In this memoir, we study the existence and regularity of the quasilinear parabolic equations:
u
t
−
div
(
A
(
x
,
t
,
∇
u
)
)
=
B
(
u
,
∇
u
)
+
μ
,
\begin{equation*} u_t-\operatorname {div}(A(x,t,\nabla u))=B(u,\nabla u)+\mu , \end{equation*}
in either
R
N
+
1
\mathbb {R}^{N+1}
or
R
N
×
(
0
,
∞
)
\mathbb {R}^N\times (0,\infty )
or on a bounded domain
Ω
×
(
0
,
T
)
⊂
R
N
+
1
\Omega \times (0,T)\subset \mathbb {R}^{N+1}
where
N
≥
2
N\geq 2
. We shall assume that the nonlinearity
A
A
fulfills standard growth conditions, the function
B
B
is a continuous and
μ
\mu
is a radon measure. Our first task is to establish the existence results with
B
(
u
,
∇
u
)
=
±
|
u
|
q
−
1
u
B(u,\nabla u)=\pm |u|^{q-1}u
, for
q
>
1
q>1
. We next obtain global weighted-Lorentz, Lorentz-Morrey and Capacitary estimates on gradient of solutions with
B
≡
0
B\equiv 0
, under minimal conditions on the boundary of domain and on nonlinearity
A
A
. Finally, due to these estimates, we solve the existence problems with
B
(
u
,
∇
u
)
=
|
∇
u
|
q
B(u,\nabla u)=|\nabla u|^q
for
q
>
1
q>1
.