Abstract
In this paper, we study the “triply” degenerate problem:
b
(
v
)
t
−
Δ
g
(
v
)
+
div
Φ
(
v
)
=
f
on
Q
:
=
(
0
,
T
)
×
Ω
,
b
(
v
(
0
,
⋅
)
)
=
b
(
v
0
)
on
Ω and “
g
(
v
)
=
g
(
a
)
on some part of the boundary
(
0
,
T
)
×
∂
Ω
,” in the case of continuous nonhomogeneous and nonstationary boundary data
a. The functions
b
,
g
are assumed to be continuous, locally Lipschitz, nondecreasing and to verify the normalization condition
b
(
0
)
=
g
(
0
)
=
0
and the range condition
R
(
b
+
g
)
=
R
. Using monotonicity and penalization methods, we prove existence of a weak renormalized entropy solution in the spirit of [K. Ammar, J. Carrillo, P. Wittbold, Scalar conservation laws with general boundary condition and continuous flux function, J. Differential Equations 228 (2006) 111–139].