In this paper we consider the following initial value problem:
\[
{
∂
u
∂
t
=
−
H
u
+
V
(
x
)
u
a
m
p
;
in
R
N
×
(
0
,
T
)
,
u
(
x
,
0
)
=
u
0
(
x
)
≥
0
a
m
p
;
on
R
N
×
{
t
=
0
}
,
\begin {cases} \frac {\partial u}{\partial t}=-Hu+V(x)u & \text {in $\mathbb {R}^N\times (0,T)$},\\ u(x,0) = u_0 (x)\geq 0 & \text {on $\mathbb {R}^N \times \{t=0\}$}, \end {cases}
\]
where
H
=
−
Δ
−
β
|
x
|
2
sin
(
1
|
x
|
α
)
H=-\Delta -\frac {\beta }{|x|^2}\sin (\frac {1}{|x|^{\alpha }})
and
0
≤
V
∈
L
loc
1
(
R
N
)
0\le V\in L_{\text {loc}}^1(\mathbb {R}^N)
. Nonexistence of positive solutions is analyzed.