ the associated solution of \eqref{1.1} defined 
in $[0,\infty)$. If there exists a constant $K>0$ such that 
$\|u(t)\|_{\lambda}\leq K$, then
	$$
 \nu t \leq C(1+\|u_t(t)\|_2\|\nabla u(t)\|_2), 	
	$$
	for all $t\in [0,\infty)$, where $\nu$ and $C$ are positive constants.
\end{lemma}

\begin{proof} 
As $u_0\in W_2$, by the uniqueness of solution, $u(t)$ is a radial function 
for all $t\geq 0$, namely, $u(x,t)=u(r,t)$, where $r=|x|$. Therefore, 
 equation \eqref{1.1}$_1$ becomes
\begin{equation}\label{3.24}
	u_{tt}-\frac{N-1}{r}u'-u''+\lambda u=af(u).
\end{equation}
Multiplying  \eqref{3.24} by $\frac{\Psi_{\varepsilon}(r)u'r^{N-1}}{N-1}$ and 
integrating over $[0,\infty)$ we obtain
\begin{equation}\label{3.25}
\begin{aligned}
&\int_0^{\infty} u_{tt} u' \Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr
	-\int_0^{\infty} |u'|^2 \Psi_{\varepsilon}(r)r^{N-2}dr\\
&-\int_0^{\infty}u''u' \Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr
+\lambda \int_0^{\infty}uu' \Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr\\
&=\int_0^{\infty}a(r)f(u)u'\Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr.
\end{aligned}
\end{equation}
Now  estimate the terms of the above equation.
\begin{gather}\label{3.26}
\begin{aligned}
&\int_0^{\infty} u_{tt} u' \Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr \\
&=\frac{\partial}{\partial t}\int_0^{\infty} u_{t} u' \Psi_{\varepsilon}(r)
 \frac{r^{N-1}}{N-1}dr
	-\int_0^{\infty} u_{t} u_t' \Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr \\
&=\frac{\partial}{\partial t}\int_0^{\infty} u_{t} 
 u' \Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr
-\frac{1}{2}\int_0^{\infty} (u_{t}^2)' \Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr\\
&=\frac{\partial}{\partial t}\int_0^{\infty} u_{t} u' \Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr
	+\frac{1}{2}\int_0^{\infty} u_{t}^2 \Phi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr;
\end{aligned} \\
\label{3.27}
\begin{aligned}
&-\int_0^{\infty}u''u' \Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr\\
&=-\frac{1}{2}\int_0^{\infty}(|u'|^2)' \Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr \\
&=\frac{1}{2}\int_0^{\infty}|u'|^2 \Big(\Psi_{\varepsilon}'(r)\frac{r^{N-1}}{N-1}
 +\Psi_{\varepsilon}(r)r^{N-2}\Big)dr;
\end{aligned} \\
\label{3.28}
\begin{aligned}
\lambda \int_0^{\infty}uu' \Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr
&=\frac{\lambda}{2} \int_0^{\infty}(u^2)' \Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr\\
&=-\frac{\lambda}{2} \int_0^{\infty}u^2 \Phi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr;
\end{aligned} \\
\begin{aligned}
&\int_0^{\infty}a(r)F(u)\Phi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr\\
&=\int_0^{\infty}a(r)F(u)\Big(\Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}\Big)'dr\\
&=-\int_0^{\infty}(a'(r)F(u)+a(r)f(u)u')\Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr.
\end{aligned} \nonumber
\end{gather}
From here,
\begin{equation}\label{3.29}
\begin{aligned}
&\int_0^{\infty}a(r)f(u)u'\Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr\\
&=-\int_0^{\infty}a(r)F(u)\Phi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr
-\int_0^{\infty}a'(r)F(u)\Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr.
\end{aligned}
\end{equation}
Substituting \eqref{3.26}--\eqref{3.29} in \eqref{3.25} we obtain
\begin{equation}\label{3.30}
\begin{aligned}
&\frac{\partial}{\partial t}\int_0^{\infty} u_{t} u' \Psi_{\varepsilon}(r)
 \frac{r^{N-1}}{N-1}dr\\
&+\frac{1}{2}\int_0^{\infty} u_{t}^2 \Phi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr
	-\int_0^{\infty} |u'|^2 \Psi_{\varepsilon}(r)r^{N-2}dr\\
&+\frac{1}{2}\int_0^{\infty}|u'|^2 \Big(\Psi_{\varepsilon}'(r)\frac{r^{N-1}}{N-1}+\Psi_{\varepsilon}(r)r^{N-2}\Big)dr
	-\frac{\lambda}{2} \int_0^{\infty}u^2 \Phi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr\\
&=-\int_0^{\infty}a(r)F(u)\Phi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr
	-\int_0^{\infty}a'(r)F(u)\Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr.
\end{aligned}
\end{equation}
Define
\begin{equation}\label{3.31}
\begin{aligned}
J_T(u)
&= \frac{N-2}{2N}\int_0^{\infty}|u'|^2 \Phi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr
	-\int_0^{\infty}a(r)F(u)\Phi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr \\
&\quad +\frac{\lambda}{2} \int_0^{\infty}u^2 \Phi_{\varepsilon}(r)
 \frac{r^{N-1}}{N-1}dr
-\frac{1}{N}\int_0^{\infty}a'(r)rF(u)\Phi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr.
\end{aligned}
\end{equation}
From \eqref{3.30} and \eqref{3.31} we have
\begin{equation}\label{3.32}
\begin{aligned}
&-\frac{\partial}{\partial t}\int_0^{\infty} u_{t} 
 u' \Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr \\
&=\underbrace{\frac{1}{2}\int_0^{\infty} u_{t}^2 \Phi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr}_{\geq 0}
	-\int_0^{\infty} |u'|^2 \Psi_{\varepsilon}(r)r^{N-2}dr \\
&\quad 	+\frac{1}{2}\int_0^{\infty}|u'|^2 
 \Big(\Psi_{\varepsilon}'(r)\frac{r^{N-1}}{N-1}+\Psi_{\varepsilon}(r)r^{N-2}\Big)dr \\
&\quad +\frac{N-2}{2N}\int_0^{\infty}|u'|^2 \Phi_{\varepsilon}(r)
\frac{r^{N-1}}{N-1}dr \\
&\quad-\frac{1}{N}\int_0^{\infty}a'(r)rF(u)\Phi_{\varepsilon}(r)
 \frac{r^{N-1}}{N-1}dr \\
&\quad +\int_0^{\infty}a'(r)F(u)\Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr-J_T(u).
\end{aligned}
\end{equation}
As
\begin{align*}
&\int_0^{\infty}|u'|^2 \Phi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr \\
&=\int_0^{\infty}|u'|^2 \Big(\Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}\Big)'dr \\
&=\int_0^{\infty}|u'|^2 \Big(\Psi_{\varepsilon}'(r)\frac{r^{N-1}}{N-1}
+\Psi_{\varepsilon}(r)r^{N-2}\Big)dr,
\end{align*}
we obtain
\begin{align*}
&-\int_0^{\infty} |u'|^2 \Psi_{\varepsilon}(r)r^{N-2}dr
	+\frac{1}{2}\int_0^{\infty}|u'|^2 \Big(\Psi_{\varepsilon}'(r)
\frac{r^{N-1}}{N-1}+\Psi_{\varepsilon}(r)r^{N-2}\Big)dr \\
&+\frac{N-2}{2N}\int_0^{\infty}|u'|^2 \Phi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr\\
&=\frac{1}{N}\int_0^{\infty}|u'|^2 \Big(\Psi_{\varepsilon}'(r)
-\frac{1}{r}\Psi_{\varepsilon}(r)\Big)r^{N-1}dr.
\end{align*}
From this identity, the assumption $\|u(t)\|_{\lambda}\leq K$ and
 \eqref{3.21} we conclude that
\begin{align*}
&\Big|-\int_0^{\infty} |u'|^2 \Psi_{\varepsilon}(r)r^{N-2}dr
+\frac{1}{2}\int_0^{\infty}|u'|^2 \Big(\Psi_{\varepsilon}'(r)\frac{r^{N-1}}{N-1}
+\Psi_{\varepsilon}(r)r^{N-2}\Big)dr \\
& +\frac{N-2}{2N}\int_0^{\infty}|u'|^2 \Phi_{\varepsilon}(r)
\frac{r^{N-1}}{N-1}dr\Big|\leq \varepsilon,
\end{align*}
\begin{equation}\label{3.33}
\begin{aligned}
&-\int_0^{\infty} |u'|^2 \Psi_{\varepsilon}(r)r^{N-2}dr
	+\frac{1}{2}\int_0^{\infty}|u'|^2 
\Big(\Psi_{\varepsilon}'(r)\frac{r^{N-1}}{N-1}+\Psi_{\varepsilon}(r)r^{N-2}\Big)dr\\
&+\frac{N-2}{2N}\int_0^{\infty}|u'|^2 \Phi_{\varepsilon}(r)
\frac{r^{N-1}}{N-1}dr\geq -\varepsilon.
\end{aligned}
\end{equation}
On the other hand,
\begin{align*}
&-\frac{1}{N}\int_0^{\infty}a'(r)rF(u)\Phi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr
	+\int_0^{\infty}a'(r)F(u)\Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr 
\\
&-\frac{1}{N}\int_0^{\infty}a'(r)rF(u)\Big(\Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}
 \Big)'dr
	+\int_0^{\infty}a'(r)F(u)\Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr
\\
&=\frac{1}{N-1}\int_0^{\infty}a'(r)rF(u)\Big(\frac{1}{r}
 \Psi_{\varepsilon}(r)-\frac{1}{N}\Psi_{\varepsilon}'(r)
 -\frac{N-1}{rN}\Psi_{\varepsilon}(r)\Big)r^{N-1}dr
\\
&=-\frac{1}{N(N-1)}\int_0^{\infty}a'(r)rF(u)\Big(\Psi_{\varepsilon}'(r)
 -\frac{1}{r}\Psi_{\varepsilon}(r)\Big)r^{N-1}dr.
\end{align*}
Using the assumptions that $a'(r)r\to 0$, when $r\to\infty$, $|F(u)|\leq C u^2$, 
$\|u(t)\|_{\lambda}\leq K$ and \eqref{3.21}, we obtain
$$
\Big|-\frac{1}{N}\int_0^{\infty}a'(r)rF(u)\Phi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr
	+\int_0^{\infty}a'(r)F(u)\Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr\Big|
< \varepsilon,
$$
or
\begin{equation}\label{3.34}
	-\frac{1}{N}\int_0^{\infty}a'(r)rF(u)\Phi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr
	+\int_0^{\infty}a'(r)F(u)\Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr
>-\varepsilon.
\end{equation}
Substituting \eqref{3.33} and \eqref{3.34} in \eqref{3.32} we have
\begin{equation}\label{3.35}	
	-\frac{\partial}{\partial t}\int_0^{\infty} u_{t} u' \Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr
	>-J_T(u)-2\varepsilon.
\end{equation}
We know that $J_T(u)\to J(u)$, when $\varepsilon\to 0$, therefore for all 
$\mu>0$, there exists $\gamma>0$ such that $|J_T(u)-J(u)|<\mu$, when 
$\varepsilon<\gamma$. From here,
\begin{equation}\label{3.36}
	-J(u)-\mu<-J_T(u),\quad \text{when }\varepsilon<\gamma.
\end{equation}
Inequalities \eqref{3.35} and \eqref{3.36} give us
$$	
	-\frac{\partial}{\partial t}\int_0^{\infty} u_{t} 
u' \Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr
	>-J(u)-\mu-2\varepsilon.
$$
Taking $\mu=\delta/4$, where $\delta$ was given by Lemma \ref{Lemma3.4},
 $\varepsilon=\min\{\frac{\delta}{8},\frac{\gamma}{2}\}$ and using 
Lemma \ref{Lemma3.4} we obtain
$$	
-\frac{\partial}{\partial t}\int_0^{\infty} u_{t} 
u' \Psi_{\varepsilon}(r)\frac{r^{N-1}}{N-1}dr
>\frac{\delta}{2},
$$
integrating over $(0,\infty)$ we conclude that
$$
	\delta t < C(1+ \|u_t(t)\|_2\|\nabla u(t)\|_2),
$$
for all $t\geq 0$, this inequality allows us to complete the proof.
\end{proof}

\begin{theorem}[Blow up]
	Suppose that {\rm (A1)--(A8)} hold. Then for all 
$(u_{0},u_{1})\in W_2\times L^2(\mathbb{R}^N)$ such that $E(0)<c$ 
there exists $0<T\leq \infty$ and a unique function 
$u:\mathbb{R}^N\times [0,T)\to\mathbb{R}$ solution of \eqref{1.1} in the class
	$$
 u\in C([0,T]; H^1(\mathbb{R}^N))\cap C^1([0,T]; L^2(\mathbb{R}^N)).
	$$
such that $u(t)\in W_2$ for all $t\in (0,T)$ and either
\begin{itemize}	
\item[(a)] the solution exists locally, i.e. 
$T<\infty$, and there exists a sequence $(t_k)_{k\in\mathbb{N}}
\subset (0,T)$ with $t_k\to T^-$ such that
 $$
\|u(t_k)\|_{\lambda}\to\infty, \:\:\text{when }  t_k\to T^-;
$$
 or

\item[(b)] the solution exists globally on $[0,\infty)$ and there exists 
a sequence $(t_k)_{k\in\mathbb{N}}\subset (0,\infty)$ with 
$t_k\to \infty$ such that
 $$
\|u(t_k)\|_{\lambda}\to\infty, \:\:\text{when }  t_k\to \infty.
$$
\end{itemize}
\end{theorem}

\begin{proof} 
The blow up result of the item (a) is a consequence of $T<\infty$, 
see Georgiev and Todorova \cite{Georgiev-Todorova} and Segal \cite{Segal}. 
Suppose that $T=\infty$. We will prove by contradiction. Suppose that there 
exists a constant $k_1>0$ such that
\begin{equation}\label{3.37}
	\lambda \|u(t)\|_2^2+\|\nabla u(t)\|_2^2\leq \|u(t)\|_{\lambda}^2\leq k_1
\end{equation}
for all $t\geq 0$. By the identity \eqref{3.9} we have
\begin{equation}\label{3.38}
	\frac{1}{2}\|u_t(t)\|_2^2=E(0)-\frac{1}{2}\|u(t)\|_{\lambda}^2+\int_{\mathbb{R}^N}a(x)F(u)dx.
\end{equation}
From \eqref{3.37} and \eqref{3.38} we conclude that there exists a constant $k_2>0$ such that
\begin{equation}\label{3.39}
	\frac{1}{2}\|u_t(t)\|_2^2\leq k_2,
\end{equation}
for all $t\geq 0$. But \eqref{3.37} and \eqref{3.39} give a contradiction 
with Lemma \ref{Lemma3.6}.
\end{proof}

\subsection*{Acknowledgments}
 Andr\'e Vicente is partially supported by Funda\c{c}\~ao
Arauc\'aria conv. 151/2014 and 547/2014.

\begin{thebibliography}{00}

\bibitem{Alves-Cavalcanti}
C. O. Alves, M. M. Cavalcanti;
\emph{On existence, uniform decay rates and blow up for solutions of the 
2-D wave equation with exponential source},
Calculus of Variations. 34 (2009), 377-411.

\bibitem{Alves-Cavalcanti-Domingos Cavalcanti-Rammaha-Toundykov}
C. O. Alves, M. M. Cavalcanti, V. N. Domingos Cavalcanti, M. A. Rammaha, D. Toundykov;
\emph{On Existence, uniform decay and blow up for solutions of systems of nonlinear wave equations with damping and source terms},
Discrete and Continuous Dynamical Systems Series S. 2. 3 (2009), 583-608.

\bibitem{Ambrosetti-Rabinowitz}
A. Ambrosetti, P.H. Rabinowitz;
\emph{Dual variational methods in critical point theory and applications},
Journal of Functional Analysis. 14 (1973), 349-381.

\bibitem{Berestycki-Lions} H. Berestycki, P. L. Lions;
\emph{Nonlinear scalar field equations. I. Existence of a ground state},
Archive for Rational Mechanics and Analysis. 82(4) (1983), 313-345.

\bibitem{Brezis} H. Brezis;
\emph{Op\'erateurs maximaux monotones et semi-groupes de contractions dans 
les espaces de Hilbert}, North-Holland Publishing Co., Amsterdam, 1973.

\bibitem{Gan-Guo-Zhang} Z. Gan, B. Guo, J. Zhang;
\emph{Sharp Threshold of Global Existence for the Klein-Gordon Equations
 with Critical Nonlinearity},
Acta Mathematicae Applicatae Sinica. 25(2) (2009), 273-282.

\bibitem{Georgiev-Todorova} V. Georgiev, G. Todorova;
\emph{Existence of a solution of the wave equation with nonlinear damping and 
source terms}, Journal of Differential Equations. 109 (1994), 295-308.

\bibitem{Jeanjean-LeCoz} L. Jeanjean, S. Le Coz;
\emph{Instability for Standing Waves of Nonlinear Klein-Gordon Equations 
via Mountain-Pass Arguments},
Transactions of the American Mathematical Society. 361(10) (2009), 5401-5416.

\bibitem{Jeanjean-Tanaka} L. Jeanjean, K. Tanaka;
\emph{Singularly perturbed elliptic problems with superlinear or asymptotically 
linear nonlinearities},
Calculus of Variations and Partial Differential Equations. 21 (2004), 287-318.

\bibitem{Jeanjean-Tanaka1} L. Jeanjean, K. Tanaka;
\emph{A remark on least energy solutions in $\mathbb{R}^N$},
Proceedings of the American Mathematical Society. 131(8) (2002), 2399-2408.

\bibitem{Kaitai-Quande} Li Kaitai, Zhang Quande;
\emph{Existence and Nonexistence of Global Solutions for the Equation of 
Dislocation of Crystals}, Journal of Differential Equations. 146 (1998), 5-21.

\bibitem{Lehrer-Maia} R. Lehrer, L. A. Maia;
\emph{Positive solutions of asymptotically linear equations via Pohozaev manifold},
Journal of Functional Analysis. 266 (2014), 213-246.

\bibitem{Levine-Todorova} H. A. Levine, G. Todorova;
\emph{Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy},
Proceedings of the American Mathematical Society. 129(3) (2000), 793-805.

\bibitem{Ohta-Todorova} M. Ohta, G. Todorova;
\emph{Strong instability of standing waves for nonlinear Klein-Gordon equation and Klein-Gordon-Zakharov system},
SIAM Journal on Mathematical Analysis. 38 (2007), 1912-1931.

\bibitem{Serrin-Todorova-Vitillaro} J. Serrin, G. Todorova, E. Vitillaro;
\emph{Existence for a nonlinear wave equation with damping and source terms},
Differential and Integral Equations. 16(1) (2003), 13-50.

\bibitem{Shatah} J. Shatah;
\emph{Unstable ground state of nonlinear Klein-Gordon Equations},
Transactions of the American Mathematical Society. 290(2) (1985), 701-710.

\bibitem{Segal} I. Segal;
\emph{Non-Linear Semi-Groups},
Annals of Mathematics. 78(2) (1963), 339-364.

\bibitem{Todorova} G. Todorova;
\emph{Stable and Unstable Sets for the Cauchy Problem for a Nonlinear Wave 
Equation with Nonlinear Damping and Source Terms},
Journal of Mathematical Analysis and Applications. 239 (1999), 213-226.

\bibitem{Wang} Y. Wang;
\emph{A sufficient condition for finite time blow up of the nonlinear 
Kklein-Gordon equations with arbitrarily positive initial energy},
Proceedings of the American Mathematical Society. 136(10) (2008), 3477-3482.

\bibitem{Willem} M. Willem;
\emph{Minimax Theorems}, Birkhauser,  Basel, 1996.

\end{thebibliography}

\end{document}