232
vm.
Degenerate
and singular parabolic
systems
IIDullp,K:l.
We
will do this in a qualitative way and with no precise specification
of
the functional dependence.
We
will use such qualitative infonnation to prove
still
qualitatively that
/Du/
E
Ll:
c
{!1
T
),
with bounds only dependent on local
V-nonns
of
/Du/.
Finally, in
the
next section, we will tum such qualitative in-
fonnation into precise
quantitative estimates
of
IIDulloo,K:
o
over compact subsets
lCoc{h.
LEMMA
4.1. Let u be a local weak solution of(1.1). Moreover in the singular
case
1 < p < 2 let the approximation assumption (1.9) be in force. Then
/Du/
E Lloc{{h),
forevery
q E [1,00).
PROOF:
Consider first the degenerate case p >
2.
Let Q
(6,
p)
c
{h
and let (
be a standard non-negative cutoff function vanishing
on
the parabolic boundary
of
Q
(6,
p). Thus, in particular,
(.,
-6)
=0.
In (3.7), take J(v)
=v
P
,
where
P?O
is
to
be chosen. Proceeding fonnally we obtain
(4.1)
sup
Iv
2
+
pe
(x,
t)dx
$
'Y
jr
f
(1
+
vP+
lt
)
dxdr
-9<t<O
J
- -
Kp
. Q(9,p)
o
(4.2)
IIIDv£¥r
dxdr
$
'Y
II
(1
+
vP+
P
) dxdT,
-9K
p
Q(9,p)
where
'Y
=
'Y
(N,p,
p,
(t,
D().
These are rigorous
if
the right hand side is finite.
We
apply the embedding Theorem 2.1
of
Chap. I to the functions
x
-+
(v£¥()
(x, t), a.e. t E
(-6,0),
over the cubes Kp. It suffices to consider the case N >
2.
Indeed
if
N = 1,
2,
we
may consider
u as a vector field defmed in
RN
N ? 3, up to a localisation, and
deduce inequalities
(4.1 )-(4.2) for it. Let 6 be a positive number to
be
chosen. Then
by Corollary
2.1
of
Chap. I and HOlder's inequality
We integrate over
(-6,0),
to obtain