Abstract
We consider the semilinear heat equation
∂
t
u
-
Δ
u
=
f
(
u
)
,
(
x
,
t
)
∈
R
N
×
[
0
,
T
)
,
(
1
)
with
f
(
u
)
=
|
u
|
p
-
1
u
log
a
(
2
+
u
2
)
, where
p
>
1
is Sobolev subcritical and
a
∈
R
. We first show an upper bound for any blow-up solution of (1). Then, using this estimate and the logarithmic property, we prove that the exact blow-up rate of any singular solution of (1) is given by the ODE solution associated with (1), namely
u
′
=
|
u
|
p
-
1
u
log
a
(
2
+
u
2
)
. In other words, all blow-up solutions in the Sobolev subcritical range are Type I solutions. To the best of our knowledge, this is the first determination of the blow-up rate for a semilinear heat equation where the main nonlinear term is not homogeneous.