AbstractWe study the existence of weak solutions to a Newtonian fluid∼non-Newtonian fluid mixed-type equation $$ {u_{t}}= \operatorname{div} \bigl(b(x,t){ \bigl\vert {\nabla A(u)} \bigr\vert ^{p(x) - 2}}\nabla A(u)+\alpha (x,t)\nabla A(u) \bigr)+f(u,x,t). $$
u
t
=
div
(
b
(
x
,
t
)
|
∇
A
(
u
)
|
p
(
x
)
−
2
∇
A
(
u
)
+
α
(
x
,
t
)
∇
A
(
u
)
)
+
f
(
u
,
x
,
t
)
.
We assume that $A'(s)=a(s)\geq 0$
A
′
(
s
)
=
a
(
s
)
≥
0
, $A(s)$
A
(
s
)
is a strictly increasing function, $A(0)=0$
A
(
0
)
=
0
, $b(x,t)\geq 0$
b
(
x
,
t
)
≥
0
, and $\alpha (x,t)\geq 0$
α
(
x
,
t
)
≥
0
. If $$ b(x,t)=\alpha (x,t)=0,\quad (x,t)\in \partial \Omega \times [0,T], $$
b
(
x
,
t
)
=
α
(
x
,
t
)
=
0
,
(
x
,
t
)
∈
∂
Ω
×
[
0
,
T
]
,
then we prove the stability of weak solutions without the boundary value condition.
$L^{\infty }$ decay estimates of solutions of nonlinear parabolic equation
