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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 68, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/68\hfil Blow up of solutions to ODEs]
{Blow up of solutions to ordinary differential equations arising
 in nonlinear dispersive problems}

\author[M. Dimova, N. Kolkovska, N. Kutev \hfil EJDE-2018/68\hfilneg]
{Milena Dimova, Natalia Kolkovska, Nikolai Kutev}

\address{Milena Dimova (corresponding author)\newline
University of National and World Economy,
Students' Town, 1700 Sofia, Bulgaria. \newline
Institute of Mathematics and Informatics,
Bulgarian Academy of Sciences, Sofia, Bulgaria}
\email{mdimova@unwe.bg}

\address{Natalia Kolkovska \newline
Institute of Mathematics and Informatics,
Bulgarian Academy of Sciences,
Acad. G. Bonchev Str., Bl.8, 1113 Sofia, Bulgaria}
\email{natali@math.bas.bg}

\address{Nikolai Kutev \newline
Institute of Mathematics and Informatics,
Bulgarian Academy of Sciences,
Acad. G. Bonchev Str., Bl.8, 1113 Sofia, Bulgaria}
\email{kutev@math.bas.bg}

\dedicatory{Communicated by Jerry Bona}

\thanks{Submitted August 2, 2017. Published March 14, 2018.}
\subjclass[2010]{35B44, 34A40, 35A24, 35L75}
\keywords{Finite time blow up; concavity method;  Klein-Gordon equation;
\hfill\break\indent double dispersive equation}

\begin{abstract}
 We study a new class of ordinary differential equations with blow up
 solutions. Necessary and sufficient conditions for finite blow up time
 are proved. Based on the new differential equation, a revised version of
 the concavity method of Levine is proposed.  As an application
 we investigate the non-existence of global solutions to the Cauchy
 problem of Klein-Gordon, and to the double dispersive equations.
 We obtain necessary and sufficient condition for finite time blow up
 with arbitrary positive energy.
 A very general sufficient condition for blow up is also given.
\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}
%\newtheorem{corollary}[theorem]{Corollary}
%\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks


\section{Introduction}\label{sec1}

The  finite time blow up of the solutions to nonlinear dispersive equations
has been intensively investigated in the previous decades. The blow up
phenomena for  semilinear wave equations, generalized Boussinesq equation,
double dispersive equation and others have been studied basically by
means of the concavity Levine's method. The main idea in Levine's method
\cite{Levine} is based on the fact that  if a twice
 continuously differentiable function $z(t)$ is a concave function, i.e.
\begin{equation} \label{z}
z''(t)\leq 0,\; t>0\quad\text{and}\quad z(0)>0,\; z'(0)<0,
\end{equation}
 then there exists $T_\ast$, $0 < T_\ast <\infty$ such that
\begin{equation}\label{zz}
z(t)\to 0\quad \text{as }t\to T_\ast, \quad t<T_\ast.
\end{equation}
To apply  Levine's method to global non-existence of solutions to a
nonlinear evolution equation one has to find a positive, smooth
function $\Psi(t)$, such that
$z(t)=\Psi^{1-\gamma}(t)$ for some $\gamma>1$ satisfies \eqref{z}
or equivalently    $\Psi(t)$ is a solution to the problem
\begin{equation} \label{1.1}
\Psi''(t) \Psi(t) - \gamma \Psi'^2(t)\geq 0,\quad t> 0, \quad
\gamma>1,\quad \Psi(0)>0, \quad \Psi'(0)>0.
\end{equation}
Then $\Psi(t)$ tends to infinity for a finite  time $T_\ast$.

In these applications $\Psi(t)$ is a nonnegative functional of the solution
to the corresponding nonlinear dispersive equation.
For example, for  semilinear wave equations $\Psi(t)$ is defined as
 $\Psi(t) =\int_{\mathbb{R}} u^2(t,x) \, dx$; while for fourth and sixth
order differential equations $\Psi(t)$ is a more complicated functional
including the $\mathrm{H}^1$ norm of the solution.

In a modification of the concavity method suggested in \cite{K-L}
the authors used that if the function $z(t)$,
instead of \eqref{z},  satisfies the second order differential
inequality
\begin{equation} \label{zzz}
z''(t) + \delta z'(t) + \mu z(t)\leq 0,\quad t>0,\quad  \delta\geq 0, \quad\mu\geq 0
\end{equation}
with suitable   initial data, then there exists $T_\ast$ such that
\eqref{zz} holds.
In this case  if $\Psi(t)$ is a solution to the inequality
$$
\Psi''(t) \Psi(t) - \gamma \Psi'^2(t)\geq -2\delta \Psi(t)\Psi'(t)
 - \mu \Psi^2(t),\quad t> 0,\quad \gamma>1,\quad \delta\geq 0, \quad\mu\geq 0
$$
equipped with appropriate initial conditions, then the function
$z(t)=\Psi^{1-\gamma}(t)$ satisfies \eqref{zzz}.

Another application  of the concavity method is done
in \cite{Korpusov,Straughan}, where   the   inequality
\begin{equation} \label{1.3}
\Psi''(t) \Psi(t) - \gamma \Psi'^2(t)\geq -\beta \Psi(t),\quad
 t> 0,\quad \gamma>1,\quad\beta>0
\end{equation}
is proposed. Then the function $z(t)=\Psi^{1-\gamma}(t)$  satisfies
\eqref{z} for a spacial choice of $\Psi(0)$ and $\Psi'(0)$.

In our previous paper \cite{KKD-MMAS} we suggest a new
 inequality
\begin{equation} \label{1.4}
\Psi''(t) \Psi(t) - \gamma \Psi'^2(t)\geq \alpha \Psi^2(t)-\beta \Psi(t),
\quad t>0,\quad \gamma>1,\quad\alpha>0,\quad\beta>0.
\end{equation}
Note, that for suitable chosen initial data  $\Psi(0)$, $\Psi'(0)$,
 the function $z(t)=\Psi^{1-\gamma}(t)$ fulfills  \eqref{z}.
In comparison with \eqref{1.3} the new inequality \eqref{1.4} includes
an additional positive term $\alpha \Psi^2(t)$ on the  right-hand side.
This term   naturally appears in the investigation of some nonlinear
dispersive equations as Klein-Gordon equation, double dispersive equation
with linear restoring force and others.

In  \cite{KKD-MMAS} the finite time blow up of the solutions to
inequality \eqref{1.4} is proved
under very general conditions on the initial data. However, these conditions
are only sufficient and not necessary  ones.

Let us mention that in the concavity method there is no precise formulation
of the  blow up result of the solutions to \eqref{1.1} and its generalizations
(see \cite{K-L,Korpusov,Levine,Straughan}). Namely,
the main assumption in this method is that $\Psi(t)$ is a twice
continuously  differentiable function for every $t\geq 0$.
However, under some conditions on the initial data,  it follows, that
 $\Psi(t)$ blows up for a finite time,
i.e.  $\Psi(t)$ is not defined for every $t\geq 0$.

 To give a rigorous formulation of  blow up  for $\Psi(t)$, we  replace
inequality \eqref{1.4} by  the corresponding differential equation
\begin{equation} \label{21}
\begin{gathered}
\Psi''(t) \Psi(t) - \gamma \Psi'^2(t)
= \alpha\Psi^2(t)-\beta\Psi(t)+H(t), \quad t\in [0,T_m),\quad 0<T_m\leq \infty, \\
\gamma>1,\quad \alpha>0,\;\beta>0.
\end{gathered}
\end{equation}
Here $\Psi(t)\in C^2([0,T_m))$ is a nonnegative solution to \eqref{21}
defined in  the maximal existence time interval
$[0,T_m)$, $0<T_m\leq\infty$ and
\begin{equation} \label{4}
H(t)\in C([0,T_m)), \quad H(t)\geq 0 \quad\text{for } t\in[0,T_m).
\end{equation}
Note, that in the analysis of nonlinear dispersive equations $\Psi(t)$
is a solution to  \eqref{21} with some function $H(t)$, see e.g. Lemma~\ref{aux}
below for Klein-Gordon equation.
 Usually  $H(t)$ can not be expressed explicitly by $\Psi(t)$.
That is why, up to now,   the nonnegative term  $H(t)$ has been neglected
and the corresponding inequality  \eqref{1.4} has been investigated.

By equation \eqref{21} we are able to formulate and to prove
  \emph{necessary and sufficient condition} for  blow up of  $\Psi(t)$
 at  the right-end point $T_m$ (see Theorem~\ref{Th3}).
Moreover, we prove that the blow up time $T_m$ is finite (see Theorem~\ref{Th0}).
In Theorem~\ref{Th3.1} we give new easy checkable sufficient condition
for finite time blow up of the solutions to \eqref{21}.
This condition generalizes the corresponding ones for blow up of
the solutions to inequality \eqref{1.4}, given  in \cite{KKD-FILOMAT,KKD-MMAS}.
The necessary and sufficient condition \eqref{22} sheds light on the reasons
 for blow up of the solutions to \eqref{21} and gives a better understanding
of the different sufficient conditions and their interrelations.

We apply the results for   ordinary differential equation \eqref{21}
to  Klein-Gordon equation
\begin{gather}
u_{tt} -  u_{xx} +  u = f(u),\quad (t,x)\in\mathbb{R}^+\times\mathbb{R}, \label{k1}\\
u(0,x)=u_0(x), \quad
u_t(0,x)=u_1(x),  \quad x \in \mathbb{R}, \label{k2} \\
u_0(x) \in \mathrm{H}^1(\mathbb{R}), \quad u_1(x)\in \mathrm{L}^2(\mathbb{R})
 \label{k3}
\end{gather}
and to double the dispersive equation
with linear restoring force
\begin{gather}
 u_{tt} - u_{xx} -  u_{ttxx} + u_{xxxx} + u +f(u)_{xx}=0, \quad
  (t,x)\in\mathbb{R}^+\times\mathbb{R},\label{d1} \\
u(0,x)=u_0(x), \quad u_t(0,x)=u_1(x),  \quad x \in \mathbb{R}, \label{d2} \\
\begin{gathered}
u_0\in \mathrm{H}^1(\mathbb{R})\cap \dot{\mathrm{H}}^{-1}(\mathbb{R}),
\quad\text{i.e. } u_0 \in \mathrm{H}^1(\mathbb{R}), \;
 (-\Delta)^{-1/2 } u_0 \in \mathrm{L}^2(\mathbb{R}), \\
u_1\in \mathrm{L}^2(\mathbb{R})\cap \dot{\mathrm{H}}^{-1}(\mathbb{R}), \quad
\text{i.e. } u_1\in \mathrm{L}^2(\mathbb{R}), \quad (-\Delta)^{-1/2 }u_1\in
\mathrm{L}^2(\mathbb{R}).
\end{gathered}\label{d3}
\end{gather}
Here $(-\Delta)^{-s}u=\mathcal{F}^{-1}\left(|\xi|^{-2s} \mathcal{F}(u)\right)$
for $s>0$, $\mathcal{F}(u)$, $\mathcal{F}^{-1}(u)$ are the Fourier transform
and the inverse Fourier transform, respectively.

The nonlinearity $f(u)$ in \eqref{k1} and \eqref{d1}  has one of the
following two forms:
\begin{equation} \label{4.4}
\begin{gathered}
f(u)=\sum_{k=1}^l a_k |u|^{p_k-1}u - \sum_{j=1}^s b_j |u|^{q_j-1}u,\\
f(u)=a_1|u|^{p_1} + \sum_{k=2}^l a_k |u|^{p_k-1}u
 - \sum_{j=1}^s b_j |u|^{q_j-1}u,
\end{gathered}
\end{equation}
where the constants $a_k$, $p_k$ $(k=1,2,\ldots,l)$ and $b_j$, $q_j$
$(j=1,2,\ldots,s)$ fulfill the conditions
\begin{equation} \label{4.5}
\begin{gathered}
 a_1>0,\quad  a_k\geq 0, \quad b_j\ge 0\quad\text{for }k=2,\ldots,l,\; j=1,\ldots,s, \\
 1<q_s<q_{s-1}<\cdots <q_1<p_1<p_2<\cdots<p_l<5.
\end{gathered}
\end{equation}
For example, the nonlinear term \eqref{4.4}-\eqref{4.5} includes the
quadratic-cubic nonlinearity ($f(u) = u^2 + u^3$), which  appears in a
number of mathematical models of physical processes, e.g. dislocations of
crystals \cite{Crystals}, propagation of longitudinal strain waves in an
isotropic cylindrical compressible
elastic rod \cite{Porubov,Samsonov} and others.

Let us recall that in the case of subcritical  initial energy ($0<E(0)< d$)
the global behaviour of the  solutions  is fully investigated by means of
 the potential well method, suggested in \cite{P-S} for semilinear wave equation.
Further on, this method has been applied to Klein-Gordon equation  \cite{Xu},
double dispersive equation  \cite{kkd-amitans2014,Xu-3-5}, more general classes
of double dispersive nonlocal wave equations \cite{Erbay} and others.
According to the potential well method, the global existence or finite
time blow up of the solutions with subcritical initial energy is proved when
the sign of  the Nehari functional  $I(0)$ is positive or negative, respectively.

For the supercritical initial energy ($E(0)> d$) there are  a few results
for  finite time blow up of the solutions to \eqref{k1}--\eqref{k3}
 (see \cite{Korpusov,Wang-KG,Xu-Ding}) and  \eqref{d1}--\eqref{d3}
(see \cite{kkd-amitans2014}).

In this article we give,  for the first time in the literature,
necessary and sufficient condition for finite time blow up of the solutions
to \eqref{k1}--\eqref{k3} and  \eqref{d1}--\eqref{d3} with  arbitrary positive
initial energy, see Theorem~\ref{Th8ii} and Theorem~\ref{ThDD}.
Although the results in Theorem~\ref{Th8ii} and Theorem~\ref{ThDD} are theoretical,
they  can be applied to the  numerical study  of nonlinear dispersive equations.
More precisely, one can check  whether
the necessary and sufficient condition \eqref{4.7} is satisfied at some time $t=b>0$
by means of some reliable numerical approach.

Moreover, we find out very general sufficient condition on $u_0$, $u_1$ for
which the corresponding solution $u(t,x)$  blows up for a finite time.
We demonstrate that all previous sufficient conditions in
 \cite{KKD-FILOMAT,Korpusov,KKD-MMAS,Wang-KG,Xu-Ding} for finite time blow
up of $u(t,x)$  can  be obtained as  a consequence from this new sufficient condition.

The necessary and sufficient conditions
 \eqref{4.7} and  \eqref{4.7d} reveal the genesis of the finite time
 blow up of the solutions to \eqref{k1}--\eqref{k3} and \eqref{d1}--\eqref{d3},
respectively.
For example, when \eqref{4.7} is fulfilled  at the initial time $t=0$, we get
the well known in the applications sufficient condition for finite time blow
up of the solutions to Klein-Gordon equation \cite{Wang-KG,Xu-Ding}
(for the nonlinear  wave equation see also  \cite{Gazzola}).
Note, that condition \eqref{4.7} at $t=0$ is only sufficient and not necessary one.
Our research shows that any sufficient condition, prescribed at $t=0$, ensures
the satisfaction of  condition \eqref{4.7} at  some  later time $t=b>0$.

This article is organized in the following way.
In Section~2 necessary and sufficient condition for finite time blow up of
the solutions to \eqref{21} is proved. Easy checkable sufficient condition
for finite time blow up of the solutions to \eqref{21} is given in Section~3.
 Section~4 deals with  applications of the results from Section~2 and Section~3
to Klein-Gordon and double dispersive equations with linear restoring force.

\section{Main results} \label{sec3}

We recall the definition for finite time blow up of a nonnegative smooth function.

\begin{definition} \label{def1} \rm 
The nonnegative function $\Psi(t)\in C^1([0,T_m))$,  $0<T_m\leq \infty$, blows up
at $T_m$ if
\begin{equation}\label{1}
\limsup_{t\to T_m, t<T_m} \Psi(t)=\infty.
\end{equation}
\end{definition}

Below we formulate a simple property of functions that blow up.

\begin{lemma}\label{Lm1}
Suppose $\Psi(t)\in C^1([0,T_m))$,  $0<T_m\leq\infty$, is a nonnegative
function and $M$ is an arbitrary constant. If $\Psi(t)$ blows up at $T_m$
then  there exists $t_0$, $t_0\in[0,T_m)$ such that $\Psi(t_0)\geq M$ and
$\Psi'(t_0)>0$.
\end{lemma}

\begin{proof}
If $\Psi(0)\geq M$ and $\Psi'(0)>0$, then Lemma~\ref{Lm1} holds for $t_0=0$.
Otherwise, from Definition \ref{def1} it follows that there exist
$t_1, t_3\in(0,T_m)$, $t_3>t_1$ such that  $\Psi(t_3)>\Psi(t_1)>M$.
We denote by $(t_2,t_3)\subset (t_1,t_3)$ the maximal interval where
$\Psi(t)>\Psi(t_1)$ for every $t\in (t_2,t_3)$.
From the mean value theorem there exists $t_0\in(t_2,t_3)$ such that
$$
\Psi(t_3)-\Psi(t_2)=(t_3-t_2) \Psi'(t_0).
$$
Since $\Psi(t_3)>\Psi(t_2)$ and $t_3>t_2 $ we obtain that $\Psi'(t_0)>0$.
From the choice of the interval $(t_2,t_3)$ it follows that $\Psi(t_0)\geq M$.
\end{proof}


The following theorem shows that blow up of the solutions to \eqref{21}
 under assumption \eqref{4} does not occur at infinity, i.e. only finite
time blow up is possible.

\begin{theorem} \label{Th0}
Suppose  $\Psi(t)\in  C^2([0,T_m))$ is a nonnegative solution of the equation
\begin{gather*}
\Psi''(t) \Psi(t) - \gamma \Psi'^2(t) = \alpha\Psi^2(t)-\beta\Psi(t)+H(t),
\quad t\in [0,T_m), \\
\gamma>1,\quad \alpha>0,\quad \beta>0,
\end{gather*}
where $[0,T_m)$, $0<T_m\leq\infty$ is the maximal existence time interval
for $\Psi(t)$ and \eqref{4} holds.
 If $\Psi(t)$ blows up at $T_m$ then $T_m<\infty$.
\end{theorem}

\begin{proof}
Suppose $\Psi(t)$ blows at $T_m$. From Lemma~\ref{Lm1} it follows that for
$M=\beta/\alpha$ there exist $b\in[0,T_m)$ such that
\begin{equation} \label{bb}
\Psi(b)\geq \beta/\alpha\quad~\text{and}\quad~\Psi'(b) >0.
\end{equation}
 From equation \eqref{21}  we have
$$
\Psi''(b)=\frac{\gamma \Psi'^2(b)}{\Psi(b)} +\alpha \Psi(b)
- \beta +\frac{H(b)}{\Psi(b)}\geq \frac{\gamma \Psi'^2(b)}{\Psi(b)}>0,
$$
thus $\Psi'(t)>\Psi'(b)>0$ for $t\in[b,b+\varepsilon)$ for some sufficiently
small $\varepsilon >0$.

 We will show  that $\Psi'(t)>0$ for every $t\in[b,T_m)$. If not, then
there exists an interval $(b,t_0)$, $t_0\in (b,T_m)$, such that
$\Psi'(t)>0$ for $t\in [b,t_0)$ and $\Psi'(t_0)=0$.  Since $\Psi(t)$ is
a strictly monotone increasing function for $t\in [b,t_0]$
 it follows that $\Psi(t)>\Psi(b)\geq \beta/\alpha$ for every $t\in (b,t_0]$.
 Moreover, from \eqref{21} and \eqref{bb} we have
$$
\Psi''(t)=\frac{\gamma \Psi'^2(t)}{\Psi(t)} +\alpha \Psi(t) - \beta
+\frac{H(t)}{\Psi(t)}> \alpha \Psi(b)-\beta\geq 0
$$
for every $t\in (b,t_0]$. Hence $\Psi'(t)$ is a strictly increasing function
for  $t\in (b,t_0]$ and we get the following impossible chain of inequalities
$$
0=\Psi'(t_0)>\Psi'(b)> 0.
$$
Thus $\Psi'(t)> 0$ for $t\in [b,T_m)$ and consequently
\begin{equation} \label{b}
\Psi(t)>\Psi(b)\geq\beta/\alpha>0\quad \text{for } t\in (b,T_m).
 \end{equation}
We define a function
$$
z(t)=\Psi^{1-\gamma}(t) \quad\text{for }  t\in [b,T_m),
$$
that satisfies
\begin{equation}\label{10}
z'(t)=(1-\gamma)\Psi^{-\gamma}(t)\Psi'(t),\quad
z''(t)=(1-\gamma)\Psi^{-1-\gamma}[\Psi''(t)\Psi(t)-\gamma\Psi'^2(t)].
\end{equation}
Function  $z(t)$ is a solution to the  initial value problem
\begin{equation}\label{11}
\begin{gathered}
z''(t)=-(\gamma-1)\left(\alpha z(t)-\beta z^{\frac{\gamma}{\gamma-1}}(t)
 + H(t) z^{\frac{\gamma+1}{\gamma-1}}(t)\right)\quad\text{for } t\in [b,T_m)\\
z(b)>0,\quad z'(b)<0.
\end{gathered}
\end{equation}


To prove that $T_m<\infty$ we assume by contradiction that  $T_m=\infty$.
 From \eqref{4}, \eqref{b} and  \eqref{10} it follows that
$$
z''(t)=-(\gamma-1) \Psi^{-\gamma}(t)\Big(\alpha \Psi(t) - \beta
+\frac{H(t)}{\Psi(t)}\Big)\leq 0
$$
for $t\geq b$.
Integrating $z''(t)\leq 0$ twice from $b$ to $t> b$, we get
$$
z'(t)\leq z'(b),\quad z(t)\leq z'(b)(t-b) +z(b).
$$
Consequently, there exists $T_\ast>b$ such that $z(T_\ast)=0$, or equivalently
$\Psi(T_\ast)=\infty$ for
\begin{equation} \label{12}
T_\ast\leq b-\frac{z(b)}{z'(b)}=b+\frac{\Psi(b)}{(\gamma-1)\Psi'(b)}<\infty,
\end{equation}
which contradicts our assumption. Thus it follows that $T_m<\infty$ and
Theorem~\ref{Th0} is proved.
\end{proof}

The following theorem is one of the the main results in this article.
We formulate and prove  necessary and sufficient condition for blow up
of the solutions to \eqref{21} at the right-end point of the existence
time interval.

\begin{theorem} \label{Th3}
Suppose  $\Psi(t)\in  C^2([0,T_m))$ is a nonnegative solution of the equation
\begin{gather*}
\Psi''(t) \Psi(t) - \gamma \Psi'^2(t)
 = \alpha\Psi^2(t)-\beta\Psi(t)+H(t), \quad t\in [0,T_m), \\
\gamma>1,\quad \alpha>0,\quad~\beta>0,
\end{gather*}
where $[0,T_m)$, $0<T_m\leq\infty$ is the maximal existence time interval
for $\Psi(t)$,  $H(t)\in C([0,\infty))$, and $H(t)\geq 0$ for $t\in[0,\infty)$.
 Then $\Psi(t)$ blows up at $T_m$  if and only if
\begin{equation}\label{22}
\text{there exists $b\in[0,T_m)$ such that $\beta\leq \alpha\Psi(b)$
and $\Psi'(b) >0$}.
\end{equation}
Moreover,
\begin{equation}\label{23}
T_m\leq b+ \frac{\Psi(b)}{(\gamma-1)\Psi'(b)}<\infty\,.
\end{equation}
\end{theorem}

\begin{proof} (Necessity)
Suppose $\Psi(t)$ blows up at $T_m$. Then  condition \eqref{22} holds
from Lemma~\ref{Lm1}  for $M=\beta/\alpha$ and $b=t_0$.

(Sufficiency) Suppose \eqref{22} holds.
From the proof of Theorem~\ref{Th0} it follows that
 $T_m<\infty$.
 Moreover, $\Psi(t)$ is a strictly increasing  function for $t\in[b,T_m)$.

If we assume that $\psi(t)$ does not blow up at $T_m$, i.e. \eqref{1}
 fails, then
$$
\limsup_{t\to T_m, t<T_m} \Psi(t)<\infty.
$$
From the monotonicity and boundedness of  $\Psi(t)$  for $t\in[b,T_m)$ we get
\begin{equation}\label{15}
\lim_{t\to T_m} \Psi(t)=\Psi(T_m)<\infty.
\end{equation}
As in the proof of Theorem~\ref{Th0} after the substitution
$z(t)=\Psi^{1-\gamma}(t)$, $t\in [b,T_m)$ we get that  $z(t)$ satisfies
 problem \eqref{11}.
Integrating the equation in \eqref{11} from $b$ to $t< T_m$ we get
$$
z'(t)=z'(b)-(\gamma-1)\int_b^t \left(\alpha z(s)-\beta z^{\frac{\gamma}{\gamma-1}}(s)
 + H(s) z^{\frac{\gamma+1}{\gamma-1}}(s)\right )\, ds
$$
or equivalently, from \eqref{10},
\begin{equation}\label{16}
\Psi'(t)=\Psi^\gamma(t) \Big[ \frac{\Psi'(b)}{\Psi^\gamma(b)}
+\int_b^t \left(\alpha\Psi^{1-\gamma}(s)-\beta\Psi^{-\gamma}(s)
+  H(s) \Psi^{-\gamma-1}(s) \right) \,ds\Big].
\end{equation}
Thus from \eqref{b}, \eqref{15} and \eqref{16} we have
\begin{align*}
\lim_{t\to T_m} \Psi'(t)
=&\Psi^{\gamma}(T_m)\Big[ \frac{\Psi'(b)}{\Psi^\gamma(b)}+
\int_b^{T_m} \left(\alpha\Psi^{1-\gamma}(s)-\beta\Psi^{-\gamma}(s)
 +  H(s) \Psi^{-\gamma-1}(s) \right) \,ds\Big]\\
=&\Psi'(T_m), \quad 0<\Psi'(T_m)<\infty.
\end{align*}

According to the theory of the initial value problems for ordinary differential
 equations, the problem
\begin{gather*}
\tilde{\Psi}''(t)\tilde{\Psi}(t)-\gamma\tilde{\Psi}'^2(t)
=\alpha\tilde{\Psi}^2(t)-\beta\tilde{\Psi}(t)+H(t)\quad \text{for }t\geq T_m,\\
\tilde{\Psi}(T_m)=\Psi(T_m),\quad\tilde{\Psi}'(T_m)=\Psi'(T_m)
\end{gather*}
has a classical solution $\tilde{\Psi}$ in the interval $[T_m,T_m+\delta)$,
 where $\delta>0$ is a sufficiently small number. Hence the function
$$
\hat{\Psi }(t)
=\begin{cases}
\Psi(t) & \text{for } t\in[0,T_m),\\[2pt]
\tilde{\Psi}(t) &\text{for }t\in[T_m, T_m+\delta),
\end{cases}
$$
$\hat{\Psi }(t)\in C^2([0,T_m+\delta))$,
$\hat{\Psi }(t)\geq 0$ and is a classical nonnegative solution
of \eqref{21} in the interval $[0,T_m+\delta)$ which contradicts the choice
of $T_m$. Hence $\Psi(t)$ blows up at $T_m$ and from \eqref{12} it follows
that   $T_m$ satisfies \eqref{23}. Thus Theorem~\ref{Th3} is proved.
\end{proof}

\section{Sufficient conditions for finite time blow up}

In this section we give some easy checkable sufficient condition on the
initial data $\Psi(0)$ and $\Psi'(0)$ for finite time blow up of the solutions
to \eqref{21}. This result is  important for the applications  of
Theorem~\ref{Th3} to nonlinear dispersive equations.

\begin{theorem}\label{Th3.1}
Suppose  $\Psi(t)\in  C^2([0,T_m))$ is a nonnegative solution of \eqref{21}
 in the maximal existence time interval
 $[0,T_m)$, $0<T_m\leq\infty$,   $H(t)\in C([0,\infty))$ and
$H(t)\geq 0$ for $t\in[0,\infty)$.
If
\begin{gather} \label{3.3i}
\beta<\frac{2\gamma-1}{2}\frac{\Psi'^2(0)}{\Psi(0)}
+\frac{\alpha (2\gamma-1)}{2(\gamma-1)}\Psi(0)
-\frac{\alpha^{2\gamma-1}\Psi^{2\gamma-1}(0)}{2(\gamma-1) \beta^{2\gamma-2}}, \\
\label{3.3ii}
\Psi'(0)>0,
\end{gather}
then $\Psi(t)$ blows up at $T_m <\infty$.
\end{theorem}

\begin{proof}
Firstly, we will show that $\Psi'(t)>0$ for every $t\in[0,T_m)$.
If not, from \eqref{3.3ii} there exist an interval $[0,t_0)$ such that
$\Psi'(t)>0$ for $t\in [0,t_0)$ and $\Psi'(t_0)=0$.
From \eqref{21} and \eqref{3.3ii} it follows that $\Psi(0)>0$.
Taking into account the monotonicity of $\Psi(t)$ in the interval $[0,t_0)$,
we conclude that $\Psi(t)>0$ for $t\in [0,t_0]$.
After the change  $z(t)=\Psi^{1-\gamma}(t)$ for $t\in [0,t_0]$ and using identities
\eqref{10}, we obtain the equation
\begin{equation}\label{3.4}
z''(t)=-(\gamma-1)\left(\alpha z(t)-\beta z^{\frac{\gamma}{\gamma-1}}(t)
+ H(t) z^{\frac{\gamma+1}{\gamma-1}}(t)\right)\quad
\text{for } t\in [0,t_0].
\end{equation}
Multiplying \eqref{3.4} by $z'(t)$ and integrating from $0$ to  $t\in(0,t_0]$
we get
\begin{equation} \label{3.5}
\begin{aligned}
z'^2(t)&=-\alpha(\gamma-1) z^2(t) + \frac{2\beta(\gamma-1)^2}{2\gamma-1}
 z^{\frac{2\gamma-1}{\gamma-1}}(t) \\
&\quad  - 2(\gamma-1)\int_0^{t} H(s) z'(s) z^{\frac{\gamma+1}{\gamma-1}}(s)\,ds
+ \tilde{C}.
\end{aligned}
\end{equation}
From \eqref{10} we obtain
\[
\tilde{C}=z'^2(0)+\alpha(\gamma-1) z^2(0)
-\frac{2\beta(\gamma-1)^2}{2\gamma-1} z^{\frac{2\gamma-1}{\gamma-1}}(0)
= \frac{2(\gamma-1)^2}{2\gamma-1} \Psi^{1-2\gamma}(0) C,
\]
where
\begin{equation}\label{3.8}
C=\frac{2\gamma-1}{2}\frac{\Psi'^2(0)}{\Psi(0)}
+\frac{\alpha (2\gamma-1)}{2(\gamma-1)}\Psi(0)-\beta.
\end{equation}
From \eqref{3.3i} we have $C>0$.

By \eqref{10},  equation \eqref{3.5} is equivalent to
\begin{align*}
\Psi'^2(t)=&-\frac{\alpha}{\gamma-1} \Psi^2(t)
 + \frac{2\beta}{2\gamma-1} \Psi(t)
 + \frac{2C }{2\gamma-1} \Psi^{1-2\gamma}(0) \Psi^{2\gamma}(t) \\
&+ 2\Psi^{2\gamma}(t) \int_0^{t} H(s) \Psi'(s) \Psi^{-2\gamma-1}(s)\,ds.
\end{align*}
Hence the inequality
\begin{align*}
\Psi'^2(t)
&\geq \Big(-\frac{\alpha}{\gamma-1} \Psi(t) + \frac{2\beta}{2\gamma-1}
 + \frac{2C}{2\gamma-1} \Psi^{1-2\gamma}(0)  \Psi^{2\gamma-1}(t)\Big) \Psi(t)\\
& =G(\Psi) \Psi(t)
\end{align*}
holds,  where
$$
G(y)=-\frac{\alpha}{\gamma-1} y + \frac{2\beta}{2\gamma-1}
+ \frac{2C}{2\gamma-1} \Psi^{1-2\gamma}(0) y^{2\gamma-1}.
$$
From the identities
\begin{gather*}
\frac{\partial G}{\partial y} = -\frac{\alpha}{\gamma-1}
 + 2 C \Psi^{1-2\gamma}(0)  y^{2\gamma-2}, \\
\frac{\partial^2 G}{\partial y^2}
=  4(\gamma-1) C\Psi^{1-2\gamma}(0) y^{2\gamma-3}>0 \quad\text{for } y>0
\end{gather*}
it follows that $G(y)$ has a minimum in $[0,\infty)$ at the point
$$
y_0=\Big(\frac{\alpha \Psi^{2\gamma-1}(0)}{2(\gamma-1)C}
 \Big)^{\frac{1}{2\gamma-2}}.
$$
Simple computations give  us
$$
G(y_0)=\frac{2}{2\gamma-1} \Big[\beta
- \alpha\Big(\frac{\alpha \Psi^{2\gamma-1}(0)}{2(\gamma-1) C}
\Big)^{\frac{1}{2\gamma-2}}\Big].
$$
From \eqref{3.3i} and \eqref{3.8} it follows that $G(y_0)>0$, hence
\begin{equation} \label{3.10}
\Psi'^2(t)\geq G(\Psi)\Psi(t)\geq G(y_0)\Psi(t)>0\quad\text{for } t\in[0,t_0].
\end{equation}
For $t=t_0$ we obtain the following impossible chain of inequalities
$$
0=\Psi'^2(t_0)\geq G(y_0)\Psi(t_0)>0.
$$
Thus $\Psi'(t)>0$ and consequently $\Psi(t)>0$ for every $t\in[0,T_m)$.
Moreover, from \eqref{3.10} we get
\begin{equation}\label{3.11}
\Psi'(t)\geq \sqrt{G(y_0)\Psi(t)}.
\end{equation}
Integrating \eqref{3.11} we obtain
$$
\Psi(t)\geq \Big(\frac{t}{2}\sqrt{G(y_0)}  + \sqrt{\Psi(0)}\Big)^2.
$$
For
$$
t=b=\max\Big( 2\Big(\sqrt{\beta/\alpha}-\sqrt{\Psi(0)}\Big) G^{-1/2}(y_0), 0\Big)
$$
we get that $\Psi(b)\geq \beta/\alpha$. Since $\Psi'(b)>0$ condition \eqref{22}
of Theorem~\ref{Th3} is satisfied. Hence the function $\Psi(t)$ blows up at
$T_m<\infty$.
\end{proof}

Let us recall that  sufficient conditions for  finite time blow up of the
solutions to \eqref{1.4} are obtained   in \cite{KKD-MMAS} and \cite{KKD-FILOMAT}.
Note, that every solution $\Psi(t)$ of equation
 \eqref{21} is also a solution to inequality \eqref{1.4}. Hence,
the sufficient conditions for blow up of the solutions  to \eqref{21} are
also sufficient ones for blow up of the solutions  to
\eqref{1.4}.
This allows us to compare the result in \cite{KKD-FILOMAT,KKD-MMAS}
with the result in the present paper.

Below we will show   that the proofs of the results in
\cite{KKD-FILOMAT,KKD-MMAS} follow from Theorem~\ref{Th3.1} and Theorem~\ref{Th3}.
 In this way we get a unified  approach  for proving  blow up of solution
to \eqref{1.4}.

\begin{theorem}\label{Cor}
Suppose  $\Psi(t)\in  C^2([0,T_m))$ is a nonnegative solution of \eqref{21}
 in the maximal existence time interval
 $[0,T_m)$, $0<T_m\leq\infty$,
$H(t)\in C([0,\infty))$ and $H(t)\geq 0$ for $t\in[0,\infty)$.
 If $\Psi'(0)>0$ and  one of the following conditions
\begin{itemize}
\item[(i)]
$\beta<\alpha \Psi(0)$;
\item[(ii)]  \cite{KKD-MMAS}
\begin{equation}\label{3.13}
\beta<\frac{2\gamma-1}{2}\frac{\Psi'^2(0)}{\Psi(0)} +\alpha \Psi(0);
\end{equation}

\item[(iii)] \cite{KKD-FILOMAT}
$$
\beta<\frac{2\gamma-1}{2}\frac{\Psi'^2(0)}{\Psi(0)} +\alpha \Psi(0)
+ \frac{\alpha \Psi(0)}{2(\gamma-1)}(1-A^{2-2\gamma}), \quad
A=\frac{\gamma-1}{\alpha}\frac{\Psi'^2(0)}{\Psi^2(0)}+1
$$
\end{itemize}
is satisfied, then $\Psi(t)$ blows up at $T_m<\infty$.
\end{theorem}


\begin{proof}
The proof of (i) follows  directly from Theorem~\ref{Th3} for $b=0$.

(ii) \textbf{Case 1:}
If $\beta< \alpha \Psi(0)$, then (ii)  follows from Theorem~\ref{Cor}(i).
\smallskip

\noindent \textbf{Case 2:}
Suppose that
\begin{equation}\label{3.14}
\alpha\Psi(0)\leq \beta<\frac{2\gamma-1}{2}\frac{\Psi'^2(0)}{\Psi(0)}
+\alpha \Psi(0)\,.
\end{equation}
Straightforward computations show that \eqref{3.3i} also holds.
Indeed, from \eqref{3.14} we get
\begin{align*}
&\beta^{2\gamma-2}\Big(\frac{2\gamma-1}{2}\frac{\Psi'^2(0)}{\Psi(0)}
 +\frac{\alpha (2\gamma-1)}{2(\gamma-1)}\Psi(0)-\beta\Big)\\
&>\beta^{2\gamma-2}\frac{\alpha\Psi(0) }{2(\gamma-1)}\\
&\geq \alpha^{2\gamma-2} \Psi^{2\gamma-2}(0)
 \frac{\alpha\Psi(0)}{2(\gamma-1)}
=\frac{\alpha^{2\gamma-1}\Psi^{2\gamma-1}(0)}{2(\gamma-1)},
\end{align*}
which is equivalent to \eqref{3.3i}.
Thus   (ii) follows from Theorem~\ref{Th3.1}


(iii) \textbf{Case 1:}
If \eqref{3.13} holds,  then  (iii) follows from Theorem~\ref{Cor}(ii).
\smallskip

\noindent\textbf{Case~2:}
Suppose that
$$
\frac{2\gamma-1}{2}\frac{\Psi'^2(0)}{\Psi(0)} +\alpha \Psi(0)
\leq\beta<\frac{2\gamma-1}{2}\frac{\Psi'^2(0)}{\Psi(0)}
+\alpha \Psi(0)+ \frac{\alpha \Psi(0)}{2(\gamma-1)}(1-A^{2-2\gamma})\,.
$$
We will show that \eqref{3.3i} is also fulfilled. Indeed, direct
computations give us
\begin{align*}
&\beta^{2\gamma-2}\Big(\frac{2\gamma-1}{2}\frac{\Psi'^2(0)}{\Psi(0)}
+\frac{\alpha (2\gamma-1)}{2(\gamma-1)}\Psi(0)-\beta\Big)\\
&>\beta^{2\gamma-2}\frac{\alpha\Psi(0)}{2(\gamma-1)} A^{2-2\gamma}\\
&\geq\Big(\frac{2\gamma-1}{2}\frac{\Psi'^2(0)}{\Psi(0)}
 +\alpha \Psi(0)\Big)^{2\gamma-2} \frac{\alpha\Psi(0)}{2(\gamma-1)} A^{2-2\gamma}\\
&=\frac{\alpha^{2\gamma-1}\Psi^{2\gamma-1}(0)}{2(\gamma-1)}
 \Big[1+\frac{1}{2(\gamma-1)}\big(1-\frac{1}{A}\big)\Big]^{2\gamma-2}\\
&>\frac{\alpha^{2\gamma-1}\Psi^{2\gamma-1}(0)}{2(\gamma-1)},
\end{align*}
which is equivalent to \eqref{3.3i}.
Thus  (iii)  follows from Theorem~\ref{Th3.1}.
\end{proof}

\section{Applications to nonlinear dispersive equation}

In this section we consider the Cauchy problem for the Klein-Gordon equation
\eqref{k1}--\eqref{k3} and double dispersive equation with linear
restoring force \eqref{d1}-\eqref{d3}.
For functions depending on $t$ and $x$ we use the following short notation:
\begin{gather*}
\|u\|=\|u(t,\cdot)\|_{\mathrm{L}^2(\mathbb{R})}, \quad
\|u\|_1=\|u(t,\cdot)\|_{\mathrm{H}^1(\mathbb{R})},\\
(u,v)=(u(t,\cdot),v(t,\cdot))
=\int_{\mathbb{R}} u(t,x)v(t,x) \, d x.
\end{gather*}
We recall the definition for blow up of the solutions to
 \eqref{k1}-\eqref{k3} and  \eqref{d1}-\eqref{d3}.

\begin{definition} \label{def2} \rm
Suppose $u(t,x)$ is a solution to \eqref{k1}-\eqref{k3} or
 \eqref{d1}-\eqref{d3} in the maximal existence time interval $[0,T_m)$,
$0<T_m\leq\infty$. Then $u(t,x)$ blows up at $T_m$ if
\begin{equation}\label{blow}
\limsup_{t\to T_m,\, t<T_m} \|u\|_1=\infty.
\end{equation}
\end{definition}

\subsection{Klein-Gordon equation}
We use  the following well known local existence result for problem
\eqref{k1}-\eqref{k3}.

\begin{theorem}\label{Th7}
Problem \eqref{k1}-\eqref{k3} admits a unique local solution
$u(t,x)$ that belongs to $\mathrm{C}([0,T_m); \mathrm{H}^1(\mathbb{R}))
\cap \mathrm{C}^1([0,T_m); \mathrm{L}^2(\mathbb{R})) \cap \mathrm{C}^2([0,T_m);
 \mathrm{H}^{-1}(\mathbb{R})$
 on a maximal existence time interval $[0,T_m)$, $T_m\leq \infty$. Moreover:
\begin{itemize}
\item[(i)] If $\limsup_{t\to T_m, t<T_m} \|u\|_1 <\infty$,
then $T_m=\infty$;

\item[(ii)] For every $t\in[0,T_m)$ the solution  $u(t,x)$ satisfies the
conservation law $E(t)=E(0)$, where
\begin{equation} \label{4.6}
E(t):=E(u(t,\cdot))=\frac{1}{2}\left((u_t,u_t) +(u,u) +\|u_x\|^2 \right)
- \int_{\mathbb{R}} \int_0^u f(y) \, dy \, d x.
\end{equation}
\end{itemize}
\end{theorem}

The following lemma  gives an  equivalent form of Definition \ref{def2} for blow up
of the solutions to \eqref{k1}-\eqref{k3}
using the subquintic growth of the  nonlinearity term \eqref{4.4}, \eqref{4.5}
i.e. $p_l<5$.
Let us underline, that the restriction on the growth of the nonlinear
term is essential for the result in Lemma~\ref{new} as well as in the
further statements.

\begin{lemma}\label{new}
Suppose $u(t,x)$ is the solution to \eqref{k1}-\eqref{k3}  in the maximal
existence time interval $[0,T_m)$, $0<T_m\leq\infty$.
Then the blow up of the $H^1$ norm of $u(t,x)$ is equivalent to the blow up of the
$L^2$ norm of  $u(t,x)$ at $T_m$, i.e. \\
$\limsup_{t\to T_m,\, t<T_m} \|u\|_1=\infty$ if and only if
\begin{equation}\label{equiv}
\limsup_{t\to T_m, t<T_m} \|u\|=\infty.
\end{equation}
\end{lemma}

\begin{proof} (Necessity)
Suppose \eqref{blow} holds. From the definition of $H^1$ norm it follows that
 either
$\limsup_{t\to T_m, t<T_m} \|u\|=\infty$, i.e. \eqref{equiv} is satisfied,
 or
\begin{equation}\label{nx}
\limsup_{t\to T_m, t<T_m} \|u_x\|=\infty
\end{equation}
and $\limsup_{t\to T_m, t<T_m} \|u\| <\infty$. Hence
\begin{equation}\label{c}
\|u(t,\cdot)\|\leq C_1
\end{equation}
holds for every $t\in[0,T_m)$ and some constant $C_1$.
From the Garliardo-Nirenberg inequality we get
\begin{equation}\label{GN}
\|u\|_{\mathrm{L}^{p+1}(\mathbb{R})}
\leq C_p \|u_x\|^{\frac{p-1}{2(p+1)}} \|u\|^{\frac{p+3}{2(p+1)}}
\leq C_p C_1^{\frac{p+3}{2(p+1)}}\|u_x\|^{\frac{p-1}{2(p+1)}}
\end{equation}
for every $p>1$ and some constant $C_p$ depending only on $p$.

By  Young's inequality for every $\varepsilon >0$  we have
\begin{equation}\label{Y}
\int_{\mathbb{R}} |u(t,x)|^{p+1} \, d x
\leq \varepsilon \|u_x\|^2 + \frac{5-p}{4} \big(\frac{p-1}{4}\big)^{\frac{p-1}{5-p}}
C_p^{\frac{4(p+1)}{5-p}}C_1^{\frac{2(p+3)}{5-p}} \varepsilon^{-\frac{p-1}{5-p}}.
\end{equation}
Applying \eqref{Y} for $p=p_k<5$, $k=1,2,\ldots, l$, and
$\varepsilon =\big(4\sum_{k=1}^{l}\frac{a_k}{p_k+1}\big)^{-1}$
we obtain from the conservation law  \eqref{4.6}
the estimate
\begin{align*}
E(0)
=& \frac{1}{2}\left((u_t,u_t) +(u,u) +\|u_x\|^2 \right)
- \sum_{k=1}^{l}\frac{a_k}{p_k+1} \int_{\mathbb{R}} |u|^{p_k+1}\, d x \\
&+ \sum_{j=1}^{s}\frac{b_j}{q_j+1} \int_{\mathbb{R}} |u|^{q_j+1}\, d x
 \geq \frac{1}{2}\|u_x\|^2 -\frac{1}{4}\|u_x\|^2 - C_2,
\end{align*}
where
$$
C_2=\sum_{k=1}^{l}\frac{a_k}{p_k+1}\frac{5-p_k}{4}
 \big(\frac{p_k-1}{4}\big)^{\frac{p_k-1}{5-p_k}} C_{p_k}^{\frac{4(p_k+1)}{5-p_k}}
 C_1^{\frac{2(p_k+3)}{5-p_k}}
\Big(4\sum_{k=1}^{l}\frac{a_k}{p_k+1}\Big)^{\frac{p_k-1}{5-p_k}} <\infty.
$$
Hence $\|u_x\|^2\leq 4(E(0)+C_2)<\infty$ for every $t\in[0,T_m)$ which
contradicts \eqref{nx}. Thus  $\limsup_{t\to T_m, t<T_m} \|u\| =\infty$.
\smallskip

\noindent (Sufficiency)
Suppose \eqref{equiv} holds. Then from the inequality $\|u\|\leq\|u\|_1$
it is obvious  that \eqref{blow} is satisfied.
The proof is complete.
\end{proof}

Later on we need the following auxiliary result.

\begin{lemma}\label{aux}
Suppose $u(t,x)$ is the  solution to \eqref{k1}-\eqref{k3}  defined in the
maximal existence time interval
 $[0,T_m)$, $0<T_m\leq\infty$. Then  the function $\Psi(t)=(u,u)$
satisfies the equation
\begin{equation} \label{4.10}
\Psi''(t) \Psi(t) - \frac{p_1 +3}{4} \Psi'^2(t) = (p_1 -1) \Psi^2(t)
-2(p_1+1)E(0)\Psi(t)+H(t),
\end{equation}
where
\begin{equation} \label{4.11}
\begin{split}
H(t)=&(p_1+3) \left[( u_t,u_t) (u, u)-(u, u_t)^2\right] \\
&+ \left[2(p_1+1)B(t)+(p_1-1)\|u_x\|^2) \right] (u,u)\geq 0
\end{split}
\end{equation}
and
\begin{equation} \label{4.9}
B(t)=\sum_{k=2}^l\frac{a_k(p_k-p_1)}{(p_k+1)(p_1+1)}
\int_{\mathbb{R}} |u|^{p_k+1} \, d x
+ \sum_{j=1}^s\frac{b_j(p_1-q_j)}{(q_j+1)(p_1+1)}
\int_{\mathbb{R}} |u|^{q_j+1} \, d x.
\end{equation}
\end{lemma}

\begin{proof}
By \eqref{k1} and \eqref{4.6},  we get the following identities for  $\Psi(t)$:
\begin{align*}
\Psi'(t)=2(u, u_t), \\
\Psi''(t)
&=2(u_t, u_t)+ 2 ( u, u_{tt})
 =2(u_t, u_t) - 2\|u\|_1^2  + 2 \int_{\mathbb{R}} u f(u) \, d x \\
&=(p_1+3)(u_t, u_t) -2(p_1+1)E(0)+(p_1-1)(u, u) \\
&\quad + (p_1-1)\|u_x\|^2 +2(p_1+1)B(t),
\end{align*}
where $B(t)$  is given by \eqref{4.9}.
From \eqref{4.5} we have
\begin{equation} \label{Bt}
B(t)\geq 0\quad \text{for }t\in[0,T_m).
\end{equation}
Substituting $\Psi'(t)$ and $\Psi''(t)$ in the left-hand side
 of \eqref{4.10}, we get that $\Psi(t)$ is a solution to \eqref{4.10}.
 Here $H(t)$ is given in \eqref{4.11} and $H(t)\geq 0$ from \eqref{Bt} and the Cauchy - Schwarz inequality. Lemma~\ref{aux} is proved.
\end{proof}

\begin{theorem}\label{Th8i}
Suppose $u(t,x)$ is the  solution to \eqref{k1}-\eqref{k3} defined in the maximal existence time interval
 $[0,T_m)$, $0<T_m\leq\infty$ and $E(0)>0$. If $u(t,x)$ blows up at $T_m$, then $T_m<\infty$.
\end{theorem}

\begin{proof}
From Lemma~\ref{aux} we get that the
 function $\Psi(t)=(u,u)$ satisfies in $[0,T_m)$  equation
\eqref{4.10}.
Hence,  $\Psi(t)$  is a solution to  \eqref{21} for
\begin{equation}\label{4.12}
\alpha=p_1-1,\quad\quad\quad~\beta=2(p_1+1)E(0)>0,\quad\quad\quad~\gamma=\frac{p_1+3}{4}>1
\end{equation}
and $H(t)$ defined in \eqref{4.11}.
From Lemma~\ref{new} it follows  that $\Psi(t)=(u,u)$ blows at $T_m$.
 Applying Theorem~\ref{Th0}  we get that $T_m<\infty$. Thus  the solution $u(t,x)$ blows up for a finite time $T_m<\infty$. Theorem~\ref{Th8i} is proved.
\end{proof}

\begin{theorem}\label{Th8ii}
Suppose $u(t,x)$ is the solution to \eqref{k1}-\eqref{k3}  defined in the
 maximal existence time interval
 $[0,T_m)$, $0<T_m\leq\infty$ and  $E(0)>0$. Then $u(t,x)$ blows up at
 $T_m$ if and only if  there exists $b\in[0,T_m)$ such that
\begin{equation}\label{4.7}
E(0)\leq\frac{p_1-1}{2(p_1+1)} (u(b,\cdot),u(b,\cdot))\quad \text{and}\quad
(u(b,\cdot),u_t(b,\cdot))>0.
\end{equation}
Moreover,
$$
T_m\leq b+ \frac{2}{(p_1-1)}\frac{(u(b,\cdot),u(b,\cdot))}{( u(b,\cdot),u_t(b,
\cdot) )}<\infty\,.
$$
\end{theorem}

\begin{proof} (Necessity)
Suppose $u(t,x)$ blows up at $T_m$, i.e. \eqref{blow} holds.
By Lemma~\ref{new} it follows that
$\limsup_{t\to T_m, t<T_m} \|u\|=\infty$, i.e.
 $\Psi(t)=(u,u)$ blows up at $T_m$.
 Then from Lemma~\ref{Lm1} for $M=2(p_1+1)E(0)/(p_1-1)$ and $b=t_0$
condition \eqref{4.7} is satisfied.
\smallskip

\noindent (Sufficiency)
Suppose \eqref{4.7} holds. We assume by contradiction that $u(t,x)$ does not
blow up at $T_m$, i.e according to Definition \ref{def2}  we have
\begin{equation}\label{bd}
\limsup_{t\to T_m, t<T_m} \|u(t,\cdot)\|_1<\infty.
\end{equation}
From the local existence result in Theorem~\ref{Th7}(i) it follows that
 $T_m=\infty$. Then  $\Psi(t)= ( u,u)$
satisfies   \eqref{21} in $[0,\infty)$ for $\alpha$, $\beta$, $\gamma$
defined in \eqref{4.12}. Note, that now $H(t)$, given in \eqref{4.11},
is a nonnegative function for every $t\geq 0$.
Moreover, condition \eqref{22} in Theorem~\ref{Th3} is fulfilled
from \eqref{4.7}.
Applying  Theorem \ref{Th3} we get that
 $\Psi(t)=(u,u)$ blows up at $T_m$, which contradicts our assumption
\eqref{bd}. The proof is complete.
\end{proof}

In the following theorems we give general sufficient conditions on the
initial data $u_0$ and $u_1$,  which guarantee finite time blow up of the
solutions to problem \eqref{k1}-\eqref{k3}.

\begin{theorem}\label{Th4.4}
Suppose $u(t,x)$ is the  solution to \eqref{k1}-\eqref{k3} with $E(0)>0$
 defined in the maximal existence time interval
 $[0,T_m)$, $0<T_m\leq\infty$. If $(u_0,u_1)>0$ and
\[
E(0)<\frac{1}{2}\frac{(u_0,u_1)^2}{(u_0,u_0)}+\frac{1}{2}(u_0,u_0)
- \big(\frac{p_1-1}{2}\big)^{\frac{p_1-1}{2}}
\big(\frac{( u_0,u_0)}{p_1+1}\big)^{\frac{p_1+1}{2}} E^{\frac{1-p_1}{2}}(0),
\]
then $u(t,x)$ blows up at $T_m<\infty$.
\end{theorem}


\begin{theorem}\label{Cor2}
Suppose $u(t,x)$ is the  solution to \eqref{k1}-\eqref{k3}  defined in the
 maximal existence time interval
 $[0,T_m)$, $0<T_m\leq\infty$ and $E(0)>0$. If $( u_0,u_1) >0$ and one of
the following conditions
\begin{itemize}
\item[(i)]  \cite{Wang-KG,Xu-Ding}
\[
E(0)< \frac{p_1-1}{2(p_1+1)}( u_0,u_0)\,,
\]
\item[(ii)]  \cite{KKD-MMAS}
\[
E(0)< \frac{1}{2}\frac{(u_0,u_1)^2}{(u_0,u_0)}+\frac{p_1-1}{2(p_1+1)}(u_0,u_0)\,,
\]
\item[(iii)]  \cite{KKD-FILOMAT}
\[
E(0) <\frac{1}{2}\frac{( u_0,u_1)^2}{(u_0,u_0)}+\frac{p_1-1}{2(p_1+1)}(u_0,u_0)
 +\frac{(u_0,u_0)}{p_1+1}
 \Big[1-\Big(1+\frac{( u_0,u_1)^2}{(u_0,u_0)^2}\Big)^{\frac{1-p_1}{2}}\Big]
\]
\end{itemize}
is satisfied, then $u(t,x)$ blows up at $T_m<\infty$.
\end{theorem}

Theorems \ref{Th4.4} and \ref{Cor2} follow from
Theorems \ref{Th3.1} and \ref{Cor}, respectively,
for $\alpha$, $\beta$, $\gamma$ defined in \eqref{4.12} and $\Psi(t)= (u,u)$,
$\Psi(0)= (u_0,u_0)$, $\Psi'(0)= 2(u_0,u_1)$.


\begin{remark} \rm
From  Lemma~\ref{aux}, neglecting the nonnegative terms $(p_1-1)\Psi^2(t)$
and $H(t)$ in \eqref{4.10}, it follows that $\Psi(t)=(u,u)$ satisfies
\eqref{1.3} for $\gamma=\frac{p_1+3}{4}>1$ and  $\beta=2(p_1+1)E(0)>0$.
By the idea of the proof of Theorem~\ref{Th3.1} one can get the result
in \cite{Korpusov,Straughan}, i.e. under the conditions:
\begin{equation}\label{str}
(u_0,u_1)>0, \quad 0<E(0)< \frac{1}{2}\frac{(u_0,u_1)^2}{(u_0,u_0)}
\end{equation}
the solution to \eqref{k1}-\eqref{k3} blows up for a finite time.
\end{remark}


\begin{remark} \rm
For the first time  condition (i) in Theorem~\ref{Cor2} was proposed in
\cite{Gazzola} for proving blow up of the  solution to semilinear wave
 equation with arbitrary high  initial energy.
Let us emphasize that this sufficient condition coincides with the necessary
 and sufficient one \eqref{4.7} formulated for $b=0$.
The other sufficient conditions (ii), (iii) in Theorem~\ref{Cor2} and \eqref{str}
guarantee the validity of condition \eqref{4.7} for some later time $t=b$.
We  can conclude that for any other sufficient condition for finite time blow up,
given at $t=0$,  the corresponding solution must satisfy \eqref{4.7} for
some $t=b>0$.
\end{remark}

\subsection{Double dispersive equation with linear restoring force}

In the space $\mathrm{L}^2(\mathbb{R})\cap \dot{\mathrm{H}}^{-1}(\mathbb{R})$
we define  the scalar product
\begin{equation} \label{br}
\langle u, v\rangle = \langle u(t,\cdot),v(t,\cdot)\rangle
= (u,v)+ \left((-\Delta)^{-1/2}u,(-\Delta)^{-1/2}v\right).
\end{equation}
Under the regularity conditions \eqref{d3} problem \eqref{d1}-\eqref{d3} has
 a unique solution
$$
u(t,x)\in \mathrm{C}([0,T_m); \mathrm{H}^1(\mathbb{R}))
\cap \mathrm{C}^1([0,T_m); \mathrm{L}^2(\mathbb{R}))
\cap \mathrm{C}^2([0,T_m); \mathrm{H}^{-1}(\mathbb{R}))
$$
on a maximal existence time interval $[0,T_m)$, $T_m\leq \infty$, and
if
$$
\limsup_{t\to T_m, t<T_m} \|u\|_1 <\infty\quad \text{then}\quad T_m=\infty.
$$
Moreover, for every $t\in[0,T_m)$ the solution  $u(t,x)$ to \eqref{d1}-\eqref{d3}
satisfies the conservation law $E(t)=E(0)$, where
\begin{equation}\label{cl}
E(t):=E(u(\cdot,t))
=\frac{1}{2}\left( \langle u_t, u_t \rangle + \langle u, u\rangle +\|u_x\|^2
\right)- \int_{\mathbb{R}} \int_0^u f(y) \, dy \, d x.
\end{equation}

The results for  double dispersive equation with linear restoring force
\eqref{d1}-\eqref{d3} are identical with the results for Klein-Gordon equation
\eqref{k1}-\eqref{k2}, proved in Subsection~4.1.
 The main deference is that the standard scalar product $(\cdot,\cdot)$ has
to be replaced with the scalar product $\langle \cdot,\cdot \rangle$ given
in \eqref{br}.
In particular, Lemma~\ref{new} holds also for the solutions to
\eqref{d1}-\eqref{d3}. However, in addition to Lemma~\ref{new},
 we need the following equivalence of  the blow up of $H^1$ norm of the
solution $u(t,x)$ to \eqref{d1}-\eqref{d3}   at $T_m$
and the blow up of $\langle u, u\rangle$  at $T_m$.

\begin{lemma}\label{neww}
Suppose $u(t,x)$ is the solution to \eqref{d1}-\eqref{d3}  in the maximal
existence time interval $[0,T_m)$, $0<T_m\leq\infty$. Then the blow up of $H^1$
norm of $u(t,x)$ at $T_m$ is equivalent to the blow up of
$\langle u(t,\cdot),u(t,\cdot)\rangle$  at $T_m$, i.e.
$\limsup_{t\to T_m, t<T_m} \|u\|_1=\infty$ if and only if
\begin{equation}\label{equivv}
\limsup_{t\to T_m, t<T_m} \langle u(t,\cdot),u(t,\cdot)\rangle=\infty.
\end{equation}
\end{lemma}

\begin{proof}
If $\limsup_{t\to T_m}\|u\|_1=\infty$ from Lemma~\ref{new} and definition
\eqref{br} it follows that $\langle u(t,\cdot),u(t,\cdot)\rangle$
blows up at $T_m$. Conversely, suppose that \eqref{equivv} holds but
\begin{equation}\label{bbb}
\limsup_{t\to T_m}\|u\|_1<\infty.
\end{equation}
From  definition \eqref{br} we get
$\limsup_{t\to T_m}\left((-\Delta)^{-1/2}u,(-\Delta)^{-1/2}u\right)=\infty$.
By the conservation law \eqref{cl} it follows that at least one of the norms
$\|u\|_{\mathrm{L}^{p_k}}$ tends to infinity for $t\to T_m$.
Hence, from the embedding of $\mathrm{H}^1$ into $\mathrm{L}^{p_k}$, $p_k>2$
we get that $\|u\|_1$  blows up at $T_m$, which contradict \eqref{bbb}.
The proof is complete.
\end{proof}

For a function $\Psi(t)=\langle u, u\rangle$ the statements in
Lemma~\ref{aux}, Theorem~\ref{Th8i},   Theorem~\ref{Th8ii},
and Theorem~\ref{Th4.4} are true for the solutions to problem
\eqref{d1}-\eqref{d3} with  the formal change of notation $(\cdot,\cdot)$
by $\langle \cdot,\cdot \rangle$. Below we
only formulate the corresponding  results without proofs.

\begin{theorem}
Suppose $u(t,x)$ is the solution to \eqref{d1}-\eqref{d3} defined in the
maximal existence time interval
 $[0,T_m)$, $0<T_m\leq\infty$ and $E(0)>0$. If $u(t,x)$ blows up at $T_m$
then $T_m<\infty$.
\end{theorem}

\begin{theorem} \label{ThDD}
Suppose $u(t,x)$ is the  solution to \eqref{d1}-\eqref{d3}  defined in the
maximal existence time interval
 $[0,T_m)$, $0<T_m\leq\infty$ and $E(0)>0$.
Then $u(t,x)$ blows up at $T_m$ if and only if there exists $b\in[0,T_m)$
such that
\begin{equation} \label{4.7d}
 E(0)\leq\frac{p_1-1}{2(p_1+1)}\langle u(b,\cdot),u(b,\cdot) \rangle\quad
\text{and}\quad \langle u(b,\cdot),u_t(b,\cdot) \rangle>0.
\end{equation}
Moreover,
$$
T_m\leq b+ \frac{2}{(p_1-1)}\frac{\langle u(b,\cdot),u(b,\cdot)
\rangle}{\langle u(b,\cdot),u_t(b,\cdot) \rangle}<\infty\,.
$$
\end{theorem}

\begin{theorem}
Suppose $u(t,x)$ is the  solution to \eqref{d1}-\eqref{d3} defined in the
maximal existence time interval  $[0,T_m)$, $0<T_m\leq\infty$ and $E(0)>0$.
If $\langle u_0,u_1\rangle >0$ and
\begin{align*}
E(0)<\frac{1}{2}\frac{\langle u_0,u_1\rangle^2}{\langle u_0,u_0\rangle}
+\frac{1}{2}\langle u_0,u_0\rangle
- \Big(\frac{p_1-1}{2}\Big)^{\frac{p_1-1}{2}}
\Big(\frac{\langle u_0,u_0\rangle}{p_1+1}\Big)^{\frac{p_1+1}{2}}
 E^{\frac{1-p_1}{2}}(0),
\end{align*}
then $u(t,x)$ blows up at $T_m<\infty$.
\end{theorem}

\subsection*{Acknowledgments}
This work is partially  supported by the Bulgarian Science Fund under
grant DFNI I-02/9. The authors thank to the anonymous referees for helpful
comments and suggestions.

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\end{document}










