# Blow-up solutions of damped Klein-Gordon equation on the Heisenberg group

Michael Ruzhansky  Michael Ruzhansky: Department of Mathematics: Analysis,
Logic and Discrete Mathematics Ghent University, Belgium and School of
Mathematical Sciences Queen Mary University of London United Kingdom E-mail
address<EMAIL_ADDRESS>and  Bolys Sabitbek  Bolys Sabitbek:
School of Mathematical Sciences Queen Mary University of London United Kingdom
and Al-Farabi Kazakh National University Almaty, Kazakhstan E-mail address
<EMAIL_ADDRESS>
###### Abstract.

In this note, we prove the blow-up of solutions of the semilinear damped
Klein-Gordon equation in a finite time for arbitrary positive initial energy
on the Heisenberg group. This work complements the paper [21] by the first
author and Tokmagambetov, where the global in time well-posedness was proved
for the small energy solutions.

###### Key words and phrases:

Blow-up, sub-Laplacian, Heisenberg group, damped Klein-Grodon equation

###### 1991 Mathematics Subject Classification:

35L71; 35R03, 35B44.

The first and second authors were supported by EPSRC grant EP/R003025/2. The
first author was also supported by FWO Odysseus 1 grant G.0H94.18N: Analysis
and Partial Differential Equations and the Methusalem programme of the Ghent
University Special Research Fund (BOF) (Grant number 01M01021).

##  1\. Introduction

###  1.1. Setting of the problem

This note is devoted to study the blow up of solutions of the Cauchy problem
for the semilinear damped Klein-Gordon equation for the sub-Laplacian
$\mathcal{L}$ on the Heisenberg group $\mathbb{H}^{n}$:

$\displaystyle\begin{cases}u_{tt}(t)-\mathcal{L}u(t)+bu_{t}(t)+mu(t)=f(u),&t>0,\\\
u(x,0)=u_{0}(x),\,\,\,&u_{0}\in H^{1}_{\mathcal{L}}(\mathbb{H}^{n}),\\\
u_{t}(x,0)=u_{1}(x),\,\,\,&u_{1}\in L^{2}(\mathbb{H}^{n}),\end{cases}$ (1.1)

with the damping term determined by $b>0$ and the mass $m>0$. A total energy
of problem (1.1) is defined as

$\displaystyle E(t)$
$\displaystyle=\frac{1}{2}||u_{t}||^{2}_{L^{2}(\mathbb{H}^{n})}+\frac{m}{2}||u||^{2}_{L^{2}(\mathbb{H}^{n})}+\frac{1}{2}||\nabla_{H}u||^{2}_{L^{2}(\mathbb{H}^{n})}-\int_{\mathbb{H}^{n}}F(u)dx,$

where we assume the following

$\displaystyle g:[0,\infty]\rightarrow\mathbb{R},$ $\displaystyle
F(z)=g(|z|)\,\,\text{ for }\,\,z\in\mathbb{C}^{n},$ (1.2) $\displaystyle
f(z)=\frac{g^{\prime}(|z|)z}{|z|}.$

Then we have that

$\displaystyle\frac{\partial}{\partial\varepsilon}F(z+\varepsilon\xi)|_{\varepsilon=0}$
$\displaystyle=\frac{\partial}{\partial\varepsilon}g(|z+\varepsilon\xi|)|_{\varepsilon=0}$
$\displaystyle=g^{\prime}(|z+\varepsilon\xi|)\frac{\partial}{\partial\varepsilon}(|z+\varepsilon\xi|)|_{\varepsilon=0}$
$\displaystyle=\frac{g^{\prime}(|z|)}{|z|}\frac{1}{2}(\overline{z}\xi+z\overline{\xi})$
$\displaystyle={\rm Re}\left(f(z)\overline{\xi}\right),$

and

$\displaystyle\frac{\partial}{\partial x_{j}}F(u(x))$
$\displaystyle=g^{\prime}(|u(x)|)\frac{\partial|u(x)|}{\partial x_{j}}$
$\displaystyle=\frac{g^{\prime}(|u(x)|)}{2|u(x)|}\left(u(x)\frac{\partial\overline{u}(x)}{\partial
x_{j}}+\frac{\partial u(x)}{\partial x_{j}}\overline{u}(x)\right)$
$\displaystyle={\rm Re}\left(f(u(x))\frac{\partial\overline{u}(x)}{\partial
x_{j}}\right).$

The conservation of energy law follows from

$\displaystyle\frac{\partial E(t)}{\partial t}$
$\displaystyle=\frac{\partial}{\partial
t}\left[\frac{1}{2}||u_{t}||^{2}_{L^{2}(\mathbb{H}^{n})}+\frac{m}{2}||u||^{2}_{L^{2}(\mathbb{H}^{n})}+\frac{1}{2}||\nabla_{H}u||^{2}_{L^{2}(\mathbb{H}^{n})}-\int_{\mathbb{H}^{n}}F(u)dx\right]$
$\displaystyle={\rm
Re}\int_{\mathbb{H}^{n}}\overline{u}_{t}[u_{tt}+mu-\mathcal{L}u-f(u)]dx$
$\displaystyle=-b\int_{\mathbb{H}^{n}}|u_{t}|^{2}dx,$

this gives

$E(t)+b\int_{0}^{t}||u_{s}(s)||^{2}_{L^{2}(\mathbb{H}^{n})}ds=E(0),$ (1.3)

where

$E(0):=\frac{1}{2}||u_{1}||^{2}_{L^{2}(\mathbb{H}^{n})}+\frac{m}{2}||u_{0}||^{2}_{L^{2}(\mathbb{H}^{n})}+\frac{1}{2}||\nabla_{H}u_{0}||^{2}_{L^{2}(\mathbb{H}^{n})}-\int_{\mathbb{H}^{n}}F(u_{0})dx.$

Also, let us define the Nehari functional

$\displaystyle
I(u)=m||u||^{2}_{L^{2}(\mathbb{H}^{n})}+||\nabla_{H}u||^{2}_{L^{2}(\mathbb{H}^{n})}-{\rm
Re}\int_{\mathbb{H}^{n}}f(u)\overline{u}dx.$

We assume that the nonlinear term $f(u)$ satisfies the condition

$f(0)=0,\,\,\,\text{ and }\,\,\alpha F(u)\leq{\rm Re}[f(u)\overline{u}],$

where $\alpha>2$. In particular, this includes the case

$f(u)=|u|^{p-1}u\,\,\,\text{ for }p>1.$

###  1.2. Literature overview

The study of the damped wave equation on the Heisenberg group started in
Bahouri-Gerrard-Xu [1] to prove the dispersive and Strichartz inequalities
based on the analysis in Besov-type spaces. Later, Greiner-Holcman-Kannai [4]
explicitly computed the wave kernel for the class of second-order subelliptic
operators, where their class contains degenerate elliptic and hypoelliptic
operators such as the sub-Laplacian and the Grushin operator. Also, Müller-
Stein [15] established $L^{p}$-estimates for the wave equation on the
Heisenberg group. Recently, Müller-Seeger [16] obtained the sharp version of
$L^{p}$ estimates on the $H$-type groups.

The blow-up solutions of evolution equations on the Heisenberg group were
considered by Georgiev-Palmieri [5] where they proved the global existence and
nonexistence results of the Cauchy problem for the semilinear damped wave
equation on the Heisenberg group with the power nonlinear term. The proof of
blow-up solutions is based on the test function method. The first author and
Yessirkegenov [22] established the existence and non-existence of global
solutions for semilinear heat equations and inequalities on sub-Riemannian
manifolds. In [23], by using the comparison principle they obtain blow-up type
results and global in $t$-boundedness of solutions of nonlinear equations for
the heat $p$-sub-Laplacian on the stratified Lie groups. The global existence
and nonexistence for the nonlinear porous medium equation were studied by the
authors in [19] on the stratified Lie groups.

This work is motivated by the paper [21] of the first author and Tokmagambetov
where the global existence of solutions for small data of problem (1.1) was
shown on the Heisenberg group and on general graded Lie groups. In the sense
of the potential wells theory, we can understand this result in the sense that
when the initial energy is less than the mountain pass level $E(0)<d$ and the
Nehari functional is positive $I(u_{0})>0$, there exists a global solution of
the problem (1.1). A natural question arises when the solution of problem
(1.1) blows up in a finite time or $E(0)>0$ and $I(u_{0})<0$.

The main aim of this paper is to obtain the blow-up solutions of problem (1.1)
in a finite time for arbitrary positive initial energy. Our proof is based on
an adopted concavity method, which was introduced by Levine [9] to establish
the blow-up solutions of the abstract wave equation of the form
$Pu_{tt}=-Au+F(u)$ (including the Klein-Gordon equation) for the negative
initial energy. It was also used for parabolic type equations (see [10, 12,
11, 13, 14]). Modifying the concavity method, Wang [29] proved the
nonexistence of global solutions to nonlinear damped Klein-Gordon equation for
arbitrary positive initial energy under sufficient conditions. Later, Yang-Xu
[27] extended this result by introducing a new auxiliary function and the
adopted concavity method.

###  1.3. Preliminaries on the Heisenberg group

Let us give a brief introduction of the Heisenberg group. Let $\mathbb{H}^{n}$
be the Heisenberg group, that is, the set $\mathbb{R}^{2n+1}$ equipped with
the group law

$\xi\circ\widetilde{\xi}:=(x+\widetilde{x},y+\widetilde{y},t+\widetilde{t}+2\sum_{i=1}^{n}(\widetilde{x}_{i}y_{i}-x_{i}\widetilde{y}_{i})),$

where $\xi:=(x,y,t)\in\mathbb{H}^{n}$, $x:=(x_{1},\ldots,x_{n})$,
$y:=(y_{1},\ldots,y_{n})$, and $\xi^{-1}=-\xi$ is the inverse element of $\xi$
with respect to the group law. The dilation operation of the Heisenberg group
with respect to the group law has the following form (see e.g. [7], [20])

$\delta_{\lambda}(\xi):=(\lambda x,\lambda
y,\lambda^{2}t)\,\,\text{for}\,\,\lambda>0.$

The Lie algebra $\mathfrak{h}$ of the left-invariant vector fields on the
Heisenberg group $\mathbb{H}^{n}$ is spanned by

$X_{i}:=\frac{\partial}{\partial x_{i}}+2y_{i}\frac{\partial}{\partial
t}\,\,\text{for}\,\,1\leq i\leq n,$ $Y_{i}:=\frac{\partial}{\partial
y_{i}}-2x_{i}\frac{\partial}{\partial t}\,\,\text{for}\,\,1\leq i\leq n,$

and with their (non-zero) commutator

$[X_{i},Y_{i}]=-4\frac{\partial}{\partial t}.$

The horizontal gradient of $\mathbb{H}^{n}$ is given by

$\nabla_{H}:=(X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{n}),$

so the sub-Laplacian on $\mathbb{H}^{n}$ is given by

$\mathcal{L}:=\sum_{i=1}^{n}\left(X_{i}^{2}+Y_{i}^{2}\right).$

######  Definition 1.1 (Weak solution).

A function

$\displaystyle u\in C([0,T_{1});H^{1}_{\mathcal{L}}(\mathbb{H}^{n}))\cap
C^{1}([0,T_{1});L^{2}(\mathbb{H}^{n})),$ $\displaystyle u_{t}\in
L^{2}([0,T_{1});H^{1}_{\mathcal{L}}(\mathbb{H}^{n})),$ $\displaystyle
u_{tt}\in L^{2}([0,T_{1});H^{-1}_{\mathcal{L}}(\mathbb{H}^{n})),$

satisfying

${\rm Re}\langle u_{tt},v\rangle+{\rm
Re}\int_{\mathbb{H}^{n}}\nabla_{H}u\cdot\nabla_{H}vdx+m{\rm
Re}\int_{\mathbb{H}^{n}}uvdx+b{\rm Re}\int_{\mathbb{H}^{n}}u_{t}vdx\ ={\rm
Re}\int_{\mathbb{H}^{n}}f(u)vdx,$ (1.4)

for all $v\in H^{1}_{\mathcal{L}}(\mathbb{H}^{n})$ and a.e. $t\in[0,T_{1})$
with $u(0)=u_{0}(x)$ and $u_{t}(0)=u_{1}(x)$ represents a weak solution of
problem (1.1).

Note that $T_{1}$ denotes the lifespan of the solution $u(x,t)$ and
$\langle\cdot,\cdot\rangle$ is the duality between
$H^{-1}_{\mathcal{L}}(\mathbb{H}^{n})$ and
$H^{1}_{\mathcal{L}}(\mathbb{H}^{n})$. Here
$H^{1}_{\mathcal{L}}(\mathbb{H}^{n})$ denotes the sub-Laplacian Sobolev space,
analysed by Folland [6], see also [7].

##  2\. Main Result

We now present the main result of this paper.

######  Theorem 2.1.

Let $b>0$, $m>0$ and $\mu=\max\\{b,m,\alpha\\}$. Assume that nonlinearity
$f(u)$ satisfies

$\alpha F(u)\leq{\rm Re}[f(u)\overline{u}]\,\,\,\text{ for }\,\,\alpha>2,$
(2.1)

where $F(u)$ is as in (1.1). Assume that the Cauchy data $u_{0}\in
H_{\mathcal{L}}^{1}(\mathbb{H}^{n})$ and $u_{1}\in L^{2}(\mathbb{H}^{n})$
satisfy

$I(u_{0})=m||u_{0}||^{2}_{L^{2}(\mathbb{H}^{n})}+||\nabla_{H}u_{0}||^{2}_{L^{2}(\mathbb{H}^{n})}-{\rm
Re}\int_{\mathbb{H}^{n}}\overline{u}_{0}f(u_{0})dx<0,$ (2.2)

and

${\rm
Re}(u_{0},u_{1})_{L^{2}(\mathbb{H}^{n})}\geq\frac{\alpha(\mu+1)}{m(\alpha-2)}E(0).$
(2.3)

Then the solution of equation (1.1) blows up in finite time $T^{*}$ such that

$0<T^{*}\leq\frac{2(\mu+1)(bT_{0}+1)}{(\alpha-2)(\mu+1-m)}\frac{||u_{0}||^{2}_{L^{2}(\mathbb{H}^{n})}}{{\rm
Re}(u_{0},u_{1})},$

where the blow-up time $T^{*}\in(0,T_{0})$ with $T_{0}<+\infty$.

######  Remark 2.2.

  * (i)

Note that we have times $T^{*}$, $T_{0}$ and $T_{1}$. The relationship between
this times is the blow-up time $T^{*}\in(0,T_{0})\subset(0,T_{1})$ where
$T_{0}<+\infty$ and $T_{1}=+\infty$.

  * (ii)

The local existence for the Klein-Gordon equation was shown in [2] and [3].
The global in time well-posedness of problem (1.1) was proved by the first
author and Tokmagambetov [21] for the small energy solutions and the
nonlinearity $f(u)$ satisfying

$\displaystyle|f(u)-f(v)|\leq C(|u|^{p-1}+|v|^{p-1})|u-v|,$

with $1<p\leq 1+1/n$.

###### Proof of Theorem 2.1 .

First, recall the Nehari functional

$\displaystyle
I(u)=m||u||^{2}_{L^{2}(\mathbb{H}^{n})}+||\nabla_{H}u||^{2}_{L^{2}(\mathbb{H}^{n})}-{\rm
Re}\int_{\mathbb{H}^{n}}\overline{u}f(u)dx.$

Then the proof includes two steps.

Step I. In this step, we claim that

$I(u(t))<0,\,\,\,\text{ and
}\,\,\,A(t)>\frac{2\alpha(\mu+1)}{m(\alpha-2)}E(0),$

for $0\leq t<T_{1}$ where $\mu=\max\\{b,m,\alpha\\}$ and

$\displaystyle A(t)=2{\rm Re}(u,u_{t})+b||u||^{2}_{L^{2}(\mathbb{H}^{n})}.$

By using (1.4) along with $v=\overline{u}$ we get

$\displaystyle A^{\prime}(t)$
$\displaystyle=2||u_{t}||^{2}_{L^{2}(\mathbb{H}^{n})}+2{\rm Re}\langle
u_{tt},u\rangle+2b{\rm Re}\int_{\mathbb{H}^{n}}\overline{u}u_{t}dx$
$\displaystyle=2||u_{t}||^{2}_{L^{2}(\mathbb{H}^{n})}-2I(u),\,\,\,0\leq
t<T_{1}.$ (2.4)

In the last line we have used that

$\displaystyle{\rm Re}\langle u_{tt},u\rangle$ $\displaystyle={\rm
Re}\int_{\mathbb{H}^{n}}f(u)\overline{u}dx-\int_{\mathbb{H}^{n}}|\nabla_{H}u|^{2}dx-m\int_{\mathbb{H}^{n}}|u|^{2}dx-b{\rm
Re}\int_{\mathbb{H}^{n}}\overline{u}u_{t}dx$ $\displaystyle=-I(u)-b{\rm
Re}\int_{\mathbb{H}^{n}}\overline{u}u_{t}dx.$

Now let us suppose by contradiction that

$I(u(t))<0\,\,\,\text{ for all }\,\,0\leq t<t_{0},$

and

$I(u(t_{0}))=0.$

Hereafter $0<t_{0}<T_{1}$. It is easy to see that $A^{\prime}(t)>0$ over
$[0,t_{0})$ and

$\displaystyle A(t)>A(0)\geq 2{\rm
Re}(u_{0},u_{1})\geq\frac{2\alpha(\mu+1)}{m(\alpha-2)}E(0).$ (2.5)

Since $u(t)$ and $u_{t}(t)$ are both continuous in $t$ that gives

$\displaystyle A(t_{0})\geq\frac{2\alpha(\mu+1)}{m(\alpha-2)}E(0).$ (2.6)

Next we need to show a contradiction to (2.6). Using (1.3) and (2.1), we have

$\displaystyle E(0)$
$\displaystyle=E(t)+b\int_{0}^{t}||u_{s}||^{2}_{L^{2}(\mathbb{H}^{n})}ds$
$\displaystyle=\frac{1}{2}||u_{t}||^{2}_{L^{2}(\mathbb{H}^{n})}+\frac{m}{2}||u||^{2}_{L^{2}(\mathbb{H}^{n})}+\frac{1}{2}||\nabla_{H}u||^{2}_{L^{2}(\mathbb{H}^{n})}$
$\displaystyle-\int_{\mathbb{H}^{n}}F(u)dx+b\int_{0}^{t}||u_{s}||^{2}_{L^{2}(\mathbb{H}^{n})}ds$
$\displaystyle\geq\frac{1}{2}||u_{t}||^{2}_{L^{2}(\mathbb{H}^{n})}+\frac{m}{2}||u||^{2}_{L^{2}(\mathbb{H}^{n})}+\frac{1}{2}||\nabla_{H}u||^{2}_{L^{2}(\mathbb{H}^{n})}$
$\displaystyle-\frac{1}{\alpha}{\rm
Re}\int_{\mathbb{H}^{n}}\overline{u}f(u)dx+b\int_{0}^{t}||u_{s}||^{2}_{L^{2}(\mathbb{H}^{n})}ds$
$\displaystyle=\frac{1}{2}||u_{t}||^{2}_{L^{2}(\mathbb{H}^{n})}+\frac{1}{\alpha}I(u)+\left(\frac{m}{2}-\frac{m}{\alpha}\right)||u||^{2}_{L^{2}(\mathbb{H}^{n})}$
$\displaystyle+\left(\frac{\alpha-2}{2\alpha}\right)||\nabla_{H}u||^{2}_{L^{2}(\mathbb{H}^{n})}+b\int_{0}^{t}||u_{s}||^{2}_{L^{2}(\mathbb{H}^{n})}ds.$

If we use $I(u(t_{0}))=0$ and $\frac{m(\alpha-2)}{\alpha(\mu+1)}<1$, then

$\displaystyle E(0)$
$\displaystyle\geq\frac{1}{2}||u_{t}(t_{0})||^{2}_{L^{2}(\mathbb{H}^{n})}+\frac{m(\alpha-2)}{2\alpha}||u(t_{0})||^{2}_{L^{2}(\mathbb{H}^{n})}$
$\displaystyle\geq\frac{m(\alpha-2)}{2\alpha(\mu+1)}\left(||u_{t}(t_{0})||^{2}_{L^{2}(\mathbb{H}^{n})}+(\mu+1)||u(t_{0})||^{2}_{L^{2}(\mathbb{H}^{n})}\right)$
$\displaystyle>\frac{m(\alpha-2)}{2\alpha(\mu+1)}\left(2{\rm
Re}(u(t_{0}),u_{t}(t_{0}))+\mu||u(t_{0})||^{2}_{L^{2}(\mathbb{H}^{n})}\right)$
$\displaystyle\geq\frac{m(\alpha-2)}{2\alpha(\mu+1)}A(t_{0}).$ (2.7)

Note that for the strict inequality above we use that the assumption (2.2)
implies that $||u_{0}||_{L^{2}(\mathbb{H}^{n})}\neq 0$. We have also used the
fact $a^{2}+b^{2}-2ab\geq 0$, where
$a=||u_{t}(t_{0})||_{L^{2}(\mathbb{H}^{n})}$ and
$b=||u(t_{0})||_{L^{2}(\mathbb{H}^{n})}$. It gives the contradiction to (2.6).
This proves our claim.

Step II. Define the functional

$\displaystyle
M(t)=||u||^{2}_{L^{2}(\mathbb{H}^{n})}+b\int_{0}^{t}||u(s)||^{2}_{L^{2}(\mathbb{H}^{n})}ds+b(T_{0}-t)||u_{0}||^{2}_{L^{2}(\mathbb{H}^{n})},$

for $0\leq t\leq T_{0}$. Then

$\displaystyle M^{\prime}(t)$ $\displaystyle=2{\rm
Re}(u,u_{t})+b||u(t)||^{2}_{L^{2}(\mathbb{H}^{n})}-b||u_{0}||^{2}_{L^{2}(\mathbb{H}^{n})}$
$\displaystyle=2{\rm Re}(u,u_{t})+2b\int_{0}^{t}{\rm Re}(u(s),u_{s}(s))ds,$

since

$\displaystyle\int_{0}^{t}\frac{d}{ds}||u(s)||_{L^{2}(\mathbb{H}^{n})}^{2}ds=||u(t)||_{L^{2}(\mathbb{H}^{n})}^{2}-||u(0)||_{L^{2}(\mathbb{H}^{n})}^{2}.$

We observe the following estimates

$\displaystyle|{\rm Re}(u,u_{t})|^{2}$
$\displaystyle\leq||u_{t}||^{2}_{L^{2}(\mathbb{H}^{n})}||u||^{2}_{L^{2}(\mathbb{H}^{n})},$
$\displaystyle\left(\int_{0}^{t}|{\rm Re}(u(s),u_{s}(s))|ds\right)^{2}$
$\displaystyle\leq\left(\int_{0}^{t}||u(s)||^{2}_{L^{2}(\mathbb{H}^{n})}ds\right)\left(\int_{0}^{t}||u_{s}(s)||^{2}_{L^{2}(\mathbb{H}^{n})}ds\right),$

and

$\displaystyle 2{\rm Re}(u,u_{t})\int_{0}^{t}{\rm Re}(u(s),u_{s}(s))ds$
$\displaystyle\leq
2||u||_{L^{2}(\mathbb{H}^{n})}||u_{t}||_{L^{2}(\mathbb{H}^{n})}$
$\displaystyle\times\left(\int_{0}^{t}||u(s)||^{2}_{L^{2}(\mathbb{H}^{n})}ds\right)^{1/2}\left(\int_{0}^{t}||u_{s}(s)||^{2}_{L^{2}(\mathbb{H}^{n})}ds\right)^{1/2}$
$\displaystyle\leq||u||^{2}_{L^{2}(\mathbb{H}^{n})}$
$\displaystyle\int_{0}^{t}||u_{s}(s)||^{2}_{L^{2}(\mathbb{H}^{n})}ds+||u_{t}||^{2}_{L^{2}(\mathbb{H}^{n})}\int_{0}^{t}||u(s)||^{2}_{L^{2}(\mathbb{H}^{n})}ds.$

Using the above inequalities, we calculate

$\displaystyle(M^{\prime}(t))^{2}$ $\displaystyle=4\left(|{\rm
Re}(u,u_{t})|^{2}+2b{\rm Re}(u,u_{t})\int_{0}^{t}{\rm
Re}(u(s),u_{s}(s))ds+b^{2}\left(\int_{0}^{t}{\rm
Re}(u(s),u_{s}(s))ds\right)^{2}\right)$ $\displaystyle\leq
4\left(||u||^{2}_{L^{2}(\mathbb{H}^{n})}+b\int_{0}^{t}||u(s)||^{2}_{L^{2}(\mathbb{H}^{n})}ds\right)\left(||u_{t}||^{2}_{L^{2}(\mathbb{H}^{n})}+\int_{0}^{t}||u_{s}(s)||^{2}_{L^{2}(\mathbb{H}^{n})}ds\right),$

for all $0\leq t\leq T_{0}$. The second derivate with respect to time of
$M(t)$ is

$\displaystyle
M^{\prime\prime}(t)=2||u_{t}||^{2}_{L^{2}(\mathbb{H}^{n})}-2I(u),$

for all $0\leq t\leq T_{0}$, where we used the equality from (2). Then we
construct the differential inequality as follows

$\displaystyle
M^{\prime\prime}(t)M(t)-\frac{\omega+3}{4}(M^{\prime}(t))^{2}\geq
M(t)\left(M^{\prime\prime}(t)-(\omega+3)\left(||u_{t}||^{2}+b\int_{0}^{t}||u_{s}(s)||^{2}_{L^{2}(\mathbb{H}^{n})}ds\right)\right)$
$\displaystyle=M(t)\left(-(\omega+1)||u_{t}||^{2}_{L^{2}(\mathbb{H}^{n})}-(\omega+3)b\int_{0}^{t}||u_{s}(s)||^{2}_{L^{2}(\mathbb{H}^{n})}ds-2I(u)\right),$

where we assume that $\omega>1$. We shall now show that the following term is
nonnegative

$\displaystyle\eta(t)$
$\displaystyle=-(\omega+1)||u_{t}||^{2}_{L^{2}(\mathbb{H}^{n})}-(\omega+3)b\int_{0}^{t}||u_{s}(s)||^{2}_{L^{2}(\mathbb{H}^{n})}ds-2I(u)$
$\displaystyle\geq(\alpha-\omega-1)||u_{t}||^{2}_{L^{2}(\mathbb{H}^{n})}+b(2\alpha-\omega-3)\int_{0}^{t}||u_{s}(s)||^{2}_{L^{2}(\mathbb{H}^{n})}ds$
$\displaystyle+m(\alpha-2)||u||^{2}_{L^{2}(\mathbb{H}^{n})}+(\alpha-2)||\nabla_{H}u||^{2}_{L^{2}(\mathbb{H}^{n})}-2\alpha
E(0)$
$\displaystyle=(\alpha-\omega-1)\left[||u_{t}||^{2}_{L^{2}(\mathbb{H}^{n})}+(b+1)||u||^{2}_{L^{2}(\mathbb{H}^{n})}\right]+(\alpha-2)||\nabla_{H}u||^{2}_{L^{2}(\mathbb{H}^{n})}-2\alpha
E(0)$
$\displaystyle+b(2\alpha-\omega-3)\int_{0}^{t}||u_{s}(s)||^{2}_{L^{2}(\mathbb{H}^{n})}ds+(m(\alpha-2)-(b+1)(\alpha-\omega-1)|)||u||^{2}_{L^{2}(\mathbb{H}^{n})}$
$\displaystyle\geq(\alpha-\omega-1)\left[2{\rm
Re}(u,u_{t})+b||u||^{2}_{L^{2}(\mathbb{H}^{n})}\right]+(\alpha-2)||\nabla_{H}u||^{2}_{L^{2}(\mathbb{H}^{n})}-2\alpha
E(0)$
$\displaystyle+b(2\alpha-\omega-3)\int_{0}^{t}||u_{s}(s)||^{2}_{L^{2}(\mathbb{H}^{n})}ds+(m(\alpha-2)-(b+1)(\alpha-\omega-1)|)||u||^{2}_{L^{2}(\mathbb{H}^{n})}.$

In the second line that we have used (2). By selecting
$\omega=\alpha-1-\frac{m(\alpha-2)}{\mu+1}$ which satisfies $\omega>1$ since
$\mu+1>m$ and using the argument from Step I, we obtain

$\displaystyle\eta(t)$ $\displaystyle>\frac{m(\alpha-2)}{\mu+1}(2{\rm
Re}(u,u_{t})+b||u||^{2}_{L^{2}(\mathbb{H}^{n})})-2\alpha E(0)$
$\displaystyle>\frac{m(\alpha-2)}{\mu+1}(2{\rm
Re}(u_{0},u_{1})+b||u_{0}||^{2}_{L^{2}(\mathbb{H}^{n})})-2\alpha E(0)$
$\displaystyle>\left(\frac{m(\alpha-2)}{\mu+1}\right)2{\rm
Re}(u_{0},u_{1})-2\alpha E(0)$ $\displaystyle\geq 0,$

Note that we have used the fact $A^{\prime}(t)>0$ and the expression (2.5)
with $A(t)=2{\rm Re}(u,u_{t})+b||u||^{2}_{L^{2}(\mathbb{H}^{n})}$, and the
condition (2.3) in the last line, respectively. So we obtain the inequality

$M^{\prime\prime}(t)M(t)-\frac{\omega+3}{4}(M^{\prime}(t))^{2}>0.$

Then

$\frac{d}{dt}\left[\frac{M^{\prime}(t)}{M^{\frac{\omega+3}{4}}(t)}\right]>0\Rightarrow\begin{cases}M^{\prime}(t)\geq\left[\frac{M^{\prime}(0)}{M^{\frac{\omega+3}{4}}(0)}\right]M^{\frac{\omega+3}{4}}(t),\\\
M(0)=(bT_{0}+1)||u_{0}||_{L^{2}(\mathbb{H}^{n})}^{2}.\end{cases}$

Let us denote $\sigma=\frac{\omega-1}{4}$. Then we have

$-\frac{1}{\sigma}\left[M^{-\sigma}(t)-M^{-\sigma}(0)\right]\geq\frac{M^{\prime}(0)}{M^{\sigma+1}(0)}t,$

that gives

$\displaystyle M(t)\geq\left(\frac{1}{M^{\sigma}(0)}-\frac{\sigma
M^{\prime}(0)}{M^{\sigma+1}(0)}t\right)^{-\frac{1}{\sigma}}.$

Then the blow-up time $T^{*}$ satisfies

$0<T^{*}\leq\frac{M(0)}{\sigma M^{\prime}(0)},$

where $M^{\prime}(0)=2{\rm Re}(u_{0},u_{1})$. This completes the proof. ∎

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