AbstractThe initial-boundary value problem of a degenerate parabolic equation arising from double phase convection is considered. Let $a(x)$
a
(
x
)
and $b(x)$
b
(
x
)
be the diffusion coefficients corresponding to the double phase respectively. In general, it is assumed that $a(x)+b(x)>0$
a
(
x
)
+
b
(
x
)
>
0
, $x\in \overline{\Omega }$
x
∈
Ω
‾
and the boundary value condition should be imposed. In this paper, the condition $a(x)+b(x)>0$
a
(
x
)
+
b
(
x
)
>
0
, $x\in \overline{\Omega }$
x
∈
Ω
‾
is weakened, and sometimes the boundary value condition is not necessary. The existence of a weak solution u is proved by parabolically regularized method, and $u_{t}\in L^{2}(Q_{T})$
u
t
∈
L
2
(
Q
T
)
is shown. The stability of weak solutions is studied according to the different integrable conditions of $a(x)$
a
(
x
)
and $b(x)$
b
(
x
)
. To ensure the well-posedness of weak solutions, the classical trace is generalized, and that the homogeneous boundary value condition can be replaced by $a(x)b(x)|_{x\in \partial \Omega }=0$
a
(
x
)
b
(
x
)
|
x
∈
∂
Ω
=
0
is found for the first time.
On the uniqueness for weak solutions of steady double-phase fluids
