In this paper we study the first initial-boundary value problem for
u
t
=
Δ
u
+
u
p
{u_t} = \Delta u + {u^p}
in conical domains
D
=
(
0
,
∞
)
×
Ω
⊂
R
N
D = (0,\infty ) \times \Omega \subset {R^N}
where
Ω
⊂
S
N
−
1
\Omega \subset {S^{N - 1}}
is an open connected manifold with boundary. We obtain some extensions of some old results of Fujita, who considered the case
D
=
R
N
D = {R^N}
. Let
λ
=
−
γ
−
\lambda = - {\gamma _ - }
where
γ
−
{\gamma _ - }
is the negative root of
γ
(
γ
+
N
−
2
)
=
ω
1
\gamma (\gamma + N - 2) = {\omega _1}
and where
ω
1
{\omega _1}
is the smallest Dirichlet eigenvalue of the Laplace-Beltrami operator on
Ω
\Omega
. We prove: If
1
>
p
>
1
+
2
/
(
2
+
λ
)
1 > p > 1 + 2/(2 + \lambda )
, there are no nontrivial global solutions. If
1
>
p
>
1
+
2
/
λ
1 > p > 1 + 2/\lambda
, there are no stationary solutions in
D
−
{
0
}
D - \{ 0\}
except
u
≡
0
u \equiv 0
. If
1
+
2
/
λ
>
p
>
(
N
+
1
)
/
(
N
−
3
)
1 + 2/\lambda > p > (N + 1)/(N - 3)
(if
N
>
3
N > 3
, arbitrary otherwise) there are singular stationary solutions
u
s
{u_s}
. If
u
(
x
,
0
)
⩽
u
s
(
x
)
u(x,0) \leqslant {u_s}(x)
, the solutions are global. If
1
+
2
/
λ
>
p
>
(
N
+
2
)
/
(
N
−
2
)
1 + 2/\lambda > p > (N + 2)/(N - 2)
and
u
(
x
,
0
)
⩽
u
s
u(x,0) \leqslant {u_s}
, with
u
(
x
,
0
)
∈
C
(
D
¯
)
u(x,0) \in C(\overline D )
, the solutions decay to zero. If
1
+
2
/
N
>
p
1 + 2/N > p
, there are global solutions. For
1
>
p
>
∞
1 > p > \infty
, there are
L
∞
{L^\infty }
data of arbitrarily small norm, decaying exponentially fast at
r
=
∞
r = \infty
, for which the solution is not global. We show that if
D
D
is the exterior of a bounded region, there are no global, nontrivial, positive solutions if
1
>
p
>
1
+
2
/
N
1 > p > 1 + 2/N
and that there are such if
p
>
1
+
2
/
N
p > 1 + 2/N
. We obtain some related results for
u
t
=
Δ
u
+
|
x
|
σ
u
p
{u_t} = \Delta u + |x{|^\sigma }{u^p}
in the cone.