We investigate the behavior of the solution
u
(
x
,
t
)
u(x,t)
of
\[
{
∂
u
∂
t
=
Δ
u
+
u
p
a
m
p
;
in
R
n
×
(
0
,
T
)
,
u
(
x
,
0
)
=
φ
(
x
)
a
m
p
;
in
R
n
,
\left \{ {\begin {array}{*{20}{c}} {\frac {{\partial u}} {{\partial t}} = \Delta u + {u^p}} \hfill & {{\text {in}}\;{\mathbb {R}^n} \times (0,T),} \hfill \\ {u(x,0) = \varphi (x)} \hfill & {{\text {in}}\;{\mathbb {R}^n},} \hfill \\ \end {array} } \right .
\]
where
Δ
=
∑
i
=
1
n
∂
2
/
∂
x
i
2
\Delta = \sum \nolimits _{i = 1}^n {{\partial ^2}/\partial _{{x_i}}^2}
is the Laplace operator,
p
>
1
p > 1
is a constant,
T
>
0
T > 0
, and
φ
\varphi
is a nonnegative bounded continuous function in
R
n
{\mathbb {R}^n}
. The main results are for the case when the initial value
φ
\varphi
has polynomial decay near
x
=
∞
x = \infty
. Assuming
φ
∼
λ
(
1
+
|
x
|
)
−
a
\varphi \sim \lambda {(1 + |x|)^{ - a}}
with
λ
\lambda
,
a
>
0
a > 0
, various questions of global (in time) existence and nonexistence, large time behavior or life span of the solution
u
(
x
,
t
)
u(x,t)
are answered in terms of simple conditions on
λ
\lambda
,
a
a
,
p
p
and the space dimension
n
n
.