# The semilinear Klein-Gordon equation
in de Sitter spacetime

Karen Yagdjian Correspondence: Department of Mathematics, University of Texas-
Pan American 1201 W. University Drive, Edinburg, TX 78541-2999, USA;
<EMAIL_ADDRESS>
###### Abstract

In this article we study the blow-up phenomena for the solutions of the
semilinear Klein-Gordon equation $\Box_{g}\phi-m^{2}\phi=-|\phi|^{p}$ with the
small mass $m\leq n/2$ in de Sitter space-time with the metric $g$. We prove
that for every $p>1$ the large energy solution blows up, while for the small
energy solutions we give a borderline $p=p(m,n)$ for the global in time
existence. The consideration is based on the representation formulas for the
solution of the Cauchy problem and on some generalizations of the Kato’s
lemma.

##  1 Introduction

In this article we study the blow-up phenomena for the solutions of the
semilinear Klein-Gordon equation $\Box_{g}\phi-m^{2}\phi=-|\phi|^{p}$ with the
small mass $m\leq n/2$ in de Sitter space-time.

In the model of the universe proposed by de Sitter the line element has the
form

$ds^{2}=-\left(1-\frac{2M_{bh}}{r}-\frac{\Lambda
r^{2}}{3}\right)c^{2}dt^{2}+\left(1-\frac{2M_{bh}}{r}-\frac{\Lambda
r^{2}}{3}\right)^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}d\phi^{2}).$

The constant $M_{bh}$ may have a meaning of the “mass of the black hole”. The
corresponding metric with this line element is called the Schwarzschild - de
Sitter metric.

The Cauchy problem for the semilinear Klein-Gordon equation in Minkowski
spacetime ($M_{bh}=\Lambda=0$) is well investigated. (See, e.g., [7] and
references therein.) In particular, Keel and Tao [7] for the semilinear
equation $u_{tt}-\Delta u=F(u)$, $u(0,x)=\varepsilon\varphi_{0}(x)$,
$u_{t}(0,x)=\varepsilon\varphi_{1}(x)$ proved that if $n=1,2,3$ and
$1<p<1+2n$, then there exists a (non-Hamiltonian) nonlinearity $F$ satisfying
$|D^{\alpha}F(u)|\leq C|u|^{p-|\alpha|}$ for $0\leq\alpha\leq[p]$ and such
that there is no finite energy global solution supported in the forward light
cone, for any nontrivial smooth compactly supported $\varphi_{0}$ and
$\varphi_{1}$ and for any $\varepsilon>0$. There is an interesting question of
instability of the ground state standing solutions $e^{i\omega
t}\phi_{\omega}(x)$ for nonlinear Klein-Gordon equation
$\partial_{t}^{2}u-\Delta u+u=|u|^{p-1}u$. Here $\phi_{\omega}$ is a ground
state of the equation $-\Delta\phi+(1-\omega^{2})\phi=|\phi|^{p-1}\phi,$ while
$0<p-1<4/(N-2)$ and $0\leq|\omega|<1$. Ohta and Todorova [9] showed that
instability occurs in the very strong sense that an arbitrarily small
perturbation of the initial data can make the perturbed solution blow up in
finite time.

The Cauchy problem for the linear wave equation without source term on the
maximally extended Schwarzschild - de Sitter spacetime in the case of non-
extremal black-hole corresponding to parameter values
$0<M_{bh}<\frac{1}{3\sqrt{\Lambda}}$, is considered by Dafermos and Rodnianski
[3]. They proved that in the region bounded by a set of black/white hole
horizons and cosmological horizons, solutions converge pointwise to a constant
faster than any given polynomial rate, where the decay is measured with
respect to natural future-directed advanced and retarded time coordinates. The
bounds on decay rates for solutions to the wave equation in the Schwarzschild
- de Sitter spacetime is a first step to a mathematical understanding of non-
linear stability problems for spacetimes containing black holes.

Catania and Georgiev [2] studied the Cauchy problem for the semilinear wave
equation $\Box_{g}\phi=|\phi|^{p}$ in the Schwarzschild metric
$(3+1)$-dimensional space-time, that is the case of $\Lambda=0$ in
$0<M_{bh}<\frac{1}{3\sqrt{\Lambda}}$. They established that the problem in the
Regge-Wheeler coordinates is locally well-posed in $H^{\sigma}$ for any
$\sigma\in[1,p+1)$. Then for the special choice of the initial data they
proved the blow-up of the solution in two cases: (a) $p\in(1,1+\sqrt{2})$ and
small initial data supported far away from the black hole; (b)
$p\in(2,1+\sqrt{2})$ and large data supported near the black hole. In both
cases, they also gave an estimate from above for the lifespan of the solution.

In the present paper we focus on the another limit case as $M_{bh}\to 0$ in
$0<M_{bh}<\frac{1}{3\sqrt{\Lambda}}$, namely, we set $M_{bh}=0$ to ignore
completely influence of the black hole. Thus, the line element in de Sitter
spacetime has the form

$ds^{2}=-\left(1-\frac{r^{2}}{R^{2}}\right)c^{2}\,dt^{2}+\left(1-\frac{r^{2}}{R^{2}}\right)^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2})\,.$

The Lamaître-Robertson transformation [8]

$r^{\prime}=\frac{r}{\sqrt{1-r^{2}/R^{2}}}e^{-ct/R}\,,\quad
t^{\prime}=t+\frac{R}{2c}\ln\left(1-\frac{r^{2}}{R^{2}}\right)\,,\quad\theta^{\prime}=\theta\,,\quad\phi^{\prime}=\phi,$

leads to the following form for the line element:
$ds^{2}=-c^{2}\,d{t^{\prime}}^{2}+e^{2ct^{\prime}/R}(d{r^{\prime}}^{2}+r^{\prime
2}\,d{\theta^{\prime}}^{2}+r^{\prime
2}\sin^{2}\theta^{\prime}\,d{\phi^{\prime}}^{2})$. By defining coordinates
$x^{\prime}$, $y^{\prime}$, $z^{\prime}$ connected with $r^{\prime}$,
$\theta^{\prime}$, $\phi^{\prime}$ by the usual equations connecting Cartesian
coordinates and polar coordinates in a Euclidean space, the line element may
be written [8, Sec.134]

$ds^{2}=-c^{2}\,d{t^{\prime}}^{2}+e^{2ct^{\prime}/R}(d{x^{\prime}}^{2}+d{y^{\prime}}^{2}+d{z^{\prime}}^{2})\,.$

The new coordinates $r^{\prime}$, $\theta^{\prime}$, $\phi^{\prime}$,
$t^{\prime}$ can take all values from $-\infty$ to $\infty$. Here $R$ is the
“radius” of the universe.

In this paper we study blow-up phenomena for semilinear equation by applying
the Lamaître-Robertson transformation and by employing the fundamental
solutions for some model linear hyperbolic equation with variable speed of
propagation. In [16] the Klein-Gordon operator in Robertson-Walker spacetime,
that is ${\mathcal{S}}:=\partial_{t}^{2}-e^{-2t}\bigtriangleup+M^{2}$, is
considered. The fundamental solution $E=E(x,t;x_{0},t_{0})$, that is solution
of ${\mathcal{S}}E=\delta(x-x_{0},t-t_{0})$, with a support in the forward
light cone $D_{+}(x_{0},t_{0})$, $x_{0}\in{\mathbb{R}}^{n}$,
$t_{0}\in{\mathbb{R}}$, and the fundamental solution with a support in the
backward light cone $D_{-}(x_{0},t_{0})$, $x_{0}\in{\mathbb{R}}^{n}$,
$t_{0}\in{\mathbb{R}}$, defined by
$D_{\pm}(x_{0},t_{0}):=\big{\\{}(x,t)\in{\mathbb{R}}^{n+1}\,;\,|x-x_{0}|$
$\leq\pm(e^{-t_{0}}-e^{-t})\,\big{\\}}$, are constructed. These fundamental
solutions have been used to represent solutions of the Cauchy problem and to
prove $L^{p}-L^{q}$ estimates for the solutions of the equation with and
without a source term that provide with some necessary tools for the studying
semilinear equations.

In the Robertson-Walker spacetime [5], one can choose coordinates so that the
metric has the form

$ds^{2}=-dt^{2}+S^{2}(t)d\sigma^{2}\,.$

In particular, the metric in de Sitter and anti-de Sitter spacetime in the
Lamaître-Robertson coordinates [8] has this form with $S(t)=e^{t}$ and
$S(t)=e^{-t}$, respectively. The matter waves in the de Sitter spacetime are
described by the function $\phi$, which satisfies equations of motion. In the
de Sitter universe the equation for the scalar field with mass $m$ and
potential function $V$ is the covariant Klein-Gordon equation

$\square_{g}\phi-m^{2}\phi=V^{\prime}(\phi)\quad\mbox{\rm
or}\quad\frac{1}{\sqrt{|g|}}\frac{\partial}{\partial
x^{i}}\left(\sqrt{|g|}g^{ik}\frac{\partial\phi}{\partial
x^{k}}\right)-m^{2}\phi=V^{\prime}(\phi)\,,$

with the usual summation convention. Written explicitly in coordinates in the
de Sitter spacetime it, in particular, for $V^{\prime}(\phi)=-|\phi|^{p}$ has
the form

$\phi_{tt}+n\phi_{t}-e^{-2t}\Delta\phi+m^{2}\phi=|\phi|^{p}\,.$ (1)

In this paper we restrict ourselves with consideration of the semilinear
equation for particle with small mass $m$, that is $0\leq m\leq n/2$. If we
introduce the new unknown function $u=e^{\frac{n}{2}t}\phi$, then it takes the
form of the semilinier Klein-Gordon equation for $u$ on de Sitter spacetime

$u_{tt}-e^{-2t}\bigtriangleup u-M^{2}u=e^{-\frac{n(p-1)}{2}t}|u|^{p},$ (2)

where non-negative curved mass $M\geq 0$ is defined as follows:

$M^{2}:=\frac{n^{2}}{4}-m^{2}\geq 0\,.$

The equation (2) can be regarded as Klein-Gordon equation with imaginary mass.
Equations with imaginary mass appear in several physical models such as
$\phi^{4}$ field model, tachion (super-light) fields, Landau-Ginzburg-Higgs
equation and others. To solve the Cauchy problem for semilinear equation we
use fundamental solution of the corresponding linear operator. We denote by
$G$ the resolving operator of the problem

$u_{tt}-e^{-2t}\bigtriangleup u-M^{2}u=f,\quad
u(x,0)=0,\quad\partial_{t}u(x,0)=0\,.$ (3)

Thus, $u=G[f]$. The equation of (3) is strictly hyperbolic. This implies the
well-posedness of the Cauchy problem (3) in the different functional spaces.
Consequently, the operator is well-defined in those functional spaces.

Then, the speed of propagation is variable, namely, it is equal to $e^{-t}$.
The second-order strictly hyperbolic equation (3) possesses two fundamental
solutions resolving the Cauchy problem without source term $f$. They can be
written in terms of the Fourier integral operators, which give complete
description of the wave front sets of the solutions. Moreover, the
integrability of the characteristic roots,
$\int_{0}^{\infty}|\lambda_{i}(t,\xi)|dt<\infty$, $i=1,2$, leads to the
existence of the so-called “horizon” for that equation. More precisely, any
signal emitted from the spatial point $x_{0}\in{\mathbb{R}}^{n}$ at time
$t_{0}\in{\mathbb{R}}$ remains inside the ball
$B^{n}_{t_{0}}(x_{0}):=\\{x\in{\mathbb{R}}^{n}\,|\,|x-x_{0}|<e^{-t_{0}}\\}$
for all time $t\in(t_{0},\infty)$. In particular, it can cause a nonexistence
of the $L^{p}-L^{q}$ decay for the solutions. In [13] this phenomenon is
illustrated by a model equation with permanently bounded domain of influence,
power decay of characteristic roots, and without $L^{p}-L^{q}$ decay. The
above mentioned $L^{p}-L^{q}$ decay estimates are one of the important tools
for studying nonlinear problems (see, e.g. [11]). In this paper we show that
this phenomenon causes the blow up of the solution. The equation (3) is
neither Lorentz invariant nor invariant with respect to usual scaling and that
creates additional difficulties.

Operator $G$ is constructed in [16] for the case of the large mass $m\geq
n/2$. The analytic continuation of this operator in parameter $M$ into
${\mathbb{C}}$ allows us to use $G$ also in the case of small mass $0\leq
m\leq n/2$. More precisely, we define the operator $G$ acting on $f(x,t)\in
C^{\infty}({\mathbb{R}}\times[0,\infty))$ by

$\displaystyle G[f](x,t)$ $\displaystyle:=$
$\displaystyle\int_{0}^{t}db\int_{x-(e^{-b}-e^{-t})}^{x+e^{-b}-e^{-t}}dy\,f(y,b)(4e^{-b-t})^{-M}\Big{(}(e^{-t}+e^{-b})^{2}-(x-y)^{2}\Big{)}^{-\frac{1}{2}+M}$
$\displaystyle\hskip 71.13188pt\times
F\Big{(}\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-(x-y)^{2}}{(e^{-b}+e^{-t})^{2}-(x-y)^{2}}\Big{)},$

where $F\big{(}a,b;c;\zeta\big{)}$ is the hypergeometric function.(See, e.g.,
[1]. For analytic continuation see , e.g., [12, Sec. 1.8] .) If $n$ is odd,
$n=2m+1$, $m\in{\mathbb{N}}$, then for $f\in
C^{\infty}({\mathbb{R}}^{n}\times[0,\infty))$, we define

$\displaystyle G[f](x,t)$ $\displaystyle:=$ $\displaystyle
2\int_{0}^{t}db\int_{0}^{e^{-b}-e^{-t}}dr_{1}\,\left(\frac{\partial}{\partial
r}\Big{(}\frac{1}{r}\frac{\partial}{\partial
r}\Big{)}^{\frac{n-3}{2}}\frac{r^{n-2}}{\omega_{n-1}c_{0}^{(n)}}\\!\\!\int_{S^{n-1}}f(x+ry,b)\,dS_{y}\right)_{r=r_{1}}$
$\displaystyle\hskip
71.13188pt\times(4e^{-b-t})^{-M}\left((e^{-t}+e^{-b})^{2}-r_{1}^{2}\right)^{-\frac{1}{2}+M}$
$\displaystyle\hskip 71.13188pt\times
F\left(\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-r_{1}^{2}}{(e^{-b}+e^{-t})^{2}-r_{1}^{2}}\right)\\!\\!,$

where $c_{0}^{(n)}=1\cdot 3\cdot\ldots\cdot(n-2)$. Constant $\omega_{n-1}$ is
the area of the unit sphere $S^{n-1}\subset{\mathbb{R}}^{n}$.

If $n$ is even, $n=2m$, $m\in{\mathbb{N}}$, then for $f\in
C^{\infty}({\mathbb{R}}^{n}\times[0,\infty))$, the operator $G$ is given by
the next expression

$\displaystyle G[f](x,t)$ $\displaystyle:=$ $\displaystyle
2\int_{0}^{t}db\int_{0}^{e^{-b}-e^{-t}}dr_{1}\,\left(\frac{\partial}{\partial
r}\Big{(}\frac{1}{r}\frac{\partial}{\partial
r}\Big{)}^{\frac{n-2}{2}}\frac{2r^{n-1}}{\omega_{n-1}c_{0}^{(n)}}\\!\\!\int_{B_{1}^{n}(0)}\frac{f(x+ry,b)}{\sqrt{1-|y|^{2}}}\,dV_{y}\right)_{r=r_{1}}$
$\displaystyle\hskip
71.13188pt\times(4e^{-b-t})^{-M}\left((e^{-t}+e^{-b})^{2}-r_{1}^{2}\right)^{-\frac{1}{2}+M}$
$\displaystyle\hskip 71.13188pt\times
F\left(\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-r_{1}^{2}}{(e^{-b}+e^{-t})^{2}-r_{1}^{2}}\right)\\!\\!.$

Here $B_{1}^{n}(0):=\\{|y|\leq 1\\}$ is the unit ball in ${\mathbb{R}}^{n}$,
while $c_{0}^{(n)}=1\cdot 3\cdot\ldots\cdot(n-1)$. Thus, in both cases, of
even and odd $n$, one can write

$\displaystyle u(x,t)$ $\displaystyle=$ $\displaystyle
2\int_{0}^{t}db\int_{0}^{e^{-b}-e^{-t}}dr\,v(x,r;b)(4e^{-b-t})^{-M}\left((e^{-t}+e^{-b})^{2}-r^{2}\right)^{-\frac{1}{2}+M}$
$\displaystyle\hskip 71.13188pt\times
F\left(\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-r^{2}}{(e^{-b}+e^{-t})^{2}-r^{2}}\right),$

where the function $v(x,t;b)$ is a solution to the Cauchy problem for the wave
equation

$v_{tt}-\bigtriangleup v=0\,,\quad v(x,0;b)=f(x,b)\,,\quad v_{t}(x,0;b)=0\,.$

It can be proved that if $n\big{(}1-\frac{2}{q}\big{)}\leq 1$, then for every
given $T>0$ the operator $G$ can be extended to the bounded operator:

$G\,:\,C([0,T];L^{q^{\prime}}({\mathbb{R}}^{n}))\longrightarrow
C([0,T];L^{q}({\mathbb{R}}^{n}))\,.$

Consequently the operator $G$ maps

$G\,:\,C([0,\infty);L^{q^{\prime}}({\mathbb{R}}^{n}))\longrightarrow
C([0,\infty);L^{q}({\mathbb{R}}^{n})),$

in the corresponding topologies. Moreover,

$G\,:\,C([0,\infty);L^{q^{\prime}}({\mathbb{R}}^{n}))\longrightarrow
C^{1}([0,\infty);{\mathcal{D}}^{\prime}({\mathbb{R}}^{n})).$

Let $u_{0}=u_{0}(x,t)$ be a solution of the Cauchy problem

$\partial_{t}^{2}u_{0}-e^{-2t}\bigtriangleup u_{0}-M^{2}u_{0}=0,\quad
u_{0}(x,0)=\varphi_{0}(x),\quad\partial_{t}u_{0}(x,0)=\varphi_{1}(x)\,.$ (4)

Then any solution $u=u(x,t)$ of the equation (2) which takes initial value
$u(x,0)=\varphi_{0}(x),\quad\partial_{t}u(x,0)=\varphi_{1}(x)$, solves also
integral equation

$u(x,t)=u_{0}(x,t)+G[e^{-\frac{n(p-1)}{2}\cdot}|u|^{p}](x,t)\,.$ (5)

Let $\Gamma\in C([0,\infty))$. For every given function $u_{0}\in
C([0,T];L^{q^{\prime}}({\mathbb{R}}^{n}))$ we consider integral equation (5)

$u(x,t)=u_{0}(x,t)+G\left[\Gamma(\cdot)\left(\int_{{\mathbb{R}}^{n}}|u(y,\cdot)|^{p}dy\right)^{\beta}|u(y,\cdot)|^{p}\right](x,t)\,,$
(6)

for the function $u\in C([0,T];L^{q}({\mathbb{R}}^{n}))\cap
C([0,T];L^{p}({\mathbb{R}}^{n}))$. Here $q^{\prime}\geq q>1$, $p\geq 1$. The
last integral equation corresponds to the slightly more general equation than
(2), namely, to the nonlocal equation

$u_{tt}-e^{-2t}\bigtriangleup
u-M^{2}u=\Gamma(t)\left(\int_{{\mathbb{R}}^{n}}|u(y,t)|^{p}dy\right)^{\beta}|u|^{p}\,.$
(7)

If $u_{0}$ is generated by the Cauchy problem (4), then the solution
$u=u(x,t)$ of (6) is said to be a weak solution of the Cauchy problem with the
initial conditions

$u(x,0)=\varphi_{0}(x),\quad\partial_{t}u(x,0)=\varphi_{1}(x)\,,$

for the equation (7). In the present paper we are looking for the conditions
on the function $\Gamma$, on constants $M$, $n$, $p$, and $\beta$ that
guarantee a non-existence of global in time weak solution, namely, the blow-up
phenomena. We are especially interested in the scale of functions
$\Gamma(t)=(1+t)^{d_{1}}e^{d_{0}t}$, where $d_{0},d_{1}\in{\mathbb{R}}$. The
function $\,e^{-\frac{n(p-1)}{2}t}$ with $d_{0}=-n(p-1)/2$ and $d_{1}=0$ is in
that scale and represents equation (2) if $\beta=0$. In particular, we find in
the next theorem the upper bound for $d_{0}$ with an existence of the global
solution for small initial data. For equation (7) in that scale the bound is
given by $d_{0}\geq-M(p(\beta+1)-1)$ and $d_{1}>2$ if $M>0$.

###### Theorem 1.1

Suppose that function $\Gamma\in C^{1}([0,\infty))$ is either non-decreasing
or non-increasing, and if $M>0$ then

$\displaystyle\Gamma(t)\geq
ce^{-M(p(\beta+1)-1)t}t^{2+\varepsilon}\quad\mbox{\rm for all}\quad
t\in[0,\infty),$

with the numbers $\varepsilon>0$ and $c>0$, while for $M=0$ it satisfies

$\displaystyle\Gamma(t)\geq ct^{-1-p(\beta+1)}\,.$

Then, for every $p>1$, $N$, and $\varepsilon$ there exists $u_{0}\in
C^{\infty}({\mathbb{R}}^{n}\times[0,\infty))$ which for any given slice of
constant time $t=const\geq 0$ has a compact support in $x$, such that
$u_{0}(x,0),\partial_{t}u_{0}(x,0)\in C^{\infty}_{0}({\mathbb{R}}^{n})$, and

$\|u_{0}(x,0)\|_{C^{N}({\mathbb{R}}^{n})}+\|\partial_{t}u_{0}(x,0)\|_{C^{N}({\mathbb{R}}^{n})}<\varepsilon$

but a global in time solution $u\in C([0,\infty);L^{q}({\mathbb{R}}^{n}))$ of
the equation (6) with permanently bounded support does not exist for all
$q\in[2,\infty)$ and $\beta>1/p-1$. More precisely, there is $T>0$ such that

$\lim_{t\nearrow T}\int_{{\mathbb{R}}^{n}}u(x,t)dx=\infty\,.$

This theorem shows that instability of the trivial solution occurs in the very
strong sense, that is, an arbitrarily small perturbation of the initial data
can make the perturbed solution blowing up in finite time.

If we allow large initial data, then according to the next theorem, for every
$d_{0}\in{\mathbb{R}}$ and $M>0$ the solution blows up in finite time.

###### Theorem 1.2

Suppose that function $\Gamma(t)=e^{\gamma t}$, where $\gamma\in{\mathbb{R}}$
and that the curved mass is positive, $M>0$. Then, for every $p>1$ and $n$
there exists $u_{0}\in C^{\infty}({\mathbb{R}}^{n}\times[0,\infty))$ which for
any given slice of constant time $t=const\geq 0$ has a compact support in $x$,
such that $u_{0}(x,0),\partial_{t}u_{0}(x,0)\in
C^{\infty}_{0}({\mathbb{R}}^{n})$ but a global in time solution $u\in
C([0,\infty);L^{q}({\mathbb{R}}^{n}))$ of the equation (6) with permanently
bounded support does not exist for all $q\in[2,\infty)$ and $\beta>1/p-1$.
More precisely, there is $T>0$ such that

$\lim_{t\nearrow T}\int_{{\mathbb{R}}^{n}}u(x,t)\,dx=\infty\,.$

Thus, for every $p>1$ the large energy classical solution of the Cauchy for
equation (1) blows up. We will prove global existence of the small energy
solution in a forthcoming paper.

The remaining part of this paper is organized as follows. In Section 2 we
prove some auxiliary integral representations for the function $\sinh(t)$ and
the linear function via Gauss’s hypergeometric function and multidimensional
integrals involving also fundamental solution of the Cauchy problem for wave
equation in Minkowski spacetime. In Section 3 we suggest two simple
generalizations of Kato’s lemma, which allow us to handle the case of
differential inequalities with exponentially decreasing kernels. In Section 4
we complete the proofs of both theorems.

##  2 Integral representations of function $M^{-1}\sinh(M(t-b))$ involving
hypergeometric function

In [1, Sec. 2.4] one can find one-dimensional integrals involving
hypergeometric function. In this section we present one more example of such
integral and also examples of multidimensional integrals appearing in the
fundamental solutions for the Klein-Gordon equation in de Sitter spacetime.
More examples related to the Tricomi and Gellerstedt equations one can find in
[14].

###### Proposition 2.1

The function $M^{-1}\sinh(M(t-b))$ with $t\geq b\geq 0$, can be represented as
follows:
(i) The one-dimensional integral

$\displaystyle\frac{1}{M}\sinh(M(t-b))$ $\displaystyle=$
$\displaystyle\int_{-(e^{-b}-e^{-t})}^{e^{-b}-e^{-t}}(4e^{-b-t})^{-M}\Big{(}(e^{-t}+e^{-b})^{2}-z^{2}\Big{)}^{-\frac{1}{2}+M}$
$\displaystyle\times
F\Big{(}\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-z^{2}}{(e^{-b}+e^{-t})^{2}-z^{2}}\Big{)}\,dz;$

(ii) If $n$ is odd, $n=2m+1$, $m\in{\mathbb{N}}$, then with
$c_{0}^{(n)}=1\cdot 3\cdot\ldots\cdot(n-2)$,

$\displaystyle\frac{1}{M}\sinh(M(t-b))$ $\displaystyle=$ $\displaystyle
2\int_{0}^{e^{-b}-e^{-t}}dr_{1}\,\left(\frac{\partial}{\partial
r}\Big{(}\frac{1}{r}\frac{\partial}{\partial
r}\Big{)}^{\frac{n-3}{2}}\frac{r^{n-2}}{\omega_{n-1}c_{0}^{(n)}}\\!\\!\int_{S^{n-1}}\,dS_{y}\right)_{r=r_{1}}\\!\\!(4e^{-b-t})^{-M}$
$\displaystyle\times\left((e^{-t}+e^{-b})^{2}-r_{1}^{2}\right)^{-\frac{1}{2}+M}F\left(\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-r_{1}^{2}}{(e^{-b}+e^{-t})^{2}-r_{1}^{2}}\right);$

(iii) If $n$ is even, $n=2m$, $m\in{\mathbb{N}}$, then with
$c_{0}^{(n)}=1\cdot 3\cdot\ldots\cdot(n-1)$,

$\displaystyle\frac{1}{M}\sinh(M(t-b))$ $\displaystyle=$ $\displaystyle
2\int_{0}^{e^{-b}-e^{-t}}\\!\\!dr_{1}\\!\\!\left(\frac{\partial}{\partial
r}\Big{(}\frac{1}{r}\frac{\partial}{\partial
r}\Big{)}^{\frac{n-2}{2}}\frac{2r^{n-1}}{\omega_{n-1}c_{0}^{(n)}}\\!\\!\int_{B_{1}^{n}(0)}\frac{1}{\sqrt{1-|y|^{2}}}dV_{y}\right)_{r=r_{1}}$
$\displaystyle\hskip
71.13188pt\times(4e^{-b-t})^{-M}\left((e^{-t}+e^{-b})^{2}-r_{1}^{2}\right)^{-\frac{1}{2}+M}$
$\displaystyle\hskip 71.13188pt\times
F\left(\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-r_{1}^{2}}{(e^{-b}+e^{-t})^{2}-r_{1}^{2}}\right)\\!\\!.$

Here the constant $\omega_{n-1}$ is the area of the unit sphere
$S^{n-1}\subset{\mathbb{R}}^{n}$.

Proof. First we consider case (i). According to Theorem 0.3 [16] for every
function $f\in$ $C^{\infty}({\mathbb{R}}\times[0,\infty))$, which for any
given slice of constant time $t=const\geq 0$ has a compact support in $x$, the
function

$\displaystyle v(x,t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}db\int_{x-(e^{-b}-e^{-t})}^{x+e^{-b}-e^{-t}}dy\,(4e^{-b-t})^{-M}\Big{(}(e^{-t}+e^{-b})^{2}-(x-y)^{2}\Big{)}^{-\frac{1}{2}+M}$
$\displaystyle\hskip 65.44142pt\times
F\Big{(}\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-(x-y)^{2}}{(e^{-b}+e^{-t})^{2}-(x-y)^{2}}\Big{)}f(y,b)$

is a unique $C^{\infty}$-solution to the Cauchy problem

$\partial_{t}^{2}v-e^{-2t}\bigtriangleup v-M^{2}v=f,\quad
v(x,0)=0,\quad\partial_{t}v(x,0)=0$ (9)

with $n=1$. It follows

$\displaystyle\int_{-\infty}^{\infty}v(x,t)dx$ $\displaystyle=$
$\displaystyle\int_{0}^{t}db\Big{(}\int_{-\infty}^{\infty}f(x,b)dx\Big{)}\int_{-(e^{-b}-e^{-t})}^{e^{-b}-e^{-t}}dz(4e^{-b-t})^{-M}\Big{(}(e^{-t}+e^{-b})^{2}-z^{2}\Big{)}^{-\frac{1}{2}+M}$
$\displaystyle\hskip 71.13188pt\times
F\Big{(}\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-z^{2}}{(e^{-b}+e^{-t})^{2}-z^{2}}\Big{)}.$

On the other hand, from the linear Klein-Gordon equation (9) and the vanishing
initial data, we obtain

$\displaystyle\int_{-\infty}^{\infty}v(x,t)dx-M^{2}\int_{0}^{t}d\tau\int_{0}^{\tau}db\int_{-\infty}^{\infty}v(x,b)dx$
$\displaystyle=$
$\displaystyle\int_{0}^{t}d\tau\int_{0}^{\tau}db\int_{-\infty}^{\infty}e^{-2b}\partial_{x}^{2}v(x,b)dx+\int_{0}^{t}d\tau\int_{0}^{\tau}db\int_{-\infty}^{\infty}f(x,b)\,dx\,,$

that is

$\displaystyle\int_{-\infty}^{\infty}v(x,t)dx-M^{2}\int_{0}^{t}d\tau\int_{0}^{\tau}db\int_{-\infty}^{\infty}v(x,b)dx$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\Big{(}\int_{-\infty}^{\infty}f(x,b)\,dx\Big{)}(t-b)db.$

Denote

$\displaystyle
V(t)=\int_{0}^{t}d\tau\int_{0}^{\tau}db\int_{-\infty}^{\infty}v(x,b)dx,\qquad
F(t):=\int_{0}^{t}\Big{(}\int_{-\infty}^{\infty}f(x,b)\,dx\Big{)}(t-b)db\,,$
(12)

then (2) and (12) imply

$V_{tt}-M^{2}V=F,\qquad V(0)=V_{t}(0)=0\,.$

We easily find

$\displaystyle V(t)$ $\displaystyle=$
$\displaystyle\frac{1}{M}\int_{0}^{t}F(\tau)\sinh(M(t-\tau))d\tau\,.$

Then (2) implies

$\displaystyle\int_{-\infty}^{\infty}v(x,t)dx$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\Big{(}\int_{-\infty}^{\infty}f(x,b)\,dx\Big{)}(t-b)db+M\int_{0}^{t}F(\tau)\sinh(M(t-\tau))d\tau$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\Big{(}\int_{-\infty}^{\infty}f(x,b)\,dx\Big{)}(t-b)db$
$\displaystyle+M\int_{0}^{t}db\Big{(}\int_{-\infty}^{\infty}f(x,b)\,dx\Big{)}\int_{b}^{t}d\tau(\tau-b)\sinh(M(t-\tau)).$

On the other hand

$\int_{b}^{t}d\tau(\tau-b)\sinh(M(t-\tau))=-\frac{1}{M}(t-b)+\frac{1}{M^{2}}\sinh(M(t-b))$

implies

$\displaystyle\int_{-\infty}^{\infty}v(x,t)dx$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\Big{(}\int_{-\infty}^{\infty}f(x,b)\,dx\Big{)}(t-b)db$
$\displaystyle+M\int_{0}^{t}db\Big{(}\int_{-\infty}^{\infty}f(x,b)\,dx\Big{)}\Big{(}-\frac{1}{M}(t-b)+\frac{1}{M^{2}}\sinh(M(t-b)))\Big{)}$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\Big{(}\int_{-\infty}^{\infty}f(x,b)\,dx\Big{)}\Big{(}\frac{1}{M}\sinh(M(t-b))\Big{)}db\,.$

Thus, for the arbitrary function $f\in
C^{\infty}({\mathbb{R}}\times[0,\infty))$ for all $t$ due to (2) one has

$\displaystyle\int_{0}^{t}\Big{(}\int_{-\infty}^{\infty}f(x,b)\,dx\Big{)}\Big{(}\frac{1}{M}\sinh(M(t-b))\Big{)}db$
$\displaystyle=$
$\displaystyle\int_{0}^{t}db\Big{(}\int_{-\infty}^{\infty}f(x,b)\,dx\Big{)}\int_{-(e^{-b}-e^{-t})}^{e^{-b}-e^{-t}}dz\,(4e^{-b-t})^{-M}\Big{(}(e^{-t}+e^{-b})^{2}-z^{2}\Big{)}^{-\frac{1}{2}+M}$
$\displaystyle\hskip 71.13188pt\times
F\Big{(}\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-z^{2}}{(e^{-b}+e^{-t})^{2}-z^{2}}\Big{)}\,.$

It follows (2.1). Thus (i) is proved.

To prove case (ii) with $n$ is odd, $n=2m+1$, $m\in{\mathbb{N}}$, we use the
identity

$1=\frac{\partial}{\partial r}\Big{(}\frac{1}{r}\frac{\partial}{\partial
r}\Big{)}^{\frac{n-3}{2}}\frac{r^{n-2}}{\omega_{n-1}c_{0}^{(n)}}\int_{S^{n-1}}\,dS_{y}$

and take into consideration that the kernel

$(4e^{-b-t})^{-M}\left((e^{-t}+e^{-b})^{2}-r_{1}^{2}\right)^{-\frac{1}{2}+M}F\left(\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-r_{1}^{2}}{(e^{-b}+e^{-t})^{2}-r_{1}^{2}}\right)\\!\\!$

is an even function of $r_{1}$. In the case of (iii) when $n$ is even, $n=2m$,
$m\in{\mathbb{N}}$, we apply the identity

$1=\frac{\partial}{\partial r}\Big{(}\frac{1}{r}\frac{\partial}{\partial
r}\Big{)}^{\frac{n-2}{2}}\frac{2r^{n-2}}{\omega_{n-1}c_{0}^{(n)}}\int_{B_{1}^{n}(0)}\frac{1}{\sqrt{1-|y|^{2}}}\,dV_{y}\,.$

The proposition is proven. $\Box$

If we set in the above integrals $b=0$ then we get integral representations of
the function $\sinh(Mt)$ depending on parameter $M>0$. We also note that both
sides of these formulas are translation invariant in $t$. By passing to the
limit as $M\to 0$ we arrive at the following corollary.

###### Corollary 2.2

The function $t-b$ with $t\geq b\geq 0$, can be represented as follows:
(i) The one-dimensional integral

$\displaystyle t-b$ $\displaystyle=$
$\displaystyle\int_{-(e^{-b}-e^{-t})}^{e^{-b}-e^{-t}}\Big{(}(e^{-t}+e^{-b})^{2}-z^{2}\Big{)}^{-\frac{1}{2}}F\Big{(}\frac{1}{2},\frac{1}{2};1;\frac{(e^{-b}-e^{-t})^{2}-z^{2}}{(e^{-b}+e^{-t})^{2}-z^{2}}\Big{)}\,dz;$

(ii) If $n$ is odd, $n=2m+1$, $m\in{\mathbb{N}}$, then with
$c_{0}^{(n)}=1\cdot 3\cdot\ldots\cdot(n-2)$,

$\displaystyle t-b$ $\displaystyle=$ $\displaystyle
2\int_{0}^{e^{-b}-e^{-t}}dr_{1}\,\left(\frac{\partial}{\partial
r}\Big{(}\frac{1}{r}\frac{\partial}{\partial
r}\Big{)}^{\frac{n-3}{2}}\frac{r^{n-2}}{\omega_{n-1}c_{0}^{(n)}}\\!\\!\int_{S^{n-1}}\,dS_{y}\right)_{r=r_{1}}$
$\displaystyle\times\left((e^{-t}+e^{-b})^{2}-r_{1}^{2}\right)^{-\frac{1}{2}}F\left(\frac{1}{2},\frac{1}{2};1;\frac{(e^{-b}-e^{-t})^{2}-r_{1}^{2}}{(e^{-b}+e^{-t})^{2}-r_{1}^{2}}\right);$

(iii) If $n$ is even, $n=2m$, $m\in{\mathbb{N}}$, then with
$c_{0}^{(n)}=1\cdot 3\cdot\ldots\cdot(n-1)$,

$\displaystyle t-b$ $\displaystyle=$ $\displaystyle
2\int_{0}^{e^{-b}-e^{-t}}dr_{1}\,\left(\frac{\partial}{\partial
r}\Big{(}\frac{1}{r}\frac{\partial}{\partial
r}\Big{)}^{\frac{n-2}{2}}\frac{2r^{n-1}}{\omega_{n-1}c_{0}^{(n)}}\\!\\!\int_{B_{1}^{n}(0)}\frac{1}{\sqrt{1-|y|^{2}}}\,dV_{y}\right)_{r=r_{1}}$
$\displaystyle\times\left((e^{-t}+e^{-b})^{2}-r_{1}^{2}\right)^{-\frac{1}{2}}F\left(\frac{1}{2},\frac{1}{2};1;\frac{(e^{-b}-e^{-t})^{2}-r_{1}^{2}}{(e^{-b}+e^{-t})^{2}-r_{1}^{2}}\right)\\!\\!.$

##  3 The second order differential inequalities

The second order differential inequality with the power decreasing kernel play
key role in proving blow-up of the solutions of the semilinear equations.
Kato’s lemma [6] allows us to derive from inequality

$\ddot{w}\geq bt^{-1-p}w^{p},\qquad p>1,\,\,b>0,\quad t\,\,\,\mbox{\rm large}$

a boundedness of the life-span of solution with property $w_{t}\geq a>0$. For
the equation in de Sitter spacetime the kernel of the corresponding ordinary
differential inequality decreases exponentially:

$\ddot{w}\geq be^{-Mt}w^{p},\qquad p>1,\,\,b>0,\,M>0,\quad t\,\,\,\mbox{\rm
large}.$

There is a global solution to the last inequality. Hence, in order to reach
exact conditions on the involving functions we have to generalize Kato’s
lemma. It is done in two following lemmas.

###### Lemma 3.1

Suppose $F(t)\in C^{2}([a,b))$, and

$\displaystyle F(t)\geq 0\,,\qquad\dot{F}(t)\geq
0\,,\qquad\ddot{F}(t)\geq\Gamma(t)F(t)^{p}\quad\mbox{\rm for all}\quad
t\in[a,b)\,,$

where $\Gamma\in C^{1}([a,\infty))$ is non-negative function, $\Gamma(a)>0$,
and $p>1$. Assume that for all $t\in[a,b)$ either

$\displaystyle\dot{\Gamma}(t)\leq 0\quad\mbox{\rm or }\quad\Gamma(t)\geq
const>0\,.$

If there exists $a_{1}\in(a,b)$ such that

$\frac{1}{\sqrt{p+1}}\int_{a}^{a_{1}}\Gamma(s)^{1/2}ds>\frac{\sqrt{2}}{p-1}F(a)^{(1-p)/2}\,,\quad\dot{F}(a)^{2}\geq\frac{2}{p+1}\Gamma(a)F(a)^{p+1},$
(13)

then $b$ must be finite unless $\lim_{t\to\infty}F(t)$ is finite.

Proof. First we consider the case of $\dot{\Gamma}\leq 0$. The conditions of
the lemma imply that derivative of the energy density function is non-
negative,

$\frac{d}{dt}\left(F_{t}(t)^{2}-\frac{2}{p+1}\Gamma(t)F(t)^{p+1}\right)\geq
0\quad\mbox{\rm for all}\quad t\in[a,b)\,.$

We integrate the last inequality and obtain

$F_{t}(t)^{2}\geq\frac{2}{p+1}\Gamma(t)F(t)^{p+1}+F_{t}(a)^{2}-\frac{2}{p+1}\Gamma(a)F(a)^{p+1}\quad\mbox{\rm
for all}\quad t\in[a,b)\,.$

In fact, according to the second inequality of the condition (13) we have

$F_{t}(t)^{2}\geq\frac{2}{p+1}\Gamma(t)F(t)^{p+1}\quad\mbox{\rm for all}\quad
t\in[a,b)\,.$

Hence,

$F_{t}(t)\geq\sqrt{\frac{2}{p+1}}\Gamma(t)^{1/2}F(t)^{(p+1)/2}\quad\mbox{\rm
for all}\quad t\in[a,b)\,.$

It follows

$\frac{d}{dt}\left(\frac{2}{1-p}F^{1-(p+1)/2}(t)\right)\geq\sqrt{\frac{2}{p+1}}\Gamma(t)^{1/2}\quad\mbox{\rm
for all}\quad t\in[a,b)\,.$

Consequently,

$\frac{2}{1-p}F^{(1-p)/2}(t)-\frac{2}{1-p}F^{(1-p)/2}(a)\geq\sqrt{\frac{2}{p+1}}\int_{a}^{t}\Gamma(s)^{1/2}ds\,.$

According to the first inequality of the condition (13) there exists $a_{1}>a$
such that

$\displaystyle\frac{2}{1-p}F^{(1-p)/2}(t)$ $\displaystyle\geq$
$\displaystyle\sqrt{\frac{2}{p+1}}\int_{a_{1}}^{t}\Gamma(s)^{1/2}ds+\sqrt{\frac{2}{p+1}}\int_{a}^{a_{1}}\Gamma(s)^{1/2}ds-\frac{2}{p-1}F^{(1-p)/2}(a)$
$\displaystyle\geq$
$\displaystyle\sqrt{\frac{2}{p+1}}\int_{a_{1}}^{t}\Gamma(s)^{1/2}ds$

for all $t\in[a_{1},b)$. Thus, for large $t$ we get contradiction. The case of
uniformly positive function $\Gamma$ follows from Kato’s Lemma [6]. Lemma is
proven. $\Box$

Next we turn to the case of the small energy and exponentially decreasing
$\Gamma(t)$.

###### Lemma 3.2

Suppose $F(t)\in C^{2}([a,b))$, and

$F(t)\geq c_{0}A(t),\quad F_{t}(t)\geq 0,\quad
F_{tt}(t)\geq\gamma(t)A(t)^{-p}F(t)^{p}\quad\mbox{\rm for all}\,\,t\in[a,b),$
(14)

where $A,\gamma\in C^{1}([a,\infty))$ are non-negative functions and $p>1$,
$c_{0}>0$. Assume that

$\displaystyle\lim_{t\to\infty}A(t)=\infty\,,$ (15)

and that

$\displaystyle\frac{d}{dt}\left(\gamma(t)A(t)^{-p}\right)\leq 0\quad\mbox{\rm
for all}\quad t\in[a,b)\,.$ (16)

If there exist $\varepsilon>0$ and $c>0$ such that

$\displaystyle\gamma(t)\geq cA(t)(\ln A(t))^{2+\varepsilon}\quad\mbox{\rm for
all}\quad t\in[a,b),$ (17)

then $b$ must be finite.

Proof. There is a point $a_{1}\geq a$ such that ${F}_{t}(a_{1})>0$. Then
$F_{t}(t)\geq F_{t}(a_{1})$ for all $t\geq a_{1}$ and consequently

$\displaystyle{F}(t)\geq\frac{1}{2}{F}_{t}(a_{1})t\quad\mbox{\rm for all}\quad
t\in[a_{2},b)\,,$

for sufficiently large $a_{2}$. Furthermore, according to (16) for the energy
density function we have

$\frac{d}{dt}\left(F_{t}(t)^{2}-2\frac{1}{p+1}\gamma(t)A(t)^{-p}F(t)^{p+1}\right)\geq
0\quad\mbox{\rm for all}\quad t\in[a_{1},b)\,.$

The last inequality implies

$F_{t}(t)^{2}\geq
2\frac{1}{p+1}\gamma(t)A(t)^{-p}F(t)^{p+1}+F_{t}(a_{1})^{2}-2\frac{1}{p+1}\gamma(a_{1})A(a_{1})^{-p}F(a_{1})^{p+1}$

for all $t\in[a_{1},b)$. For sufficiently large $a_{2}\geq a_{1}$ using
conditions (14), (15), and (17) we derive

$\displaystyle\frac{1}{p+1}\gamma(t)A(t)^{-p}F(t)^{p+1}$ $\displaystyle\geq$
$\displaystyle\frac{1}{p+1}c_{0}^{p+1}\gamma(t)A(t)$ $\displaystyle\geq$
$\displaystyle\frac{1}{p+1}cc_{0}^{p+1}A(t)^{2}(\ln A(t))^{2+\varepsilon}$
$\displaystyle\geq$ $\displaystyle
F_{t}(a_{1})^{2}-2\frac{1}{p+1}\gamma(a_{1})A(a_{1})^{-p}F(a_{1})^{p+1}$

for all $t\in[a_{2},b)$. Hence,

$F_{t}(t)\geq\sqrt{\frac{1}{p+1}}\gamma(t)^{1/2}A(t)^{-p/2}F(t)^{(p+1)/2}\quad\mbox{\rm
for all}\quad t\in[a_{2},b)\,.$

It follows

$F_{t}(t)\geq\delta\gamma(t)^{1/2}A(t)^{-p/2}F(t)^{(p-1)/2}(\ln
F(t))^{-1-\varepsilon/2}F(t)(\ln F(t))^{1+\varepsilon/2}$

for all $t\in[a_{2},b)$. But with sufficiently large $a_{2}\geq a_{1}$ we
obtain

$F(t)^{(p-1)/2}(\ln F(t))^{-1-\varepsilon/2}\geq\delta A(t)^{(p-1)/2}(\ln
A(t))^{-1-\varepsilon/2}\quad\mbox{\rm for all}\quad t\in[a_{2},b)\,.$

Hence,

$F_{t}(t)\geq\delta\left(\gamma(t)A(t)^{-1}(\ln
A(t))^{-2-\varepsilon}\right)^{1/2}F(t)(\ln
F(t))^{1+\varepsilon/2}\quad\mbox{\rm for all}\quad t\in[a_{2},b)$

implies

$F_{t}(t)\geq\delta cF(t)(\ln F(t))^{1+\varepsilon/2}\quad\mbox{\rm for
all}\quad t\in[a_{2},b)\,.$

The last nonlinear differential inequality does not have global solution with
$F>0$. Lemma is proven. $\Box$

###### Remark 3.3

We note here that the equation

$\displaystyle\ddot{F}(t)=e^{-dt}F(t)^{p}\,,\quad d>0,$

has a global solution $F(t)=c_{F}e^{\frac{d}{p-1}t}$, where
$c_{F}=\left({d}/(p-1)\right)^{2/(p-1)}$, while corresponding
$A(t)=c_{A}e^{at}$, $a>0$, and $\gamma(t)=c_{\gamma}e^{(pa-d)t}$. The
condition (17) implies $a>d/(p-1)$. On the other hand, the first inequality of
(14) holds only if $a\leq d/(p-1)$.

##  4 Nonexistence of the global solution for the integral equation associated
with the Klein-Gordon equation

Since $G$ is a fundamental solution of the strictly hyperbolic operator, for
every given function $u_{0}\in C([0,T];$ $L^{q}({\mathbb{R}}^{n}))\cap
C^{\infty}([0,T]\times{\mathbb{R}}^{n})$ there exist $T_{0}>0$ and solution
$u\in C([0,T_{0}];L^{q}({\mathbb{R}}^{n}))$. Moreover, for every given $T$ one
can prove existence of the solution $u\in C([0,T];L^{q}({\mathbb{R}}^{n}))$
provided that $\sup_{t\in[0,T]}\|u_{0}(\cdot,t)\|_{L^{q}({\mathbb{R}}^{n})}$
is small enough. Theorem 1.1 shows that the set of such $T$, in general, is
bounded.

Proof of Theorem 1.1. Let $u_{0}\in
C^{\infty}([0,\infty)\times{\mathbb{R}}^{n})$ be a function with the
permanently bounded support, that is
supp$\,u_{0}(\cdot,t)\subset\\{\,x\in{\mathbb{R}}^{n}\,;\,|x|\leq
constant\,\\}$ for all $t\geq 0$. We denote $\varphi_{0}(x):=u_{0}(x,0)$ and
$\varphi_{1}(x):=\partial_{t}u_{0}(x,0)$. One can find $u_{0}$ such that

$\int_{{\mathbb{R}}^{n}}u_{0}(x,t)dx=C_{0}\cosh(Mt)+C_{1}\frac{1}{M}\sinh(Mt)\qquad\mbox{\rm
for all}\quad t\geq 0\,,$ (18)

where

$C_{0}:=\int_{{\mathbb{R}}^{n}}\varphi_{0}(x)dx,\quad
C_{1}:=\int_{{\mathbb{R}}^{n}}\varphi_{1}(x)dx\,.$ (19)

The solution of the problem (4) with the data $\varphi_{0}(x)$,
$\varphi_{1}(x)\in C^{\infty}_{0}({\mathbb{R}}^{n})$ exemplifies such
function. Indeed, this unique smooth solution obeys finite propagation speed
property that implies
supp$\,u_{0}(\cdot,t)\subset\\{\,x\in{\mathbb{R}}^{n}\,;\,|x|\leq
R_{0}+1-e^{-t}\leq R_{0}+1\,\\}$ if supp$\,\varphi_{0}$,
supp$\,\varphi_{1}\subset\\{\,x\in{\mathbb{R}}^{n}\,;\,|x|\leq R_{0}\,\\}$. In
order to check (18) for that solution $u_{0}$ we integrate (4) with respect to
$x$ over ${\mathbb{R}}^{n}$ and then solve the initial problem with data (19)
for the obtained ordinary differential equation.

Suppose that $u\in C([0,\infty);L^{q}({\mathbb{R}}^{n}))$ with permanently
bounded support is a solution to (6) generated by $u_{0}$. According to the
definition of the solution, for every given $T>0$ we have

$G\left[\Gamma(\cdot)\left(\int_{{\mathbb{R}}^{n}}|u(y,\cdot)|^{p}dy\right)^{\beta}|u|^{p}\right]\in
C([0,T];L^{q}({\mathbb{R}}^{n}))$

and $u(x,0)=\varphi_{0}(x)\,,\quad u_{t}(x,0)=\varphi_{1}(x)\,.$ Then $u\in
C([0,\infty);L^{1}({\mathbb{R}}^{n}))$ and we can integrate equation (6):

$\int_{{\mathbb{R}}^{n}}u(x,t)\,dx=\int_{{\mathbb{R}}^{n}}u_{0}(x,t)\,dx+\int_{{\mathbb{R}}^{n}}G\left[\Gamma(\cdot)\left(\int_{{\mathbb{R}}^{n}}|u(y,\cdot)|^{p}dy\right)^{\beta}|u|^{p}\right](x,t)\,dx.$
(20)

In particular,

$\displaystyle\int_{{\mathbb{R}}^{n}}u(x,0)\,dx=\int_{{\mathbb{R}}^{n}}\varphi_{0}(x)\,dx,\quad\int_{{\mathbb{R}}^{n}}u_{t}(x,0)\,dx=\int_{{\mathbb{R}}^{n}}\varphi_{1}(x)\,dx\,.$

To evaluate the last term of (20) we apply Proposition 2.1. Consider the case
of odd $n\geq 3$. Then, for the smooth function $u=u(x,t)$ we obtain

$\displaystyle\int_{{\mathbb{R}}^{n}}G[\Gamma(\cdot)|u|^{p}](x,t)\,dx$
$\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}^{n}}\,dx2\int_{0}^{t}db\int_{0}^{e^{-b}-e^{-t}}dr_{1}\Big{(}\frac{\partial}{\partial
r}\Big{(}\frac{1}{r}\frac{\partial}{\partial
r}\Big{)}^{\frac{n-3}{2}}\frac{r^{n-2}}{\omega_{n-1}c_{0}^{(n)}}$
$\displaystyle\times\int_{S^{n-1}}\Big{[}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}|u(z,b)|^{p}dz\Big{)}^{\beta}|u(x+ry,b)|^{p}\Big{]}\,dS_{y}\Big{)}_{r=r_{1}}$
$\displaystyle\times(4e^{-b-t})^{-M}\left((e^{-t}+e^{-b})^{2}-r_{1}^{2}\right)^{-\frac{1}{2}+M}$
$\displaystyle\times
F\left(\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-r_{1}^{2}}{(e^{-b}+e^{-t})^{2}-r_{1}^{2}}\right)\,.$

Therefore,

$\displaystyle\int_{{\mathbb{R}}^{n}}G[\Gamma(\cdot)|u|^{p}](x,t)\,dx$
$\displaystyle=$ $\displaystyle
2\int_{0}^{t}db\int_{0}^{e^{-b}-e^{-t}}dr_{1}\Big{(}\frac{\partial}{\partial
r}\Big{(}\frac{1}{r}\frac{\partial}{\partial
r}\Big{)}^{\frac{n-3}{2}}\frac{r^{n-2}}{\omega_{n-1}c_{0}^{(n)}}$
$\displaystyle\times\int_{S^{n-1}}\Big{[}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}|u(z,b)|^{p}dz\Big{)}^{\beta}\left(\int_{{\mathbb{R}}^{n}}|u(x+ry,b)|^{p}\,dx\right)\Big{]}\,dS_{y}\Big{)}_{r=r_{1}}$
$\displaystyle\times(4e^{-b-t})^{-M}\left((e^{-t}+e^{-b})^{2}-r_{1}^{2}\right)^{-\frac{1}{2}+M}$
$\displaystyle\times
F\left(\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-r_{1}^{2}}{(e^{-b}+e^{-t})^{2}-r_{1}^{2}}\right)$

implies,

$\displaystyle\int_{{\mathbb{R}}^{n}}G[\Gamma(\cdot)|u|^{p}](x,t)\,dx$
$\displaystyle=$ $\displaystyle
2\int_{0}^{t}db\int_{0}^{e^{-b}-e^{-t}}dr_{1}\,\Big{(}\frac{\partial}{\partial
r}\Big{(}\frac{1}{r}\frac{\partial}{\partial
r}\Big{)}^{\frac{n-3}{2}}\frac{r^{n-2}}{\omega_{n-1}c_{0}^{(n)}}$
$\displaystyle\times\int_{S^{n-1}}\Big{[}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}|u(z,b)|^{p}dz\Big{)}^{\beta+1}\Big{]}\,dS_{y}\Big{)}_{r=r_{1}}$
$\displaystyle\times(4e^{-b-t})^{-M}\left((e^{-t}+e^{-b})^{2}-r_{1}^{2}\right)^{-\frac{1}{2}+M}$
$\displaystyle\times
F\left(\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-r_{1}^{2}}{(e^{-b}+e^{-t})^{2}-r_{1}^{2}}\right)\,.$

We obtain,

$\displaystyle\int_{{\mathbb{R}}^{n}}G[\Gamma(\cdot)|u|^{p}](x,t)dx$
$\displaystyle=$
$\displaystyle\int_{0}^{t}db\Big{[}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}|u(z,b)|^{p}dz\Big{)}^{\beta+1}\Big{]}\int_{0}^{e^{-b}-e^{-t}}dr_{1}$
$\displaystyle\times 2\left(\frac{\partial}{\partial
r}\Big{(}\frac{1}{r}\frac{\partial}{\partial
r}\Big{)}^{\frac{n-3}{2}}\frac{r^{n-2}}{\omega_{n-1}c_{0}^{(n)}}\\!\\!\int_{S^{n-1}}\,dS_{y}\right)_{r=r_{1}}$
$\displaystyle\times(4e^{-b-t})^{-M}\left((e^{-t}+e^{-b})^{2}-r_{1}^{2}\right)^{-\frac{1}{2}+M}$
$\displaystyle\times
F\left(\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-r_{1}^{2}}{(e^{-b}+e^{-t})^{2}-r_{1}^{2}}\right)$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}|u(z,b)|^{p}dz\Big{)}^{\beta+1}\frac{1}{M}\sinh(M(t-b))\,db.$

Thus, for the solution $u=u(x,t)$ we have proven

$\displaystyle\int_{{\mathbb{R}}^{n}}G[\Gamma(\cdot)|u|^{p}](x,t)\,dx$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}|u(z,b)|^{p}dz\Big{)}^{\beta+1}\frac{1}{M}\sinh(M(t-b))\,db\,.$

Hence (20) reads

$\displaystyle\int_{{\mathbb{R}}^{n}}u(x,t)\,dx$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}^{n}}u_{0}(x,t)\,dx+\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}|u(z,b)|^{p}dz\Big{)}^{\beta+1}\frac{1}{M}\sinh(M(t-b))\,db\,.$

Taking into account (18) and (19) we derive

$\displaystyle\int_{{\mathbb{R}}^{n}}u(x,t)\,dx$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(C_{0}+\frac{C_{1}}{M}\right)e^{Mt}+\frac{1}{2}\left(C_{0}-\frac{C_{1}}{M}\right)e^{-Mt}$
$\displaystyle+\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}|u(z,b)|^{p}dz\Big{)}^{\beta+1}\frac{1}{M}\sinh(M(t-b))\,db\,.$

We discuss separately two cases: with positive curved mass, $M>0$, and
vanishing curved mass, $M=0$, respectively.

In the case of $M>0$ we obtain

$\displaystyle\int_{{\mathbb{R}}^{n}}u(x,t)\,dx$ $\displaystyle=$
$\displaystyle C_{0}\cosh(Mt)+\frac{C_{1}}{M}\sinh(Mt)$ $\displaystyle+$
$\displaystyle\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}|u(z,b)|^{p}dz\Big{)}^{\beta+1}\frac{1}{M}\sinh(M(t-b))\,db\,.$

Denote

$\displaystyle F(t)$ $\displaystyle:=$
$\displaystyle\int_{{\mathbb{R}}^{n}}u(x,t)\,dx\,,$

then the function $F(t)$ is

$\displaystyle F(t)$ $\displaystyle=$ $\displaystyle
C_{0}\cosh(Mt)+\frac{C_{1}}{M}\sinh(Mt)$
$\displaystyle+\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}|u(z,b)|^{p}dz\Big{)}^{\beta+1}\frac{1}{M}\sinh(M(t-b))\,db\,.$

It follows $F\in C^{2}([0,\infty))$. More precisely,

$\displaystyle\dot{F}(t)$ $\displaystyle=$ $\displaystyle
C_{1}\cosh(Mt)+MC_{0}\sinh(Mt)$
$\displaystyle+\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}|u(z,b)|^{p}dz\Big{)}^{\beta+1}\cosh(M(t-b))\,db\,,$
$\displaystyle\ddot{F}(t)$ $\displaystyle=$ $\displaystyle
M^{2}F(t)+\Gamma(t)\Big{(}\int_{{\mathbb{R}}^{n}}|u(z,t)|^{p}dz\Big{)}^{\beta+1}\,.$
(22)

In particular, since $\Gamma(t)\geq 0$, we obtain

$F(t)\geq C_{0}\cosh(Mt)+\frac{C_{1}}{M}\sinh(Mt)\,\,\mbox{\rm
and}\,\,\ddot{F}(t)\geq\Gamma(t)\Big{(}\int_{{\mathbb{R}}^{n}}|u(z,t)|^{p}dz\Big{)}^{\beta+1}.$
(23)

On the other hand, since the solution $u=u(x,t)$ has permanently bounded
support, then supp$\,u(\cdot,t)\subset\\{\,x\in{\mathbb{R}}^{n}\,;\,|x|\leq
R\,\\}$ for some positive number $R$. Using the compact support of
$u(\cdot,t)$ and Hölder’s inequality we get with $\tau_{n}$ the volume of the
unit ball in ${\mathbb{R}}^{n}$,

$\displaystyle\left|\int_{{\mathbb{R}}^{n}}u(x,t)\,dx\right|^{p}$
$\displaystyle\leq$ $\displaystyle\left(\int_{|x|\leq
R}1\,dx\right)^{p-1}\left(\int_{|x|\leq R}|u(x,t)|^{p}\,dx\right)$
$\displaystyle=$
$\displaystyle\tau_{n}\Gamma(t)^{-1/(\beta+1)}R^{n(p-1)}\left(\Gamma(t)^{1/(\beta+1)}\int_{{\mathbb{R}}^{n}}|u(x,t)|^{p}\,dx\right)$
$\displaystyle=$
$\displaystyle\tau_{n}\Gamma(t)^{-1/(\beta+1)}R^{n(p-1)}\Big{(}\ddot{F}(t)-M^{2}F(t)\Big{)}^{1/(\beta+1)}$
$\displaystyle\leq$
$\displaystyle\tau_{n}\Gamma(t)^{-1/(\beta+1)}R^{n(p-1)}\ddot{F}(t)^{1/(\beta+1)}\,.$

Here we assume $\Gamma(t)>0$. Thus

$\ddot{F}(t)\geq\tau_{n}^{-(\beta+1)}R^{-n(p-1)(\beta+1)}\Gamma(t)|F(t)|^{p(\beta+1)}\quad\mbox{\rm
for all}\quad t\in[0,\infty)\,.$

By means of the inequality

$MC_{0}+C_{1}>0$

we conclude that $F(t)\geq 0$ and that

$\ddot{F}(t)\geq\delta_{0}\Gamma(t)F(t)^{p(\beta+1)}\qquad\mbox{\rm for
large}\,\,\,t\quad\mbox{\rm with}\,\,\,\delta_{0}>0\,.$

Hence, for appropriate $C_{0}$ and $C_{1}$ the last inequality together with
(4) to (23) implies the following system of the ordinary differential
inequalities

$\left\\{\begin{array}[]{ccccc}\displaystyle
F(t)&\geq&C_{0}\cosh(Mt)+\frac{C_{1}}{M}\sinh(Mt)&\mbox{\rm for all}&\quad
t\in[a,b),\\\
\displaystyle\dot{F}(t)&\geq&C_{1}\cosh(Mt)+MC_{0}\sinh(Mt)&\mbox{\rm for
all}&\quad t\in[a,b),\\\
\displaystyle\ddot{F}(t)&\geq&\delta_{0}\Gamma(t)F(t)^{p(\beta+1)}&\mbox{\rm
for all}&\quad t\in[a,b).\end{array}\right.$

The Lemma 3.1 shows that if $F(t)\in C^{2}([0,b))$ and the energy of particle
is large, then $b$ must be finite.

The conditions of the Lemma 3.1 are fulfilled on $(0,b)$ for the function

$\Gamma(t)=\delta_{0}e^{\gamma t},\quad\gamma\in{\mathbb{R}},$

with $\gamma>0$ without any condition on the energy. They are fulfilled with
$\gamma<0$ if the kinetic energy and the potential energy are sufficiently
large, that is $C_{0}>0$, $C_{1}>0$, and

$C_{1}\geq\sqrt{\frac{2\delta_{0}}{p+1}}C_{0}^{(p+1)/2}\quad\mbox{\rm
and}\quad C_{0}^{p-1}>\frac{\gamma^{2}(p+1)}{\delta_{0}(p-1)}\,.$

Next we turn to the case of the small energy and exponentially decreasing
$\Gamma(t)$. We apply Lemma 3.2 with $A(t)=e^{Mt}$ and $p$ replaced with
$p(\beta+1)$. More precisely, if we set

$A(t)=e^{Mt},\qquad\gamma(t)=\Gamma(t)e^{Mp(\beta+1)t},$

then the conditions of the last lemma read:

$p(\beta+1)>1\quad\mbox{\rm and }\quad\Gamma_{t}(t)\leq 0\quad\mbox{\rm for
all }\quad t\in[0,\infty).$

The last inequality follows from the monotonicity of $\Gamma(t)$. By the
condition of the theorem, there exist $\varepsilon>0$ and $c>0$ such that

$\displaystyle\Gamma(t)\geq
ce^{-M(p(\beta+1)-1)t}t^{2+\varepsilon}\quad\mbox{\rm for all}\quad
t\in[a,b),$

that coincides with (17). The case of $M>0$ is proved.

Now consider the case of $M=0$. Let

$\displaystyle C_{0}:=\int_{{\mathbb{R}}^{n}}\varphi_{0}(x)dx,\quad
C_{1}:=\int_{{\mathbb{R}}^{n}}\varphi_{1}(x)dx,\qquad C_{1}>0.$

Then Corollary 2.2 allows us to write

$\displaystyle\int_{{\mathbb{R}}^{n}}G[\Gamma(\cdot)|u|^{p}](x,t)\,dx$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}|u(z,b)|^{p}dz\Big{)}^{\beta+1}(t-b)\,db.$

Hence (20) reads:

$\int_{{\mathbb{R}}^{n}}u(x,t)\,dx=\int_{{\mathbb{R}}^{n}}u_{0}(x,t)\,dx+\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}|u(z,b)|^{p}dz\Big{)}^{\beta+1}\,.$

Now we choose a function $u_{0}\in
C^{\infty}([0,\infty)\times{\mathbb{R}}^{n})$ such that

$\int_{{\mathbb{R}}^{n}}u_{0}(x,t)dx=C_{0}+C_{1}t\,.$

The solution of the problem (4) with $M=0$ exemplifies such functions. Thus

$\displaystyle\int_{{\mathbb{R}}^{n}}u(x,t)\,dx$ $\displaystyle=$
$\displaystyle
C_{0}+C_{1}t+\,\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}|u(z,b)|^{p}dz\Big{)}^{\beta+1}(t-b)\,db\,.$

Denote

$\displaystyle F(t)$ $\displaystyle:=$
$\displaystyle\int_{{\mathbb{R}}^{n}}u(x,t)\,dx\,,$

then

$\displaystyle F(t)$ $\displaystyle=$ $\displaystyle
C_{0}+C_{1}t+\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}|u(z,b)|^{p}dz\Big{)}^{\beta+1}(t-b)\,db\,.$

It follows $F\in C^{2}([0,\infty))$. More precisely,

$\displaystyle\dot{F}(t)$ $\displaystyle=$ $\displaystyle
C_{1}+\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}|u(z,b)|^{p}dz\Big{)}^{\beta+1}\,db\,,$
$\displaystyle\ddot{F}(t)$ $\displaystyle=$
$\displaystyle\Gamma(t)\Big{(}\int_{{\mathbb{R}}^{n}}|u(z,t)|^{p}dz\Big{)}^{\beta+1}\,.$
(24)

In particular,

$\displaystyle F(t)\geq C_{0}+C_{1}t\quad\mbox{\rm
and}\quad\ddot{F}(t)=\Gamma(t)\Big{(}\int_{{\mathbb{R}}^{n}}|u(z,t)|^{p}dz\Big{)}^{\beta+1}\,.$
(25)

On the other hand according to (24) we obtain

$\displaystyle\left|\int_{{\mathbb{R}}^{n}}u(x,t)\,dx\right|^{p}$
$\displaystyle\leq$ $\displaystyle\left(\int_{|x|\leq
R}1\,dx\right)^{p-1}\left(\int_{|x|\leq R}|u(x,t)|^{p}\,dx\right)$
$\displaystyle=$
$\displaystyle\tau_{n}\Gamma(t)^{-1/(\beta+1)}R^{n(p-1)}\left(\Gamma(t)^{1/(\beta+1)}\int_{{\mathbb{R}}^{n}}|u(x,t)|^{p}\,dx\right)$
$\displaystyle\leq$
$\displaystyle\tau_{n}\Gamma(t)^{-1/(\beta+1)}R^{n(p-1)}\ddot{F}(t)^{1/(\beta+1)}\,.$

Thus

$\ddot{F}(t)\geq\tau_{n}^{-(\beta+1)}R^{-n(p-1)(\beta+1)}\Gamma(t)|F(t)|^{p(\beta+1)}$

for all $t$ in $[0,\infty)$. By means of the condition $C_{1}>0$ we conclude

$\ddot{F}(t)\geq C\Gamma(t)F(t)^{p(\beta+1)}\qquad\mbox{\rm for
large}\,\,t\quad\mbox{\rm with}\,\,C>0\,.$

But for appropriate $C_{0}$ and $C_{1}$ one has $F(t)>0$ and the last
inequality together with (25) implies

$\displaystyle\left\\{\begin{array}[]{ccccc}\displaystyle
F(t)&\geq&C_{0}+C_{1}t&\mbox{\rm for all}&t\in[a,b),\\\
\displaystyle\ddot{F}(t)&\geq&\delta_{0}\Gamma(t)F(t)^{p(\beta+1)}&\mbox{\rm
for all}&t\in[a,b).\end{array}\right.$

The Kato’s Lemma 2 [6] shows that if $F(t)\in C^{2}([0,b))$ and $\Gamma(t)\geq
t^{-1-p(\beta+1)}$ with $p(\beta+1)>1$, then $b$ must be finite. Theorem is
proven. $\square$

###### Remark 4.1

In fact, we have proved that any solution $u=u(x,t)$ with permanently bounded
support blows up if either $MC_{0}+C_{1}>0$ and $M>0$ or $C_{1}>0$ and $M=0$.

Proof of Theorem 1.2. The case of $\gamma\geq 0$ is covered by Theorem 1.1 and
implies a blow-up even for the small data. Therefore, we restrict ourselves to
the case of $\gamma<0$. Then, with a special choice of $C_{0}$ and $C_{1}$
after arguments have been used in the proof of Theorem 1.1 we arrive at the
following system of the ordinary differential inequalities

$\displaystyle\left\\{\begin{array}[]{ccccc}\displaystyle
F(t)&\geq&Ce^{Mt}&\mbox{\rm for all}&t\in[0,b),\\\
\displaystyle\dot{F}(t)&\geq&Ce^{Mt}&\mbox{\rm for all}&t\in[0,b),\\\
\displaystyle\ddot{F}(t)&\geq&\delta_{0}e^{\gamma
t}F(t)^{p(\beta+1)}&\mbox{\rm for all}&t\in[0,b),\end{array}\right.$

where $C>0$ and $\delta_{0}>0$. We claim that $b<\infty$. Indeed, we check
conditions of Lemma 3.1 with

$\Gamma(t)=\delta_{0}e^{\gamma t}\,.$

The condition (13),

$\displaystyle\frac{1}{\sqrt{p+1}}\int_{0}^{a_{1}}\Gamma(s)^{1/2}ds>\frac{\sqrt{2}}{p-1}F^{(1-p)/2}(0)\,,\quad\dot{F}^{2}(0)\geq\frac{2}{p+1}\Gamma(0)F(0)^{p+1},$

reads:

$\displaystyle\frac{1}{\sqrt{p+1}}\int_{0}^{a_{1}}\delta_{0}^{1/2}e^{\gamma
s/2}ds>\frac{\sqrt{2}}{p-1}C_{0}^{(1-p)/2}\,,\quad
C_{1}^{2}\geq\frac{2}{p+1}\delta_{0}C_{0}^{p+1}.$

The first inequality is fulfilled if $C_{0}$, that is the initial potential
energy, is sufficiently large, while the second one is fulfilled if $C_{1}$,
that is the initial kinetic energy, is large enough. Theorem is proven.
$\square$

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