# Semilinear damped wave equations on the Heisenberg group with initial data
from Sobolev spaces of negative order

Aparajita Dasgupta  Aparajita Dasgupta Department of Mathematics Indian
Institute of Technology Delhi Delhi, 110016 India<EMAIL_ADDRESS>,  Vishvesh Kumar  Vishvesh Kumar Department of Mathematics: Analysis, Logic
and Discrete Mathematics Ghent University Krijgslaan 281, Building S8, B 9000
Ghent, Belgium<EMAIL_ADDRESS>and<EMAIL_ADDRESS>,  Shyam
Swarup Mondal  Shyam Swarup Mondal Department of Mathematics: Analysis, Logic
and Discrete Mathematics Ghent University Krijgslaan 281, Building S8, B 9000
Ghent Belgium<EMAIL_ADDRESS>and  Michael Ruzhansky  Michael
Ruzhansky Department of Mathematics: Analysis, Logic and Discrete Mathematics
Ghent University Krijgslaan 281, Building S8, B 9000 Ghent Belgium and School
of Mathematical Sciences Queen Mary University of London, United Kingdom
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we focus on studying the Cauchy problem for semilinear damped
wave equations involving the sub-Laplacian $\mathcal{L}$ on the Heisenberg
group $\mathbb{H}^{n}$ with power type nonlinearity $|u|^{p}$ and initial data
taken from Sobolev spaces of negative order homogeneous Sobolev space
$\dot{H}^{-\gamma}_{\mathcal{L}}(\mathbb{H}^{n}),\gamma>0$, on
$\mathbb{H}^{n}$. In particular, in the framework of Sobolev spaces of
negative order, we prove that the critical exponent is the exponent
$p_{\text{crit}}(Q,\gamma)=1+\frac{4}{Q+2\gamma},$ for some
$\gamma\in(0,\frac{Q}{2})$, where $Q:=2n+2$ is the homogeneous dimension of
$\mathbb{H}^{n}$. More precisely, we establish

  * •

a global-in-time existence of small data Sobolev solutions of lower regularity
for $p>p_{\text{crit}}(Q,\gamma)$ in the energy evolution space; l

  * •

a finite time blow-up of weak solutions for $1<p<p_{\text{crit}}(Q,\gamma)$
under certain conditions on the initial data by using the test function
method.

Furthermore, to precisely characterize the blow-up time, we derive sharp upper
bound and lower bound estimates for the lifespan in the subcritical case.

###### Key words and phrases:

Heisenberg group, Semilinear damped wave equation, Critical exponent, Negative
order Sobolev spaces, Global existence, Finite blow-up

###### 2020 Mathematics Subject Classification:

Primary 43A80, 35L71, 35A01; Secondary 35B33, 35B44,

###### Contents

  1. 1 Introduction and discussion on main results
    1. 1.1 Description of problem and background
    2. 1.2 Main results: Comprehensive review
  2. 2 Preliminaries: Analysis on the Heisenberg group $\mathbb{H}^{n}$
    1. 2.1 Notations
    2. 2.2 The Hermite operator
    3. 2.3 The Heisenberg group
    4. 2.4 Fourier analysis on the Heisenberg group $\mathbb{H}^{n}$
  3. 3 Linear damped wave equations: $\dot{H}_{\mathcal{L}}^{s}(\mathbb{H}^{n})$-norm estimates
  4. 4 Global-in-time well-posedness
  5. 5 Blow-up analysis of solutions: a test function method
  6. 6 Sharp lifespan estimates of solutions

##  1\. Introduction and discussion on main results

###  1.1. Description of problem and background

In this study, our main aim is to determine a new critical exponent for the
Cauchy problem for a semilinear damped wave equation with the power type
nonlinearities as follows:

$\displaystyle\begin{cases}u_{tt}-\mathcal{L}u+u_{t}=|u|^{p},&g\in\mathbb{H}^{n},~{}t>0,\\\
u(0,g)=\varepsilon u_{0}(g),&g\in\mathbb{H}^{n},\\\ u_{t}(0,g)=\varepsilon
u_{1}(g),&g\in\mathbb{H}^{n},\end{cases}$ (1.1)

where $\mathcal{L}$ is the sub-Laplacian on the Heisenberg group
$\mathbb{H}^{n}$, $1<p<\infty$, and the initial data $(u_{0},u_{1})$ with its
size parameter $\varepsilon>0$ belongs to subelliptic (or Folland-Stein)
homogeneous Sobolev spaces of negative order
$\left(u_{0},u_{1}\right)\in\dot{H}_{\mathcal{L}}^{-\gamma}(\mathbb{H}^{n})\times\dot{H}_{\mathcal{L}}^{-\gamma}(\mathbb{H}^{n})$
with $\gamma>0$. In other words, we study the global-in-time existence of
small data solutions and the blow-up in finite time of solutions to the Cauchy
problem (1.1).

To discuss the classical Euclidean scenario, let us consider the semilinear
damped wave equation on $\mathbb{R}^{n}$ with the power type nonlinearities as
follows:

$\displaystyle\begin{cases}u_{tt}-\Delta
u+u_{t}=|u|^{p},&x\in\mathbb{R}^{n},~{}t>0,\\\
u(0,x)=u_{0}(x),&x\in\mathbb{R}^{n},\\\
u_{t}(0,x)=u_{1}(x),&x\in\mathbb{R}^{n},\end{cases}$ (1.2)

where $1<p<\infty$ and $\Delta$ is the Laplacian on $\mathbb{R}^{n}$. When the
initial data belongs additionally to $L^{1}$-space, the global existence or a
blow-up result to (1.2), depending on the critical exponent has been studied
in [20, 23, 37, 35] and references therein. The critical exponent signifies
the threshold condition on the exponent $p$ for the global-in-time Sobolev
solutions and the blow-up of local-in-time weak solutions with small data. The
critical exponent for solutions to (1.2) is the so-called Fujita exponent
given by $p_{\text{Fuji}}(n):=1+\frac{2}{n}$ (see [20, 35, 37, 23]). More
precisely,

  * •

When $n=1,2$, Matsumura in his seminal paper [23] proved the global-in-time
existence of small data solutions for $p>p_{\text{Fuji}}(n)$.

  * •

For any $n\geq 1$, a global existence for $p>p_{\text{Fuji}}(n)$ (by assuming
compactly supported initial data) and blow-up of the local-in-time solutions
in the subcritical case $1<p<p_{\text{Fuji}}(n)$ was explored by Todorova and
Yordanov [35].

  * •

For $p=p_{\text{Fuji}}(n),$ the blow-up result was obtained by Zhang [37].

Later, Ikehata and Tanizawa [20] removed the restriction of compactly
supported data for the supercritical case $p>p_{\text{Fuji}}(n)$. Moreover,
the sharp lifespan estimates for the Cauchy problem (1.2) with additional
$L^{1}$-data assumption are given by

$T_{\varepsilon}\left\\{\begin{array}[]{ll}=\infty&\text{ if
}p>p_{\mathrm{Fuji}}(n),\\\
\simeq\exp\left(C\varepsilon^{-(p-1)}\right)&\text{ if
}p=p_{\mathrm{Fuji}}(n),\\\ \simeq
C\varepsilon^{-\frac{2(p-1)}{2-n(p-1)}}&\text{ if
}p<p_{\mathrm{Fuji}}(n),\end{array}\right.$

where $C$ is a positive constant independent of $\varepsilon$. We cite [18,
16, 17, 22] for a detailed study on the sharp lifespan estimates for the
Cauchy problem (1.2). We also refer to the excellent book [8] for the global-
in-time small data solutions for the semilinear damped wave equations in the
Euclidean framework.

In recent years, considerable attention has been devoted by numerous
researchers to finding new critical exponents for the classical semilinear
damped wave equations on $\mathbb{R}^{n}$ in different contexts. For instance,
considering the Cauchy problem (1.2) with initial data additionally belonging
to $L^{m}$-spaces with $m\in(1,2)$, the critical exponent is changed and the
new modified Fujita exponent becomes

$p_{\text{Fuji}}(n):=p_{\text{Fuji }}\left(\frac{n}{m}\right)=1+\frac{2m}{n}.$

Unlike the $L^{1}$-data case, here with additional $L^{m}$-regular data, the
global-in-time solution exists uniquely at the critical point
$p_{\text{Fuji}}\left(\frac{n}{m}\right)=1+\frac{2m}{n}$. This is the main
difference between $L^{1}$ and $L^{m}$ regular data. We refer to [19, 15, 25]
and references therein for a detailed study related to the critical exponent
$p_{\text{Fuji}}\left(\frac{n}{m}\right)$ for the solutions to the semilinear
wave equations with the $L^{m}$-regular data.

The study of the semilinear damped wave equation has also been extended in the
non-Euclidean framework. Several papers have studied nonlinear PDEs in non-
Euclidean settings in the last decades. For example, the semilinear wave
equation with or without damping has been investigated for the Heisenberg
group in [24, 31, 11]. In the case of graded groups, we refer to the recent
works [30, 33, 31, 12]. We refer to [26, 28, 27, 10, 4, 7] concerning the
damped wave equation on compact Lie groups and [21, 2, 3, 36] on a Riemannian
symmetric space of non-compact type. In particular, for the semilinear damped
wave equation on the Heisenberg group $\mathbb{H}^{n}$, namely

$\displaystyle\begin{cases}u_{tt}-\mathcal{L}u+u_{t}=|u|^{p},&g\in\mathbb{H}^{n},~{}t>0,\\\
u(0,g)=u_{0}(g),&g\in\mathbb{H}^{n},\\\
u_{t}(0,g)=u_{1}(g),&g\in\mathbb{H}^{n},\end{cases}$ (1.3)

where $p>1$, $\mathcal{L}$ is the sub-Laplacian on the Heisenberg group
$\mathbb{H}^{n}$, it was shown in [11] that the critical exponent is given by

$\displaystyle p_{\text{Fuji}}(Q):=1+\frac{2}{Q},$ (1.4)

where $Q:=2n+2$ represents the homogeneous dimension of $\mathbb{H}^{n}$. It
is interesting to note that (1.4) is also the Fujita exponent for the
semilinear heat equation

$\displaystyle\begin{cases}u_{t}-\mathcal{L}u=|u|^{p},&g\in\mathbb{H}^{n},~{}t>0,\\\
u(0,g)=u_{0}(g),&g\in\mathbb{H}^{n},\\\ \end{cases}$

on $\mathbb{H}^{n},$ where $p>1$. This topic has been discussed in [12, 32,
11]. However, in the case of a compact Lie group $\mathbb{G}$, it has been
shown in [26] that $p_{\text{Fuji}}(0):=\infty$ is the critical exponent for
the solution to the semilinear wave equation on $\mathbb{G}$. We cite [31] for
a global existence result with small data in the more general setting of
graded Lie groups for the semilinear damped wave equation involving a Rockland
operator with an additional mass term.

Recently, Chen and Reissig [6] considered the following semilinear damped wave
equation

$\displaystyle\begin{cases}u_{tt}-\Delta
u+u_{t}=|u|^{p},&x\in\mathbb{R}^{n},~{}t>0,\\\
u(0,x)=u_{0}(x),&x\in\mathbb{R}^{n},\\\
u_{t}(0,x)=u_{1}(x),&x\in\mathbb{R}^{n},\end{cases}$ (1.5)

on $\mathbb{R}^{n},$ where $p>1$ and the initial data additionally belonging
to homogeneous Sobolev spaces of negative order
$\dot{H}^{-\gamma}(\mathbb{R}^{n})$ with $\gamma>0$. They obtained a new
critical exponent $p=p_{\text{crit}}(n,\gamma):=1+\frac{4}{n+2\gamma}$ for
some $\gamma\in(0,\frac{n}{2})$ in this framework. More specifically, the
authors proved that:

  * •

For $p>p_{\text{crit}}(n,\gamma)$, the problem (1.5) admits a global-in-time
Sobolev solution for sufficiently small data of lower regularity.

  * •

For $1<p<p_{\text{crit}}(n,\gamma)$, the solutions to (1.5) blow-up in a
finite time. In other words, there exists $T>0$ such that the solution to
(1.5) satisfies
$\left\|u\left(\cdot,t_{m}\right)\right\|_{\infty}\rightarrow\infty$ as
$t_{m}\rightarrow T$.

Further, the authors also investigated sharp lifespan estimates for weak
solutions to (1.5), in which the sharpness of lifespan estimates is given by

$T_{\varepsilon}\left\\{\begin{array}[]{ll}=\infty&\text{ if }p>p_{\text{crit
}}(n,\gamma),\\\ \simeq
C\varepsilon^{-\frac{2}{2p^{\prime}-2-\frac{n}{2}-\gamma}}&\text{ if
}p<p_{\text{crit }}(n,\gamma),\end{array}\right.$

where $C$ is a positive constant independent of $\varepsilon$ and $p^{\prime}$
is the Lebesgue exponent conjugate of $p$ such that
$\frac{1}{p}+\frac{1}{p^{\prime}}=1$.

To the best of our knowledge, in the non-Euclidean framework, the subelliptic
damped wave equation on the Heisenberg group with initial data localized in
Sobolev spaces of negative order has not been considered in the literature so
far, even for the linear Cauchy problem. Therefore, an interesting and viable
problem is to study several qualitative properties such as global-in-time
well-posedness, blow-up criterion, decay rate, asymptotic profiles to
solutions for the subelliptic damped wave equations on the Heisenberg group
$\mathbb{H}^{n}$ with initial data additionally belonging to subelliptic
Sobolev space $\dot{H}_{\mathcal{L}}^{-\gamma}(\mathbb{H}^{n})$ with
$\gamma>0$.

The main aim of this paper is to investigate and determine a critical exponent
for the Cauchy problem for semilinear damped wave equation (1.1) with the
initial data additionally belonging to subelliptic homogeneous Sobolev spaces
$\dot{H}_{\mathcal{L}}^{-\gamma}(\mathbb{H}^{n})$ of negative order
$\gamma>0$. More specifically,

  * •

Under additional assumptions for the initial data in
$\dot{H}_{\mathcal{L}}^{-\gamma}(\mathbb{H}^{n})$, we obtain a new critical
exponent to (1.1) given by

$\displaystyle p_{\text{crit }}(Q,\gamma):=1+\frac{4}{Q+2\gamma},$ (1.6)

with
$\gamma\in\bigg{(}0,\min\bigg{\\{}\frac{Q}{2},\frac{\sqrt{Q^{2}+16Q}}{4}-\frac{Q}{4}\bigg{\\}}\bigg{)}.$

  * •

We derive sharp lifespan estimates for weak solutions to the semilinear Cauchy
problem (1.1). Define the lifespan $T_{\varepsilon}$ as the maximal existence
time for solution of (1.1), i.e.,

$\displaystyle T_{\varepsilon}:=\Big{\\{}T>0~{}$ $\displaystyle:~{}\text{there
exists a unique local-in-time solution to the Cauchy}$ $\displaystyle\text{
problem (\ref{eq0010}) on $[0,T)$ with a fixed
parameter}~{}\varepsilon>0\Big{\\}}.$ (1.7)

If the initial data is from $\dot{H}_{\mathcal{L}}^{-\gamma}(\mathbb{H}^{n})$
with $\gamma\in\left(0,2\right)$ and the exponent $p$ satisfies
$1+\frac{2\gamma}{Q}\leq p\leq\frac{Q}{Q-2}$, then the new sharp lifespan
estimates are given by

$T_{\varepsilon}\left\\{\begin{array}[]{ll}=\infty&\text{ if }p>p_{\text{crit
}}(Q,\gamma),\\\ \simeq
C\varepsilon^{-\frac{2}{2p^{\prime}-2-\frac{Q}{2}-\gamma}}&\text{ if
}p<p_{\text{crit }}(Q,\gamma),\end{array}\right.$

where the positive constant $C$ is independent of $\varepsilon.$

###  1.2. Main results: Comprehensive review

Throughout the paper, we denote $L^{q}(\mathbb{H}^{n}),1\leq q<\infty$, the
space of $q$-integrable functions on $\mathbb{H}^{n}$ with respect to the Haar
measure $dg$ on $\mathbb{H}^{n},$ which is nothing but the Lebesgue measure of
$\mathbb{R}^{2n+1},$ and the space of all essentially bounded functions on
$\mathbb{H}^{n}$ for $q=\infty$. The fractional subelliptic (Folland-Stein)
Sobolev space $H_{\mathcal{L}}^{s}(\mathbb{H}^{n}),s\in\mathbb{R}$, associated
to the sub-Laplacian $\mathcal{L}$ on $\mathbb{H}^{n}$, is defined as

$H_{\mathcal{L}}^{s}\left(\mathbb{H}^{n}\right):=\left\\{f\in\mathcal{D}^{\prime}\left(\mathbb{H}^{n}\right):(I-\mathcal{L})^{s/2}f\in
L^{2}\left(\mathbb{H}^{n}\right)\right\\},$

equipped with the norm

$\|f\|_{H_{\mathcal{L}}^{s}\left(\mathbb{H}^{n}\right)}:=\left\|(I-\mathcal{L})^{s/2}f\right\|_{L^{2}\left(\mathbb{H}^{n}\right)}.$

Similarly, we denote by $\dot{H}_{\mathcal{L}}^{s}(\mathbb{H}^{n}),$ the
homogeneous Sobolev defined as the space of all
$f\in\mathcal{D}^{\prime}(\mathbb{H}^{n})$ such that
$(-\mathcal{L})^{{s}/{2}}f\in L^{2}(\mathbb{H}^{n})$. At times, we also write
$H_{\mathcal{L}}^{s}$ and $\dot{H}_{\mathcal{L}}^{s}$ for
$H_{\mathcal{L}}^{s}\left(\mathbb{H}^{n}\right)$ and
$\dot{H}_{\mathcal{L}}^{s}\left(\mathbb{H}^{n}\right),$ respectively from here
on.

For
$(u_{0},u_{1})\in\mathcal{A}_{s}^{\mathcal{L}}:=(H^{s}_{\mathcal{L}}\cap\dot{H}^{-\gamma}_{\mathcal{L}})\times(L^{2}\cap\dot{H}^{-\gamma}_{\mathcal{L}})$,
we denote
$\left\|\left(u_{0},u_{1}\right)\right\|_{\mathcal{A}_{s}^{\mathcal{L}}}$ as

$\left\|\left(u_{0},u_{1}\right)\right\|_{\mathcal{A}_{s}^{\mathcal{L}}}:=\|u_{0}\|_{H^{s}_{\mathcal{L}}\cap\dot{H}^{-\gamma}_{\mathcal{L}}}+\|u_{1}\|_{L^{2}\cap\dot{H}^{-\gamma}_{\mathcal{L}}}.$

Utilizing techniques derived from the noncommutative Fourier analysis on the
Heisenberg group $\mathbb{H}^{n}$, we present our initial finding regarding
time decay estimates in the $\dot{H}_{\mathcal{L}}^{s}$-norm of solutions to
the homogeneous version of the Cauchy problem (1.1). This result is detailed
below.

######  Theorem 1.1.

Let $\mathbb{H}^{n}$ be the Heisenberg group with the homogeneous dimension
$Q.$ Let
$\left(u_{0},u_{1}\right)\in\left(H_{\mathcal{L}}^{s}\cap\dot{H}_{\mathcal{L}}^{-\gamma}\right)\times\left(H_{\mathcal{L}}^{s-1}\cap\dot{H}_{\mathcal{L}}^{-\gamma}\right)$
such that $s\geq 0$ and $\gamma\in\mathbb{R}$ such that $s+\gamma\geq 0$.
Consider the following linear Cauchy problem

$\displaystyle\begin{cases}u_{tt}-\mathcal{L}u+u_{t}=0,&g\in\mathbb{H}^{n},t>0,\\\
u(0,g)=u_{0}(g),&g\in\mathbb{H}^{n},\\\
u_{t}(0,g)=u_{1}(g),&g\in\mathbb{H}^{n}.\end{cases}$ (1.8)

Then, the solution $u$ satisfies the following
$\dot{H}_{\mathcal{L}}^{s}$-decay estimate

$\displaystyle\|u(t,\cdot)\|_{\dot{H}_{\mathcal{L}}^{s}}\lesssim(1+t)^{-\frac{s+\gamma}{2}}\left(\left\|u_{0}\right\|_{H^{s}_{\mathcal{L}}\cap\dot{H}^{-\gamma}_{\mathcal{L}}}+\left\|u_{1}\right\|_{H_{\mathcal{L}}^{s-1}\cap\dot{H}_{\mathcal{L}}^{-\gamma}}\right),$
(1.9)

for any $t\geq 0$.

The next result is about the global-in-time well-posedness of the Cauchy
problem (1.1) in the energy evolution space
$\mathcal{C}\left([0,T],H^{s}_{\mathcal{L}}\right),s\in(0,1]$. In this case, a
version of a Gagliardo-Nirenberg type inequality on $\mathbb{H}^{n}$ (see
Section 2) will play a crucial role in estimating the power type nonlinearity
in $L^{2}(\mathbb{H}^{n})$. Let us first make clear what do we mean by a
solution of (1.1). For the global existence result we will work with the mild
solutions of (1.1).

We say that a function $u$ is a mild solution to (1.1) on $[0,T]$ if $u$ is a
fixed point for the integral operator $N:u\in X_{s}(T)\mapsto Nu(t,g),$ given
by

$\displaystyle Nu(t,g):=u^{\text{lin}}(t,g)+u^{\text{non}}(t,g),$ (1.10)

in the energy evolution space
$X_{s}(T)\doteq\mathcal{C}\left([0,T],H_{\mathcal{L}}^{s}(\mathbb{H}^{n})\right),s\in(0,1]$,
equipped with the norm

$\displaystyle\|u\|_{X_{s}(T)}$
$\displaystyle:=\sup\limits_{t\in[0,T]}\left((1+t)^{\frac{\gamma}{2}}\|u(t,\cdot)\|_{L^{2}}+(1+t)^{\frac{s+\gamma}{2}}\|u(t,\cdot)\|_{\dot{H}^{s}_{\mathcal{L}}}\right),$

with $\gamma>0,$ where

$u^{\text{lin}}(t,g)=u_{0}(g)*_{(g)}E_{0}(t,g)+u_{1}(g)*_{(g)}E_{1}(t,g)$

is the solution to the corresponding linear Cauchy problem (1.8) and

$u^{\text{non}}(t,g)=\int_{0}^{t}|u(s,g)|^{p}*_{(g)}E_{1}(t-s,g)\;ds.$

Here $*_{(g)}$ is the group convolution product on $\mathbb{H}^{n}$ with
respect to the $g$ variable and $E_{0}(t,g)$ and $E_{1}(t,g)$ represent the
fundamental solutions to the homogeneous problem (1.8) with initial data
$\left(u_{0},u_{1}\right)=\left(\delta_{0},0\right)$ and
$\left(u_{0},u_{1}\right)=$ $\left(0,\delta_{0}\right)$, respectively.

We prove the global-in-time existence and uniqueness of small data Sobolev
solutions to the semilinear damped wave equation (1.1) of low regularity by
finding a unique fixed point to the operator $N$, i.e., $Nu\in X_{s}(T)$ for
all $T>0$. It means that there exists a unique global solution $u$ to the
equation $Nu=u\in X_{s}(T)$ which also gives the solution to (1.1). In order
to prove that $N$ has a uniquely determined fixed point, we use Banach’s fixed
point argument with respect to the norm on $X_{s}(T)$ as defined above.

Keeping $p_{\text{crit }}(Q,\gamma)=1+\frac{4}{Q+2\gamma}$ in mind, we state
the following global-in-time existence result.

######  Theorem 1.2.

Let $s\in(0,1]$ and $\gamma\in\left(0,\frac{Q}{2}\right)$. Assume that the
exponent $p$ satisfies $1<p\leq\frac{Q}{Q-2s}$ and

$\displaystyle p\left\\{\begin{array}[]{ll}>p_{\text{crit }}(Q,\gamma)&\text{
if }\gamma\leq\tilde{\gamma},\\\ \geq 1+\frac{2\gamma}{Q}&\text{ if
}\gamma>\tilde{\gamma},\end{array}\right.$ (1.13)

where $\tilde{\gamma}$ denotes the positive root of the quadratic equation
$2\tilde{\gamma}^{2}+Q\tilde{\gamma}-2Q=0$. In addition to this, we also
assume that there exists a small parameter $\varepsilon>0$ such that
$\left(u_{0},u_{1}\right)\in\mathcal{A}_{s}^{\mathcal{L}}$ satisfies
$\left\|\left(u_{0},u_{1}\right)\right\|_{\mathcal{A}_{s}^{\mathcal{L}}}\leq\varepsilon.$
Then, there exists a small positive constant $\varepsilon_{0}$ such that the
Cauchy problem for the semilinear damped wave equation (1.1) has a uniquely
determined Sobolev solution

$u\in\mathcal{C}\left([0,\infty),H^{s}_{\mathcal{L}}\right),s\in(0,1],$

for every $0<\varepsilon\leq\varepsilon_{0}$. Therefore, the lifespan of the
solution is given by $T_{\varepsilon}=\infty$. Moreover, the solution
satisfies the following two estimates listed below:

$\|u(t,\cdot)\|_{L^{2}}\lesssim(1+t)^{-\frac{\gamma}{2}}\left\|\left(u_{0},u_{1}\right)\right\|_{\mathcal{A}_{s}^{\mathcal{L}}},$

and

$\|u(t,\cdot)\|_{\dot{H}^{s}_{\mathcal{L}}}\lesssim(1+t)^{-\frac{s+\gamma}{2}}\left\|\left(u_{0},u_{1}\right)\right\|_{\mathcal{A}_{s}^{\mathcal{L}}}.$

It is crucial to emphasize that there is no loss in the decay of the solution
when transitioning from the linear to the nonlinear problem. In other words,
the decay rates outlined in the preceding theorem align precisely with the
decay rates established for the corresponding linearized damped wave equation,
as presented in Theorem 1.1. However, the restriction $1<p\leq\frac{Q}{Q-2s}$
in the above theorem is necessary in order to apply a Gagliardo-Nirenberg type
inequality on $\mathbb{H}^{n}$.

######  Remark 1.3.

Some examples for the admissible range of exponents $p$ for the global-in-time
existence result in the low dimensions Heisenberg group $\mathbb{H}^{n}$ with
$n=1$ and $2$, that is, $Q=4$ and $6$, respectively, are as follows:

  * •

When $Q=4$, we take $s\in(0,1]$ and $\gamma\in(0,2)$ and the exponent
satisfies

$\begin{array}[]{l}\vspace{0.3cm}1+\frac{2}{2+\gamma}<p\leq\frac{2}{2-s}~{}\quad\text{
if }~{}0<\gamma\leq\tilde{\gamma}=1.24,\\\ 1+\frac{\gamma}{2}\leq
p\leq\frac{2}{2-s}~{}\quad\quad\text{ if }~{}1.24<\gamma<2.\end{array}$

  * •

When $Q=6$, we take $s\in(0,1]$ and $\gamma\in(0,3)$ and the exponent
satisfies

$\begin{array}[]{l}\vspace{0.3cm}1+\frac{2}{3+\gamma}<p\leq\frac{3}{3-s}~{}\quad\text{
if }~{}0<\gamma\leq\tilde{\gamma}=1.37,\\\ 1+\frac{\gamma}{3}\leq
p\leq\frac{3}{3-s}~{}\quad\quad\text{ if }~{}1.37<\gamma<3.\end{array}$

Note that the positive root $\tilde{\gamma}$ of
$2\tilde{\gamma}^{2}+Q\tilde{\gamma}-2Q=0$ is always strictly less than 2 for
any homogeneous dimension $Q$.

Based on the aforementioned illustrations, it is clear that the global
existence result, as stated in Theorem 1.2, is only pertinent to specific
lower homogeneous dimensions. This constraint is due to the technical
stipulation that $1<p\leq\frac{Q}{Q-2s}$. For global existence results in
higher homogeneous dimensions, one can study Sobolev solutions by considering
initial data from subelliptic Sobolev spaces with an appropriate degree of
higher/large regularity.

Our next result is about a blow-up (in-time) result in the subcritical case to
the Cauchy problem (1.1) for certain values of $p$ regardless of the size of
the initial data. Before the blow-up result, we first introduce a suitable
notion of a weak solution to the Cauchy problem (1.1).

######  Definitions 1.4.

For any $T>0$, a weak solution of the Cauchy problem (1.1) in
$[0,T)\times\mathbb{H}^{n}$ is a function $u\in L_{\text{loc
}}^{p}\left([0,T)\times\mathbb{H}^{n}\right)$ that satisfies the following
integral relation:

$\displaystyle\int_{0}^{T}\int_{\mathbb{H}^{n}}u(t,g)\left(\partial_{t}^{2}\phi(t,g)-\mathcal{L}\phi(t,g)-\partial_{t}\phi(t,g)\right)\;dg\;dt-\varepsilon\int_{\mathbb{H}^{n}}u_{0}(g)\phi(0,g)\;dg$
$\displaystyle-\varepsilon\int_{\mathbb{H}^{n}}u_{1}(g)\phi(0,g)\;dg+\varepsilon\int_{\mathbb{H}^{n}}u_{0}(g)\partial_{t}\phi(0,g)\;dg=\int_{0}^{T}\int_{\mathbb{H}^{n}}|u(t,g)|^{p}\phi(t,g)\;dg\;dt,$
(1.14)

for any $\phi\in\mathcal{C}_{0}^{\infty}([0,T)\times\mathbb{H}^{n})$ and any
$t\in(0,T)$. If $T=\infty$, we call $u$ to be a global-in-time weak solution
to (1.1), otherwise $u$ is said to be a local in time weak solution to (1.1).

Indeed, under some additional assumption on the initial data, we have the
following blow-up result.

######  Theorem 1.5.

Let $\gamma\in\left(0,\frac{Q}{2}\right)$ and let the exponent $p$ satisfy
$1<p<p_{\text{crit }}(Q,\gamma)$. We also assume that the non-negative initial
data
$\left(u_{0},u_{1}\right)\in\dot{H}_{\mathcal{L}}^{-\gamma}\times\dot{H}_{\mathcal{L}}^{-\gamma}$
satisfies

$\displaystyle u_{0}(g)+u_{1}(g)$ $\displaystyle\geq C_{0}\langle
g\rangle^{-Q\left(\frac{1}{2}+\frac{\gamma}{Q}\right)}(\log(e+|g|))^{-1},\quad
g\in\mathbb{H}^{n},$ (1.15)

where $C_{0}>0$ is a fixed constant. Then, there is no global (in time) weak
solution to the Cauchy problem (1.1). Moreover, the lifespan
$T_{w,\varepsilon}$ of local (in time) weak solutions to the Cauchy problem
(1.1) satisfies the following upper bound estimate

$T_{w,\varepsilon}\leq
C\varepsilon^{-\frac{2}{2p^{\prime}-2-\frac{Q}{2}-\gamma}},$

where $p^{\prime}$ is the conjugate exponent to $p$ and $C$ is a positive
constant independent of $\varepsilon$.

From Theorems 1.2 and Theorem 1.5, we can conclude that the critical exponent
for the semilinear damped wave equation (1.1) is $p_{\text{crit
}}(Q,\gamma):=1+\frac{4}{Q+2\gamma}$ (see (1.6)) when initial data are
additionally taken from the negative order Sobolev space
$\dot{H}_{\mathcal{L}}^{-\gamma},\gamma>0$. It gives us a new way to look at
the critical exponent for the semilinear damped wave equation in Sobolev space
with a negative order. Further, if we take $\gamma=\frac{Q}{2}$, then the
critical exponent becomes $p_{\text{crit }}(Q,\frac{Q}{2})=1+\frac{2}{Q},$
which is the well-known Fujita exponent $p_{\text{Fuji }}(Q)$ on the
Heisenberg group $\mathbb{H}^{n}$. In another way, one may represent our new
critical exponent as $p_{\text{crit }}(Q,\gamma)=p_{\text{Fuji
}}(\frac{Q}{2}+\gamma)$.

The behavior of the system at the critical case $p=p_{\text{crit }}(Q,\gamma)$
remains uncertain, as it is unclear whether a global (in time) small data
Sobolev solutions exist or the weak solutions will blow-up.

In the next remark, particularly, we consider $s=1$ for Theorem 1.2 and 1.5 to
give a rough idea about the critical exponent.

######  Remark 1.6.

Due to the applications of Gagliardo-Nirenberg inequality, we have the
technical restriction on the exponent $1<p\leq\frac{Q}{Q-2}$. Consequently,
the ranges of exponent $p$ for global-in-time existence and blow-up are as
follows.

  * •

When $Q=4,6$

    * –

blow-up of weak solutions if $1<p<p_{\text{crit }}(Q,\gamma)$,

    * –

and global-in-time existence of Sobolev solutions if

$p_{\text{crit
}}(Q,\gamma)<p\leq\frac{Q}{Q-2},\qquad\text{for}~{}0<\gamma\leq\tilde{\gamma},$

as well as

$1+\frac{2\gamma}{Q}\leq
p\leq\frac{Q}{Q-2},\qquad\text{for}~{}\tilde{\gamma}<\gamma<\frac{Q}{2}.$

  * •

When $Q\geq 8$, from Theorem 1.2, the set of admissible exponents $p$ is
completely empty.

$\gamma$$p$$0$$\frac{Q}{Q-2}$${\tilde{\gamma}}$$\frac{Q}{2}$$1$$\longleftarrow$
$p=1+\frac{2\gamma}{Q}$$p=p_{\mathrm{crit}}(Q,\gamma)$$\longrightarrow$$\nearrow$$\nwarrow$

For $Q=4,6$$\square$ Range for global existence of solution

$\square$ Range for blow-up of solution

Figure 1. Description of the critical exponent in the $(\gamma,p)$ plane

For a detailed analysis of the critical exponent which depends on the
parameter $\gamma$ and $Q$, we can describe it by the $(\gamma,p)$ plane in
Figure 1. In Figure 1, with an increase of the homogeneous dimension $Q$ of
$\mathbb{H}^{n}$, the curve $p=p_{\text{crit }}(Q,\gamma)$ and the segment
$p=1+\frac{2\gamma}{Q}$ will move following the direction of $\nearrow$ and
$\nwarrow$ lines arrows, respectively.

The formulations of Theorem 1.2 and Theorem 1.5 for $s=1$ also give the
critical index for regularity of initial data that belongs additionally to
$\dot{H}_{\mathcal{L}}^{-\gamma}$ with $0<\gamma<\tilde{\gamma}$.
Specifically, for $1+\frac{2}{Q}<p<1+\frac{4}{Q}$, the critical index is given
by $\gamma_{\text{crit }}(p,Q):=\frac{2}{p-1}-\frac{Q}{2}$. Regarding the
initial data that also additionally belongs to
$\dot{H}_{\mathcal{L}}^{-\gamma}$ with $\gamma>0$,

  * •

then local (in-time) weak solutions in general blow up in finite time for
$0<\gamma<\gamma_{\text{crit }}(p,Q)$;

  * •

then the global (in-time) small data Sobolev solutions exists uniquely for
$\gamma_{\text{crit }}(p,Q)<\gamma<\min\\{\tilde{\gamma},\frac{Q}{2}\\}$.

From Theorem 1.5, for $1<p<p_{\text{crit }}(Q,\gamma)$, we know that the non-
trivial local (in time) weak solution blow up in finite time and we have the
following upper bound estimates for the lifespan

$T_{\varepsilon}\leq T_{w,\varepsilon}\leq
C\varepsilon^{-\frac{2}{2p^{\prime}-2-\frac{Q}{2}-\gamma}}.$

Now, it is important to investigate the lower bound estimates for the
lifespan.

To thoroughly examine the lifespan, it is essential to consider the precise
definition of mild solutions to the Cauchy problem (1.1) on $[0,T)$ with $T>0$
for $u\in C([0,T),H_{\mathcal{L}}^{1})$. Let $T_{m,\varepsilon}$ be the
lifespan of a mild solution $u$. Then, we have the following result regarding
the lower bound for the lifespan.

######  Theorem 1.7.

Let $\gamma\in(0,2)$ and let the exponent $p$ satisfy $1<p<p_{\text{crit
}}(Q,\gamma)$ such that

$\displaystyle 1+\frac{2\gamma}{Q}\leq p\leq\frac{Q}{Q-2}.$ (1.16)

We also assume that
$\left(u_{0},u_{1}\right)\in\mathcal{A}_{1}^{\mathcal{L}}$. Then, there exists
a constant $\varepsilon_{0}$ such that for every
$\varepsilon\in\left(0,\varepsilon_{0}\right]$, the lifespan
$T_{m,\varepsilon}$ of mild solutions $u$ to the Cauchy problem (1.1)
satisfies the following lower bound condition:

$T_{m,\varepsilon}\geq
D\varepsilon^{-\frac{2}{2p^{\prime}-2-\frac{Q}{2}-\gamma}},$

where $D$ is a positive constant independent of $\varepsilon$, but may depends
on $p,Q,\gamma$ as well as
$\left\|\left(u_{0},u_{1}\right)\right\|_{\mathcal{A}_{1}^{\mathcal{L}}}$.

Since $u$ in Theorem 1.2 is a mild solution to (1.1), this mild solution is
also a weak solution to (1.1) according to a density argument with timespan
$T_{m,\varepsilon}\leq T_{\varepsilon}$. Once more drawing on Theorems 1.5 and
1.7, for $1<p<p_{\text{crit }}(Q,\gamma)$, we claim a sharp estimate for the
lifespan $T_{\varepsilon}$ as stated below

$T_{\varepsilon}\simeq
C\varepsilon^{-\frac{2}{2p^{\prime}-2-\frac{Q}{2}-\gamma}}=C\varepsilon^{-\frac{2(p-1)}{2-\left(\frac{Q}{2}+\gamma\right)(p-1)}},$

for $\gamma\in(0,\min\\{2,\frac{Q}{2}\\})$. Moreover, the lifespan estimate
mentioned above agrees with the one for additional $L^{1}$-regular data when
particularly we consider $\gamma=\frac{Q}{2}$.

######  Remark 1.8.

One interesting observation is to discuss the acceptable ranges for $p$ so
that we can get sharp lifespan estimates. This can be summed up as follows:

  * •

when $Q=4$, for $1+\frac{2\gamma}{Q}\leq p<p_{\text{crit }}(Q,\gamma)$ and
$0<\gamma\leq\frac{-Q+\sqrt{Q^{2}+16Q}}{4}$, we can achieve sharp lifespan
estimates;

  * •

When $Q=6$, for $1+\frac{2\gamma}{Q}\leq p<p_{\text{crit }}(Q,\gamma)$ with
$p\leq\frac{Q}{Q-2}$ and $0<\gamma\leq\frac{-Q+\sqrt{Q^{2}+16Q}}{4}$, sharp
lifespan estimates can be achieved;

  * •

when $Q\geq 8$, for $1+\frac{2\gamma}{Q}\leq p\leq\frac{Q}{Q-2}$ and
$0<\gamma\leq\frac{Q}{Q-2}$, we can achieve sharp lifespan estimates.

We conclude the introduction with a brief outline of the organization of the
paper. In Section 1, we discuss the main results and their explanations
related to the critical exponent $p_{\text{crit}}(Q,\gamma)$ and sharp
lifespan estimates for weak solutions to Cauchy problem (1.1) when initial
data taken additionally from $\dot{H}_{\mathcal{L}}^{-\gamma}$. Section 2 is
devoted to recalling some basics of the Fourier analysis on the Heisenberg
group $\mathbb{H}^{n}$ to make the paper self-contained. Using the Fourier
analysis on the Heisenberg group $\mathbb{H}^{n}$, we derive
$\dot{H}^{s}_{\mathcal{L}}(\mathbb{H}^{n})$-decay estimates for the solution
to the linear damped wave equation (1.1) with vanishing right-hand side in
Section 3. Using the $\dot{H}^{s}_{\mathcal{L}}(\mathbb{H}^{n})$-decay
estimates for the solution to a linear damped wave equation, we demonstrate
global-in-time well-posedness for the semilinear Cauchy problem (1.1) if
$p>p_{\text{crit }}(Q,\gamma)$ with the help of Banach’s fixed point argument
in Section 4. In order to estimate the nonlinear term in
$\dot{H}_{\mathcal{L}}^{-\gamma}(\mathbb{H}^{n})$, we use Sobolev inequality
and the Gagliardo-Nirenberg inequality on the Heisenberg group
$\mathbb{H}^{n}$. In Section 5, by employing the test function method, we
conclude the optimality by showing the blow-up of weak solutions even for
small data for the range $1<p<p_{\text{crit }}(Q,\gamma)$. As a byproduct, we
also obtain upper bound estimates for the lifespan. We conclude our paper by
deriving the sharp lower bound estimates for the lifespan of mild solutions in
Section 6 by using the method of contradiction.

##  2\. Preliminaries: Analysis on the Heisenberg group $\mathbb{H}^{n}$

In this section, we recall some basics of the Fourier analysis on the
Heisenberg groups $\mathbb{H}^{n}$ to make the manuscript self-contained. A
complete account of the representation theory on $\mathbb{H}^{n}$ can be found
in [11, 34, 31, 9, 12, 29]. However, we mainly adopt the notation and
terminology given in [9] for convenience. We commence this section by
establishing the notations that will be employed consistently throughout the
paper.

###  2.1. Notations

Throughout the article, we use the following notations:

  * •

$f\lesssim g:$ There exists a positive constant $C$ (whose value may change
from line to line in this manuscript) such that $f\leq Cg.$

  * •

$f\simeq g$: Means that $f\lesssim g$ and $g\lesssim f$.

  * •

$\mathbb{H}^{n}:$ The Heisenberg group.

  * •

$Q$: The homogeneous dimension of $\mathbb{H}^{n}$.

  * •

$dg:$ The Haar measure on the Heisenberg group $\mathbb{H}^{n}.$

  * •

$\mathcal{L}:$ The sub-Laplacian on $\mathbb{H}^{n}.$

  * •

${H}_{\mathcal{L}}^{-\gamma}$: The subelliptic Sobolev spaces of negative
order with $\gamma>0$ on $\mathbb{H}^{n}$.

###  2.2. The Hermite operator

In this subsection, we recall some definitions and properties of Hermite
functions which we will use frequently in order to study Schrödinger
representations and sub-Laplacian on the Heisenberg group $\mathbb{H}^{n}$. We
start with the definition of Hermite polynomials on $\mathbb{R}.$

Let $H_{k}$ denote the Hermite polynomial on $\mathbb{R}$, defined by

$H_{k}(x)=(-1)^{k}\frac{d^{k}}{dx^{k}}(e^{-x^{2}})e^{x^{2}},\quad
k=0,1,2,\cdots,$

and $h_{k}$ denote the normalized Hermite functions on $\mathbb{R}$ defined by

$h_{k}(x)=(2^{k}\sqrt{\pi}k!)^{-\frac{1}{2}}H_{k}(x)e^{-\frac{1}{2}x^{2}},\quad
k=0,1,2,\cdots,$

The Hermite functions $\\{h_{k}\\}$ are the eigenfunctions of the Hermite
operator (or the one-dimensional harmonic oscillator)
$H=-\frac{d^{2}}{dx^{2}}+x^{2}$ with eigenvalues $2k+1,k=0,1,2,\cdots$. These
functions form an orthonormal basis for $L^{2}(\mathbb{R})$. The higher
dimensional Hermite functions denoted by $e_{k}$ are then obtained by taking
tensor products of one dimensional Hermite functions. Thus for any multi-index
$k=(k_{1},\cdots,k_{n})\in\mathbb{N}_{0}^{n}$ and
$x=(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}$, we define
$e_{k}(x)=\prod_{j=1}^{n}h_{k_{j}}(x_{j}).$ The family
$\\{e_{k}\\}_{k\in\mathbb{N}^{n}_{0}}$ is then an orthonormal basis for
$L^{2}(\mathbb{R}^{n})$. They are eigenfunctions of the Hermite operator
$\mathrm{H}=-\Delta+|x|^{2}$, namely, we have an ordered set of positive
numbers $\\{\mu_{k}\\}_{k\in\mathbb{N}_{0}^{n}}$ such that

$\mathrm{H}e_{k}(x)=\mu_{k}e_{k}(x),\quad x\in\mathbb{R}^{n},$

for all $k\in\mathbb{N}_{0}^{n}$ and $x\in\mathbb{R}^{n}$. More precisely,
$\mathrm{H}$ has eigenvalues

$\mu_{k}=\sum_{j=1}^{n}(2k_{j}+1)=2|k|+n,$

corresponding to the eigenfunction $e_{k}$ for $k\in\mathbb{N}_{0}^{n}.$

###  2.3. The Heisenberg group

One of the simplest examples of a non-commutative and non-compact group is the
famous Heisenberg group $\mathbb{H}^{n}$. The theory of the Heisenberg group
plays a crucial role in several branches of mathematics and physics. The
Heisenberg group $\mathbb{H}^{n}$ is a nilpotent Lie group whose underlying
manifold is $\mathbb{R}^{2n+1}$ and the group operation is defined by

$(x,y,t)\circ(x^{\prime},y^{\prime},t^{\prime})=(x+x^{\prime},y+y^{\prime},t+t^{\prime}+\frac{1}{2}(xy^{\prime}-x^{\prime}y)),$

where $(x,y,t)$, $(x^{\prime},y^{\prime},t^{\prime})$ are in
$\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}$ and $xy^{\prime}$ denotes
the standard scalar product in $\mathbb{R}^{n}$. Moreover, $\mathbb{H}^{n}$ is
a unimodular Lie group on which the left-invariant Haar measure $dg$ is the
usual Lebesgue measure $\,dx\,dy\,dt.$

The canonical basis for the Lie algebra $\mathfrak{h}_{n}$ of $\mathbb{H}^{n}$
is given by the left-invariant vector fields:

$\displaystyle X_{j}$
$\displaystyle=\partial_{x_{j}}-\frac{y_{j}}{2}\partial_{t},$ $\displaystyle
Y_{j}=\partial_{y_{j}}+\frac{x_{j}}{2}\partial_{t},\quad j=1,2,\dotsc
n,~{}\mathrm{and}~{}T=\partial_{t},$ (2.1)

which satisfy the commutator relations
$[X_{i},Y_{j}]=\delta_{ij}T,\quad\text{for}~{}i,j=1,2,\dotsc n.$

Moreover, the canonical basis for $\mathfrak{h}_{n}$ admits the decomposition
$\mathfrak{h}_{n}=V\oplus W,$ where
$V=\operatorname{span}\\{X_{j},Y_{j}\\}_{j=1}^{n}$ and
$W=\operatorname{span}\\{T\\}$. Thus, the Heisenberg group $\mathbb{H}^{n}$ is
a step 2 stratified Lie group and its homogeneous dimension is $Q:=2n+2$. The
sublaplacian $\mathcal{L}$ on $\mathbb{H}^{n}$ is defined as

$\displaystyle\mathcal{L}$
$\displaystyle=\sum_{j=1}^{n}(X_{j}^{2}+Y_{j}^{2})=\sum_{j=1}^{n}\bigg{(}\bigg{(}\partial_{x_{j}}-\frac{y_{j}}{2}\partial_{t}\bigg{)}^{2}+\bigg{(}\partial_{y_{j}}+\frac{x_{j}}{2}\partial_{t}\bigg{)}^{2}\bigg{)}$
$\displaystyle=\Delta_{\mathbb{R}^{2n}}+\frac{1}{4}(|x|^{2}+|y|^{2})\partial_{t}^{2}+\sum_{j=1}^{n}\left(x_{j}\partial_{y_{j}t}^{2}-y_{j}\partial_{x_{j}t}^{2}\right),$

where $\Delta_{\mathbb{R}^{2n}}$ is the standard Laplacian on
$\mathbb{R}^{2n}.$

###  2.4. Fourier analysis on the Heisenberg group $\mathbb{H}^{n}$

We start this subsection by recalling the definition of the operator-valued
group Fourier transform on $\mathbb{H}^{n}$. By Stone-von Neumann theorem, the
only infinite-dimensional unitary irreducible representations (up to unitary
equivalence) are given by $\pi_{\lambda}$, $\lambda$ in $\mathbb{R}^{*}$,
where the mapping $\pi_{\lambda}$ is a strongly continuous unitary
representation defined by

$\pi_{\lambda}(g)f(u)=e^{i\lambda(t+\frac{1}{2}xy)}e^{i\sqrt{\lambda}yu}f(u+\sqrt{|\lambda|x}),\quad
g=(x,y,t)\in\mathbb{H}^{n},$

for all $f\in L^{2}(\mathbb{R}^{n}).$ We use the convention

$\sqrt{\lambda}:={\rm
sgn}(\lambda)\sqrt{|\lambda|}=\begin{cases}\sqrt{\lambda},&\lambda>0,\\\
-\sqrt{|\lambda|},&\lambda<0.\end{cases}$

For each ${\lambda}\in\mathbb{R}^{*}$, the group Fourier transform of $f\in
L^{1}(\mathbb{H}^{n})$ is a bounded linear operator on $L^{2}(\mathbb{R}^{n})$
defined by

$\displaystyle\widehat{f}(\lambda)\equiv\pi_{\lambda}(f)=\int_{\mathbb{H}^{n}}f(g)\pi_{\lambda}^{*}(g)\,dg.$

Let $B(L^{2}(\mathbb{R}^{n}))$ be the set of all bounded operators on
$L^{2}(\mathbb{R}^{n})$. As the Schrödinger representations are unitary, for
any $\lambda\in\mathbb{N}^{*}$, we have

$\|\widehat{f}\|_{B(L^{2}(\mathbb{R}^{n}))}\leq\|f\|_{L^{1}(\mathbb{H}^{n})}.$

If $f\in L^{2}(\mathbb{H}^{n})$, then $\widehat{f}(\lambda)$ is a Hilbert-
Schmidt operator on $L^{2}(\mathbb{R}^{n})$ and satisfies the following
Plancherel formula

$\|{f}\|_{L^{2}(\mathbb{H}^{n})}^{2}=\int_{\mathbb{R}^{*}}\|\widehat{f}(\lambda)\|_{S_{2}}^{2}\,d\mu(\lambda),$

where $\|.\|_{S_{2}}$ stands for the norm in the Hilbert space $S_{2}$, the
set of all Hilbert-Schmidt operators on $L^{2}(\mathbb{R}^{n})$ and
$d\mu(\lambda)=c_{n}{|\lambda|}^{n}\,d\lambda$ with $c_{n}$ being a positive
constant.

Therefore, with the help of the orthonormal basis
$\left\\{e_{k}\right\\}_{k\in\mathbb{N}_{0}^{n}}$ for
$L^{2}\left(\mathbb{R}^{n}\right)$ and the definition of the Hilbert-Schmidt
norm, we have

$\displaystyle\|\widehat{f}(\lambda)\|_{S_{2}}^{2}$
$\displaystyle\doteq\operatorname{Tr}\left[(\pi_{\lambda}(f))^{*}\pi_{\lambda}(f)\right]$
$\displaystyle=\sum_{k\in\mathbb{N}_{0}^{n}}\|\widehat{f}(\lambda)e_{k}\|_{L^{2}\left(\mathbb{R}^{n}\right)}^{2}=\sum_{k,\ell\in\mathbb{N}^{n}_{0}}|(\widehat{f}(\lambda)e_{k},e_{\ell})_{L^{2}\left(\mathbb{R}^{n}\right)}|^{2}.$

The above expression allows us to write the Plancherel formula in the
following way:

$\|{f}\|_{L^{2}(\mathbb{H}^{n})}^{2}=\int_{\mathbb{R}^{*}}\sum_{k,\ell\in\mathbb{N}^{n}_{0}}|(\widehat{f}(\lambda)e_{k},e_{\ell})_{L^{2}\left(\mathbb{R}^{n}\right)}|^{2}\,d\mu(\lambda).$

For $f\in\mathcal{S}(\mathbb{H}^{n})$, the space of all Schwartz class
functions on $\mathbb{H}^{n},$ the Fourier inversion formula takes the form

$\displaystyle f(g)$
$\displaystyle:=\int_{\mathbb{R}^{*}}\operatorname{Tr}[\pi_{\lambda}(g)\widehat{f}(\lambda)]\,d\mu(\lambda),\quad
g\in\mathbb{H}^{n},$

where $\operatorname{Tr(A)}$ denotes the trace of the operator $A$.

Furthermore, the action of the infinitesimal representation $d\pi_{\lambda}$
of $\pi_{\lambda}$ on the generators of the first layer of the Lie algebra
$\mathfrak{h}_{n}$ is given by

$\displaystyle\;d\pi_{\lambda}\left(X_{j}\right)$
$\displaystyle=\sqrt{|\lambda|}\partial_{x_{j}}$ $\displaystyle\text{ for
}j=1,\cdots,n,$ $\displaystyle\mathrm{~{}d}\pi_{\lambda}\left(Y_{j}\right)$
$\displaystyle=i\operatorname{sign}(\lambda)\sqrt{|\lambda|}x_{j}$
$\displaystyle\text{ for }j=1,\cdots,n.$

Since the action of $\;d\pi_{\lambda}$ can be extended to the universal
enveloping algebra of $\mathfrak{h}_{n}$, combining the above two expressions,
we obtain

$\;d\pi_{\lambda}\left(\mathcal{L}\right)=\;d\pi_{\lambda}\bigg{(}\sum_{j=1}^{n}(X_{j}^{2}+Y_{j}^{2})\bigg{)}=|\lambda|\sum_{j=1}^{n}\big{(}\partial_{x_{j}}^{2}-x_{j}^{2}\big{)}=-|\lambda|\mathrm{H},$

where $\mathrm{H}=-\Delta+|x|^{2}$ is the Hermite operator on
$\mathbb{R}^{n}$. Thus the symbol $\sigma_{\mathcal{L}}(\lambda)$ of
$\mathcal{L}$ acting on $L^{2}(\mathbb{R}^{n})$ takes the form
$\sigma_{\mathcal{L}}(\lambda)=-|\lambda|\mathrm{H}.$

The Sobolev spaces $H_{\mathcal{L}}^{s},s\in\mathbb{R}$, associated to the
sublaplacian $\mathcal{L}$, are defined as

$H_{\mathcal{L}}^{s}\left(\mathbb{H}^{n}\right):=\left\\{f\in\mathcal{D}^{\prime}\left(\mathbb{H}^{n}\right):(I-\mathcal{L})^{s/2}f\in
L^{2}\left(\mathbb{H}^{n}\right)\right\\},$

with the norm

$\|f\|_{H_{\mathcal{L}}^{s}\left(\mathbb{H}^{n}\right)}:=\left\|(I-\mathcal{L})^{s/2}f\right\|_{L^{2}\left(\mathbb{H}^{n}\right)}.$

Similarly, we denote by $\dot{H}_{\mathcal{L}}^{s}(\mathbb{H}^{n}),$ the
homogeneous Sobolev defined as the space of all
$f\in\mathcal{D}^{\prime}(\mathbb{H}^{n})$ such that
$(-\mathcal{L})^{{s}/{2}}f\in L^{2}(\mathbb{H}^{n})$. More generally, we
define $\dot{H}_{\mathcal{L}}^{p,s}(\mathbb{H}^{n})$ as the homogeneous
Sobolev defined as the space of all $f\in\mathcal{D}^{\prime}(\mathbb{H}^{n})$
such that $(-\mathcal{L})^{{s}/{2}}f\in L^{p}(\mathbb{H}^{n})$ for
$s<\frac{Q}{p}$. Then we recall the following important inequalities, see e.g.
[31, 9, 5, 13, 14] for a more general graded Lie group framework. However, we
will state those in the Heisenberg group setting.

######  Theorem 2.1 (Sobolev inequality).

Let $\mathbb{H}^{n}$ be the Heisenberg group with the homogeneous dimension
$Q$. Let $a\geq 0$ and $1<p\leq q<\infty$ be such that

$\frac{a}{Q}=\frac{1}{p}-\frac{1}{q}.$

Then we have the following inequality

$\|f\|_{\dot{H}_{\mathcal{L}}^{q,-a}(\mathbb{H}^{n})}\lesssim\|f\|_{L^{p}(\mathbb{H}^{n})}.$

We have the Gagliardo-Nirenberg inequality on $\mathbb{H}^{n}$ as follows:

######  Theorem 2.2 (Gagliardo-Nirenberg inequality).

Let $Q$ be the homogeneous dimension on the Heisenberg group $\mathbb{H}^{n}$.
Assume that

$s\in(0,1),1<r<\frac{Q}{s},\text{ and }~{}2\leq q\leq\frac{rQ}{Q-sr}.$

Then we have the following Gagliardo-Nirenberg type inequality,

$\|u\|_{L^{q}(\mathbb{H}^{n})}\lesssim\|u\|_{\dot{H}_{\mathcal{L}}^{r,s}(\mathbb{H}^{n})}^{\theta}\|u\|_{L^{2}(\mathbb{H}^{n})}^{1-\theta},$

for
$\theta=\frac{Q}{s}\left(\frac{1}{2}-\frac{1}{q}\right)/{\left(\frac{s}{Q}+\frac{1}{2}-\frac{1}{r}\right)}\in[0,1]$,
provided $\frac{s}{Q}+\frac{1}{2}\neq\frac{1}{r}$.

##  3\. Linear damped wave equations:
$\dot{H}_{\mathcal{L}}^{s}(\mathbb{H}^{n})$-norm estimates

In this section, as a preliminary step in preparing to investigate the local
and global well-posedness of the nonlinear Cauchy problem (1.1), we examine
its associated linear counterpart, which involves a vanishing right-hand side.
Specifically, our attention is directed towards the decay properties of
solutions, in which our proofs are slightly different from the known results.
Let us consider the Cauchy problem

$\displaystyle\begin{cases}\partial^{2}_{t}u-\mathcal{L}u+\partial_{t}u=0,&g\in\mathbb{H}^{n},t>0,\\\
u(0,g)=u_{0}(g),&g\in\mathbb{H}^{n},\\\
\partial_{t}u(0,g)=u_{1}(g),&g\in\mathbb{H}^{n},\end{cases}$ (3.1)

where $u_{0}(g)$ and $u_{1}(g)$ are the initial data additionally belonging to
subelliptic Sobolev space $\dot{H}_{\mathcal{L}}^{-\gamma}(\mathbb{H}^{n})$ of
negative order. We will derive
$\dot{H}_{\mathcal{L}}^{s}(\mathbb{H}^{n})$-norm estimates for the solution
$u(t,\cdot)$ to the homogeneous problem (3.1). We make use of the group
Fourier transform on the Heisenberg group $\mathbb{H}^{n}$, specifically with
respect to the spatial variable $g$, and combine it with the Plancherel
identity to estimate the $\dot{H}_{\mathcal{L}}^{s}(\mathbb{H}^{n})$-norm.

Given that the group Fourier transform of a function $f\in
L^{2}\left(\mathbb{H}^{n}\right)$ is no longer a function but a family of
bounded linear operators
$\\{\widehat{f}(\lambda)\\}_{\lambda\in\mathbb{R}^{*}}$ on
$L^{2}\left(\mathbb{R}^{n}\right)$, the proofs are more involved and do not
follow as in the Euclidean setup. We overcome this obstacle by using a trick
to project these operators by using the orthonormal basis
$\left\\{e_{k}\right\\}_{k\in\mathbb{N}^{n}}$ of $L^{2}(\mathbb{R}^{n})$,
namely, by working with the Fourier coefficients
$\left\\{\left(\widehat{f}(\lambda)e_{k},e_{\ell}\right)_{L^{2}\left(\mathbb{R}^{n}\right)}\right\\}_{k,\ell\in\mathbb{N}^{n}}$
for the Fourier transform $\widehat{f}(\lambda)$ for each
$\lambda\in\mathbb{R}^{*}.$

Invoking the group Fourier transform with respect to $g$ on (3.1), we get a
Cauchy problem related to a parameter-dependent functional differential
equation for $\widehat{u}(t,\lambda),$ namely,

$\displaystyle\begin{cases}\partial^{2}_{t}\widehat{u}(t,\lambda)-\sigma_{\mathcal{L}}(\lambda)\widehat{u}(t,\lambda)+\partial_{t}\widehat{u}(t,\lambda)=0,&\lambda\in\mathbb{R}^{*},~{}t>0,\\\
\widehat{u}(0,\lambda)=\widehat{u}_{0}(\lambda),&\lambda\in\mathbb{R}^{*},\\\
\partial_{t}\widehat{u}(0,\lambda)=\widehat{u}_{1}(\lambda),&\lambda\in\mathbb{R}^{*},\end{cases}$
(3.2)

where $\sigma_{\mathcal{L}}(\lambda)$ is the symbol of the sub-Laplacian
$\mathcal{L}$ on $\mathbb{H}^{n}$. In fact, we know that
$\sigma_{\mathcal{L}}(\lambda)=-|\lambda|\mathrm{H}$. For any
$k,\ell\in\mathbb{N}^{n}$, let us introduce the notation

$\widehat{u}(t,\lambda)_{k,\ell}\doteq\left(\widehat{u}(t,\lambda)e_{k},e_{\ell}\right)_{L^{2}\left(\mathbb{R}^{n}\right)},$

where $\left\\{e_{k}\right\\}_{k\in\mathbb{N}^{n}}$ is the system of Hermite
functions forming an orthonormal basis of $L^{2}(\mathbb{R}^{n})$. Since
$\mathrm{H}e_{k}=\mu_{k}e_{k}$, $\widehat{u}(t,\lambda)_{k,\ell}$ solves an
ordinary differential equation with respect to the variable $t$ depending on
parameters $\lambda\in\mathbb{R}^{*}$ and $k,\ell\in\mathbb{N}^{n},$

$\displaystyle\begin{cases}\partial^{2}_{t}\widehat{u}(t,\lambda)_{kl}+\partial_{t}\widehat{u}(t,\lambda)_{kl}+\beta_{k,\lambda}^{2}\widehat{u}(t,\lambda)_{kl}=0,&\lambda\in\mathbb{R}^{*},~{}t>0,\\\
\widehat{u}(0,\lambda)_{kl}=\widehat{u}_{0}(\lambda)_{kl},&\lambda\in\mathbb{R}^{*},\\\
\partial_{t}\widehat{u}(0,\lambda)_{kl}=\widehat{u}_{1}(\lambda)_{kl},&\lambda\in\mathbb{R}^{*},\end{cases}$
(3.3)

where $\beta_{k,\lambda}^{2}=|\lambda|\mu_{k}.$ Then, the characteristic
equation of (3.3) is given by

$m^{2}+m+\beta_{k,\lambda}^{2}=0,$

and so the characteristic roots $m_{1}$ and $m_{2}$ are
$\frac{-1-\sqrt{1-4\beta_{k,\lambda}^{2}}}{2}$ and
$\frac{-1+\sqrt{1-4\beta_{k,\lambda}^{2}}}{2},$ respectively.

Observe that, for $|\beta_{k,\lambda}|\ll 1$, we get

$\displaystyle
m_{1}=\frac{-1-\sqrt{1-4\beta_{k,\lambda}^{2}}}{2}=\frac{-1-(1-4{\beta_{k,\lambda}}^{2})^{\frac{1}{2}}}{2}=-1+\mathcal{O}\left(\beta_{k,\lambda}^{2}\right),$
$\displaystyle
m_{2}=\frac{-1+(1-4{\beta_{k,\lambda}}^{2})^{\frac{1}{2}}}{2}=-\beta_{k,\lambda}^{2}+\mathcal{O}\left(\beta_{k,\lambda}^{4}\right).$

Again, for $|\beta_{k,\lambda}|\gg 1$, we obtain

$\displaystyle m_{1}=\frac{-1-(1-4\beta_{k,\lambda}^{2})^{\frac{1}{2}}}{2}$
$\displaystyle=\frac{-1-2i|\beta_{k,\lambda}|(1-(4\beta_{k,\lambda}^{2})^{-1})^{\frac{1}{2}}}{2}$
$\displaystyle=-i|\beta_{k,\lambda}|-\frac{1}{2}+\mathcal{O}\left(|\beta_{k,\lambda}|^{-1}\right),$
$\displaystyle m_{2}=\frac{-1+(1-4\beta_{k,\lambda}^{2})^{\frac{1}{2}}}{2}$
$\displaystyle=\frac{-1+2i|\beta_{k,\lambda}|(1-(4\beta_{k,\lambda}^{2})^{-1})^{\frac{1}{2}}}{2}$
$\displaystyle=i|\beta_{k,\lambda}|-\frac{1}{2}+\mathcal{O}\left(|\beta_{k,\lambda}|^{-1}\right).$

We collect the above easy observations as the following relations:

  * •

$m_{1}=-1+\mathcal{O}\left(\beta_{k,\lambda}^{2}\right),m_{2}=-\beta_{k,\lambda}^{2}+\mathcal{O}\left(\beta_{k,\lambda}^{4}\right)$
for $|\beta_{k,\lambda}|<\varepsilon\ll 1$;

  * •

$m_{1}=-i|\beta_{k,\lambda}|-\frac{1}{2}+\mathcal{O}\left(|\beta_{k,\lambda}|^{-1}\right)$
and
$m_{2}=i|\beta_{k,\lambda}|-\frac{1}{2}+\mathcal{O}\left(|\beta_{k,\lambda}|^{-1}\right)$
for $|\beta_{k,\lambda}|>N\gg 1$;

  * •

$\text{Re}(m_{1})>0$ and $\text{Re}(m_{2})>0$ for
$\varepsilon\leq|\beta_{k,\lambda}|\leq N$.

Therefore, the solution to the homogeneous problem (3.3) is given by

$\displaystyle\widehat{u}(t,\lambda)_{kl}$
$\displaystyle=A_{0}(t,\lambda)_{k\ell}\widehat{u}_{0}(\lambda)_{kl}+A_{1}(t,\lambda)_{k\ell}\widehat{u}_{1}(\lambda)_{k\ell},$
(3.4)

where

$\displaystyle
A_{0}(t,\lambda)_{k\ell}=\frac{m_{1}{e}^{m_{2}t}-m_{2}{e}^{m_{1}t}}{m_{1}-m_{2}}$
$\displaystyle=\begin{cases}\frac{\left(-1+\mathcal{O}(\beta_{k,\lambda}^{2})\right){e}^{\left(-\beta_{k,\lambda}^{2}+\mathcal{O}(\beta_{k,\lambda}^{4})\right)t}-\left(-\beta_{k,\lambda}^{2}+\mathcal{O}(\beta_{k,\lambda}^{4})\right){e}^{\left(-1+\mathcal{O}(\beta_{k,\lambda}^{2})\right)t}}{-1+\mathcal{O}\left(\beta_{k,\lambda}^{2}\right)}&\text{
for }|\beta_{k,\lambda}|<\varepsilon,\\\ \\\
\frac{\left(i|\beta_{k,\lambda}|-\frac{1}{2}+\mathcal{O}\left(|\beta_{k,\lambda}|^{-1}\right)\right){e}^{\left(-i|\beta_{k,\lambda}|-\frac{1}{2}+\mathcal{O}\left(|\beta_{k,\lambda}|^{-1}\right)\right)t}}{2i|\beta_{k,\lambda}|+\mathcal{O}(1)}&\\\
\\\
\qquad-\frac{\left(-i|\beta_{k,\lambda}|-\frac{1}{2}+\mathcal{O}\left(|\beta_{k,\lambda}|^{-1}\right)\right){e}^{\left(i|\beta_{k,\lambda}|-\frac{1}{2}+\mathcal{O}\left(|\beta_{k,\lambda}|^{-1}\right)\right)t}}{2i|\beta_{k,\lambda}|+\mathcal{O}(1)}&\text{
for }|\beta_{k,\lambda}|>N.\end{cases}$ (3.6)

and

$\displaystyle
A_{1}(t,\lambda)_{k\ell}=\frac{{e}^{m_{1}t}-{e}^{m_{2}t}}{m_{1}-m_{2}}$
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{{e}^{\left(-1+\mathcal{O}\left(\beta_{k,\lambda}^{2}\right)\right)t}-{e}^{\left(-\beta_{k,\lambda}^{2}+\mathcal{O}\left(\beta_{k,\lambda}^{4}\right)\right)t}}{-1+\mathcal{O}\left(\beta_{k,\lambda}^{2}\right)}&\text{
for }|\beta_{k,\lambda}|<\varepsilon,\\\ \\\
\frac{{e}^{\left(i|\beta_{k,\lambda}|-\frac{1}{2}+\mathcal{O}\left(|\beta_{k,\lambda}|^{-1}\right)\right)t}-{e}^{\left(-i|\beta_{k,\lambda}|-\frac{1}{2}+\mathcal{O}\left(|\beta_{k,\lambda}|^{-1}\right)\right)t}}{2i|\beta_{k,\lambda}|+\mathcal{O}(1)}&\text{
for }|\beta_{k,\lambda}|>N.\end{array}\right.$ (3.11)

Thus from the above collected observations and asymptotic expression, we
deduce the following pointwise estimates

$\displaystyle\left|A_{0}(t,\lambda)_{k\ell}\right|\lesssim\left\\{\begin{array}[]{ll}|\beta_{k,\lambda}|^{2}{e}^{-ct}+{e}^{-ct\beta_{k,\lambda}^{2}}&\text{
for }|\beta_{k,\lambda}|<\varepsilon\ll 1,\\\ \\\ \mathrm{e}^{-ct}&\text{ for
}\varepsilon\leq|\beta_{k,\lambda}|\leq N,\\\ \\\ \mathrm{e}^{-ct}&\text{ for
}|\beta_{k,\lambda}|>N\gg 1,\end{array}\right.$ (3.17)

and

$\displaystyle\left|A_{1}(t,\lambda)_{k\ell}\right|\lesssim\left\\{\begin{array}[]{ll}{e}^{-ct}+{e}^{-ct\beta_{k,\lambda}^{2}}&\text{
for }|\beta_{k,\lambda}|<\varepsilon\ll 1,\\\ \\\ {e}^{-ct}&\text{ for
}\varepsilon\leq|\beta_{k,\lambda}|\leq N,\\\ \\\
|\beta_{k,\lambda}|^{-1}{e}^{-ct}&\text{ for }|\beta_{k,\lambda}|>N\gg
1,\end{array}\right.$ (3.23)

for some suitable positive constant $c$.

Before finding the Sobolev norm of $u(t,\cdot)$, first we notice that, for
$\beta_{k,\lambda}^{2}=|\lambda|\mu_{k}<\varepsilon^{2}$, we have

$\displaystyle\left|(|\lambda|\mu_{k})^{s+\gamma}{e}^{-ct|\lambda|\mu_{k}}\right|\lesssim(1+t)^{-(s+\gamma)},\qquad\text{for~{}}s+\gamma\geq
0,$ (3.24)

by using the fact that $(\varepsilon^{2}+x)^{s+\gamma}e^{-cx}$ is uniformly
bounded for large $x$.

Now by using the Plancherel formula and the fact that
$\\{e_{k}\\}_{k\in\mathbb{N}^{n}}$ is an orthonormal basis of
$L^{2}(\mathbb{R}^{n})$, we have

$\displaystyle\|u(t,\cdot)\|_{\dot{H}^{s}_{\mathcal{L}}}^{2}=\int_{\mathbb{R}^{*}}\|(-\sigma_{\mathcal{L}}(\lambda))^{\frac{s}{2}}\widehat{u}(t,\lambda)\|_{S_{2}}^{2}\,d\mu(\lambda)$
$\displaystyle\qquad=\int_{\mathbb{R}^{*}}\sum_{k,\ell\in\mathbb{N}^{n}}(|\lambda|\mu_{k})^{s}|\widehat{u}(t,\lambda)_{k\ell}|^{2}\,d\mu(\lambda)$
$\displaystyle\qquad\lesssim\int_{\mathbb{R}^{*}}\sum_{k,\ell\in\mathbb{N}^{n}}(|\lambda|\mu_{k})^{s}\left[|A_{0}(t,\lambda)_{k\ell}|^{2}|\widehat{u}_{0}(\lambda)_{kl}|^{2}+|A_{1}(t,\lambda)_{k\ell}|^{2}|\widehat{u}_{1}(\lambda)_{k\ell}|^{2}\right]d\mu(\lambda)$
$\displaystyle\qquad=(I)+(II),$ (3.25)

where

$(I):=\int_{\mathbb{R}^{*}}\sum_{k,\ell\in\mathbb{N}^{n}}(|\lambda|\mu_{k})^{s}|A_{0}(t,\lambda)_{k\ell}|^{2}|\widehat{u}_{0}(\lambda)_{kl}|^{2}d\mu(\lambda),$

and

$(II):=\int_{\mathbb{R}^{*}}\sum_{k,\ell\in\mathbb{N}^{n}}(|\lambda|\mu_{k})^{s}|A_{1}(t,\lambda)_{k\ell}|^{2}|\widehat{u}_{1}(\lambda)_{k\ell}|^{2}d\mu(\lambda).$

Case 1: When $|\beta_{k,\lambda}|<\varepsilon\ll 1$: Recalling that
$\beta_{k,\lambda}^{2}=|\lambda|\mu_{k}$ and using (3.17) and (3.24), we get

$\displaystyle(I)$
$\displaystyle=\int_{\mathbb{R}^{*}}\sum_{k,\ell\in\mathbb{N}^{n}}(|\lambda|\mu_{k})^{s+\gamma}|A_{0}(t,\lambda)_{k\ell}|^{2}(|\lambda|\mu_{k})^{-\gamma}|\widehat{u}_{0}(\lambda)_{kl}|^{2}d\mu(\lambda)$
$\displaystyle\leq\int_{\mathbb{R}^{*}}\sum_{k,\ell\in\mathbb{N}^{n}}(|\lambda|\mu_{k})^{s+\gamma}\\{|\beta_{k,\lambda}|^{2}{e}^{-ct}+{e}^{-ct\beta_{k,\lambda}^{2}}\\}^{2}(|\lambda|\mu_{k})^{-\gamma}|\widehat{u}_{0}(\lambda)_{kl}|^{2}d\mu(\lambda)$
$\displaystyle\leq\int_{\mathbb{R}^{*}}\sum_{k,\ell\in\mathbb{N}^{n}}(|\lambda|\mu_{k})^{s+\gamma}{e}^{-2ct\beta_{k,\lambda}^{2}}\\{|\beta_{k,\lambda}|^{2}{e}^{-ct(1-\beta_{k,\lambda}^{2})}+1\\}^{2}(|\lambda|\mu_{k})^{-\gamma}|\widehat{u}_{0}(\lambda)_{kl}|^{2}d\mu(\lambda)$
$\displaystyle\lesssim(1+t)^{-(s+\gamma)}\int_{\mathbb{R}^{*}}\sum_{k,\ell\in\mathbb{N}^{n}}(|\lambda|\mu_{k})^{-\gamma}|\widehat{u}_{0}(\lambda)_{kl}|^{2}d\mu(\lambda)$
$\displaystyle=(1+t)^{-(s+\gamma)}\|{u}_{0}\|_{\dot{H}^{-\gamma}_{\mathcal{L}}}^{2}.$
(3.26)

Similarly, using (3.23) and (3.24) we can find that

$\displaystyle(II)$
$\displaystyle=\int_{\mathbb{R}^{*}}\sum_{k,\ell\in\mathbb{N}^{n}}(|\lambda|\mu_{k})^{s}|A_{1}(t,\lambda)_{k\ell}|^{2}|\widehat{u}_{1}(\lambda)_{k\ell}|^{2}d\mu(\lambda)$
$\displaystyle\lesssim(1+t)^{-(s+\gamma)}\|{u}_{1}\|_{\dot{H}^{-\gamma}_{\mathcal{L}}}^{2}.$
(3.27)

Therefore, from (3), (3), and (3), we obtain

$\displaystyle\|u(t,\cdot)\|_{\dot{H}^{s}_{\mathcal{L}}}^{2}\lesssim(1+t)^{-(s+\gamma)}\Big{[}\|{u}_{0}\|_{\dot{H}^{-\gamma}_{\mathcal{L}}}^{2}+\|{u}_{1}\|_{\dot{H}^{-\gamma}_{\mathcal{L}}}^{2}\Big{]}.$
(3.28)

Case 2: When $|\beta_{k,\lambda}|>N\gg 1$: Again using (3.17) we get

$\displaystyle(I)$
$\displaystyle=\int_{\mathbb{R}^{*}}\sum_{k,\ell\in\mathbb{N}^{n}}(|\lambda|\mu_{k})^{s}|A_{0}(t,\lambda)_{k\ell}|^{2}|\widehat{u}_{0}(\lambda)_{kl}|^{2}d\mu(\lambda)$
$\displaystyle\lesssim{e}^{-2ct}\int_{\mathbb{R}^{*}}\sum_{k,\ell\in\mathbb{N}^{n}}(1+|\lambda|\mu_{k})^{s}|\widehat{u}_{0}(\lambda)_{kl}|^{2}d\mu(\lambda)$
$\displaystyle={e}^{-2ct}\|{u}_{0}\|_{H_{\mathcal{L}}^{-s}}^{2}.$ (3.29)

Also, with the help of (3.23) we find that

$\displaystyle(II)$
$\displaystyle=\int_{\mathbb{R}^{*}}\sum_{k,\ell\in\mathbb{N}^{n}}(|\lambda|\mu_{k})^{s}|A_{1}(t,\lambda)_{k\ell}|^{2}|\widehat{u}_{1}(\lambda)_{k\ell}|^{2}d\mu(\lambda)$
$\displaystyle\lesssim\int_{\mathbb{R}^{*}}\sum_{k,\ell\in\mathbb{N}^{n}}(|\lambda|\mu_{k})^{s}|\beta_{k,\lambda}|^{-2}{e}^{-2ct}|\widehat{u}_{1}(\lambda)_{k\ell}|^{2}d\mu(\lambda)$
$\displaystyle={e}^{-2ct}\int_{\mathbb{R}^{*}}\sum_{k,\ell\in\mathbb{N}^{n}}(1+|\lambda|\mu_{k})^{s-1}\left(\frac{|\lambda|\mu_{k}}{1+|\lambda|\mu_{k}}\right)^{s-1}|\widehat{u}_{1}(\lambda)_{k\ell}|^{2}d\mu(\lambda)$
$\displaystyle\lesssim{e}^{-2ct}\|{u}_{1}\|_{H_{\mathcal{L}}^{s-1}}^{2}.$
(3.30)

Therefore from (3), (3), and (3), we have

$\displaystyle\|u(t,\cdot)\|_{\dot{H}^{s}_{\mathcal{L}}}^{2}\lesssim{e}^{-2ct}\Big{[}\|{u}_{0}\|_{H_{\mathcal{L}}^{s}}^{2}+\|{u}_{1}\|_{H_{\mathcal{L}}^{s-1}}^{2}\Big{]}.$
(3.31)

Case 3: When $\varepsilon<|\beta_{k,\lambda}|<N$: From (3.17) and (3.23) , we
get

$\displaystyle(I)$
$\displaystyle=\int_{\mathbb{R}^{*}}\sum_{k,\ell\in\mathbb{N}^{n}}(|\lambda|\mu_{k})^{s}|A_{0}(t,\lambda)_{k\ell}|^{2}|\widehat{u}_{0}(\lambda)_{kl}|^{2}d\mu(\lambda)$
$\displaystyle\lesssim{e}^{-2ct}\int_{\mathbb{R}^{*}}\sum_{k,\ell\in\mathbb{N}^{n}}(|\lambda|\mu_{k})^{s}|\widehat{u}_{0}(\lambda)_{kl}|^{2}d\mu(\lambda)$
$\displaystyle\lesssim{e}^{-2ct}\int_{\mathbb{R}^{*}}\sum_{k,\ell\in\mathbb{N}^{n}}(1+|\lambda|\mu_{k})^{s}|\widehat{u}_{0}(\lambda)_{kl}|^{2}d\mu(\lambda)={e}^{-2ct}\|{u}_{0}\|_{H_{\mathcal{L}}^{s}}^{2},$
(3.32)

and

$\displaystyle(II)$
$\displaystyle=\int_{\mathbb{R}^{*}}\sum_{k,\ell\in\mathbb{N}^{n}}(|\lambda|\mu_{k})^{s}|A_{1}(t,\lambda)_{k\ell}|^{2}|\widehat{u}_{1}(\lambda)_{k\ell}|^{2}d\mu(\lambda)$
$\displaystyle\lesssim{e}^{-2ct}\int_{\mathbb{R}^{*}}\sum_{k,\ell\in\mathbb{N}^{n}}(1+|\lambda|\mu_{k})^{s}|\widehat{u}_{0}(\lambda)_{kl}|^{2}d\mu(\lambda)$
$\displaystyle\lesssim{e}^{-2ct}\int_{\mathbb{R}^{*}}\sum_{k,\ell\in\mathbb{N}^{n}}(1+|\lambda|\mu_{k})^{s-1}|\widehat{u}_{0}(\lambda)_{kl}|^{2}d\mu(\lambda)$
$\displaystyle={e}^{-2ct}\|{u}_{1}\|_{H_{\mathcal{L}}^{s-1}}^{2}.$ (3.33)

Therefore from (3), and (3), we have

$\displaystyle\|u(t,\cdot)\|_{\dot{H}_{\mathcal{L}}^{s}}^{2}\lesssim{e}^{-2ct}\Big{[}\|{u}_{0}\|_{H_{\mathcal{L}}^{s}}^{2}+\|{u}_{1}\|_{H_{\mathcal{L}}^{s-1}}^{2}\Big{]}.$
(3.34)

###### Proof of Theorem 1.1.

Combining (3.28), (3.31), and (3.34), we obtain the following
$\dot{H}_{\mathcal{L}}^{s}$-decay estimate

$\|u(t,\cdot)\|_{\dot{H}_{\mathcal{L}}^{s}}\lesssim(1+t)^{-\frac{s+\gamma}{2}}\left(\left\|u_{0}\right\|_{H^{s}_{\mathcal{L}}\cap\dot{H}^{-\gamma}_{\mathcal{L}}}+\left\|u_{1}\right\|_{H_{\mathcal{L}}^{s-1}\cap\dot{H}_{\mathcal{L}}^{-\gamma}}\right),$

for any $t\geq 0$. ∎

##  4\. Global-in-time well-posedness

In this section, we will prove Theorem 1.2, that is, the global-in-time well-
posedness of the Cauchy problem (1.1) in the energy evolution space
$\mathcal{C}\left([0,T],H^{s}_{\mathcal{L}}(\mathbb{H}^{n})\right)$.

First, we introduce some notations to present the proof of Theorem 1.2. For
$T>0$, we consider the evolution spaces

$X_{s}(T):=\mathcal{C}\left([0,T],H^{s}_{\mathcal{L}}\right)\quad\text{with}~{}s\in(0,1],$

equipped with the norm

$\displaystyle\|u\|_{X_{s}(T)}$
$\displaystyle:=\sup\limits_{t\in[0,T]}\left((1+t)^{\frac{\gamma}{2}}\|u(t,\cdot)\|_{L^{2}}+(1+t)^{\frac{s+\gamma}{2}}\|u(t,\cdot)\|_{\dot{H}^{s}_{\mathcal{L}}}\right),$
(4.1)

with $\gamma>0.$

The motivation behind the definition of time-weighted Sobolev norm (4.1) is
closely tied to the estimates for solutions to the linear Cauchy problem
(3.1), specifically, Theorem 1.1 for both the cases of $s=0$ and $\gamma>0$

Suppose that $u$ is a mild solution to (1.1). Let us define the integral
operator $N$ on the energy evolution space
$\mathcal{C}\left([0,T],H_{\mathcal{L}}^{s}\right)$ given by

$\displaystyle Nu(t,g):=u^{\text{lin}}(t,g)+u^{\text{non}}(t,g),$ (4.2)

where

$u^{\text{lin}}(t,g)=u_{0}(g)*_{(g)}E_{0}(t,g)+u_{1}(g)*_{(g)}E_{1}(t,g)$

is the solution to the corresponding linear Cauchy problem (3.1) and

$u^{\text{non}}(t,g):=\int_{0}^{t}|u(s,g)|^{p}*_{(g)}E_{1}(t-s,g)\;ds.$

In the subsequent part, our objective is to establish the global-in-time
existence and uniqueness of Sobolev solutions with low regularity for the
semilinear damped wave equation (1.1). This will be accomplished by
demonstrating the existence of a unique fixed point for the operator $N$,
implying that $Nu\in X_{s}(T)$ for all $T>0$.

###### Proof of Theorem 1.2.

Recall that for $s\geq 0$ and $\gamma\in\mathbb{R}$ such that $s+\gamma\geq
0$, from the estimate (1.9) of Theorem 1.1, we have

$\|u(t,\cdot)\|_{\dot{{H}}_{\mathcal{L}}^{s}(\mathbb{H}^{n})}\lesssim(1+t)^{-\frac{s+\gamma}{2}}\left(\left\|u_{0}\right\|_{H^{s}_{\mathcal{L}}\cap\dot{{H}}_{\mathcal{L}}^{-\gamma}}+\left\|u_{1}\right\|_{H_{\mathcal{L}}^{s-1}\cap\dot{{H}}_{\mathcal{L}}^{-\gamma}}\right).$
(4.3)

In particular, for $\gamma=0$ and $s\geq 0,$ we get

$\displaystyle\|u(t,\cdot)\|_{{\dot{H}}_{\mathcal{L}}^{s}}\lesssim(1+t)^{-\frac{s}{2}}\left(\left\|u_{0}\right\|_{H_{\mathcal{L}}^{s}}+\left\|u_{1}\right\|_{L^{2}}\right).$
(4.4)

From (4.3), in particular for $s\in[0,1]$ and $\gamma>0$ we have the following
estimate for the Sobolev solutions of the linear Cauchy problem

$\displaystyle\|u(t,\cdot)\|_{\dot{{H}}_{\mathcal{L}}^{s}}\lesssim(1+t)^{-\frac{s+\gamma}{2}}\left(\left\|u_{0}\right\|_{H^{s}_{\mathcal{L}}\cap{\dot{H}}_{\mathcal{L}}^{-\gamma}}+\left\|u_{1}\right\|_{L^{2}\cap{\dot{H}}_{\mathcal{L}}^{-\gamma}}\right),$
(4.5)

where we have used the Sobolev embedding $L^{2}\subset H_{\mathcal{L}}^{s-1}$
for $s\leq 1$. Particularly, for $s=0$ in (4.5), using the Sobolev embedding
$H_{\mathcal{L}}^{s}\subset L^{2}$ for $s\geq 0$, we obtain

$\displaystyle\|u(t,\cdot)\|_{L^{2}}$
$\displaystyle\lesssim(1+t)^{-\frac{\gamma}{2}}\left(\left\|u_{0}\right\|_{L^{2}\cap{\dot{H}}_{\mathcal{L}}^{-\gamma}}+\left\|u_{1}\right\|_{L^{2}\cap{\dot{H}}_{\mathcal{L}}^{-\gamma}}\right)$
$\displaystyle\lesssim(1+t)^{-\frac{\gamma}{2}}\left(\left\|u_{0}\right\|_{H^{s}_{\mathcal{L}}\cap{\dot{H}}_{\mathcal{L}}^{-\gamma}}+\left\|u_{1}\right\|_{L^{2}\cap{\dot{H}}_{\mathcal{L}}^{-\gamma}}\right).$
(4.6)

Now from (4.5) and (4), for $s\in(0,1]$, we can write

$\displaystyle(1+t)^{\frac{s+\gamma}{2}}\|u(t,\cdot)\|_{{\dot{H}}_{\mathcal{L}}^{s}}$
$\displaystyle+(1+t)^{\frac{\gamma}{2}}\|u(t,\cdot)\|_{L^{2}}$
$\displaystyle\lesssim\left(\left\|u_{0}\right\|_{H^{s}_{\mathcal{L}}\cap{\dot{H}}_{\mathcal{L}}^{-\gamma}}+\left\|u_{1}\right\|_{L^{2}\cap{\dot{H}}_{\mathcal{L}}^{-\gamma}}\right)=\left\|\left(u_{0},u_{1}\right)\right\|_{\mathcal{A}_{s}^{\mathcal{L}}}.$

Thus from above, we can claim that $u\in X_{s}(T)$ and

$\displaystyle\|u^{\text{lin}}\|_{X_{s}(T)}\lesssim\left\|\left(u_{0},u_{1}\right)\right\|_{\mathcal{A}_{s}^{\mathcal{L}}}.$
(4.7)

Our next aim is to prove

$\|u^{\text{non}}\|_{X_{s}(T)}\lesssim\left\|u\right\|_{X_{s}(T)}^{p},$

under some conditions for $p$. First we will calculate $L^{2}$ and
$\dot{H}^{-\gamma}_{\mathcal{L}}$ norm of $|u(t,\cdot)|^{p}$. Applying
Gagliardo-Nirenberg inequality (Theorem 2.2), we have

$\displaystyle\left\||u(\sigma,\cdot)|^{p}\right\|_{L^{2}}=\left\|u(\sigma,\cdot)\right\|_{L^{2p}}^{p}$
$\displaystyle\lesssim\|u(\sigma,\cdot)\|_{\dot{H}_{\mathcal{L}}^{s}(\mathbb{H}^{n})}^{p\theta_{1}}\|u(\sigma,\cdot)\|_{L^{2}(\mathbb{H}^{n})}^{p(1-\theta_{1})}$
$\displaystyle=(1+\sigma)^{-\frac{p}{2}(s\theta_{1}+\gamma)}\left\\{(1+\sigma)^{\frac{(s+\gamma)}{2}}\|u(\sigma,\cdot)\|_{\dot{H}_{\mathcal{L}}^{s}(\mathbb{H}^{n})}\right\\}^{p\theta_{1}}\left\\{(1+\sigma)^{\frac{\gamma}{2}}\|u(\sigma,\cdot)\|_{L^{2}(\mathbb{H}^{n})}\right\\}^{p(1-\theta_{1})}$
$\displaystyle=(1+\sigma)^{-\frac{p}{2}\left[\gamma+\frac{Q}{2}\left(1-\frac{1}{p}\right)\right]}\left\\{(1+\sigma)^{\frac{(s+\gamma)}{2}}\|u(\sigma,\cdot)\|_{\dot{H}_{\mathcal{L}}^{s}(\mathbb{H}^{n})}\right\\}^{p\theta_{1}}\left\\{(1+\sigma)^{\frac{\gamma}{2}}\|u(\sigma,\cdot)\|_{L^{2}(\mathbb{H}^{n})}\right\\}^{p(1-\theta_{1})}$
$\displaystyle\lesssim(1+\sigma)^{-p\left(\frac{\gamma}{2}+\frac{Q}{4}\right)+\frac{Q}{4}}\left\|u\right\|_{X_{s}(T)}^{p},$
(4.8)

for $\sigma\in[0,T]$ with $\theta_{1}=\frac{Q}{2s}(1-\frac{1}{p})\in[0,1]$,
provided that

$\displaystyle 1\leq p\leq\frac{Q}{Q-2s},\quad\text{if}~{}~{}Q>2s.$ (4.9)

On the other hand, using the Sobolev inequality (Theorem 2.1), we have

$\displaystyle\left\||u(\sigma,\cdot)|^{p}\right\|_{\dot{H}^{-\gamma}_{\mathcal{L}}}\lesssim\left\||u(\sigma,\cdot)|^{p}\right\|_{L^{m}}=\left\|u(\sigma,\cdot)\right\|_{L^{mp}}^{p}$

with $\frac{1}{m}-\frac{1}{2}=\frac{\gamma}{Q}$ provided that $0<\gamma<Q$ and
$1<m<2$. Since $m\in(1,2)$, we have to restrict $0<\gamma<\frac{Q}{2}$. Again,
the Gagliardo-Nirenberg inequality (Theorem 2.2) implies that

$\displaystyle\left\||u(\sigma,\cdot)|^{p}\right\|_{\dot{H}^{-\gamma}_{\mathcal{L}}}\lesssim\left\|u(\sigma,\cdot)\right\|_{L^{mp}}^{p}$
$\displaystyle\lesssim\|u(\sigma,\cdot)\|_{\dot{H}_{\mathcal{L}}^{s}(\mathbb{H}^{n})}^{p\theta_{2}}\|u(\sigma,\cdot)\|_{L^{2}(\mathbb{H}^{n})}^{p(1-\theta_{2})}$
$\displaystyle=(1+\sigma)^{-\frac{p}{2}(s\theta_{2}+\gamma)}\left\\{(1+\sigma)^{\frac{(s+\gamma)}{2}}\|u(\sigma,\cdot)\|_{\dot{H}_{\mathcal{L}}^{s}(\mathbb{H}^{n})}\right\\}^{p\theta_{1}}\left\\{(1+\sigma)^{\frac{\gamma}{2}}\|u(\sigma,\cdot)\|_{L^{2}(\mathbb{H}^{n})}\right\\}^{p(1-\theta_{1})}$
$\displaystyle=(1+\sigma)^{-\frac{p}{2}\left[\gamma+Q(\frac{1}{2}-\frac{1}{mp})\right]}\left\\{(1+\sigma)^{\frac{(s+\gamma)}{2}}\|u(\sigma,\cdot)\|_{\dot{H}_{\mathcal{L}}^{s}(\mathbb{H}^{n})}\right\\}^{p\theta_{1}}\left\\{(1+\sigma)^{\frac{\gamma}{2}}\|u(\sigma,\cdot)\|_{L^{2}(\mathbb{H}^{n})}\right\\}^{p(1-\theta_{1})}$
$\displaystyle=(1+\sigma)^{-p(\frac{\gamma}{2}+\frac{Q}{4})+\frac{\gamma}{2}+\frac{Q}{4}}\left\\{(1+\sigma)^{\frac{(s+\gamma)}{2}}\|u(\sigma,\cdot)\|_{\dot{H}_{\mathcal{L}}^{s}(\mathbb{H}^{n})}\right\\}^{p\theta_{2}}\left\\{(1+\sigma)^{\frac{\gamma}{2}}\|u(\sigma,\cdot)\|_{L^{2}(\mathbb{H}^{n})}\right\\}^{p(1-\theta_{2})}$
$\displaystyle\lesssim(1+\sigma)^{-p(\frac{\gamma}{2}+\frac{Q}{4})+\frac{\gamma}{2}+\frac{Q}{4}}\left\|u\right\|_{X_{s}(T)}^{p},$
(4.10)

for $\sigma\in[0,T]$ with
$\theta_{2}=\frac{Q}{s}(\frac{1}{2}-\frac{1}{mp})\in[0,1]$. Using the fact
that $\theta_{2}\in[0,1]$, we get

$\displaystyle p\geq\frac{2}{m}\quad\text{and}\quad
p\leq\frac{2Q}{m(Q-2s)},\quad\text{if}~{}Q>2s.$ (4.11)

Since $m=\frac{2Q}{Q+2\gamma}\in(1,2)$, from (4.11), we obtain

$\displaystyle 1<1+\frac{2\gamma}{Q}\leq p\leq\frac{2Q}{(Q-2s)}\quad\text{ if
}Q>2s.$ (4.12)

Therefore, from (4.9) and (4.12), if we consider

$\displaystyle 1+\frac{2\gamma}{Q}\leq
p\left\\{\begin{array}[]{ll}<\infty&\text{ if }Q\leq 2s,\\\
\leq\frac{Q}{(Q-2s)}&\text{ if }Q>2s,\end{array}\right.$

then from (4) and (4), for $\sigma\in[0,T]$, we assert that

$\displaystyle\left\||u(\sigma,\cdot)|^{p}\right\|_{L^{2}\cap\dot{H}^{-\gamma}_{\mathcal{L}}}\lesssim(1+\sigma)^{-p(\frac{\gamma}{2}+\frac{Q}{4})+\frac{\gamma}{2}+\frac{Q}{4}}\left\|u\right\|_{X_{s}(T)}^{p}.$
(4.13)

Using the $(L^{2}\cap\dot{H}_{\mathcal{L}}^{-\gamma})-L^{2}$ estimate given in
(4.5) (with $u_{0}=0$, $u_{1}=|u(\sigma,\cdot)|^{p}$) as well as from the
statement (4.13), we have the following $L^{2}$-estimate of the solution

$\displaystyle\left\|u^{\mathrm{non}}(t,\cdot)\right\|_{L^{2}}$
$\displaystyle=\bigg{\|}\int\limits_{0}^{t}|u(\sigma,\cdot)|^{p}*_{(g)}E_{1}(t-\sigma,\cdot)d\sigma\bigg{\|}_{L^{2}(\mathbb{H}^{n})}$
$\displaystyle\leq\int\limits_{0}^{t}(1+t-\sigma)^{-\frac{\gamma}{2}}\||u(\sigma,\cdot)|^{p}\|_{L^{2}\cap\dot{H}^{-\gamma}_{\mathcal{L}}}d\sigma$
$\displaystyle\lesssim\int_{0}^{t}(1+t-\sigma)^{-\frac{\gamma}{2}}(1+\sigma)^{-p(\frac{\gamma}{2}+\frac{Q}{4})+\frac{\gamma}{2}+\frac{Q}{4}}d\sigma~{}\left\|u\right\|_{X_{s}(T)}^{p}$
$\displaystyle\lesssim(1+t)^{-\frac{\gamma}{2}}\int_{0}^{\frac{t}{2}}(1+\sigma)^{-p(\frac{\gamma}{2}+\frac{Q}{4})+\frac{\gamma}{2}+\frac{Q}{4}}d\sigma~{}\left\|u\right\|_{X_{s}(T)}^{p}$
(4.14)
$\displaystyle\qquad+(1+t)^{-p(\frac{\gamma}{2}+\frac{Q}{4})+\frac{\gamma}{2}+\frac{Q}{4}}\int_{\frac{t}{2}}^{t}(1+t-\sigma)^{-\frac{\gamma}{2}}\mathrm{~{}d}\sigma\|u\|_{X_{s}(T)}^{p}.$
(4.15)

For $p>1+\frac{4}{Q+2\gamma},$ the integral given in (4.14) converges
uniformly over $\left[0,\frac{t}{2}\right]$. Now we will compute the integral
given in (4.15) precisely. Now

$\displaystyle\int_{\frac{t}{2}}^{t}(1+t-\sigma)^{-\frac{\gamma}{2}}\mathrm{~{}d}\sigma\lesssim\left\\{\begin{array}[]{ll}(1+t)^{1-\frac{\gamma}{2}}&\text{
if }\gamma<2,\\\ \ln(\mathrm{e}+t)&\text{ if }\gamma=2,\\\ 1&\text{ if
}\gamma>2.\end{array}\right.$ (4.19)

Thus by considering $p>1+\frac{4}{Q+2\gamma}$ if $\gamma\leq 2$ and using the
integral estimate (4.19), we get

$\displaystyle(1+t)^{-\left(\frac{\gamma}{2}+\frac{Q}{4}\right)p+\frac{\gamma}{2}+\frac{Q}{4}}\int_{\frac{t}{2}}^{t}(1+t-\sigma)^{-\frac{\gamma}{2}}{~{}d}\sigma$
$\displaystyle\leq(1+t)^{-1}\int_{\frac{t}{2}}^{t}(1+t-\sigma)^{-\frac{\gamma}{2}}{~{}d}\sigma\lesssim(1+t)^{-\frac{\gamma}{2}}.$

Again by considering $p>1+\frac{2\gamma}{Q+2\gamma}$ if $\gamma>2$ and using
the integral estimate (4.19), a simple calculation yields

$\displaystyle(1+t)^{-\left(\frac{\gamma}{2}+\frac{Q}{4}\right)p+\frac{\gamma}{2}+\frac{Q}{4}}\int_{\frac{t}{2}}^{t}(1+t-\sigma)^{-\frac{\gamma}{2}}{~{}d}\sigma\lesssim(1+t)^{-\frac{\gamma}{2}}.$

Summarizing all the estimates, it holds that

$\displaystyle(1+t)^{\frac{\gamma}{2}}\|u^{\text{non}}(t,\cdot)\|_{L^{2}}\lesssim\left\|u\right\|_{X_{s}(T)}^{p}.$
(4.20)

Now we will estimate $\dot{H}_{\mathcal{L}}^{s}$-norm of the solution. Similar
to the previous argument, we use
$\left(L^{2}\cap\dot{H}_{\mathcal{L}}^{-\gamma}\right)-\dot{H}_{\mathcal{L}}^{s}$
estimate (4.5) in $\left[0,\frac{t}{2}\right]$ and the
$L^{2}-\dot{H}_{\mathcal{L}}^{s}$ estimate (4.4) in
$\left[\frac{t}{2},t\right]$ to estimate the solution itself in
$\dot{H}_{\mathcal{L}}^{s}$. Thus

$\displaystyle\left\|u^{\mathrm{non}}(t,\cdot)\right\|_{\dot{H}_{\mathcal{L}}^{s}}$
$\displaystyle\leq\int\limits_{0}^{t}\||u(\sigma,\cdot)|^{p}*_{(g)}E_{1}(t-\sigma,\cdot)\|_{\dot{H}_{\mathcal{L}}^{s}}d\sigma$
$\displaystyle\lesssim\int_{0}^{\frac{t}{2}}(1+t-\sigma)^{-\frac{(s+\gamma)}{2}}(1+\sigma)^{-p(\frac{\gamma}{2}+\frac{Q}{4})+\frac{\gamma}{2}+\frac{Q}{4}}d\sigma~{}\left\|u\right\|_{X_{s}(T)}^{p}$
$\displaystyle\qquad+\int_{\frac{t}{2}}^{t}(1+t-\sigma)^{-\frac{s}{2}}(1+\sigma)^{-p(\frac{\gamma}{2}+\frac{Q}{4})+\frac{Q}{4}}\mathrm{~{}d}\sigma\|u\|_{X_{s}(T)}^{p}$
$\displaystyle\lesssim(1+t)^{-\frac{(s+\gamma)}{2}}\int_{0}^{\frac{t}{2}}(1+\sigma)^{-p(\frac{\gamma}{2}+\frac{Q}{4})+\frac{\gamma}{2}+\frac{Q}{4}}d\sigma~{}\left\|u\right\|_{X_{s}(T)}^{p}$
$\displaystyle\qquad+(1+t)^{-p(\frac{\gamma}{2}+\frac{Q}{4})+\frac{Q}{4}}\int_{\frac{t}{2}}^{t}(1+t-\sigma)^{-\frac{s}{2}}\mathrm{~{}d}\sigma\|u\|_{X_{s}(T)}^{p}.$
(4.21)

For $p>1+\frac{4}{Q+2\gamma}$, summarizing as previously, we have

$\displaystyle(1+t)^{\frac{s+\gamma}{2}}\left\|u^{\text{non}}(t,\cdot)\right\|_{\dot{H}_{\mathcal{L}}^{s}}\lesssim\|u\|_{X_{s}(T)}^{p}.$
(4.22)

Therefore from (4.22) and (4.20), we obtain

$\displaystyle\left\|u^{\text{non}}(t,\cdot)\right\|_{X_{s}(T)}\lesssim\|u\|_{X_{s}(T)}^{p},$

under the conditions on $p$ as follows:

  * •

$1<p\leq\frac{Q}{Q-2s}$, from the application of the Gagliardo-Nirenberg
inequality,

  * •

$p>1+\frac{4}{Q+2\gamma}$ if $\gamma\leq 2$,

  * •

$p>1+\frac{2\gamma}{Q+2\gamma}$ if $\gamma>2$, from the integrability and
decay estimates of solution.

  * •

$p\geq 1+\frac{2\gamma}{Q}$, from the application of the Gagliardo-Nirenberg
inequality.

First, we consider the case $\gamma>2.$ Then we observe that

$\max\left\\{1+\frac{2\gamma}{Q+2\gamma},1+\frac{2\gamma}{Q}\right\\}=1+\frac{2\gamma}{Q}.$

Now for $\gamma\leq 2$, we have to compare between $1+\frac{4}{Q+2\gamma}$ and
$1+\frac{2\gamma}{Q}$. Notice that $1+\frac{4}{Q+2\gamma}$ and
$1+\frac{2\gamma}{Q}$ intersects at a point $\tilde{\gamma}$, which is the
positive root of the quadratic equation $2\gamma^{2}+Q\gamma-2Q=0$. Moreover,
it is easy to check that the positive root $\tilde{\gamma}<2$ for all $Q\geq
4.$ This shows that

  * •

$\max\left\\{1+\frac{4}{Q+2\gamma},1+\frac{2\gamma}{Q}\right\\}=1+\frac{4}{Q+2\gamma}$
for $\gamma\leq\tilde{\gamma}$,

  * •

$\max\left\\{1+\frac{4}{Q+2\gamma},1+\frac{2\gamma}{Q}\right\\}=1+\frac{2\gamma}{Q}$
for $\tilde{\gamma}<\gamma\leq 2$.

Finally, for any $\gamma\in(0,\frac{Q}{2})$, the condition for the exponent
$p$ is reduced to

$\displaystyle
p\left\\{\begin{array}[]{ll}>1+\frac{4}{Q+2\gamma}=p_{\text{crit
}}(Q,\gamma)&\text{ if }\gamma\leq\tilde{\gamma},\\\ \geq
1+\frac{2\gamma}{Q}&\text{ if }\gamma>\tilde{\gamma}.\end{array}\right.$

Furthermore, we have

$\displaystyle\|Nu\|_{X_{s}(T)}\lesssim\left\|\left(u_{0},u_{1}\right)\right\|_{\mathcal{A}_{s}^{\mathcal{L}}}+\|u\|_{X_{s}(T)}^{p}.$
(4.23)

Now we will calculate $\|Nu-N\bar{u}\|_{X_{s}(T)}$. In order to do that, we
first notice that

$\|Nu-N\bar{u}\|_{X_{s}(T)}=\left\|\int_{0}^{t}E_{1}(t-\sigma,\cdot)*_{(g)}\left(|u(\sigma,\cdot)|^{p}-|\bar{u}(\sigma,\cdot)|^{p}\right){d}\sigma\right\|_{X_{s}(T)}.$

Similar to (4.13), using the Sobolev inequality (Theorem 2.1), we have

$\displaystyle\left\||u(\sigma,\cdot)|^{p}-|\bar{u}(\sigma,\cdot)|^{p}\right\|_{\dot{H}^{-\gamma}_{\mathcal{L}}}\lesssim\left\||u(\sigma,\cdot)|^{p}-|\bar{u}(\sigma,\cdot)|^{p}\right\|_{L^{m}}$

with $\frac{1}{m}-\frac{1}{2}=\frac{\gamma}{Q}$ provided that $0<\gamma<Q$ and
$1<m<2$. An application of Hölder’s inequality yields

$\displaystyle\left\||u(\sigma,\cdot)|^{p}-|\bar{u}(\sigma,\cdot)|^{p}\right\|_{L^{m}}\lesssim\|u(\sigma,\cdot)-\bar{u}(\sigma,\cdot)\|_{L^{mp}}\left(\|u(\sigma,\cdot)\|_{L^{mp}}^{p-1}+\|\bar{u}(\sigma,\cdot)\|_{L^{mp}}^{p-1}\right).$
(4.24)

Again, the Gagliardo-Nirenberg inequality (Theorem 2.2) implies that

$\displaystyle\left\|u(\sigma,\cdot)-\bar{u}(\sigma,\cdot)\right\|_{L^{mp}}$
$\displaystyle\lesssim\|u(\sigma,\cdot)-\bar{u}(\sigma,\cdot)\|_{\dot{H}_{\mathcal{L}}^{s}(\mathbb{H}^{n})}^{\theta_{3}}\|u(\sigma,\cdot)-\bar{u}(\sigma,\cdot)\|_{L^{2}(\mathbb{H}^{n})}^{(1-\theta_{3})}$
$\displaystyle=(1+\sigma)^{-\left(\frac{\gamma}{2}+\frac{s\theta_{3}}{2}\right)}\left\\{(1+\sigma)^{\frac{(s+\gamma)}{2}}\|u(\sigma,\cdot)-\bar{u}(\sigma,\cdot)\|_{\dot{H}_{\mathcal{L}}^{s}(\mathbb{H}^{n})}\right\\}^{\theta_{3}}$
$\displaystyle\qquad\times\left\\{(1+\sigma)^{\frac{\gamma}{2}}\|u(\sigma,\cdot)-\bar{u}(\sigma,\cdot)\|_{L^{2}(\mathbb{H}^{n})}\right\\}^{(1-\theta_{3})}$
$\displaystyle\lesssim(1+\sigma)^{-\left(\frac{\gamma}{2}+\frac{s\theta_{3}}{2}\right)}\left\|u(\sigma,\cdot)-\bar{u}(\sigma,\cdot)\right\|_{X_{s}(T)},$
(4.25)

and

$\displaystyle\left\|u(\sigma,\cdot)\right\|_{L^{mp}}^{p-1}$
$\displaystyle\lesssim\|u(\sigma,\cdot)\|_{\dot{H}_{\mathcal{L}}^{s}(\mathbb{H}^{n})}^{(p-1)\theta_{3}}\|u(\sigma,\cdot)\|_{L^{2}(\mathbb{H}^{n})}^{(p-1)(1-\theta_{3})}$
$\displaystyle=(1+\sigma)^{-\frac{(p-1)}{2}(s\theta_{3}+\gamma)}\left\\{(1+\sigma)^{\frac{(s+\gamma)}{2}}\|u(\sigma,\cdot)\|_{\dot{H}_{\mathcal{L}}^{s}(\mathbb{H}^{n})}\right\\}^{(p-1)\theta_{3}}$
$\displaystyle\quad\times\left\\{(1+\sigma)^{\frac{\gamma}{2}}\|u(\sigma,\cdot)\|_{L^{2}(\mathbb{H}^{n})}\right\\}^{(p-1)(1-\theta_{3})}$
$\displaystyle\lesssim(1+\sigma)^{-\frac{(p-1)}{2}(s\theta_{3}+\gamma)}\left\|u\right\|_{X_{s}(T)}^{p-1},$
(4.26)

for $\sigma\in[0,T]$ with
$\theta_{3}=\frac{Q}{s}(\frac{1}{2}-\frac{1}{mp})\in[0,1]$. Thus from (4.24),
(4), and (4), we finally have

$\displaystyle\left\||u(\sigma,\cdot)|^{p}-|\bar{u}(\sigma,\cdot)|^{p}\right\|_{\dot{H}^{-\gamma}_{\mathcal{L}}}$
$\displaystyle\lesssim(1+\sigma)^{-\left(\frac{p-1}{2}+\frac{1}{2}\right)(\gamma+{Q}(\frac{1}{2}-\frac{1}{mp}))}\left\|u(\sigma,\cdot)-\bar{u}(\sigma,\cdot)\right\|_{X_{s}(T)}\left(\|u\|_{X_{s}(T)}^{p-1}+\|\bar{u}\|_{X_{s}(T)}^{p-1}\right)$
$\displaystyle\lesssim(1+\sigma)^{-\frac{p}{2}(1-\frac{1}{p})(\gamma+\frac{Q}{2})}\left\|u(\sigma,\cdot)-\bar{u}(\sigma,\cdot)\right\|_{X_{s}(T)}\left(\|u\|_{X_{s}(T)}^{p-1}+\|\bar{u}\|_{X_{s}(T)}^{p-1}\right)$
$\displaystyle\lesssim(1+\sigma)^{-\left(\frac{\gamma}{2}+\frac{Q}{4}\right)p+\frac{\gamma}{2}+\frac{Q}{4}}\left\|u(\sigma,\cdot)-\bar{u}(\sigma,\cdot)\right\|_{X_{s}(T)}\left(\|u\|_{X_{s}(T)}^{p-1}+\|\bar{u}\|_{X_{s}(T)}^{p-1}\right).$

Now using the given range of $p$, we get

$\displaystyle\|Nu-N\bar{u}\|_{X_{s}(T)}\leq
C\|u-\bar{u}\|_{X_{s}(T)}\left(\|u\|_{X_{s}(T)}^{p-1}+\|\bar{u}\|_{X_{s}(T)}^{p-1}\right),$
(4.27)

for any $u,\bar{u}\in X_{s}(T)$ and for some $C>0.$ Also from (4.23), we
obtain

$\displaystyle\|Nu\|_{X_{s}(T)}\leq
D\left\|\left(u_{0},u_{1}\right)\right\|_{\mathcal{A}_{s}^{\mathcal{L}}}+D\|u\|_{X_{s}(T)}^{p},$
(4.28)

for some $D>0$ with initial data space
$\mathcal{A}_{s}^{\mathcal{L}}:=(H^{s}_{\mathcal{L}}\cap\dot{H}^{-\gamma}_{\mathcal{L}})\times(L^{2}\cap\dot{H}^{-\gamma}_{\mathcal{L}})$.

Consequently, for some $r>1$, we choose
$R=rD\left\|\left(u_{0},u_{1}\right)\right\|_{\mathcal{A}_{s}^{\mathcal{L}}}$
with sufficiently small
$\left\|\left(u_{0},u_{1}\right)\right\|_{\mathcal{A}_{s}^{\mathcal{L}}}<\varepsilon$
so that

$DR^{p}<\frac{R}{r}\quad\text{and}\quad 2CR^{p-1}<\frac{1}{r}.$

Then (4.28) and (4.27) reduce to

$\displaystyle\|Nu\|_{X_{s}(T)}\leq\frac{2R}{r}$ (4.29)

and

$\displaystyle\|Nu-N\bar{u}\|_{X_{s}(T)}\leq\frac{1}{r}\|u-\bar{u}\|_{X_{s}(T)},$
(4.30)

respectively for all $u,\bar{u}\in\mathcal{B}(R):=\\{u\in
X_{s}(T):\|u\|_{X_{s}(T)}\leq R\\}$.

Since
$\left\|\left(u_{0},u_{1}\right)\right\|_{\mathcal{A}_{s}^{\mathcal{L}}}<\varepsilon$
is sufficiently small, then from (4.29), we see that $Nu\in X_{s}(T)$, that
is, $N$ maps $X_{s}(T)$ into itself. Also from (4.30), it follows that the map
$N$ is a contraction mapping on the ball $\mathcal{B}(R)$ around the origin in
the Banach space $X_{s}(T)$. Therefore, by Banach’s fixed point theorem, there
exists a uniquely determined fixed point $u^{*}$ of the operator $N$, which
means $u^{*}=Nu^{*}\in X_{s}(T)$ for all positive $T$. This fixed point
$u^{*}$ will be our mild solution to (1.1) on $[0,T]$. This implies that there
exists a global (in-time) small data Sobolev solution $u^{*}$ of the equation
$u^{*}=Nu^{*}$ in $X_{s}(T)$, which also gives the solution to the semilinear
damped wave equation (1.1) and this completes the proof of the theorem. ∎

##  5\. Blow-up analysis of solutions: a test function method

Before proving the main result of this section, we recall briefly the
definition of a weak solution to (1.1).

A weak solution of the Cauchy problem (1.1) in $[0,T)\times\mathbb{H}^{n}$ is
a function $u\in L_{\text{loc }}^{p}\left([0,T)\times\mathbb{H}^{n}\right)$
that satisfies

$\displaystyle\int_{0}^{T}\int_{\mathbb{H}^{n}}|u(t,\eta)|^{p}\varphi(t,\eta)\;d\eta\;dt+\varepsilon\int_{\mathbb{H}^{n}}\left(u_{0}(\eta)+u_{1}(\eta)\right)\varphi(0,\eta)\;d\eta-\varepsilon\int_{\mathbb{H}^{n}}u_{0}(\eta)\partial_{t}\varphi(0,\eta)\;d\eta$
$\displaystyle\quad=\int_{0}^{T}\int_{\mathbb{H}^{n}}u(t,\eta)\left(\partial_{t}^{2}\varphi(t,\eta)-\mathcal{L}\varphi(t,\eta)-\partial_{t}\varphi(t,\eta)\right)\;d\eta\;dt$
(5.1)

for any
$\varphi\in\mathcal{C}_{0}^{\infty}\left([0,T)\times\mathbb{H}^{n}\right)$. If
$T=\infty$, we call $u$ a global in time weak solution to (1.1), otherwise we
call $u$ to be a local in time weak solution.

###### Proof of Theorem 1.5.

In order to prove this result, we apply the so-called test function method. By
contradiction, we assume that there exists a global in time weak solution $u$
to (1.1). Let us consider two bump functions
$\alpha\in\mathcal{C}_{0}^{\infty}(\mathbb{R}^{n})$ and
$\beta\in\mathcal{C}_{0}^{\infty}(\mathbb{R})$ such that

$\alpha=1$ on $B_{n}\left(\frac{1}{2}\right)$ with
$\operatorname{supp}\alpha\subset B_{n}(1)$

and

$\text{$\beta=1$ on $\left[-\frac{1}{4},\frac{1}{4}\right]$ with
$\operatorname{supp}\beta\subset(-1,1)$}.$

If $R>1$ is a parameter, then, we define the test function
$\varphi_{R}\in\mathcal{C}_{0}^{\infty}\left([0,\infty)\times\mathbb{R}^{2n+1}\right)$
with separate variables as

$\varphi_{R}(t,x,y,\tau)\doteq\beta\left(\frac{t}{R^{2}}\right)\alpha\left(\frac{x}{R}\right)\alpha\left(\frac{y}{R}\right)\beta\left(\frac{\tau}{R^{2}}\right),\quad(t,x,y,\tau)\in[0,\infty)\times\mathbb{R}^{2n+1}.$

Let $\mathcal{D}_{R}:=B_{n}(R)\times B_{n}(R)\times\left[-R^{2},R^{2}\right]$.
Then using support of $\alpha$ and $\beta$, we can conclude that
$\operatorname{supp}\varphi_{R}\subset\left[0,R^{2}\right]\times\mathcal{D}_{R}$.
Furthermore, a simple calculation yields

$\displaystyle\partial_{t}\varphi_{R}(t,x,y,\tau)$
$\displaystyle=R^{-2}\beta^{\prime}\left(\frac{t}{R^{2}}\right)\alpha\left(\frac{x}{R}\right)\alpha\left(\frac{y}{R}\right)\beta\left(\frac{\tau}{R^{2}}\right),$
$\displaystyle\partial_{t}^{2}\varphi_{R}(t,x,y,\tau)$
$\displaystyle=R^{-4}\beta^{\prime\prime}\left(\frac{t}{R^{2}}\right)\alpha\left(\frac{x}{R}\right)\alpha\left(\frac{y}{R}\right)\beta\left(\frac{\tau}{R^{2}}\right).$

Moreover, from the following expression of the sublaplacian

$\mathcal{L}=\Delta+\frac{1}{4}(|x|^{2}+|y|^{2})\partial_{\tau}^{2}+\sum_{j=1}^{n}\left(x_{j}\partial_{y_{j}\tau}^{2}-y_{j}\partial_{x_{j}\tau}^{2}\right),$

we have

$\displaystyle\mathcal{L}\varphi_{R}(t,x,y,\tau)$
$\displaystyle=R^{-2}\beta\left(\frac{t}{R^{2}}\right)\Delta\alpha\left(\frac{x}{R}\right)\alpha\left(\frac{y}{R}\right)\beta\left(\frac{\tau}{R^{2}}\right)$
$\displaystyle+R^{-2}\beta\left(\frac{t}{R^{2}}\right)\alpha\left(\frac{x}{R}\right)\Delta\alpha\left(\frac{y}{R}\right)\beta\left(\frac{\tau}{R^{2}}\right)$
$\displaystyle+R^{-3}\sum_{j=1}^{n}x_{j}\beta\left(\frac{t}{R^{2}}\right)\alpha\left(\frac{x}{R}\right)\partial_{j}\alpha\left(\frac{y}{R}\right)\beta^{\prime}\left(\frac{\tau}{R^{2}}\right)$
$\displaystyle-R^{-3}\sum_{j=1}^{n}y_{j}\beta\left(\frac{t}{R^{2}}\right)\partial_{j}\alpha\left(\frac{x}{R}\right)\alpha\left(\frac{y}{R}\right)\beta^{\prime}\left(\frac{\tau}{R^{2}}\right)$
$\displaystyle+\frac{1}{4}R^{-4}\left(|x|^{2}+|y|^{2}\right)\beta\left(\frac{t}{R^{2}}\right)\alpha\left(\frac{x}{R}\right)\alpha\left(\frac{y}{R}\right)\beta^{\prime\prime}\left(\frac{\tau}{R^{2}}\right),$

where $\Delta$ denotes the Laplace operator on $\mathbb{R}^{n}$.

From the condition $0\leq\alpha,\beta\leq 1$, we can get
$\alpha\leq\alpha^{\frac{1}{p}}$ and $\beta\leq\beta^{\frac{1}{p}}$. Moreover,
using the following bounds

$\displaystyle\left|\partial_{j}\alpha\right|\lesssim\alpha^{\frac{1}{p}}\quad\quad\text{
for any }1\leq j\leq n,$
$\displaystyle\left|\partial_{j}\partial_{k}\alpha\right|\lesssim\alpha^{\frac{1}{p}}\quad\text{
for any }1\leq j,k\leq n,$
$\displaystyle\left|\beta^{\prime}\right|\lesssim\beta^{\frac{1}{p}},\quad\left|\beta^{\prime\prime}\right|\lesssim\beta^{\frac{1}{p}},$

we immediately get

$\displaystyle\left|\partial_{t}\varphi_{R}\right|$ $\displaystyle\lesssim
R^{-2}\left(\varphi_{R}\right)^{\frac{1}{p}},$
$\displaystyle\left|\partial_{t}^{2}\varphi_{R}\right|$ $\displaystyle\lesssim
R^{-4}\left(\varphi_{R}\right)^{\frac{1}{p}}\lesssim
R^{-2}\left(\varphi_{R}\right)^{\frac{1}{p}},\text{and}$
$\displaystyle\left|\mathcal{L}\varphi_{R}\right|$ $\displaystyle\lesssim
R^{-2}\left(\varphi_{R}\right)^{\frac{1}{p}}.$

Let

$I_{R}=\int_{0}^{T}\int_{\mathbb{H}^{n}}|u(t,\eta)|^{p}\varphi_{R}(t,\eta)\;d\eta\;dt.$

Now from (1.4) with keeping in mind the fact $\partial_{t}\varphi_{R}\leq 0$,
an application of Hölder’s and Young’s inequalities yields

$\displaystyle\int_{0}^{T}\int_{\mathbb{H}^{n}}|u(t,g)|^{p}\varphi_{R}(t,g)\;dg\;dt+\varepsilon\int_{\mathbb{H}^{n}}\left(u_{0}(g)+u_{1}(g)\right)\varphi_{R}(0,g)\;dg$
$\displaystyle\leq\int_{0}^{T}\int_{\mathbb{H}^{n}}|u(t,g)|\left(|\partial_{t}^{2}\varphi_{R}(t,g)|+|\mathcal{L}\varphi_{R}(t,g)|+|\partial_{t}\varphi_{R}(t,g)|\right)\;dg\;dt$
$\displaystyle\lesssim
R^{-2}\int_{0}^{T}\int_{\mathbb{H}^{n}}|u(t,g)||\varphi_{R}(t,g)|^{\frac{1}{p}}\;dg\;dt$
$\displaystyle\leq
CR^{-2}\left(\int_{0}^{\infty}\int_{\mathbb{H}^{n}}|u(t,g)|^{p}\varphi_{R}(t,g)\;dg\;dt\right)^{\frac{1}{p}}\left(\int_{\left[0,R^{2}\right]\times\mathcal{D}_{R}}dt\;dg\right)^{\frac{1}{p^{\prime}}}$
(5.2)
$\displaystyle=CR^{-2}I_{R}^{\frac{1}{p}}R^{\frac{2+Q}{p^{\prime}}}=CR^{-2+\frac{Q+2}{p^{\prime}}}I_{R}^{\frac{1}{p}}$
$\displaystyle\leq\frac{I_{R}}{p}+\frac{CR^{Q+2-2p^{\prime}}}{p^{\prime}},$

where in (5), we used the fact that
$\operatorname{meas}(\mathcal{D}_{R})\approx R^{Q}$. Therefore

$\displaystyle\int_{0}^{T}\int_{\mathbb{H}^{n}}|u(t,g)|^{p}\varphi_{R}(t,g)\;dg\;dt\leq\frac{I_{R}}{p}+\frac{CR^{Q+2-2p^{\prime}}}{p^{\prime}}-\varepsilon\int_{\mathbb{H}^{n}}(u_{0}(g)+u_{1}(g))\varphi_{R}(0,g)\;dg.$
(5.3)

Now using our assumption (1.15), it remains to find estimate for

$\int_{\mathbb{H}^{n}}\left(u_{0}(g)+u_{1}(g)\right)\varphi_{R}(0,g)dg.$

Before that, we have to justify that the set of all initial data
$\left(u_{0},u_{1}\right)$ in
$\dot{H}_{\mathcal{L}}^{-\gamma}\times\dot{H}_{\mathcal{L}}^{-\gamma}$ with
the assumptions (1.15), is non-empty. For that, we denote the set
$\mathcal{D}_{Q,\gamma}$ as

$\displaystyle\mathcal{D}_{Q,\gamma}$
$\displaystyle:=\\{\left(u_{0},u_{1}\right)\in\mathcal{D}^{\prime}(\mathbb{H}^{n})\times\mathcal{D}^{\prime}(\mathbb{H}^{n}):u_{0}(g)+u_{1}(g)\geq
C_{0}\langle
g\rangle^{-Q\left(\frac{1}{2}+\frac{\gamma}{Q}\right)}(\log(e+|g|))^{-1}\\},$

then first we claim that
$\mathcal{D}_{Q,\gamma}\cap\left(L^{\frac{2Q}{Q+2\gamma}}\times
L^{\frac{2Q}{Q+2\gamma}}\right)\neq\emptyset$ for $\frac{2Q}{Q+2\gamma}>1$. In
particular, consider the functions

$\displaystyle u_{0}(g)=u_{1}(g)$ $\displaystyle=C_{0}\langle
g\rangle^{-Q\left(\frac{1}{2}+\frac{\gamma}{Q}\right)}(\log(e+|g|))^{-1}.$

Then

$\displaystyle\int_{\mathbb{H}^{n}}|u_{0}(g)|^{\frac{2Q}{Q+2\gamma}}{d}g$
$\displaystyle=C_{0}^{\frac{2Q}{Q+2\gamma}}\int_{\mathbb{H}^{n}}\langle
g\rangle^{-Q\left(\frac{1}{2}+\frac{\gamma}{Q}\right)\times\frac{2Q}{Q+2\gamma}}(\log(e+|g|))^{-\frac{2Q}{Q+2\gamma}}\;dg$
$\displaystyle=C_{0}^{\frac{2Q}{Q+2\gamma}}\int_{\mathbb{H}^{n}}\langle
g\rangle^{-Q}(\log(e+|g|))^{-\frac{2Q}{Q+2\gamma}}\;dg$ $\displaystyle\lesssim
C_{0}^{\frac{2Q}{Q+2\gamma}}\int_{0}^{\infty}\langle
r\rangle^{-Q}r^{Q-1}(\log(\mathrm{e}+r))^{-\frac{2Q}{Q+2\gamma}}dr$
$\displaystyle=C_{0}^{\frac{2Q}{Q+2\gamma}}\int_{0}^{\infty}\langle
r\rangle^{-1}(\log(\mathrm{e}+r))^{-\frac{2Q}{Q+2\gamma}}dr<\infty,$

for $\frac{2Q}{Q+2\gamma}>1$. This implies that $u_{0},u_{1}\in
L^{\frac{2Q}{Q+2\gamma}}$. Since
$\frac{Q+2\gamma}{2Q}-\frac{1}{2}=\frac{\gamma}{Q}$ with
$\gamma\in\left(0,\frac{Q}{2}\right)$, according to the Sobolev inequality
(Theorem 2.1), we have
$L^{\frac{2Q}{Q+2\gamma}}\subset\dot{H}_{\mathcal{L}}^{-\gamma}$. Therefore,
in every instance, we can deduce that

$\mathcal{D}_{Q,\gamma}\cap\left(\dot{H}_{\mathcal{L}}^{-\gamma}\times\dot{H}_{\mathcal{L}}^{-\gamma}\right)\neq\emptyset,\quad\text{for}~{}\gamma\in\left(0,\frac{Q}{2}\right).$

Now from our assumption (1.15), for $R\gg 1$, we obtain

$\displaystyle\int_{\mathbb{H}^{n}}\left(u_{0}(g)+u_{1}(g)\right)\varphi_{R}(0,g)\;dg$
$\displaystyle\geq C_{0}\int_{\mathcal{D}_{R}}\langle
g\rangle^{-Q\left(\frac{1}{2}+\frac{\gamma}{Q}\right)}(\log(e+|g|))^{-1}\,dg$
$\displaystyle\geq C_{0}R^{Q-Q(\frac{1}{2}+\frac{\gamma}{Q})}(\log
R)^{-1}=C_{0}R^{\frac{Q}{2}-\gamma}(\log R)^{-1}.$ (5.4)

Thus from (5.3) and (5), we have

$\displaystyle
I_{R}=\int_{0}^{T}\int_{\mathbb{H}^{n}}|u(t,g)|^{p}\varphi_{R}(t,g)\;dg\;dt\leq\frac{I_{R}}{p}+\frac{CR^{Q+2-2p^{\prime}}}{p^{\prime}}-\varepsilon
C_{0}R^{\frac{Q}{2}-\gamma}(\log R)^{-1},$

which further implies that

$\displaystyle
0\leq\Big{(}1-\frac{1}{p}\Big{)}I_{R}\leq\frac{CR^{Q+2-2p^{\prime}}}{p^{\prime}}-\varepsilon
C_{0}R^{\frac{Q}{2}-\gamma}(\log R)^{-1}.$ (5.5)

By the assumption $p\in\left(1,1+\frac{4}{Q+2\gamma}\right),$ we have
$Q+2-2p^{\prime}-\frac{Q}{2}+\gamma<0$ and therefore, it is easy to see that

$R^{Q+2-2p^{\prime}-\frac{Q}{2}+\gamma}\log R<\frac{\varepsilon
C_{0}p^{\prime}}{C},$ (5.6)

for $R\gg 1.$

Hence, from (5.5) and (5.6), we obtain

$0\leq\Big{(}1-\frac{1}{p}\Big{)}I_{R}\leq\frac{CR^{Q+2-2p^{\prime}}}{p^{\prime}}-\varepsilon
C_{0}R^{\frac{Q}{2}-\gamma}(\log R)^{-1}<0,$

which is a contradiction. This completes the proof of the blow-up result.

Since the scaling factor $R^{2}$, appeared in the bump function $\beta$ with
respect to the time varibale has to be dominated by the lifespan
$T_{w,\varepsilon}$ of the weak solution in order to guarantee
$\varphi_{R}\in\mathcal{C}_{0}^{\infty}([0,T)\times\mathbb{H}^{n})$, to
generate upper bound estimate for the lifespan, we consider $R\uparrow
T_{w,\varepsilon}^{\frac{1}{2}}$ in (5.5). As a result, a contradiction
(similar to (5.5)) exists if

$\frac{T_{w,\varepsilon}^{\frac{Q+2}{2}-p^{\prime}}}{p^{\prime}}<C\varepsilon
T_{w,\varepsilon}^{\frac{Q-2\gamma}{4}}(\log T_{w,\varepsilon})^{-1}.$

In other words

$T_{w,\varepsilon}\leq
C\varepsilon^{-\frac{2}{2p^{\prime}-2-\frac{Q}{2}-\gamma}},$

which is the desired lifespan for the local in time weak solutions to the
Cauchy problem (1.1), where the constant $C$ is positive and independent of
$\varepsilon,p^{\prime}$, and $p^{\prime}$ is the conjugate exponent to $p$.

∎

##  6\. Sharp lifespan estimates of solutions

The aim of this section is to find the lower bound estimates of lifespan. We
will make use of certain notations, specifically, the evolution space
$X_{1}(T)$ and the data space $\mathcal{A}_{1}^{\mathcal{L}}$, introduced in
Section 1. As we are now considering the case where
$1<p<p_{\text{crit}}(Q,\gamma)$, in order to obtain lower limit estimates for
the lifespan, we will use a different nonlinear inequality instead of (4.7).

###### Proof of Theorem 1.7.

Here we note that using the definition of mild solutions (1.10), we can
estimate and express local-in-time mild solutions directly. Proceeding in a
similar way as to establish (4.14)-(4.15) in Section 4, we get

$\displaystyle(1+t)^{\frac{\gamma}{2}}\|u(t,\cdot)\|_{L^{2}}\lesssim$
$\displaystyle\varepsilon\left\|\left(u_{0},u_{1}\right)\right\|_{\mathcal{A}_{1}^{\mathcal{L}}}+\int_{0}^{\frac{t}{2}}(1+\sigma)^{-p(\frac{\gamma}{2}+\frac{Q}{4})+\frac{\gamma}{2}+\frac{Q}{4}}d\sigma~{}\left\|u\right\|_{X_{1}(T)}^{p}$
$\displaystyle+(1+t)^{-p(\frac{\gamma}{2}+\frac{Q}{4})+\frac{\gamma}{2}+\frac{Q}{4}+1}\|u\|_{X_{1}(T)}^{p}$
(6.1)

because of $\gamma\in(0,2)$, where we restricted $p$ as

$1+\frac{2\gamma}{Q}\leq p\leq\frac{Q}{Q-2},$

for the application of the Gagliardo-Nirenberg inequality. Since
$1<p<p_{\text{crit }}(Q,\gamma)=1+\frac{4}{Q+2\gamma}$, this implies that

$-p\left(\frac{\gamma}{2}+\frac{Q}{4}\right)+\frac{\gamma}{2}+\frac{Q}{4}>-\frac{2\gamma+Q}{4}\left(1+\frac{4}{Q+2\gamma}\right)+\frac{2\gamma+Q}{4}=-1.$

Therefore, from (6), we have

$\displaystyle(1+t)^{\frac{\gamma}{2}}\|u(t,\cdot)\|_{L^{2}}\lesssim\varepsilon\left\|\left(u_{0},u_{1}\right)\right\|_{\mathcal{A}_{1}^{\mathcal{L}}}+(1+t)^{-p\left(\frac{\gamma}{2}+\frac{Q}{4}\right)+\frac{Q}{4}+\frac{\gamma}{2}+1}\|u\|_{X_{1}(T)}^{p}.$
(6.2)

Choosing $s=1$ in (4.21) leads to

$\displaystyle(1+t)^{\frac{1+\gamma}{2}}\left\|u(t,\cdot)\right\|_{\dot{H}_{\mathcal{L}}^{1}}\lesssim\varepsilon\left\|\left(u_{0},u_{1}\right)\right\|_{\mathcal{A}_{1}^{\mathcal{L}}}+(1+t)^{-p(\frac{\gamma}{2}+\frac{Q}{4})+\frac{Q}{4}+\frac{\gamma}{2}+1}\|u\|_{X_{1}(T)}^{p}.$
(6.3)

Thus from (6.2) and (6.3), we can say that

$\displaystyle\|u\|_{X_{1}(T)}\leq\varepsilon
C+D(1+t)^{\alpha(p,Q,\gamma)}\|u\|_{X_{1}(T)}^{p},$ (6.4)

where
$\alpha(p,Q,\gamma):=-p(\frac{\gamma}{2}+\frac{Q}{4})+\frac{Q}{4}+\frac{\gamma}{2}+1\in(0,1)$,
and $C,D$ are two positive constants independent of $\varepsilon$ and $T$.

Now let us now introduce

$T^{*}:=\sup\left\\{T\in\left[0,T_{m,\varepsilon}\right)\text{ such that
}\mathcal{G}(T):=\|u\|_{X_{1}(T)}\leq M\varepsilon\right\\}$

with a large enough constant $M>0$, which we shall select later. As a result,
from (6.4) and the fact that $\mathcal{G}\left(T^{*}\right)\leq M\varepsilon$,
we obtain

$\displaystyle\mathcal{G}\left(T^{*}\right)$
$\displaystyle=\|u\|_{X_{1}(T^{*})}$ $\displaystyle\leq\left(\varepsilon
C+D\left(1+T^{*}\right)^{\alpha(p,Q,\gamma)}M^{p}\varepsilon^{p}\right)$
$\displaystyle=\varepsilon\left(C+D\left(1+T^{*}\right)^{\alpha(p,Q,\gamma)}M^{p}\varepsilon^{p-1}\right)<M\varepsilon$
(6.5)

for large $M$, provided $C<M$ and

$D\left(1+T^{*}\right)^{\alpha(p,Q,\gamma)}M^{p-1}\varepsilon^{p-1}<1.$

Its important to note that the function $\mathcal{G}=\mathcal{G}(T)$ is
continuous for $T\in\left(0,T_{m,\varepsilon}\right)$. Nonetheless, (6)
demonstrates that there exists a time
$T_{0}\in\left(T^{*},T_{m,\varepsilon}\right)$ such that
$\mathcal{G}\left(T_{0}\right)\leq M\varepsilon$, which contradicts to the
assumption that $T^{*}$ is the supremum.To put it another way, we must enforce
the following condition:

$D\left(1+T^{*}\right)^{\alpha(p,Q,\gamma)}M^{p-1}\varepsilon^{p-1}\geq 1.$

This implies that

$1+T^{*}\geq
D^{-\frac{1}{\alpha(p,Q,\gamma)}}M^{-\frac{p-1}{\alpha(p,Q,\gamma)}}\varepsilon^{-\frac{p-1}{\alpha(p,Q,\gamma)}}.$

This implies that we can deduce the blow-up time estimate as

$T_{m,\varepsilon}\geq
D\varepsilon^{-\frac{p-1}{\alpha_{0}(p,Q,\gamma)}}=D\varepsilon^{-\frac{2}{2p^{\prime}-2-\frac{Q}{2}-\gamma}}.$

This completes the proof of the theorem regarding lower bound estimates of
lifespan. ∎

## Acknowledgement

AD is supported by Core Research Grant, RP03890G, Science and Engineering
Research Board (SERB), DST, India. VK and MR are supported by the FWO Odysseus
1 grant G.0H94.18N: Analysis and Partial Differential Equations, the
Methusalem program of the Ghent University Special Research Fund (BOF) (Grant
number 01M01021), and by FWO Senior Research Grant G011522N. MR is also
supported by EPSRC grants EP/R003025/2 and EP/V005529/1. SSM is supported by
post-doctoral fellowship at the Indian Institute of Technology Delhi, India.
SSM also thanks Ghent Analysis & PDE Center of Ghent University for the
financial support of his visit to Ghent University during which this work has
been completed.

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