In this article we consider the problem
(\textit {P})
{
u
t
−
Δ
u
m
a
m
p
;
=
a
m
p
;
|
∇
u
|
q
+
f
(
x
,
t
)
,
u
≥
0
a
m
p
;
in
Ω
T
≡
Ω
×
(
0
,
T
)
,
u
(
x
,
t
)
a
m
p
;
=
a
m
p
;
0
a
m
p
;
on
∂
Ω
×
(
0
,
T
)
,
u
(
x
,
0
)
a
m
p
;
=
a
m
p
;
u
0
(
x
)
a
m
p
;
in
Ω
,
\begin{equation}\tag {\textit {P}} \left \{\begin {array}{rclll} u_t-\Delta u^m&=&|\nabla u|^q +\,f(x,t),\quad u\ge 0 &\hbox { in } \Omega _T\equiv \Omega \times (0,T),\\ u(x,t)&=&0 &\quad \hbox { on } \partial \Omega \times (0,T),\\ u(x,0)&=&u_0(x)&\quad \hbox { in } \Omega , \end{array} \right . \end{equation}
where
Ω
⊂
R
N
\Omega \subset \mathbb {R}^N
is a bounded regular domain,
N
≥
1
N\ge 1
,
1
>
q
≤
2
1>q\le 2
, and
f
≥
0
f\ge 0
,
u
0
≥
0
u_0\ge 0
are in a suitable class of measurable functions.
We obtain some results for the so-called elliptic-parabolic problems with measure data related to problem
(
P
)
(P)
that we use to study the existence of solutions to problem
(
P
)
(P)
according with the values of the parameters
q
q
and
m
m
.