Of concern is the singular problem
∂
u
/
∂
t
=
Δ
u
+
(
c
/
|
x
|
2
)
u
+
f
(
t
,
x
)
,
u
(
x
,
0
)
=
u
0
(
x
)
\partial u/\partial t = \Delta u + (c/|x{|^2})\,u + f(t,x), u(x,0) = u_{0}(x)
, and its generalizations. Here
c
⩾
0
,
x
∈
R
N
,
t
>
0
c \geqslant 0,x \in {{\mathbf {R}}^N},t > 0
, and
f
f
and
u
0
{u_0}
are nonnegative and not both identically zero. There is a dimension dependent constant
C
∗
(
N
)
{C_{\ast } }(N)
such that the problem has no solution for
c
>
C
∗
(
N
)
c > {C_{\ast } }(N)
. For
c
⩽
C
∗
(
N
)
c \leqslant {C_{\ast } }(N)
necessary and sufficient conditions are found for
f
f
and
u
0
{u_0}
so that a nonnegative solution exists.