Indeed, we have
n
X
i,k=1 N
X
j=1
A
ij
x, t, u, vu
i
u
k
v
kj
=
n
X
i,k=1 N
X
j=1 N
X
m=1
a
jm
v
im
+ α
ij
u
i
u
k
v
kj
=
N
X
j,m=1
a
jm n
X
i=1
u
i
v
im n
X
k=1
u
k
v
kj
+
n
X
j=1 n
X
i=1
α
ij
u
i n
X
k=1
u
k
v
kj
≥
N
X
j=1
λ
n
X
k=1
u
k
v
kj 2
−
n
X
i=1
α
ij
u
i n
X
k=1
u
k
v
kj
≥ 0.
We follow the approach of [1, 5, 6] and start with the derivation, presented in Sec- tion 2, of a local energy estimates for weak solutions to 1. We then outline, in Sec-
tion 3 and Section 4 how the methods in [5] can be applied to obtain local integrability and boundedness.
We also remark that the techniques presented can be modified to handle doubly degenerate problems, where A
ij
x, t, u, vv
ij
≥ Φ|u||v|
p
− C
3
|u|
δ
− φ
o
x, t for some
Φ, following the same lines as the proof in [6].
2 Energy Estimates for u
2.1 Notation Preliminaries
Let x
, t ∈ Ω
T
, without loss of generality we can assume x
, t = 0, 0. For
R 0 we set Q
R
= B
R
0 × −R
p
, 0, and for −R
p
≤ τ ≤ 0, we define Q
τ R
= B
R
0 × −R
p
, τ . For a fixed 0 σ 1, we consider a function ζ ∈ C
∞
Ω
T
with 0 ≤ ζ ≤ 1, ζ = 1 in Q
τ σR
, and ζ = 0 near |x| = R or t = −R
p
. We also require that |ζ
t
| + |∇ζ|
p
≤ C
σ
R
p
= 2
1 − σ
p
R
p
. We denote by ζ
k
∈ C
∞
Q
τ R
the elements of a sequence of functions ζ
k
→ ζ uniformly in Q
τ R
. While, for η 0, we let J
η
be a smooth, symmetric, mollifying kernel in space-time, and for a given function f we use the notation f
η
≡ J
η
∗ f to represent its convolution with J
η
. Finally, for fixed ǫ
0, and κ 0, we consider the function f
s = s − κ
+
s − κ
+
+ ǫ .
3 In the following, we will use the fact that
0 ≤ f s ≤ 1, and that f
′
s =
s κ, ǫ
[s − κ
+
+ ǫ]
2
s κ EJQTDE, 2004 No. 14, p. 4
verifies 0 ≤ f
′
s ≤
s κ,
1
ǫ κ s
2κ,
1
s κ s
2κ, 4
provided 0 ǫ
1 2
. We are now ready to start the derivation of our energy estimate. Fix η
0, κ 0 and consider the test function
{u
i,η
x, tf |u
η
x, t|ζ
p k
x, t}
η
. Because this is a C
∞
function for η sufficiently small, we can substitute it into the definition of weak solution to obtain
Z Z
Ω
T
− u
i
∂ ∂t
{u
i,η
f |u
η
|ζ
p k
}
η
dx dt +
Z Z
Ω
T
A
ij
x, t, u, ∇u ∂
∂x
j
{u
i,η
f |u
η
|ζ
p k
}
η
dx dt =
Z Z
Ω
T
B
i
x, t, u, ∇u {u
i,η
f |u
η
|ζ
p k
}
η
dx dt. 5
For convenience of notation, we rewrite 5 in compact form as I
1
+ I
2
= I
3
, and discuss each of these terms in turn.
2.2 Estimate of I
1
We begin by using the symmetry of the mollifying kernel, and integration by parts to rewrite I
1
as I
1
= − Z Z
Ω
T
u
i,η
∂ ∂t
{u
i,η
f |u
η
|ζ
p k
} dx dt =
Z Z
Q
τ R
∂ ∂t
u
i,η
u
i,η
f |u
η
|ζ
p k
dx dt. We then notice that summing over the index i implies
X
i
u
i,η
∂ ∂t
u
i,η
= 1
2 X
i
∂ ∂t
u
i,η 2
= 1
2 ∂
∂t |u
η
|
2
= |u
η
| ∂
∂t |u
η
|, 6
and we derive I
1
= Z Z
Q
τ R
|u
η
| ∂
|u
η
| ∂t
f |u
η
|ζ
p k
dx dt. If we now let k
→ ∞, thanks to the uniform convergence of ζ
k
→ ζ, and the smooth- ness of the mollified functions we obtain
lim
k→∞
I
1
= Z Z
Q
τ R
|u
η
| ∂
|u
η
| ∂t
f |u
η
|ζ
p
dx dt.
EJQTDE, 2004 No. 14, p. 5
Proceeding in a standard fashion, we rewrite the integral on the right hand side as Z Z
Q
τ R
∂ ∂t
Z
|u
η
|
sf s ds
ζ
p
dx dt =
Z Z
Q
τ R
∂ ∂t
Z
|u
η
|
sf s ds
ζ
p
dx dt − p
Z Z
Q
τ R
Z
|u
η
|
sf s ds
ζ
p−1
ζ
t
dx dt, and applying integration by parts, since ζ
= 0 on t = −R
p
, we gather lim
k→∞
I
1
= Z
B
R
Z
|u
η
|
sf s ds
ζ
p
dx
t=τ
− p Z Z
Q
τ R
Z
|u
η
|
sf s ds
ζ
p−1
ζ
t
dx dt. 7
We would like to take the limit for η ↓ 0 in 7, and we are able to do so, since from
Z
|u
η
|
sf s ds −
Z
|u|
sf s ds
= Z
|u
η
| |u|
sf s ds
≤ γ
1
|u
η
|
2
− |u|
2
, with γ
1
=
1 2
max |f |, we can conclude Z
B
R
Z
|u
η
|
sf s ds −
Z
|u|
sf s ds
ζ
p
dx
t=τ
≤ γ
1
Z
B
R
|u
η
|
2
− |u|
2
dx
t=τ η↓0
−−− −−→ 0
for a.e. τ , and Z Z
Q
τ R
Z
|u
η
|
sf s ds −
Z
|u|
sf s ds
ζ
p−1
ζ
t
dx dt ≤ γ
2
Z Z
Q
τ R
|u
η
|
2
− |u|
2
dx dt
η↓0
−−− −−→ 0,
where γ
2
is a constant that depends on σ, R and p. Note that the above limits are zero due to the fact that u
∈ L
∞,loc
0, T ; L
2,loc
Ω. In conclusion, we have the following estimate
lim
η↓0
lim
k→∞
I
1
= Z
B
R
Z
|u|
sf s ds
ζ
p
dx
t=τ
− p Z Z
Q
τ R
Z
|u|
sf s ds
ζ
p−1
ζ
t
dx dt. 8
EJQTDE, 2004 No. 14, p. 6
2.3 Estimate of I
