We are primarily concerned with the absence of positive solutions of the following problem,
\[
{
∂
u
∂
t
=
Δ
(
u
m
)
+
V
(
x
)
u
m
+
λ
u
q
a
m
p
;
in
Ω
×
(
0
,
T
)
,
u
(
x
,
0
)
=
u
0
(
x
)
≥
0
a
m
p
;
in
Ω
,
∂
u
∂
ν
=
β
(
x
)
u
a
m
p
;
on
∂
Ω
×
(
0
,
T
)
,
\begin {cases} \frac {\partial u}{\partial t}=\Delta ( u^m)+V(x)u^{m}+ \lambda u^q & \text {in}\quad \Omega \times (0, T ) ,\\ u(x,0)=u_{0}(x)\geq 0 & \text {in} \quad \Omega ,\\ \frac {\partial u}{\partial \nu }=\beta (x) u & \text {on }\partial \Omega \times (0,T), \end {cases}
\]
where
0
>
m
>
1
0>m>1
,
V
∈
L
l
o
c
1
(
Ω
)
V\in L_{loc}^1(\Omega )
,
β
∈
L
l
o
c
1
(
∂
Ω
)
\beta \in L_{loc}^1(\partial \Omega )
,
λ
∈
R
\lambda \in \mathbb {R}
,
q
>
0
q>0
,
Ω
⊂
R
N
\Omega \subset \mathbb {R}^N
is a bounded open subset of
R
N
\mathbb {R}^N
with smooth boundary
∂
Ω
\partial \Omega
, and
∂
u
∂
ν
\frac {\partial u}{\partial \nu }
is the outer normal derivative of
u
u
on
∂
Ω
\partial \Omega
. Moreover, we also present some new sharp Hardy and Leray type inequalities with remainder terms that provide us concrete potentials to use in the partial differential equation of our interest.