# Instantaneous blowup and singular potentials on Heisenberg groups

Gisele R. Goldstein  Department of Mathematical Sciences, University of
Memphis, 38152 Memphis, TN, USA<EMAIL_ADDRESS>,  Jerome A. Goldstein
Department of Mathematical Sciences, University of Memphis, 38152 Memphis, TN,
USA<EMAIL_ADDRESS>,  Alessia E. Kogoj  Dipartimento di Scienze Pure e
Applicate, Università degli Studi di Urbino Carlo Bo, Piazza della Repubblica,
13, I 61029 Urbino (PU), Italy<EMAIL_ADDRESS>,  Abdelaziz Rhandi
Dipartimento di Ingegneria dell’Informazione, Ingegneria Elettrica e
Matematica Applicata, Università degli Studi di Salerno, Via Giovanni Paolo
II, 132, I 84084 Fisciano (SA), Italy<EMAIL_ADDRESS>and  Cristian Tacelli
Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni
Paolo II, 132, I 84084 Fisciano (SA), Italy<EMAIL_ADDRESS>
###### Abstract.

In this paper we generalize the instantaneous blowup result from [3] and [15]
to the heat equation perturbed by singular potentials on the Heisenberg group.

###### 2010 Mathematics Subject Classification:

47D07, 47D08; 35J10, 35K20

##  1\. Introduction

The problem of existence and nonexistence of nonnegative solutions to the heat
equation with singular potentials
$V_{c}^{*}(x)=\dfrac{c}{|x|^{2}},\,x\in\Omega_{N}$,

(1.1) $\left\\{\begin{array}[]{lll}\dfrac{\partial u}{\partial
t}(x,t)=\varDelta
u(x,t)+V_{c}^{*}(x)u(x,t)\quad(x,t)\in\Omega_{N}\times(0,\infty),\\\ \\\
u(x,0)=u_{0}(x),\quad x\in\Omega_{N},\end{array}\right.$

where $\Omega_{N}=\left\\{\begin{array}[]{ll}\mathbb{R}^{N}\,\hbox{\ if }N\geq
2,\\\ (0,\infty)\,\hbox{\ if }N=1,\end{array}\right.$ was settled and solved
by Baras and Goldstein [3]. For $\Omega_{1}=(0,\infty)$ one has to add a
Dirichlet boundary condition at $0$. For simplicity we assume in the sequel
that $N\geq 3$ and set $C_{*}(N):=\left(\frac{N-2}{2}\right)^{2}$.

Obviously, the phenomenon of existence and nonexistence is caused by the
singular potential $V_{c}^{*}$, which is controlled by Hardy’s inequality

$C_{*}(N)\int_{\mathbb{R}^{N}}\frac{|\varphi(x)|^{2}}{|x|^{2}}dx\leq\int_{\mathbb{R}^{N}}|\nabla\varphi(x)|^{2}\,dx,\quad\forall\varphi\in
C_{c}^{\infty}(\mathbb{R}^{N}),$

together with its optimal constant $C_{*}(N)$. Moreover $V_{c}^{*}$ belongs to
a borderline case where the strong maximum principle and Gaussian bounds fail,
cf. [2].

Let $W_{n}(x)=\inf\\{V_{c}^{*}(x),n\\}$ be the cutoff potential, with
$c>C_{*}(N)$. Let $u_{n}$ be the unique solution of

$\left\\{\begin{array}[]{ll}\dfrac{\partial_{t}u_{n}}{\partial t}-\varDelta
u_{n}-W_{n}u_{n}=0,\quad\hbox{\ in }{\mathbb{R}}^{N}\times(0,\infty),\\\ \\\
u_{n}(x,0)=u(x,0)=u_{0}(x)\geq 0.\end{array}\right.$

Here $0\not\equiv u_{0}\in L^{2}(\mathbb{R}^{N})$ or, more generally, $u_{0}$
grows no faster than $e^{|x|^{2-\varepsilon}}$ at infinity. Since $W_{n}$ is
bounded, $u_{n}$ exists. If a positive solution $u$ to (1.1) were to exist,
then $0<u_{n}\leq u$ which is a contradiction, since $u_{n}(x,t)$ tends to
infinity at all spatial points and at all positive times (see [3, Theorem
2.2.(ii)]). This is called instantaneous blowup.

Given nonnegative functions $u_{0}\in L^{1}(\mathbb{R}^{N}),\,0\leq V\in
L^{1}_{loc}(\mathbb{R}^{N})$ and $0\leq f\in
L^{1}(\mathbb{R}^{N}\times(0,T))$, Baras and Goldstein [3] considered the
problem of finding a function $u$ such that

$(P)\quad\left\\{\begin{array}[]{ll}0\leq u\hbox{\ on
}\mathbb{R}^{N}\times(0,T),\,V(\cdot)u\in
L^{1}_{loc}(\mathbb{R}^{N}\times(0,T)),\\\ \\\ \dfrac{\partial u}{\partial
t}=\varDelta u+Vu+f\,\hbox{\ in
}\mathcal{D}^{\prime}(\mathbb{R}^{N}\times(0,T)),\\\ \\\ \mathrm{esslim}_{t\to
0^{+}}\int_{\mathbb{R}^{N}}u(x,t)\psi(x)\,dx=\int_{\mathbb{R}^{N}}u_{0}(x)\psi(x)\,dx,\,\mbox{for
all}\ \psi\in\mathcal{D}(\mathbb{R}^{N}).\end{array}\right.$

Here
$\mathcal{D}(\mathbb{R}^{N}):=C_{c}^{\infty}(\mathbb{R}^{N}),\,\mathcal{D}(\mathbb{R}^{N}\times(0,T)):=C_{c}^{\infty}(\mathbb{R}^{N}\times(0,T))$
with the usual topology and $\mathcal{D}^{\prime}(\mathbb{R}^{N}\times(0,T))$
the dual of $\mathcal{D}(\mathbb{R}^{N}\times(0,T))$, is the space of all
distributions on $\mathbb{R}^{N}\times(0,T)$.
Consider the potential

$W_{0}(x)=\left\\{\begin{array}[]{ll}\dfrac{c}{|x|^{2}}\quad\hbox{\ if }x\in
B_{1},\\\ \\\ 0\quad\quad\hbox{\ if }x\in\mathbb{R}^{N}\setminus
B_{1}.\end{array}\right.$

Here $B_{1}$ can be replaced by $B_{\delta}$ for every fixed $\delta>0$, where
$B_{r}$ denotes the ball in $\mathbb{R}^{N}$ of center $0$ and radius $r>0$.
Baras and Goldstein [3] proved the following result.

######  Theorem 1.1.

  * (i)

Let $0\leq c\leq C_{*}(N)$ and let $V\geq 0$ be a measurable potential
satisfying $V\in L^{\infty}(\mathbb{R}^{N}\setminus B_{1})$, where $B_{1}$
denotes the unit ball in $\mathbb{R}^{N}$. Let $0\leq f\in
L^{1}(\mathbb{R}^{N}\times(0,T))$. If $V\leq W_{0}$ in $B_{1}$, then $(P)$ has
a positive solution if

(1.2)
$\int_{\mathbb{R}^{N}}|x|^{-\alpha}u_{0}(x)\,dx<\infty\,,\quad\int_{0}^{T}\int_{\mathbb{R}^{N}}f(x,s)|x|^{-\alpha}dxds<\infty,$

where $\alpha$ is the smallest root of $\alpha(N-2-\alpha)=c$. If $V\geq
W_{0}$ in $B_{1}$, and if $(P)$ has a solution u, then

$\int_{\Omega^{\prime}}|x|^{-\alpha}u_{0}(x)\,dx<\infty,\quad\int_{0}^{T-\varepsilon}\int_{\Omega^{\prime}}f(x,s)|x|^{-\alpha}dxds<\infty,$

for each $\varepsilon\in(0,T)$ and each
$\Omega^{\prime}\subset\subset\mathbb{R}^{N}$ with $\alpha$ as above. If
either $u_{0}\neq 0$ or $f\neq 0$ in $\mathbb{R}^{N}\times(0,\varepsilon)$ for
each $\varepsilon\in(0,T)$, then given
$\Omega^{\prime}\subset\subset\mathbb{R}^{N}$, there is a
$C=C(\varepsilon,\Omega^{\prime})>0$ such that

(1.3) $u(x,t)\geq\frac{C}{|x|^{\alpha}}\text{ if
}(x,t)\in\Omega^{\prime}\times[\varepsilon,T).$

  * (ii)

If $c>C_{*}(N)$, $V\geq W_{0}$ and either $u_{0}\not\equiv 0$ or $f\not\equiv
0$, then $(P)$ does not have a positive solution.

Many extensions of the above result have been done by several authors, cf.
[6], [7], [9], [10], [12], [13], [14], [15], [16], [17]. In this article we
present a new result of this type replacing the Laplacian on $\mathbb{R}^{N}$
by the sub-Laplacian $\varDelta_{\mathbb{H}}$ (also known as the Kohn
Laplacian) on the Heisenberg group $\mathbb{H}^{N}$. For the definitions see
Section 2.

For this purpose let us consider, for $w=(z,l)\in\mathbb{H}^{N}$, the problem

(1.4) $\left\\{\begin{array}[]{ll}\dfrac{\partial u}{\partial
t}(w,t)=\varDelta_{\mathbb{H}}u(w,t)+V_{*}(w)u(w,t)+f(w,t)&t>0,\,w\in\mathbb{H}^{N},\\\
\\\ u(w,0)=u_{0}(w),&w\in\mathbb{H}^{N}.\end{array}\right.$

Assume $u_{0}\geq 0$, $f\geq 0$ and as $V_{*}$ choose the corresponding
critical potential in the case of the Heisenberg group $\mathbb{H}^{N}$

$V_{*}(w)=c\frac{|z|^{2}}{|z|^{4}+l^{2}},\quad w=(z,l)\in\mathbb{H}^{N}.$

We thus look at the problem

$(P)_{\mathbb{H}^{N}}\quad\left\\{\begin{array}[]{ll}\dfrac{\partial
u}{\partial t}=\varDelta_{\mathbb{H}}u+V_{*}u+f&\text{in
}\mathbb{H}^{N}\times(0,T),\\\ \\\
u(w,0)=u_{0}(w)&w\in\mathbb{H}^{N},\end{array}\right.$

with $u_{0}\geq 0$ and $u_{0}\not\equiv 0$ a.e. Set
$V_{n}(w)=\min\\{V_{*}(w),n\\}$, $f_{n}(w,t)=\min\\{f(w,t),n\\}$. Let $u_{n}$
be the unique nonnegative solution of

$(P_{n})_{\mathbb{H}^{N}}\quad\left\\{\begin{array}[]{ll}\dfrac{\partial
u_{n}}{\partial t}=\varDelta_{\mathbb{H}}u_{n}+V_{n}u_{n}+f_{n}&\text{ in
}\mathbb{H}^{N}\times(0,T),\\\ \\\
u_{n}(w,0)=u_{0}(w)&w\in\mathbb{H}^{N},\end{array}\right.$

and assume that $u_{n}$ exists. We only need to assume that the heat equation
with no potential has a global positive solution when $u_{0}$ is the initial
value, see (2.6) and Theorem 2.2 below. It is sufficient that $u_{0}\in
L^{2}_{loc}(\mathbb{H}^{N})$ and $u_{0}$ grows no faster than
$e^{d^{2}(w)-\varepsilon}$ at infinity, where $d(\cdot)$ is the function given
by (2.4).

Let

$C^{*}(N)=N^{2}.$

We will prove, for $u_{n}$ the solution of $(P_{n})_{\mathbb{H}^{N}}$, that

  * (I)

If $0<c\leq C^{*}(N)$, then

$\lim_{n\to\infty}u_{n}(w,t)=u(w,t),\quad(w,t)\in\mathbb{H}^{N}\times(0,T),$

exists and is a solution of $(P)_{\mathbb{H}^{N}}$.

  * (II)

If $c>C^{*}(N)$, then

(1.5) $\lim_{n\to\infty}u_{n}(w,t)=+\infty$

for all $(w,t)\in\mathbb{H}^{N}\times(0,T).$

The conclusion in (II), namely (1.5), is known as instantaneous blowup, or
(IBU) for short.

In the existence case $(I)$, by the maximum principle for
$\varDelta_{\mathbb{H}}$, it is clear that we can replace $V_{n},V_{*}$ by
$\tilde{V}_{n},\tilde{V}_{*}$ where $\tilde{V}_{n}\leq V_{n}$,
$\tilde{V}_{*}\leq V_{*}$ a.e. for each $n$. Similarly, for the nonexistence
result $(II)$, we can replace $V_{n},V_{*}$ by $\tilde{V}_{n},\tilde{V}_{*}$
where $\tilde{V}_{n}\geq V_{n}$, $\tilde{V}_{*}\geq V_{*}$ a.e. (at least in a
neighborhood of the origin).

The paper is organized as follows. In the next section we recall the
definitions of the Heisenberg group $\mathbb{H}^{N}$ and the sub-Laplacian
$\varDelta_{\mathbb{H}}$ on $\mathbb{H}^{N}$. We also give some known
properties of $\varDelta_{\mathbb{H}}$ that we need in this paper. In Section
3 we state and prove the main results of this paper. In the Appendix we prove
some technical lemmas that we use in the proof of the main results.

This paper treats the same basic problem as did [15]. There, the existence
part of Theorem 3.4 was proved, using a different method. But part (ii) of
Theorem 3.4 is much stronger than the corresponding result of [15].

In 1999, X. Cabré and Y. Martel [5] gave a different approach to a more
general problem. The paper [3] treated a potential $V\geq 0$ with only one
singularity, at the origin, while [5] allowed for a much more general
potential which one takes to be $0\leq W\in
L^{1}_{loc}(\mathbb{R}^{N}\setminus\\{0\\}).$ In [5] the authors defined the
“generalized first eigenvalue” of the Schrödinger operator $-\varDelta-W$ as

$\displaystyle\sigma_{W}=\inf_{u\in
C_{c}^{1}(\mathbb{R}^{N}),\,\|u\|_{L^{2}}=1}\left\\{\int_{\mathbb{R}^{N}}(|\nabla
u(x)|^{2}-W(x)|u(x)|^{2})\ dx\right\\}$ $\displaystyle\left(\hbox{\ or
}\sigma_{W}=\inf_{u\in
C_{c}^{1}(\mathbb{R}^{N}\setminus\\{0\\}),\,\|u\|_{L^{2}}=1}\left\\{\int_{\mathbb{R}^{N}}(|\nabla
u(x)|^{2}-W(x)|u(x)|^{2})\ dx\right\\}\hbox{\ if }N\leq 2\right).$

Note that for $W(x)=\frac{c}{|x|^{2}},\,x\in\mathbb{R}^{N}$,
$\sigma_{W}=-\infty$ if $c>C_{*}(N)$ and $\sigma_{W}>-\infty$ if $c\leq
C_{*}(N).$ Roughly speaking, in [5] the existence of positive solutions, when
$\sigma_{W}>-\infty$ and for $\sigma_{W}=-\infty$, was obtained; further the
authors proved that there is no globally defined pointwise solution that is
exponentially bounded in time. This is a much weaker conclusion than the
instantaneous blowup (IBU).

In the $\mathbb{H}^{N}$ setting, the authors in [15] used the method of [5]
and proved nonexistence of globally defined (in $(x,t)$) positive solutions
that grow at most exponentially for $c>C^{*}(N).$ But the question of (IBU)
remained open until now.

##  2\. Notations and preliminaries

The Heisenberg group and its sub-Laplacian play a crucial role in several
branches of harmonic analysis, complex geometry and partial differential
equations (see e.g. [8, 11, 19, 20]; see also the survey papers [18, 21]).

The Heisenberg group $\mathbb{H}^{N},N\in\mathbb{N}$, is the stratified Lie
group of step two

(2.1) $\ (\mathbb{R}^{2N+1},\circ,D_{\lambda}).$

If we denote the generic point of $\mathbb{R}^{2N+1}$ by $w=(z,l)=(x,y,l)$,
with $x,y\in\mathbb{R}^{N}$ and $l\in\mathbb{R}$, the composition law $\circ$
is defined by

$(x,y,l)\circ(x^{\prime},y^{\prime},l^{\prime})=(x+x^{\prime},y+y^{\prime},l+l^{\prime}+2(x^{\prime}\cdot
y-y^{\prime}\cdot x)),$

where $x\cdot y$ denotes the inner product in $\mathbb{R}^{N}$.

In (2.1), $D_{\lambda}$, $\lambda>0$ denotes the anisotropic dilation

$D_{\lambda}:\mathbb{R}^{2N+1}\longrightarrow\mathbb{R}^{2N+1},D_{\lambda}(z,l)=(\lambda
z,\lambda^{2}l).$

The family $(D_{\lambda})_{\lambda>0}$ is a group of automorphisms of
$\mathbb{H}^{N}$, that is,

$D_{\lambda}((z,l)\circ(z^{\prime},l^{\prime}))=(D_{\lambda}(z,l)\circ
D_{\lambda}(z^{\prime},l^{\prime})).$

The real number

$Q:=2N+2$

is called the homogeneous dimension of $\mathbb{H}^{N}$ since it appears in
the formula

$|D_{\lambda}(A)|=\lambda^{Q}|A|,$

where $A\subseteq\mathbb{R}^{2N+1}$ is a Lebesgue measurable set and $|A|$
stands for the Lebesgue measure of $A$.

A basis for the Lie algebra of left invariant vector fields on
$\mathbb{H}^{N}$ is given by

$X_{j}=\partial_{x_{j}}+2y_{j}\partial_{l},\quad
Y_{j}=\partial_{y_{j}}-2x_{j}\partial_{l},\quad j=1,\ldots,N.$

One easily calculates that

(2.2) $\displaystyle[X_{j},X_{k}]=[Y_{j},Y_{k}]=0\ \mbox{for every}\
j,k=1,\ldots,N,\ \mbox{and}\ [X_{j},Y_{k}]=-4\delta_{jk}\partial_{l}.$

These are the canonical commutation relations of Quantum Mechanics for
position and momentum, whence $\mathbb{H}^{N}$ is called the Heisenberg group.

The subelliptic gradient is the gradient taken with respect to the horizontal
directions $\nabla_{\mathbb{H}}:=(X_{1},\ldots,X_{N},Y_{1},\ldots,Y_{N})$ and
the sub-Laplacian on $\mathbb{H}^{N}$ is

$\varDelta_{\mathbb{H}}:=\sum_{j=1}^{N}(X_{j}^{2}+Y_{j}^{2})=\nabla_{\mathbb{H}}\cdot\nabla_{\mathbb{H}},$

and it can be explicitly also written as

$\varDelta_{\mathbb{H}}=\varDelta_{z}+4|z|^{2}\partial_{l}^{2}+4\partial_{l}T,$

where

$\varDelta_{z}=\sum_{j=1}^{N}(\partial_{x_{j}}^{2}+\partial_{y_{j}}^{2})$

and

$T=\sum_{j=1}^{N}(y_{j}\partial_{x_{j}}-x_{j}\partial_{y_{j}}).$

From (2.2) it immediately follows that

$\mathrm{rank\ }\mathrm{Lie\
}(X_{1},\ldots,X_{N},Y_{1},\ldots,Y_{N})(z,l)=2N+1$

at any point $(z,l)\in\mathbb{R}^{2N+1}.$ Then, by a celebrated theorem of
Hörmander, $\varDelta_{\mathbb{H}}$ is hypoelliptic, that is, every
distributional solution of $\varDelta_{\mathbb{H}}u=f$ is smooth whenever $f$
is smooth.

The operator $\varDelta_{\mathbb{H}}$ is left translation invariant on
$\mathbb{H}^{N}$ and $D_{\lambda}$-homogeneous of degree two. Moreover
$\varDelta_{\mathbb{H}}$ has a fundamental solution (with a pole at the
origin) given by

(2.3)
$\gamma(w)=c_{N}\left(\frac{1}{d(w)}\right)^{Q-2}\\!\\!\\!\\!\\!\\!=\,c_{N}\left(\frac{1}{d(w)}\right)^{2N}\\!\\!\\!\\!\\!\\!\\!,\
\quad w\neq(0,0),$

where

(2.4) $d(w)=(|z|^{4}+l^{2})^{\frac{1}{4}}\,\hbox{\ for
}w=(z,l)\in\mathbb{H}^{N}$

defines the metric $\rho(w,\tilde{w}):=d(\tilde{w}^{-1}\circ w)$ on
$\mathbb{H}^{N}$, and $\tilde{w}^{-1}$ denotes the inverse of $\tilde{w}$ in
the group $\mathbb{H}^{N}$.

In the following lemma we summarize some properties of $d$ and its gradient
$\nabla_{\mathbb{H}}$ which one can obtain by simple computations, see [4,
Proposition 5.4.3].

######  Lemma 2.1.

For $d(w)=(|z|^{4}+l^{2})^{\frac{1}{4}},\,w=(z,l)\in\mathbb{H}^{N}$, the
following hold:

$\displaystyle|\nabla_{\mathbb{H}}d(w)|^{2}$ $\displaystyle=$
$\displaystyle|z|^{2}(|z|^{4}+l^{2})^{-\frac{1}{2}},$
$\displaystyle\varDelta_{\mathbb{H}}d(w)$ $\displaystyle=$
$\displaystyle\frac{Q-1}{d(w)}|\nabla_{\mathbb{H}}d(w)|^{2},$ (2.5)
$\displaystyle-\varDelta_{\mathbb{H}}d^{-\alpha}(w)$ $\displaystyle=$
$\displaystyle Cd^{-\alpha}(w)\frac{|z|^{2}}{|z|^{4}+l^{2}}$

for $w\in\mathbb{H}^{N}\setminus\\{(0,0)\\}$, where
$C:=\alpha(Q-2-\alpha)=\alpha(2N-\alpha)$. So,
$\varDelta_{\mathbb{H}}d^{-\alpha}\in L^{1}_{loc}(\mathbb{H}^{N})$ if and only
if $2N-\alpha>0$.

It is known that the left translation invariance of $\varDelta_{\mathbb{H}}$
implies that the semigroup $e^{t\varDelta_{\mathbb{H}}}$ is given by a right
convolution

(2.6)
$e^{t\varDelta_{\mathbb{H}}}f(w)=\int_{\mathbb{H}^{N}}f(\tilde{w})p_{t}(\tilde{w}^{-1}\circ
w)\,d\tilde{w},\quad t>0,\,w\in\mathbb{H}^{N},$

where $(w,t)\mapsto p_{t}(w)$ is the fundamental solution of
$\left(\frac{\partial}{\partial t}+\varDelta_{\mathbb{H}}\right)u=0$. Hence,
by hypoellipticity, $p_{t}(w)$ is a $C^{\infty}$ function on
$\mathbb{H}^{N}\times(0,\infty)$ and $\|p_{t}\|_{1}=1$. Moreover, $p_{t}$
satisfies the following Gaussian estimates, cf. [22, Theorem IV.4.2 and
Theorem IV.4.3].

######  Theorem 2.2.

The heat kernel $p_{t}$ satisfies

$Ct^{-\frac{Q}{2}}\exp\left(-c\frac{d^{2}(w)}{t}\right)\leq p_{t}(w)\leq
C_{\varepsilon}t^{-\frac{Q}{2}}\exp\left(\frac{-d^{2}(w)}{4(1+\varepsilon)t}\right)$

for some positive constants $C,\,c,\,C_{\varepsilon}$, any
$\varepsilon>0,\,w\in\mathbb{H}^{N}$ and $t>0$.

##  3\. The main results

In this section we make the following hypotheses.

######  Hypotheses 3.1.

  * •

$0\leq V\in L^{1}_{loc}(\mathbb{H}^{N});$

  * •

$0\leq f\in L^{1}(\mathbb{H}^{N}\times(0,T));$

  * •

$0\leq u_{0}\in L^{1}(\mathbb{H}^{N})$ (or more generally $u_{0}$ can be a
positive finite Radon measure).

We consider the problem

(3.1) $\left\\{\begin{array}[]{ll}\dfrac{\partial u}{\partial
t}=\varDelta_{\mathbb{H}}u+Vu+f&\text{ in
}{\mathcal{D}}^{\prime}(\mathbb{H}^{N}\times(0,T)),\\\ {\rm esslim}_{t\to
0^{+}}\displaystyle\int_{\mathbb{H}^{N}}u(w,t)\psi(w)\,dw=\displaystyle\int_{\mathbb{H}^{N}}u_{0}(w)\psi(w)\,dw&\text{
for all }\psi\in{\mathcal{D}}(\mathbb{H}^{N}),\\\ u\geq 0&\text{ on
}\mathbb{H}^{N}\times(0,T),\\\ \\\ Vu\in
L^{1}_{loc}(\mathbb{H}^{N}\times(0,T)).\end{array}\right.$

Here ${\mathcal{D}}(\mathbb{H}^{N})=C_{c}^{\infty}(\mathbb{H}^{N})$ (resp.
$\mathcal{D}(\mathbb{H}^{N}\times(0,T))=C_{c}^{\infty}(\mathbb{H}^{N}\times(0,T))$)
with the usual topologies and
${\mathcal{D}}^{\prime}:={\mathcal{D}}^{\prime}(\mathbb{H}^{N})$ (resp.
$\mathcal{D}^{\prime}_{T}:=\mathcal{D}^{\prime}(\mathbb{H}^{N}\times(0,T))$)
is its dual space. We also consider the approximating problem

(3.2) $\left\\{\begin{array}[]{ll}\dfrac{\partial u_{n}}{\partial
t}=\varDelta_{\mathbb{H}}u_{n}+V_{n}u_{n}+f_{n}&\text{ in
}{\mathcal{D}}^{\prime}_{T},\\\ \\\ {\rm lim}_{t\to
0^{+}}\displaystyle\int_{\mathbb{H}^{N}}u_{n}(w,t)\psi(w)\,dw=\displaystyle\int_{\mathbb{H}^{N}}u_{0}(w)\psi(w)\,dw,&\mbox{
for all }\psi\in{\mathcal{D}}(\mathbb{H}^{N}).\\\ \end{array}\right.$

Here

$\displaystyle f_{n}=\min\\{f,n\\},$ $\displaystyle V_{n}\in
L^{\infty},\,0\leq V_{n}\leq V,\,V_{n}\uparrow V\,\,a.e.$

By the variation of parameters formula, (3.2) has a unique bounded nonnegative
solution obtained by solving the integral equation

(3.3)
$u_{n}(t)=e^{t\varDelta_{\mathbb{H}}}u_{0}+\int_{0}^{t}e^{(t-s)\varDelta_{\mathbb{H}}}V_{n}(\cdot)u_{n}(s)\,ds+\int_{0}^{t}e^{(t-s)\varDelta_{\mathbb{H}}}f_{n}(s)\,ds,$

where $\left(e^{t\varDelta_{\mathbb{H}}}\right)_{t\geq 0}$ is the semigroup
generated by $\varDelta_{\mathbb{H}}$ on $\mathbb{H}^{N}$. We note that
$V_{n}$ is a bounded multiplication operator on
$L^{p}\left(\mathbb{H}^{N}\right)$ for all $p\in[1,+\infty)$. Since
$\\{V_{n}\\}$ is an increasing sequence, clearly $\\{u_{n}\\}$ is an
increasing sequence, as well.

######  Proposition 3.2.

Suppose there is a $(w_{0},t_{0})\in\mathbb{H}^{N}\times(0,T)$ with
$\lim_{n\to\infty}u_{n}(w_{0},t_{0})<\infty$. Then (3.1) has a nonnegative
solution on $\mathbb{H}^{N}\times(0,T_{0})$ for all $0<T_{0}<t_{0}$ given by

(3.4) $u(w,t)=\lim_{n\to\infty}u_{n}(w,t)\quad a.e.\text{ in
}\mathbb{H}^{N}\times(0,T_{0}).$

Moreover, if (3.1) has a nonnegative solution in $\mathbb{H}^{N}\times(0,T)$,
then $\lim_{n\to\infty}u_{n}(w,t)<\infty$ a.e. in $\mathbb{H}^{N}\times(0,T)$.

###### Proof.

Clearly, if $u\geq 0$ is a solution of (3.1), then $u_{n}\leq u$ for all $n$,
so $\lim_{n\to\infty}u_{n}(w,t)\leq u(w,t)$ a.e. in
$\mathbb{H}^{N}\times(0,T)$. This establishes the last part of the
proposition.

For the main part, we start by considering

$U_{n}=e^{t}u_{n},\quad t>0.$

Then

$\frac{\partial U_{n}}{\partial
t}=\varDelta_{\mathbb{H}}U_{n}+(V_{n}+1)U_{n}+e^{t}f_{n},$

and, using the variation of parameters formula, we obtain

(3.5)
$e^{t_{0}}u_{n}(w_{0},t_{0})\geq\int_{0}^{t_{0}}e^{s}(e^{(t_{0}-s)\varDelta_{\mathbb{H}}}(V_{n}+1)u_{n}(s))(w_{0})\,ds,\quad(w_{0},t_{0})\in\mathbb{H}^{N}\times(0,T),$

since $e^{\varDelta_{\mathbb{H}}}u_{0}\geq 0$ and $f_{n}\geq 0.$ On the other
hand, it follows from the Gaussian estimates in Theorem 2.2 that

$\displaystyle\int_{0}^{t_{0}}e^{s}(e^{(t_{0}-s)\varDelta_{\mathbb{H}}}(V_{n}+1)u_{n}(s))(w_{0})\,ds$
$\displaystyle\geq$ $\displaystyle
C\int_{0}^{t_{0}}\int_{\mathbb{H}^{N}}e^{s}(V_{n}(\tilde{w})+1)u_{n}(\tilde{w},s)(t_{0}-s)^{-\frac{Q}{2}}\exp\left(-c\frac{d^{2}(\tilde{w}^{-1}\circ
w_{0})}{t_{0}-s}\right)d\tilde{w}ds.$

So if $\Omega^{\prime}\subset\subset\mathbb{H}^{N}$ and $\varepsilon\in(0,T)$,
it follows that, for $(w_{0},t_{0})\in\mathbb{H}^{N}\times(0,T)$, there is
$c_{0}>0$ such that

(3.6)
$c_{0}\int_{0}^{t_{0}-\varepsilon}\int_{\Omega^{\prime}}V_{n}(\tilde{w})u_{n}(\tilde{w},s)\,d\tilde{w}ds+c_{0}\int_{0}^{t_{0}-\varepsilon}\int_{\Omega^{\prime}}u_{n}(\tilde{w},s)\,d\tilde{w}ds\leq
e^{t_{0}}u_{n}(w_{0},t_{0}).$

By our hypothesis $u_{n}$ increases, moreover the right hand side of (3.6) is
clearly bounded, so by the monotone convergence theorem, $u_{n}\uparrow u$ and
$V_{n}u_{n}\uparrow Vu$ in
$L^{1}\left(\Omega^{\prime}\times(0,t_{0}-\varepsilon)\right)$ and $u$ is a
solution of (3.1) in the sense of distributions. ∎

######  Remark 3.3.

Notice that the solution of (3.1) satisfies the integral equation

$\displaystyle
u(w,t)=e^{t\varDelta_{\mathbb{H}}}u_{0}(w)+\int_{0}^{t}e^{(t-s)\varDelta_{\mathbb{H}}}V(w)u(w,s)\,ds$
$\displaystyle\quad+\int_{0}^{t}e^{(t-s)\varDelta_{\mathbb{H}}}f(w,s)\,ds,\qquad(w,t)\in\mathbb{H}^{N}\times(0,t_{0}).$

Also, since $u_{n}(w,t)\to u(w,t)<\infty$ a.e. on
$\mathbb{H}^{N}\times(0,t_{0})$, we get, using (3.5),
$s\longmapsto(e^{(t_{0}-s)\varDelta_{\mathbb{H}}}V(\cdot)u(\cdot,s))(w)\in
L^{1}(0,t_{0})$ for a.e. $w\in\mathbb{H}^{N}$.

The inverse square potential in the Euclidean case of $x\in\mathbb{R}^{N}$ is
$V^{*}_{c}(x)=\dfrac{c}{|x|^{2}}$ and the critical constant is the best
constant

$C_{*}(N)=\left(\frac{N-2}{2}\right)^{2}$

in Hardy’s inequality

$\int_{\mathbb{R}^{N}}|\nabla u(x)|^{2}\ dx\geq
C_{*}(N)\int_{\mathbb{R}^{N}}\dfrac{|u(x)|^{2}}{|x|^{2}}\ dx$

for $u\in C^{1}_{c}(\mathbb{R}^{N})$ if $N\geq 3$ and for $u\in
C^{1}_{c}(\mathbb{R}^{N}\setminus\\{0\\})$ for $N=1,2.$
The multiplication operator $V^{*}_{c}$ and the Laplacian both have the same
scaling property, namely

$U(\lambda)^{-1}\mathcal{L}U(\lambda)=\lambda^{2}\mathcal{L}$

for $\mathcal{L}=V_{c}^{*}$ or $\mathcal{L}=\varDelta$, where
$U(\lambda)f(x)=\lambda^{\frac{N}{2}}f(\lambda x)$, for $\lambda>0,$ defines a
unitary operator on $L^{2}(\mathbb{R}^{N})$.

In the case of the Heisenberg group $\mathbb{H}^{N}$, the corresponding
critical potential is

$\tilde{V}^{*}_{c}(w)=\dfrac{c|z|^{2}}{|z|^{2}+l^{2}}$

for $w=(x,y,l)=(z,l)\in\mathbb{H}^{N}$ and $c>0.$ The corresponding Hardy’s
inequality, due to Garofalo and Lanconelli ([11], see also [4], [15]), is

$\int_{\mathbb{H}^{N}}|\nabla_{\mathbb{H}^{N}}u(w)|^{2}\ dw\geq
C^{*}(N)\int_{\mathbb{H}^{N}}\tilde{V}^{*}_{1}(w)|u(w)|^{2}\ dw,$

with the best constant being $C^{*}(N)=N^{2}$, for all $N\geq 1.$ Both
$\varDelta_{\mathbb{H}^{N}}$ and multiplication by $\tilde{V}^{*}_{c}$ scale
in the same way. Let

$\tilde{U}(\lambda)f(z,l)=\lambda^{N+1}f(\lambda z,\lambda^{2}l);$

$\tilde{U}(\lambda)$ is unitary on $L^{2}(\mathbb{H}^{N})$ for all $\lambda>0$
and

$\tilde{U}(\lambda)^{-1}\mathcal{L}\tilde{U}(\lambda)=\lambda^{2}\mathcal{L}$

for $\mathcal{L}=\varDelta_{\mathbb{H}^{N}}$ or
$\mathcal{L}=\tilde{V}^{*}_{c}$, and all $\lambda>0.$

As in the Euclidean case, the critical potential is
$C^{*}(N)\tilde{V}^{*}_{1}=\tilde{V}^{*}_{C^{*}(N)}$ near the origin. That is,
by localizing to the unit ball $B_{1}$ in $\mathbb{H}^{N}$ (or to $B_{\rho}$
for any $\rho>0$), let

(3.7)
$V^{*}_{0}(w)=\left\\{\begin{array}[]{ll}\dfrac{c|z|^{2}}{|z|^{4}+l^{2}}&w\in
B_{1}\\\ \\\ 0&w\in\mathbb{H}^{N}\setminus B_{1},\end{array}\right.$

where $B_{1}$ is the unit ball centered at the origin in $\mathbb{H}^{N}$ with
respect to the metric $\rho(w,w^{\prime})=d(w^{\prime-1}\circ
w),\,w,\,w^{\prime}\in\mathbb{H}^{N}$.

Finally, notice that the smallest root of

$\alpha(Q-2-\alpha)=c$

is given by

$\displaystyle\alpha=\frac{Q-2}{2}-\sqrt{\left(\frac{Q-2}{2}\right)^{2}-c}=N-\sqrt{N^{2}-c},$

when $c\leq C^{*}(N)$.

The following theorem is the main result of this paper. It is an extension of
[15, Theorem 1.1] and a generalization of [3].

######  Theorem 3.4.

  * (i)

Let $0\leq c\leq C^{*}(N)$ and let $V\geq 0$ be a measurable potential
satisfying $V\in L^{\infty}(\mathbb{H}^{N}\setminus B_{1})$. Let $0\leq f\in
L^{1}(\mathbb{H}^{N}\times(0,T))$. If $V\leq V^{*}_{0}$ in $B_{1}$, then (3.1)
has a solution if

(3.8)
$\int_{\mathbb{H}^{N}}d(w)^{-\alpha}u_{0}(w)\,dw<\infty\,,\quad\int_{0}^{T}\int_{\mathbb{H}^{N}}f(w,s)d(w)^{-\alpha}dwds<\infty,$

where $\alpha$ is the smallest root of $\alpha(2N-\alpha)=c$. If $V\geq
V^{*}_{0}$ in $B_{1}$, and if (3.1) has a solution u, then

$\int_{\mathbb{H}^{N}}d(w)^{-\alpha}u_{0}(w)\,dw<\infty,\quad\int_{0}^{T-\varepsilon}\int_{\Omega^{\prime}}f(w,s)d(w)^{-\alpha}dwds<\infty$

for each $\varepsilon\in(0,T)$ and each
$\Omega^{\prime}\subset\subset\mathbb{H}^{N}$ with $\alpha$ as above. If
either $u_{0}\not\equiv 0$ or $f\not\equiv 0$ in
$\mathbb{H}^{N}\times(0,\varepsilon)$ for each $\varepsilon\in(0,T)$, then
given $\Omega^{\prime}\subset\subset\mathbb{H}^{N}$ with
$0\in\Omega^{\prime}$, there is a constant
$C=C(\varepsilon,\Omega^{\prime})>0$ such that

(3.9)
$u(w,t)\geq\frac{C}{d^{\alpha}(\omega)},\quad(w,t)\in\Omega^{\prime}\times[\varepsilon,T].$

  * (ii)

If $c>C^{*}(N)$, $V\geq V^{*}_{0}$ and either $u_{0}\not\equiv 0$ or
$f\not\equiv 0$, then (3.1) does not have a positive solution. Moreover, we
have instantaneous blowup.

###### Proof.

(i): We can consider $V\leq V^{*}_{0}\hbox{\ in }\mathbb{H}^{N}$. Otherwise
consider $V=\tilde{V}+B=:V\chi_{B_{1}}+V\chi_{B^{c}_{1}}$ with $\tilde{V}\leq
V^{*}_{0}$ in $\mathbb{H}^{N}$, $B\in L^{\infty}(\mathbb{H}^{N})$ and use
Proposition A.5 in the Appendix.

Let $\phi(w):=d(w)^{-\alpha}$, and choose a convex function $\rho\in
C^{2}(\mathbb{R})$ with $\rho(0)=\rho^{\prime}(0)=0.$ Next, multiply (3.2) by
$\rho^{\prime}(u_{n})\phi$ and integrate over $\mathbb{H}^{N}\times[\delta,t)$
for $0<\delta<t<T.$ Then, letting $\int$ denote $\int_{\mathbb{H}^{N}}$,

$\displaystyle\int_{\delta}^{t}\int\frac{\partial u_{n}}{\partial
s}\rho^{\prime}(u_{n})\phi=\int_{\delta}^{t}\int\varDelta_{\mathbb{H}}u_{n}\rho^{\prime}(u_{n})\phi+\int_{\delta}^{t}\int(V_{n}u_{n}+f_{n})\rho^{\prime}(u_{n})\phi,$

and so

$\displaystyle\int\int_{\delta}^{t}\frac{\partial}{\partial
s}(\rho(u_{n}))\phi=-\int_{\delta}^{t}\int\nabla_{\mathbb{H}}u_{n}\cdot\nabla_{\mathbb{H}}(\rho^{\prime}(u_{n})\phi)+\int_{\delta}^{t}\int(V_{n}u_{n}+f_{n})\rho^{\prime}(u_{n})\phi.$

Then,

$\displaystyle\int\rho(u_{n}(t))\phi$
$\displaystyle=-\int_{\delta}^{t}\int\rho^{\prime\prime}(u_{n})|\nabla_{\mathbb{H}}u_{n}|^{2}\phi+(\nabla_{\mathbb{H}}u_{n}\cdot\nabla_{\mathbb{H}}\phi)\rho^{\prime}(u_{n})$
$\displaystyle\qquad+\int_{\delta}^{t}\int(V_{n}u_{n}+f_{n})\rho^{\prime}(u_{n})\phi+\int\rho(u_{n}(\delta))\phi$
$\displaystyle=\int_{\delta}^{t}\int-\rho^{\prime\prime}(u_{n})|\nabla_{\mathbb{H}}u_{n}|^{2}\phi+\int_{\delta}^{t}\int\rho(u_{n})\varDelta_{\mathbb{H}}\phi$
$\displaystyle\qquad+\int_{\delta}^{t}\int(V_{n}u_{n}+f_{n})\rho^{\prime}(u_{n})\phi\,+\int\rho(u_{n}(\delta))\phi,$

since
$\rho^{\prime}(u_{n})\nabla_{\mathbb{H}}u_{n}=\nabla_{\mathbb{H}}(\rho(u_{n})).$
Hence

(3.10)
$\int\rho(u_{n}(t))\phi\leq\int_{\delta}^{t}\int\rho(u_{n})\varDelta_{\mathbb{H}}\phi+\int_{\delta}^{t}\int(V_{n}u_{n}+f_{n})\rho^{\prime}(u_{n})\phi+\int\rho(u_{n}(\delta))\phi.$

Replace $\rho$ in (3.10) with the convex function
$\rho_{\varepsilon}(r)=\sqrt{r^{2}+\varepsilon^{2}}-\varepsilon^{2},\,r\geq
0,$ and let $\varepsilon\to 0$ to obtain, by the monotone convergence theorem,

(3.11) $\int u_{n}(t)\phi\leq\int_{\delta}^{t}\int
u_{n}\varDelta_{\mathbb{H}}\phi+\int_{\delta}^{t}\int(V_{n}u_{n}+f_{n})\phi+\int
u_{n}(\delta)\phi.$

Next we want to let $\delta\to 0.$ Notice that

(3.12) $e^{\delta\varDelta_{\mathbb{H}}}u_{0}\leq
u_{n}(\delta)=e^{\delta(\varDelta_{\mathbb{H}}+V_{n})}u_{0}+\int_{0}^{\delta}e^{(\delta-s)(\varDelta_{\mathbb{H}}+V_{n})}f_{n}(s)ds.$

Since $\|V_{n}\|_{\infty}=:c_{n}<\infty,$ it follows from the
Daletskii–Trotter product formula that

$\displaystyle
e^{\delta(\varDelta_{\mathbb{H}}+V_{n})}u_{0}=\lim_{m\to\infty}\left(e^{\delta\varDelta_{\mathbb{H}}/m}e^{\frac{\delta}{m}V_{n}}\right)^{m}u_{0}$
$\displaystyle\quad\leq e^{\delta
c_{n}}e^{\delta\varDelta_{\mathbb{H}}}u_{0},$

by the positivity of the semigroup $\\{e^{\delta\varDelta_{\mathbb{H}}}\\}.$
So (3.12) becomes

$e^{\delta\varDelta_{\mathbb{H}}}u_{0}\leq u_{n}(\delta)\leq e^{\delta
c_{n}}e^{\delta\varDelta_{\mathbb{H}}}u_{0}+\int_{0}^{\delta}e^{c_{n}(\delta-s)}e^{(\delta-s)\varDelta_{\mathbb{H}}}f_{n}(s)ds,$

and by the contractivity of $e^{t\varDelta_{\mathbb{H}}}$ we have

$\displaystyle\int\left(e^{\delta\varDelta_{\mathbb{H}}}u_{0}\right)\phi\leq\int
u_{n}(\delta)\phi\leq e^{\delta
c_{n}}\int\left(e^{\delta\varDelta_{\mathbb{H}}}u_{0}\right)\phi+e^{\delta
c_{n}}\|f_{n}\|_{\infty}\delta\int\phi.$

The strong continuity of the semigroup implies

$\lim_{\delta\to
0}\int\left(e^{\delta\varDelta_{\mathbb{H}}}u_{0}\right)\phi=\int\phi u_{0}.$

Thus we have shown that

$\lim_{\delta\to 0}\int u_{n}(\delta)\phi=\int\phi u_{0}.$

Now let $\delta\to 0$ in $\eqref{eq:16}$, using (2.1), to deduce

$\displaystyle\int u_{n}(t)\phi\leq\int_{0}^{t}\int
u_{n}\varDelta_{\mathbb{H}}\phi+\int_{0}^{t}\int
V_{n}u_{n}\phi+\int_{0}^{t}\int f_{n}\phi+\int u_{0}\phi$
$\displaystyle\quad=\int_{0}^{t}\int\left(-c\frac{|z|^{2}}{|z|^{4}+l^{2}}+V_{n}\right)u_{n}\phi+\int_{0}^{t}\int
f_{n}\phi+\int u_{0}\phi$ $\displaystyle\qquad\leq\int_{0}^{t}\int
f_{n}\phi+\int u_{0}\phi,$

since $V_{n}\leq V^{*}_{0}$. It follows that if $\int_{0}^{t}\int
f\phi+\int\phi u_{0}<\infty$, then, by Proposition 3.2, $u_{n}(w,t)\uparrow
u(w,t)(<+\infty)$ as $n\to\infty$ for all $t\in(0,T]$ and a.e.
$w\in\mathbb{H}^{N}$, which gives the first part of (i) of the theorem.

Let us now prove the second part of (i). The inequality (3.9) is proved in
Lemma 3.5 below. On the other hand, by the first part of (i) we have that, for
each $w\in\mathbb{H}^{N}\setminus\\{0\\},$ (3.1) has a solution with
$u_{0}=\phi^{-1}(w)\delta_{w}$, $f\equiv 0$ and $V=V^{*}_{0}$, where
$\delta_{w}$ denotes the Dirac measure at $w$. We denote this solution by
$u_{w}$. We define

$h_{w}(\tilde{w},t)=u_{w}(\tilde{w},t)\phi(\tilde{w})^{-1},\quad(\tilde{w},t)\in\mathbb{H}^{N}\times(0,T],$

and set $h=u\phi^{-1}$ and $h_{n}=u_{n}\phi^{-1}$ with $u$ (respectively
$u_{n}$) the solution of (3.1) (respectively (3.2)) obtained by Proposition
3.2.

We now prove

(3.13)
$h(w,t)\geq\int_{\mathbb{H}^{N}}h_{w}(\tilde{w},t)\phi(\tilde{w})u_{0}(\tilde{w})\,d\tilde{w}+\int_{0}^{t}\int_{\mathbb{H}^{N}}h_{w}(\tilde{w},t-s)f(\tilde{w},s)\phi(\tilde{w})\,d\tilde{w}\,ds$

holds for $w\in\mathbb{H}^{N}\setminus\\{0\\}$ and $t\in(0,T]$.

To this end let $u_{n}$ be the solution of (3.2), and let $v_{n}$ be the
solution of

$\left\\{\begin{array}[]{l}\dfrac{\partial v_{n}}{\partial
t}=\varDelta_{\mathbb{H}}v_{n}+V_{0,n}v_{n}\,,\\\ \\\
v_{n}(0)=\phi(w)^{-1}\delta_{w},\end{array}\right.$

where $V_{0,n}=\min\left\\{V^{*}_{0},n\right\\}$. Note that by the above
construction, $v_{n}(\tilde{w},t)\uparrow u_{w}(\tilde{w},t)$ as $n\to\infty$
for all $t\in(0,T]$ and a.e. $\tilde{w}\in\mathbb{H}^{N}$.

On the other hand, we have

$\displaystyle\frac{\partial}{\partial
s}\int_{\mathbb{H}^{N}}u_{n}(\tilde{w},s)v_{n}(\tilde{w},t-s)\,d\tilde{w}=\int_{\mathbb{H}^{N}}\left[\frac{\partial
u_{n}}{\partial
s}(\tilde{w},s)v_{n}(\tilde{w},t-s)-u_{n}(\tilde{w},s)\frac{\partial
v_{n}}{\partial s}(\tilde{w},t-s)\right]d\tilde{w}$
$\displaystyle\quad=\int_{\mathbb{H}^{N}}\left[v_{n}(\tilde{w},t-s)\varDelta_{\mathbb{H}}u_{n}(\tilde{w},s)-\varDelta_{\mathbb{H}}v_{n}(\tilde{w},t-s)u_{n}(\tilde{w},s)+f_{n}(\tilde{w},s)v_{n}(\tilde{w},t-s)\right]d\tilde{w}$
$\displaystyle\qquad+\int_{\mathbb{H}^{N}}\left(V_{n}-V_{0,n}\right)u_{n}(\tilde{w},s)v_{n}(\tilde{w},t-s)\,d\tilde{w}$
$\displaystyle\quad\geq\int_{\mathbb{H}^{N}}f_{n}(\tilde{w},s)v_{n}(\tilde{w},t-s)\,d\tilde{w}.$

Hence integrating from $\delta$ to $t-\delta$ yields

(3.14)
$\begin{array}[]{lll}\displaystyle\int_{\mathbb{H}^{N}}u_{n}(\tilde{w},t-\delta)v_{n}(\tilde{w},\delta)\,d\tilde{w}\geq\\\
\\\
\displaystyle\int_{\delta}^{t-\delta}\displaystyle\int_{\mathbb{H}^{N}}f_{n}(\tilde{w},s)v_{n}(\tilde{w},t-s)\,d\tilde{w}\,ds+\int_{\mathbb{H}^{N}}u_{n}(\tilde{w},\delta)v_{n}(\tilde{w},t-\delta)\,d\tilde{w}.\end{array}$

Letting $\delta\to 0$ in (3.14) and noting that, as $\delta\to 0$,
$u_{n}(t-\delta)\to u_{n}(t)$ weakly, $v_{n}(t-\delta)\to v_{n}(t)$ weakly,
$u_{n}(\delta)\to u_{0}$ weakly and $v_{n}(\delta)\to\phi(w)^{-1}\delta_{w}$
weakly, we get

(3.15)
$u_{n}(w,t)\phi^{-1}(w)\geq\int_{0}^{t}\int_{\mathbb{H}^{N}}f_{n}(\tilde{w},s)v_{n}(\tilde{w},t-s)\,d\tilde{w}\,ds+\int_{\mathbb{H}^{N}}v_{n}(\tilde{w},t)u_{0}(\tilde{w})\,d\tilde{w}.$

Letting $n\to\infty$ in (3.15) and noting that $u_{n}(w,t)\uparrow
u(w,t)=h(w,t)\phi(w)$ and $v_{n}(\tilde{w},t)\uparrow
u_{w}(\tilde{w},t)=h_{w}(\tilde{w},t)\phi(\tilde{w})$, we obtain (3.13).

Applying (3.9) to $u_{w}$ for a fixed $w\in\mathbb{H}^{N}\setminus\\{0\\}$, we
obtain that there exists a constant $C>0$ such that

$h_{w}(\tilde{w},t)\geq C\text{ for
}(\tilde{w},t)\in\Omega^{\prime}\times[\varepsilon,T].$

It follows from (3.13) that

$h(w,t)\geq
C\int_{\Omega^{\prime}}\phi(\tilde{w})u_{0}(\tilde{w})\,d\tilde{w}+C\int_{0}^{T-\varepsilon}\int_{\Omega^{\prime}}f(\tilde{w},s)\phi(\tilde{w})\,d\tilde{w}ds.$

If a solution $u$ exists, we must have $h(w,t)<\infty$ for a.e.
$w\in\mathbb{H}^{N}$ and all $t\in(0,T]$. Thus, necessary conditions for the
existence of a solution $u$ are

$\int_{\Omega^{\prime}}\phi(w)u_{0}(w)\,dw<\infty,\text{ and
}\int_{0}^{T-\varepsilon}\int_{\Omega^{\prime}}f(w,s)\phi(w)\,dw\,ds<\infty.$

This completes the proof of (i).

(ii): Let $c>C^{*}(N)$ and let $u\not\equiv 0$ a solution of (3.1). Then,

$\frac{\partial u}{\partial
t}-\varDelta_{\mathbb{H}}u=C^{*}(N)\frac{|z|^{2}}{|z|^{4}+l^{2}}u+(c-C^{*}(N))\frac{|z|^{2}}{|z|^{4}+l^{2}}u.$

From part $(i)$, a solution exists only if

$(c-C^{*}(N))\frac{|z|^{2}}{|z|^{4}+l^{2}}u\phi\in
L^{1}(\Omega^{\prime}\times(0,T-\varepsilon)),$

for $\Omega^{\prime}$ any compact set in $\mathbb{H}^{N}$ and $\varepsilon>0$.
(Here we have assumed $0\in\Omega^{\prime}$.)
But by the preceding proof (see (3.9) with $\alpha=N$), we have

$u\geq C_{\varepsilon}d^{-N}(w),$

in $\Omega^{\prime}\times[\varepsilon,T)$, and so we would need
$\frac{|z|^{2}}{|z|^{4}+l^{2}}d^{-N}\in L^{1}(\Omega^{\prime})$, which is
false.

∎

######  Lemma 3.5.

Assume $0\leq c\leq C^{*}(N),\,\alpha$ the smallest root of
$\alpha(2N-\alpha)=c$, and $0\leq V\in L^{\infty}(\mathbb{H}^{N}\setminus
B_{1})$ with $V\geq V^{*}_{0}$ in $B_{1}$. If $u$ is a solution of (3.1) with
$u_{0}\not\equiv 0$ in $\mathbb{H}^{N}$, then given
$\Omega^{\prime}\subset\subset\mathbb{H}^{N},\,\varepsilon\in(0,T)$, there is
$C=C(\varepsilon,\Omega^{\prime})>0$ such that (3.9) holds.

###### Proof.

Assume that $\Omega^{\prime}\subset\subset\mathbb{H}^{N}$ with
$0\in\Omega^{\prime}$. Since $u_{0}\not\equiv 0$, it follows from Theorem 2.2
that there is a constant $C_{0}>0$ with

(3.16) $e^{t\varDelta_{\mathbb{H}}}u_{0}(\tilde{w})\geq C_{0},$

for $\tilde{w}\in\Omega^{\prime}$ and $\frac{\varepsilon}{2}\leq t<T$. Since
$u\geq e^{t\varDelta_{\mathbb{H}}}u_{0}$, by the Maximum Principle, (3.9)
follows from (3.16) for the case $\alpha=0$. So from now on we assume that
$\alpha$ is strictly positive.

Let as before $V_{0,n}:=\inf\left\\{V^{*}_{0},n\right\\}$, and consider the
problems

(3.17) $\left\\{\begin{array}[]{ll}\dfrac{\partial z}{\partial
t}=\varDelta_{\mathbb{H}}z+V^{*}_{0}z&\hbox{\ in
}\mathcal{D}^{\prime}\left(\mathbb{H}^{N}\times[\frac{\varepsilon}{2},T]\right),\\\
\\\
z\left(\tilde{w},\frac{\varepsilon}{2}\right)=C_{0}\,\chi_{\Omega^{\prime}}(\tilde{w})&\tilde{w}\in\mathbb{H}^{N},\end{array}\right.$

(3.18) $\left\\{\begin{array}[]{ll}\dfrac{\partial z_{n}}{\partial
t}=\varDelta_{\mathbb{H}}z_{n}+V_{0,n}z_{n}&\hbox{\ in
}\mathcal{D}^{\prime}\left(\mathbb{H}^{N}\times[\frac{\varepsilon}{2},T]\right),\\\
\\\
z_{n}\left(\tilde{w},\frac{\varepsilon}{2}\right)=C_{0}\,\chi_{\Omega^{\prime}}(\tilde{w})&\tilde{w}\in\mathbb{H}^{N},\end{array}\right.$

and for $B_{r_{0}}\subset\Omega^{\prime}$ a ball centered at the origin with
radius $r_{0}\in(0,1),$

(3.19) $\left\\{\begin{array}[]{lll}\dfrac{\partial v_{n}}{\partial
t}=\varDelta_{\mathbb{H}}v_{n}+V_{0,n}v_{n}&\hbox{\ in
}\mathcal{D}^{\prime}\left(B_{r_{0}}\times[\frac{\varepsilon}{2},T]\right),\\\
\\\ v_{n}=0&\text{ on }\partial B_{r_{0}},\\\ \\\
v_{n}\left(\tilde{w},\frac{\varepsilon}{2}\right)=C_{0}&\tilde{w}\in
B_{r_{0}}.\end{array}\right.$

Notice that (3.18) has a unique solution $z_{n}\geq 0$, also that
$z_{n}(\tilde{w},t)\uparrow z(\tilde{w},t)$, for almost every
$(\tilde{w},t)\in\mathbb{H}^{N}\times[\frac{\varepsilon}{2},T)$, where $z$ is
the unique solution of (3.17). It is also clear that $z_{n}\geq v_{n}$, the
solution of (3.19), and that $v_{n}$ is a radial function111 Recall that a
function $g(w)$ is radial on $\mathbb{H}^{n}$ if $w=(z,l)$ and
$g(z,l)=g(|z|,l)$. In fact, our function $v_{n}$ is even more special, since
$v_{n}=v_{n}(d(w)).$ Notice that this gives
$\nabla_{\mathbb{H}}v_{n}=v^{\prime}_{n}(d(w))\nabla_{\mathbb{H}}d(w)$. .
Finally, we note that $u$ is bounded below by the solution of (3.17) since
$V\geq V^{*}_{0}$.

Multiply the equation in $\eqref{eq:23}$ by $v_{n}^{p-1}\phi^{2-p},\,p\geq 2,$
where we recall that $\phi(w)=d(w)^{-\alpha}$ only depends on $w$, and
integrate to get

$\int_{B_{r_{0}}}\frac{\partial v_{n}}{\partial
t}v_{n}^{p-1}\phi^{2-p}=\int_{B_{r_{0}}}(\varDelta_{\mathbb{H}}v_{n})v_{n}^{p-1}\phi^{2-p}+\int_{B_{r_{0}}}V_{0,n}v_{n}^{p}\phi^{2-p},$

so

(3.20) $\frac{\partial}{\partial
t}\int_{B_{r_{0}}}\frac{1}{p}\left(\frac{v_{n}}{\phi}\right)^{p}\phi^{2}=-\int_{B_{r_{0}}}\nabla_{\mathbb{H}}v_{n}\cdot\nabla_{\mathbb{H}}\left(v_{n}^{p-1}\phi^{2-p}\right)+\int_{B_{r_{0}}}V_{0,n}\left(\frac{v_{n}}{\phi}\right)^{p}\phi^{2}.$

Set $g_{n}=\frac{v_{n}}{\phi}$. Then equation (3.20) becomes

$\displaystyle\frac{\partial}{\partial
t}\int_{B_{r_{0}}}\frac{1}{p}g_{n}^{p}\phi^{2}=-\frac{4(p-1)}{p^{2}}\int_{B_{r_{0}}}|\nabla_{\mathbb{H}}g_{n}^{p/2}|^{2}\phi^{2}+\int_{B_{r_{0}}}g_{n}^{p}\left(\varDelta_{\mathbb{H}}\phi\right)\phi+\int_{B_{r_{0}}}V_{0,n}g_{n}^{p}\phi^{2}.$

Using (2.1) and the fact that $\alpha(2N-\alpha)=c$, we obtain $V_{0,n}\leq
V^{*}_{0}\leq-\frac{\varDelta_{\mathbb{H}}\phi}{\phi}$, so that
$V_{0,n}\phi^{2}\leq\left(-\varDelta_{\mathbb{H}}\phi\right)\phi$. Hence we
have shown

$\frac{\partial}{\partial t}\int_{B_{r_{0}}}g_{n}^{p}\phi^{2}\leq 0,$

and we thus have, for $\frac{\varepsilon}{2}\leq t\leq T$, that

(3.21) $\left(\int_{B_{r_{0}}}v_{n}^{p}\phi^{2-p}\right)^{1/p}\leq
C_{0}\left(\int_{B_{r_{0}}}\phi^{2-p}\right)^{1/p}.$

Letting $p\to\infty$ in (3.21) we get

(3.22) $g_{n}\leq C_{0}\hbox{\ a.e. in }B_{r_{0}},$

which is equivalent to $v_{n}\leq C_{0}\phi$ a.e. in $B_{r_{0}}$. So we can
make sense of

$v=\lim_{n\to\infty}v_{n}\text{ and }g=\lim_{n\to\infty}g_{n}.$

Now we claim that

(3.23) $0<C_{1}\leq g(w,t)\leq C_{0}$

for $t\in[\varepsilon,T]$ and a.e. $w\in B_{\frac{r_{0}}{2}}$. Once (3.23) is
proved, and since

$u\geq z\geq z_{n}\geq v_{n}=g_{n}\phi,$

(3.9) follows directly in the case $\Omega^{\prime}=B_{\frac{r_{0}}{2}}$.
Otherwise, we observe that for almost every
$\tilde{w}\in\Omega^{\prime}\setminus B_{\frac{r_{0}}{2}}$ we have

$h(\tilde{w},t)=\phi(\tilde{w})^{-1}u(\tilde{w})\geq\phi(\tilde{w})^{-1}\left(e^{t\varDelta_{\mathbb{H}}}u_{0}\right)(\tilde{w})\geq
C_{2}>0,$

from Theorem 2.2 since

$\phi(\tilde{w})^{-1}\geq C_{3}>0,$

for all $t\in[\varepsilon,T]$ and some constants $C_{2},C_{3}>0$. This
concludes the proof of (3.9).

Now we must prove (3.23). By (3.22), the right inequality is proved. For the
remaining part of (3.23), let $\mathcal{I}\in
C^{2}(\mathbb{R}_{+},\mathbb{R}_{+})$ be convex. Multiply equation (3.19) by
${\mathcal{I}}^{\prime}(g_{n}){\mathcal{I}}(g_{n})\phi\psi^{2}$ and integrate
over
$Q=B_{r_{0}}\times\left(\frac{\varepsilon}{2},t\right),\,t\in[\frac{\varepsilon}{2},T]$,
where $\psi\in{\mathcal{D}}(B_{r_{0}}\times(\frac{\varepsilon}{2},T])$, to get

$\int_{Q}\mathcal{I}^{\prime}(g_{n})\mathcal{I}(g_{n})\frac{\partial
v_{n}}{\partial
t}\phi\psi^{2}=\int_{Q}\left\\{\varDelta_{\mathbb{H}}v_{n}\mathcal{I}^{\prime}(g_{n})\mathcal{I}(g_{n})\phi\psi^{2}+V_{0,n}v_{n}\mathcal{I}^{\prime}(g_{n})\mathcal{I}(g_{n})\phi\psi^{2}\right\\},$
$\frac{1}{2}\int_{Q}\frac{\partial}{\partial
t}(\mathcal{I}(g_{n}))^{2}\phi^{2}\psi^{2}=-\int_{Q}\nabla_{\mathbb{H}}(g_{n}\phi)\cdot\nabla_{\mathbb{H}}(\mathcal{I}^{\prime}(g_{n})\mathcal{I}(g_{n})\phi\psi^{2})+\int_{Q}V_{0,n}g_{n}\mathcal{I}^{\prime}(g_{n})\mathcal{I}(g_{n})\phi^{2}\psi^{2}.$

Notice that

$\displaystyle\int_{B_{r_{0}}}\nabla_{\mathbb{H}}(g_{n}\phi)\cdot\nabla_{\mathbb{H}}(\mathcal{I}^{\prime}(g_{n})\mathcal{I}(g_{n})\phi\psi^{2})$
$\displaystyle\quad=\int_{B_{r_{0}}}\left\\{(\nabla_{\mathbb{H}}g_{n}\cdot\nabla_{\mathbb{H}}(\mathcal{I}^{\prime}(g_{n}))\mathcal{I}(g_{n})\phi\psi^{2})\phi+g_{n}(\nabla_{\mathbb{H}}\phi\cdot\nabla_{\mathbb{H}}(\mathcal{I}^{\prime}(g_{n}))\mathcal{I}(g_{n})\phi\psi^{2})\right\\}$
$\displaystyle\quad=\int_{B_{r_{0}}}\left\\{\mathcal{I}^{\prime\prime}(g_{n})|\nabla_{\mathbb{H}}g_{n}|^{2}\mathcal{I}(g_{n})\phi^{2}\psi^{2}+|\nabla_{\mathbb{H}}\mathcal{I}(g_{n})|^{2}\phi^{2}\psi^{2}\right\\}$
$\displaystyle\qquad+\int_{B_{r_{0}}}\left\\{(\nabla_{\mathbb{H}}\mathcal{I}(g_{n})\cdot\nabla_{\mathbb{H}}\phi)\psi^{2}\phi\mathcal{I}(g_{n})+(\nabla_{\mathbb{H}}\mathcal{I}(g_{n})\cdot\nabla_{\mathbb{H}}\psi^{2})\mathcal{I}(g_{n})\phi^{2}\right\\}$
$\displaystyle\qquad+\int_{B_{r_{0}}}(-\varDelta_{\mathbb{H}}\phi)g_{n}\mathcal{I}^{\prime}(g_{n})\mathcal{I}(g_{n})\phi\psi^{2}+\int_{B_{r_{0}}}-\left(\nabla_{\mathbb{H}}\phi\cdot\nabla_{\mathbb{H}}\mathcal{I}(g_{n})\right)\mathcal{I}(g_{n})\phi\psi^{2}$
$\displaystyle\quad=\int_{B_{r_{0}}}\left\\{\mathcal{I}^{\prime\prime}(g_{n})|\nabla_{\mathbb{H}}g_{n}|^{2}\mathcal{I}(g_{n})\phi^{2}\psi^{2}+|\nabla_{\mathbb{H}}\mathcal{I}(g_{n})|^{2}\phi^{2}\psi^{2}\right\\}$
$\displaystyle\qquad+\int_{B_{r_{0}}}(\nabla_{\mathbb{H}}\mathcal{I}(g_{n})\cdot\nabla_{\mathbb{H}}\psi^{2})\mathcal{I}(g_{n})\phi^{2}+\int_{B_{r_{0}}}(-\varDelta_{\mathbb{H}}\phi)g_{n}\mathcal{I}^{\prime}(g_{n})\mathcal{I}(g_{n})\phi\psi^{2},$

and so

$\displaystyle\frac{1}{2}\int_{Q}\frac{\partial}{\partial
t}\left(\mathcal{I}(g_{n})^{2}\right)\phi^{2}\psi^{2}+\int_{Q}\nabla_{\mathbb{H}}\mathcal{I}(g_{n})\cdot\nabla_{\mathbb{H}}\psi^{2}\mathcal{I}(g_{n})\phi^{2}$
$\displaystyle\quad=-\int_{Q}\mathcal{I}^{\prime\prime}(g_{n})|\nabla_{\mathbb{H}}g_{n}|^{2}\left(\mathcal{I}(g_{n})\phi^{2}\psi^{2}\right)$
(3.24)
$\displaystyle\qquad-\int_{Q}|\nabla_{\mathbb{H}}\mathcal{I}(g_{n})|^{2}\phi^{2}\psi^{2}+\int\varDelta_{\mathbb{H}}\phi
g_{n}\mathcal{I}^{\prime}(g_{n})\mathcal{I}(g_{n})\phi\psi^{2}$
$\displaystyle\qquad+\int_{Q}V_{0,n}g_{n}\mathcal{I}^{\prime}(g_{n})\mathcal{I}(g_{n})\phi^{2}\psi^{2}.$

Using Hölder’s inequality,

$\displaystyle\left|2\int_{B_{r_{0}}}\left(\nabla\mathcal{I}(g_{n})\cdot\nabla_{\mathbb{H}}\psi\right)\mathcal{I}(g_{n})\phi^{2}\psi\right|$
$\displaystyle\quad\leq\frac{1}{2}\int_{B_{r_{0}}}|\nabla_{\mathbb{H}}\mathcal{I}(g_{n})|^{2}\phi^{2}\psi^{2}+2\int_{B_{r_{0}}}|\nabla_{\mathbb{H}}\psi|^{2}|\mathcal{I}(g_{n})|^{2}\phi^{2},$

on the second term of the left hand side of (3), using the convexity
assumption on $\mathcal{I}$, and integrating by parts on the first term in
(3), we obtain

$\displaystyle\frac{1}{2}\int_{B_{r_{0}}}\left(\mathcal{I}(g_{n})^{2}\psi^{2}\phi^{2}\right)(t)+\frac{1}{2}\int_{Q}|\nabla_{\mathbb{H}}\mathcal{I}(g_{n})|^{2}\phi^{2}\psi^{2}$
(3.25)
$\displaystyle\quad\leq\int_{Q}(V_{0,n}\phi+\varDelta_{\mathbb{H}}\phi)g_{n}\mathcal{I}^{\prime}(g_{n})\mathcal{I}(g_{n})\phi\psi^{2}$
$\displaystyle\qquad+\int_{Q}\mathcal{I}(g_{n})^{2}(2|\nabla_{\mathbb{H}}\psi|^{2}+\psi\frac{\partial\psi}{\partial
t})\phi^{2}.$

Now, we make a key observation. Since $\alpha<N$, we have
$\phi\varDelta_{\mathbb{H}}\phi\in L^{1}(B_{r_{0}})$. Assume $r_{0}$ to be
sufficiently small. Since
$V^{*}_{0}=\frac{-\varDelta_{\mathbb{H}}\phi}{\phi}$, the first term on the
right hand side of (3) converges to $0$ as $n\to\infty$ by Lebesgue’s
dominated convergence theorem, since $\|g_{n}\|\leq C_{0}$ in $B_{r_{0}}$ and
$\mathcal{I}$ is convex, $C^{2}$ and nonnegative. Letting $n\to\infty$, (3)
gives

$\displaystyle\int_{B_{r_{0}}}\mathcal{I}(g)^{2}\psi^{2}\phi^{2}+\int_{Q}|\nabla_{\mathbb{H}}\mathcal{I}(g)|^{2}\psi^{2}\phi^{2}$
(3.26) $\displaystyle\qquad\leq
2\int_{Q}\mathcal{I}(g)^{2}(2|\nabla_{\mathbb{H}}\psi|^{2}+\psi\frac{\partial\psi}{\partial
t})\phi^{2}.$

Choose $\psi$ so that $0\leq\psi\leq 1$

for $s\geq\frac{\varepsilon}{2},\,r<r_{0}$ and $0<\delta<r$, and

$\psi(w,t)=\left\\{\begin{array}[]{l}1\qquad
B_{r-\delta}\times[s+\delta,T],\\\
0\qquad\left(B_{r_{0}}\times[0,s]\right)\cup\left(B_{r_{0}}\setminus
B_{r-\frac{\delta}{2}}\times[0,T]\right),\end{array}\right.$

so that

$\left|\frac{\partial\psi}{\partial
t}\right|\leq\frac{\tilde{C}}{\delta},\qquad\left|\nabla_{\mathbb{H}}\psi\right|^{2}\leq\frac{\tilde{C}}{\delta^{2}},$

for some constant $\tilde{C}$ independent of $s,\delta.$ Then for all
$s+\delta\leq t\leq T$ (3) becomes

$\displaystyle\int_{B_{r-\delta}}\mathcal{I}(g(t))^{2}\phi^{2}+\int_{s+\delta}^{T}\int_{B_{r-\delta}}|\nabla_{\mathbb{H}}\mathcal{I}(g)|^{2}\phi^{2}$
(3.27) $\displaystyle\quad\leq
6\tilde{C}\delta^{-2}\int_{s}^{T}\int_{B_{r_{0}}}\mathcal{I}(g)^{2}\phi^{2}.$

Note that for fixed $t$, $w\mapsto\mathcal{I}(g(w,t))$ is a radial function;
in fact as we noted earlier, $\mathcal{I}(g(w,t))$ is a function of $d(w)$.
Applying (A.4), with $\beta$ as in Lemma A.4, and (3), one obtains

$\displaystyle\int_{s+\delta}^{T}\int_{B_{r-\delta}}\mathcal{I}(g)^{2+2\beta}\phi^{2}$
$\displaystyle\quad\leq\hat{\hat{C}}\int_{s+\delta}^{T}\left[\left(\int_{B_{r-\delta}}|\nabla_{\mathbb{H}}\mathcal{I}(g(t))|^{2}\phi^{2}+\mathcal{I}(g(t))^{2}\phi^{2}\right)\left(\int_{B_{r-\delta}}\mathcal{I}(g(t))^{2}\phi^{2}\right)^{\beta}\right]dt$
$\displaystyle\quad\leq\hat{\hat{C}}\left[\int_{s+\delta}^{T}\int_{B_{r-\delta}}|\nabla_{\mathbb{H}}\mathcal{I}(g)|^{2}\phi^{2}+\int_{s+\delta}^{T}\int_{B_{r-\delta}}\mathcal{I}(g)^{2}\phi^{2}\right]\left(6\tilde{C}\delta^{-2}\int_{s}^{T}\int_{B_{r_{0}}}\mathcal{I}(g)^{2}\phi^{2}\right)^{\beta}$
$\displaystyle\quad\leq\hat{\hat{C}}(6\tilde{C}\delta^{-2}+1)\left(\int_{s}^{T}\int_{B_{r_{0}}}\mathcal{I}(g)^{2}\phi^{2}\right)\left(6\tilde{C}\delta^{-2}\int_{s}^{T}\int_{B_{r_{0}}}\mathcal{I}(g)^{2}\phi^{2}\right)^{\beta}.$

Since $0<\delta<r<1$, it follows that

$\displaystyle\left(\int_{s+\delta}^{T}\int_{B_{r-\delta}}\mathcal{I}(g)^{2+2\beta}\phi^{2}\right)^{\frac{1}{2+2\beta}}\leq\hat{\hat{C}}^{\frac{1}{2+2\beta}}(6\tilde{C}+1)^{1/2}\delta^{-1}\left(\int_{s}^{T}\int_{B_{r_{0}}}\mathcal{I}(g)^{2}\phi^{2}\right)^{1/2}$
(3.28)
$\displaystyle\quad\leq\overline{C}\delta^{-1}\left(\int_{s}^{T}\int_{B_{r_{0}}}\mathcal{I}(g)^{2}\phi^{2}\right)^{1/2}.$

Let $b>0$ be sufficiently small, and set

$\displaystyle\delta=\frac{b}{2^{n}},\quad
r_{n+1}=r_{n}-\frac{b}{2^{n}},\quad\mathcal{I}_{n+1}=\mathcal{I}_{n}^{1+\beta},\quad
s_{n+1}=s_{n}+\frac{b}{2^{n}},$ $\displaystyle\text{ and
}k_{n}=\left(\int_{s_{n}}^{T}\int_{B_{r_{n}}}\mathcal{I}_{n}(g)^{2}\phi^{2}\right)^{1/2}.$

Here $\mathcal{I}_{1}=\mathcal{I}$ and $r_{1},s_{1}>0$ with
$s_{1}\geq\frac{\varepsilon}{2}$ and $r_{1}<1$ are given. Applying (3) yields

(3.29) $k_{n+1}^{\frac{1}{1+\beta}}\leq\overline{C}2^{n}b^{-1}k_{n}.$

Applying Lemma A.1 in the Appendix, we have

$k_{n}^{\frac{1}{(1+\beta)^{n-1}}}\leq\left(\frac{\overline{C}}{b}\right)^{\frac{a_{n}}{(1+\beta)^{n-2}}}2^{\frac{d_{n}}{(1+\beta)^{n-2}}}k_{1},$

where $a_{n}=\sum_{j=0}^{n-2}(1+\beta)^{j}$ and
$d_{n}=\sum_{j=0}^{n-2}(j+1)(1+\beta)^{n-2-j}$ for $n\geq 2$. Letting
$n\to\infty$ we have

$s_{n}\to s_{1}+b,\,r_{n}\to
r_{1}-b,\,\frac{a_{n}}{(1+\beta)^{n-2}}\to\left(\frac{1+\beta}{\beta}\right),\,\frac{d_{n}}{(1+\beta)^{n-2}}\to\left(\frac{1+\beta}{\beta}\right)^{2}$

and taking into account that $\mathcal{I}_{n}=\mathcal{I}^{(1+\beta)^{n-1}}$
we obtain

$k_{n}^{\frac{1}{(1+\beta)^{n-1}}}=\left(\int_{s_{n}}^{T}\int_{B_{r_{n}}}\mathcal{I}(g)^{2(1+\beta)^{n-1}}\phi^{2}\right)^{\frac{1}{2(1+\beta)^{n-1}}}\to\sup_{B_{r_{1}-b}\times[s_{1}+b,T]}\mathcal{I}(g).$

Finally, we have

(3.30)
$\sup_{B_{r_{1}-b}\times[s_{1}+b,T]}\mathcal{I}(g)\leq\left(\frac{\overline{C}}{b}2^{\frac{1+\beta}{\beta}}\right)^{\frac{1+\beta}{\beta}}\left(\int_{s_{1}}^{T}\int_{B_{r_{1}}}\mathcal{I}(g)^{2}\phi^{2}\right)^{1/2}.$

Now, consider a sequence $\mathcal{I}_{n}(r)\in
C^{2}(\mathbb{R}_{+},\mathbb{R}_{+})$ of convex function converging to
$\frac{1}{r^{\gamma}}$ as $n\to\infty$, where $\gamma>0$ is a parameter to be
chosen later. Replace $\mathcal{I}$ by a $\mathcal{I}_{n}$ in (3.30), we
obtain

$\sup_{B_{r_{1}-b}\times[s_{1}+b,T]}g^{-\gamma}\leq\left(\frac{\overline{C}}{b}2^{\frac{1+\beta}{\beta}}\right)^{\frac{1+\beta}{\beta}}\left(\int_{s_{1}}^{T}\int_{B_{r_{1}}}g^{-2\gamma}\phi^{2}\right)^{1/2}.$

Set $s_{1}=\frac{3}{4}\varepsilon$, $b=\frac{\varepsilon}{4}$ and
$r_{1}<r_{0}$, where $r_{0}$ is the one chosen in the beginning of the proof.
We have

$g(\omega,t)=\frac{v}{\phi}\geq\phi^{-1}(w)\left(e^{t\varDelta_{\mathbb{H}}}C_{0}\right)(w)=C_{0}\phi^{-1}(w),$

for almost every $w\in B_{r_{0}}$ where $C_{0}$ is the constant given in
(3.16). So

$\sup_{B_{r_{1}-\varepsilon/4}\times[\varepsilon,T]}g^{-\gamma}\leq
C_{2}C_{0}^{-\gamma}\varepsilon^{-\frac{1+\beta}{\beta}}\left(\int_{\frac{3}{4}\varepsilon}^{T}\int_{B_{r_{1}}}\phi^{2+2\gamma}\right)^{1/2},$

and it follows that

(3.31) $g(w,t)\geq
C_{2}^{-\frac{1}{\gamma}}C_{0}\varepsilon^{\left(1+\frac{1}{\beta}\right)\frac{1}{\gamma}}\left(\int_{B_{r_{1}}}\phi^{2+2\gamma}\right)^{-\frac{1}{2\gamma}},$

for almost every $w\in B_{r_{1}-\frac{\varepsilon}{4}}$ and for all
$t\in[\varepsilon,T]$, where $C_{2}$ is a positive constant independent of
$\varepsilon$ and $r_{1}.$ A simple computation shows that
$\int_{B_{r_{1}}}\phi^{2+2\gamma}<\infty$ by choosing
$0<\gamma<\frac{N+1}{\alpha}-1$, which is possible since $\alpha\in(0,N]$.

Thus (3.23) follows by taking $\gamma\in(0,\frac{N+1}{\alpha}-1)$ and
$\varepsilon=2(2r_{1}-r_{0})$ with $\frac{r_{0}}{2}<r_{1}<r_{0}$.

This concludes the proof. ∎

We end this section by the following remark.

######  Remark 3.6.

The arguments used are based on the explicit form of the fundamental solution
and on the existence of an underlying group of dilations; thus, the results
would likely extend to the setting of H-type groups.

## Appendix A

In this Appendix we collect all technical lemmas that we needed for proving
the main result.

######  Lemma A.1.

For $\beta>0$ and $n\geq 2$, define $a_{n}=\sum_{j=0}^{n-2}(1+\beta)^{j}$ and
$d_{n}=\sum_{j=0}^{n-2}(j+1)(1+\beta)^{n-2-j}$ and $k_{n}\geq 0$ for $n\geq 1$
such that

$k_{n}^{\frac{1}{1+\beta}}\leq\overline{C}2^{n-1}b^{-1}k_{n-1}.$

Then

(A.1)
$k_{n}^{\frac{1}{1+\beta}}\leq\left(\frac{\overline{C}}{b}\right)^{a_{n}}2^{d_{n}}k_{1}^{(1+\beta)^{n-2}}.$

###### Proof.

We use an induction argument. Assume (A.1) is true for $1\leq k\leq n.$ We
will show

$k_{n+1}^{\frac{1}{1+\beta}}\leq\left(\frac{\overline{C}}{b}\right)^{a_{n+1}}2^{d_{n+1}}k_{1}^{(1+\beta)^{n-1}}.$

Clearly (3.29) gives (A.1) if $n=1.$ By (3.29),

$k_{n+1}^{\frac{1}{1+\beta}}\leq\overline{C}2^{n}b^{-1}k_{n}\leq\left(\frac{\overline{C}}{b}\right)2^{n}\left(\frac{\overline{C}}{b}\right)^{a_{n}(1+\beta)}2^{d_{n}(1+\beta)}k_{1}^{(1+\beta)^{n-1}},$

by the induction hypothesis. Now it is easy to check that

$a_{n}(1+\beta)+1=\sum_{j=0}^{n-2}(1+\beta)^{j+1}+1=\sum_{j=0}^{n-1}(1+\beta)^{j}=a_{n+1}$

and

$\displaystyle
d_{n}(1+\beta)+n=\sum_{j=0}^{n-2}(j+1)(1+\beta)^{n-1-j}+n=\sum_{j=0}^{n-1}(j+1)(1+\beta)^{n-1-j}=d_{n+1}.$

∎

The following two lemmas can be found in [3, Appendix].

######  Lemma A.2.

If $0\leq h\in C^{1}[0,2r]$ and $h(2r)=0$, then

(A.2) $\left(\int_{0}^{2r}h^{p}(s)s^{\gamma-1}ds\right)^{1/p}\leq
M_{0}\left(\int_{0}^{2r}\left|\frac{dh}{ds}\right|^{2}s^{\gamma-1}ds\right)^{1/2},$

where $\gamma=2N+2-2\alpha>2$, $\frac{1}{p}=\frac{1}{2}-\frac{1}{\gamma}$ and
$M_{0}$ is a constant depending only on $\gamma.$

######  Lemma A.3.

If $0<r^{\prime}\leq r\leq 1$, $0\leq h\in C^{1}[0,r]$ then

(A.3)
$\left(\int_{0}^{r}|h(s)|^{p}s^{2N-2\alpha+1}ds\right)^{2/p}\leq\hat{C}\left(\int_{0}^{r}[|h^{\prime}(s)|^{2}+|h(s)|^{2}]s^{2N-2\alpha+1}ds\right),$

where $\frac{1}{p}\geq\frac{1}{2}-\frac{1}{2N+2-2\alpha}$ and $p=\infty$ if
$N=\alpha-1$, $\hat{C}$ depends on $r^{\prime}$ but not $r.$

The following lemma is needed for the proof of Lemma 3.5.

######  Lemma A.4.

If $k\in
C^{1}(\mathbb{R}_{+},\mathbb{R}_{+}),\,\tilde{k}(w):=k(d(w)),\,\phi(w)=d(w)^{-\alpha},\,w=(z,l)\in\mathbb{H}^{N},$
and $0<\beta$ is such that $\beta+\frac{2}{p}=1$, where
$\frac{1}{p}=\frac{1}{2}-\frac{1}{2N-2\alpha+2}$, then

(A.4)
$\begin{array}[]{ll}\displaystyle\int_{B_{r}}\tilde{k}^{2+2\beta}(w)\phi^{2}(w)\,dw\leq\\\
\\\
\hat{\hat{C}}\left(\displaystyle\int_{B_{r}}\left(|\nabla_{\mathbb{H}}\tilde{k}(w)|^{2}+\tilde{k}^{2}(w)\right)\phi^{2}(w)\,dw\right)\left(\displaystyle\int_{B_{r}}\tilde{k}^{2}(w)\phi^{2}(w)\,dw\right)^{\beta}.\end{array}$

###### Proof.

We first prove that

(A.5)
$\int_{B_{r}}|\nabla_{\mathbb{H}}k(d(\omega))|^{2}\phi^{2}(d(w))\,dw=C_{N}\int_{0}^{r}\left|k^{\prime}(s))\right|^{2}s^{2N-2\alpha+1}\,ds,$

where
$C_{N}:=S_{2N-1}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos^{N}\varphi\,d\varphi$
and $S_{2N-1}$ the surface area of the unit ball in $\mathbb{R}^{2N}$.

Consider the change of variables $\rho=|z|$ in polar coordinates, and take

$\left\\{\begin{array}[]{ll}\rho^{2}=r^{2}\cos\varphi,\\\
l=r^{2}\sin\varphi\end{array}\right.$

with $\varphi\in[-\frac{\pi}{2},\frac{\pi}{2}]$. Recalling from Lemma 2.1 that
$|\nabla_{\mathbb{H}}d(w)|^{2}=|z|^{2}d(w)^{-2},\quad
w=(z,l)\in\mathbb{H}^{N}$ and since
$\nabla_{\mathbb{H}}k(d(w))=k^{\prime}(d(w))\nabla_{\mathbb{H}}d(w)$ we obtain

$\displaystyle\int_{B_{r}}|\nabla_{\mathbb{H}}k(d(w))|^{2}\phi^{2}(d(w))\,dw$
$\displaystyle=$ $\displaystyle
S_{2N-1}\int|\nabla_{\mathbb{H}}k(\sqrt[4]{\rho^{4}+l^{2}})|^{2}\phi^{2}(\sqrt[4]{\rho^{4}+l^{2}})\rho^{2N-1}\,d\rho
dl$ $\displaystyle=$ $\displaystyle
S_{2N-1}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\int_{0}^{r}|k^{\prime}(s)|^{2}(s^{2}\cos\varphi)s^{-2}s^{-2\alpha}(s\sqrt{\cos\varphi})^{2N-1}\left(\frac{s^{2}}{\sqrt{\cos\varphi}}\right)\,dsd\varphi$
$\displaystyle=$ $\displaystyle
C_{N}\int_{0}^{r}|k^{\prime}(s)|^{2}s^{2N-2\alpha+1}\,ds$

for
$C_{N}:=S_{2N-1}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos^{N}\varphi\,d\varphi$.

It follows from Hölder’s inequality, (A.5) and Lemma A.3 that

$\displaystyle\int_{B_{r}}k^{2+2\beta}(w)\phi^{2}(w)\,dw=C_{N}\int_{0}^{r}k^{2+2\beta}(s)\phi^{2}(s)s^{2N+1}ds$
$\displaystyle\quad\leq\left(C_{N}\int_{0}^{r}k^{p}(s)\phi^{2}(s)s^{2N+1}ds\right)^{2/p}\left(C_{N}\int_{0}^{r}k^{2}(s)\phi^{2}(s)s^{2N+1}ds\right)^{\beta}$
$\displaystyle\quad=\left(C_{N}\int_{0}^{r}k^{p}(s)s^{2N-2\alpha+1}ds\right)^{2/p}\left(\int_{B_{r}}k^{2}\phi^{2}\right)^{\beta}$
$\displaystyle\quad\leq\hat{C}C_{N}^{\frac{2}{p}-1}C_{N}\int_{0}^{r}\left(|k^{\prime}(s)|^{2}+k^{2}(s)\right)s^{2N-2\alpha+1}ds\left(\int_{B_{r}}k^{2}\phi^{2}\right)^{\beta}$
$\displaystyle\quad=\hat{\hat{C}}\left(\int_{B_{r}}|\nabla_{\mathbb{H}}k|^{2}\phi^{2}+\int_{B_{r}}k^{2}\phi^{2}\right)\left(\int_{B_{r}}k^{2}\phi^{2}\right)^{\beta}.$

This completes the proof. ∎

We conclude this Appendix by proving the following perturbation result.

######  Proposition A.5.

Assume that the problem (3.1) has a solution for some $u_{0}\geq 0$, $f\geq 0$
and let $B\in L^{\infty}(\mathbb{H}^{N})$. Then the problem

(A.6) $\left\\{\begin{array}[]{ll}\dfrac{\partial u}{\partial
t}=\varDelta_{\mathbb{H}}u+V(\cdot)u+B(\cdot)u+f&\text{ in
}{\mathcal{D}}^{\prime}(\mathbb{H}^{N}\times(0,T)),\\\ \\\ {\rm esslim}_{t\to
0^{+}}\displaystyle\int_{\mathbb{H}^{N}}u(w,t)\psi(w)\,dw=\int_{\mathbb{H}^{N}}u_{0}(w)\psi(w)\,dw&\text{
}\forall\ \psi\in{\mathcal{D}}(\mathbb{H}^{N}),\\\ \\\ u\geq 0&\text{ on
}\mathbb{H}^{N}\times(0,T),\\\ \\\ Vu\in
L^{1}_{loc}(\mathbb{H}^{N}\times(0,T)),\end{array}\right.$

has a solution.

###### Proof.

Let $u_{n}$ be the solution of (3.2). We know that $u_{n}\uparrow u$, $u$
being a solution of (3.1). Suppose that $v_{n}$ solves

(A.7) $\left\\{\begin{array}[]{ll}\dfrac{\partial v_{n}}{\partial
t}=\varDelta_{\mathbb{H}}v_{n}+\left(V_{n}(\cdot)+B(\cdot)\right)v_{n}+f_{n}&\text{
in }{\mathcal{D}}^{\prime}_{T},\\\ \\\ {\rm lim}_{t\to
0^{+}}\displaystyle\int_{\mathbb{H}^{N}}v_{n}(w,t)\psi(w)\,dw=\displaystyle\int_{\mathbb{H}^{N}}u_{0}(w)\psi(w)\,dw,&\forall\psi\in{\mathcal{D}}(\mathbb{H}^{N}),\\\
\end{array}\right.$

where $f_{n}=\min\\{f,n\\}$ and $V_{n}=\min\\{V,n\\}$. Fix
$\lambda\geq\|B\|_{\infty}$, and consider

$U_{n}=e^{\lambda t}u_{n}.$

So $U_{n}$ satisfies

$\frac{\partial U_{n}}{\partial
t}=\varDelta_{\mathbb{H}}U_{n}+(V_{n}+\lambda)U_{n}+e^{t}f_{n}.$

By the Maximum Principle we have

$v_{n}(w,t)\leq U_{n}(w,t)\leq e^{\lambda t}u(w,t)\,\hbox{\ for a.e.
}(w,t)\in\mathbb{H}^{N}\times(0,T).$

Clearly $\\{v_{n}\\}$ is an increasing sequence and since $u,Vu\in
L^{1}_{loc}(\mathbb{H}^{N}\times(0,T))$, it follows by the Monotone
Convergence theorem that $v_{n}\uparrow v$ and $(V_{n}+B)v_{n}\uparrow(V+B)v$
in $L^{1}_{loc}(\mathbb{H}^{N}\times(0,T))$, and $v$ gives a solution of
(A.6). ∎

## Acknowledgement

A.E. Kogoj, A. Rhandi and C. Tacelli have been partially supported by the
Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro
Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

## References

  * [1] W. Arendt, G.R. Goldstein, and J.A. Goldstein, Outgrowths of Hardy’s inequality, Contemporary Math. 412 (2006), 51–68.
  * [2] D.G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm Sup. Pisa 22 (1968), 607–694.
  * [3] P. Baras and J.A. Goldstein, The heat equation with singular potential, Trans. Amer. Math. Soc. 284 (1984), 121–139.
  * [4] A. Bonfiglioli, F. Lanconelli and F. Uguzzoni, “Stratified Lie Groups and Potential Theory for their Sub-Laplacians”, Springer Monographs in Mathematics, Berlin, 2007.
  * [5] X. Cabré and Y. Martel, Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier, C. R. Acad. Sci. Paris, 329 (1999), 973–978.
  * [6] A. Canale, F. Gregorio, A. Rhandi, C. Tacelli, _Weighted Hardy’s inequalities and Kolmogorov-type operators_ , Appl. Anal. 98 (2019), 1236-1254.
  * [7] T. Durante and A. Rhandi, _On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials_ , Discrete Contin. Dyn. Syst. Ser. S 6 (2013), 649–655.
  * [8] G.B. Folland and E.M. Stein, _Estimates for the $\bar{\partial}_{b}$ complex and analysis on the Heisenberg group_, Comm. Pure Appl. Math. 27 (1974), 429–522.
  * [9] S. Fornaro and A. Rhandi, _On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^{p}$-spaces_, Discrete Contin. Dyn. Syst. 33 (2013), 5049–5058.
  * [10] S. Fornaro, F. Gregorio and A. Rhandi, _Elliptic operators with unbounded diffusion coefficients perturbed by inverse square potentials in Lp-spaces_ , Commun. Pure Appl. Anal. 15 (2016), 2357–2372.
  * [11] N. Garofalo and E. Lanconelli, _Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation_ , Ann. Inst. Fourier (Grenoble) 40 (1990), 313–356.
  * [12] G.R. Goldstein, J.A. Goldstein and A. Rhandi, _Weighted Hardy’s inequality and the Kolmogorov equation perturbed by an inverse-square potential_ , Appl. Anal. 91 (2012), 2057–2071.
  * [13] J.A. Goldstein, D. Hauer and A. Rhandi, _Existence and nonexistence of positive solutions of $p$-Kolmogorov equations perturbed by a Hardy potential_, Nonlinear Anal. 131 (2016), 121–154.
  * [14] J.A. Goldstein and I. Kombe, Instantaneous blow up, Contemporary Math. 327 (2003), 141–150.
  * [15] J.A. Goldstein and Q.S. Zhang, On a degenerate heat equation with a singular potential, J. Funct. Anal. 186 (2001), 342–359.
  * [16] J.A. Goldstein and Q.S. Zhang, Linear parabolic equations with strong singular potentials, Trans. Amer. Math. Soc. 355 (2003), 197–211.
  * [17] D. Hauer and A. Rhandi, _A weighted Hardy inequality and nonexistence of positive solutions_ , Arch. Math. 100 (2013), 273–287.
  * [18] R. Howe, _On the role of the Heisenberg group in harmonic analysis_ , Bull. Amer. Math. Soc. (N.S.) 3 (1980), 821–843.
  * [19] D. Jerison and J.M. Lee, _Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem_ , J. Amer. Math. Soc. 1 (1988), 1–13.
  * [20] D. Jerison and J.M. Lee, _Intrinsic CR normal coordinates and the CR Yamabe problem_ , J. Differential Geom. 29 (1989), 303–343.
  * [21] S. Semmes, _An introduction to Heisenberg groups in analysis and geometry_ , Notices Amer. Math. Soc. 50 (2003), 640–646.
  * [22] N.Th. Varopoulos, L. Saloff-Coste and T. Coulhon, “Analysis and Geometry on Groups”, Cambridge University Press, 1992.