We show that if
u
u
is a solution to
∑
i
,
j
=
1
n
a
i
j
(
x
,
t
)
D
i
j
u
(
x
,
t
)
−
D
t
u
(
x
,
t
)
=
ϕ
(
x
)
\sum \nolimits _{i,j = 1}^n {{a_{ij}}(x,t){D_{ij}}u(x,t) - {D_t}u(x,t) = \phi (x)}
on a cylinder
Ω
T
=
Ω
×
(
0
,
T
)
{\Omega _T} = \Omega \times (0,T)
, where
Ω
\Omega
is a bounded open set in
R
n
,
T
>
0
{{\mathbf {R}}^n},T > 0
, and
u
u
vanishes continuously on the parabolic boundary of
Ω
T
{\Omega _T}
. Then the maximum of
u
u
on the cylinder is bounded by a constant
C
C
depending on the ellipticity of the coefficient matrix
(
a
i
j
(
x
,
t
)
)
({a_{ij}}(x,t))
, the diameter of
Ω
\Omega
, and the dimension
n
n
times the
L
n
{L^n}
norm of
ϕ
\phi
in
Ω
\Omega
.