Escher J., Guidotti P., Hieber M. et al. (editors) Parabolic Problems: The Herbert Amann Festschrift
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398 N.V. Krylov
Definition 4.1. Denote by D(S, T )thesetofallD-valued functions u (written as
u
t
(x) in a common abuse of notation) on [S, T ] ∩R such that, for any φ ∈ C
∞
0
,the
function (u
t
,φ) is measurable. Denote by D
1
(S, T ) the subset of D(S, T ) consisting
of u such that, for any φ ∈ C
∞
0
, R ∈ (0, ∞), and finite t
1
,t
2
∈ [S, T ] such that
t
1
<t
2
we have
t
2
t
1
sup
|x|≤R
|(u
t
,φ(·−x))|dt < ∞. (4.1)
Definition 4.2. Let f, u ∈ D(S, T ). We say that the equation
∂
t
u
t
(x)=f
t
(x),t∈ [S, T ] ∩R, (4.2)
holds in the sense of distributions if f ∈ D
1
(S, T )andforanyφ ∈ C
∞
0
for all finite
s, t ∈ [S, T ]wehave
(u
t
,φ)=(u
s
,φ)+
t
s
(f
r
,φ) dr.
Let x
t
be an R
d
-valued function given by
x
t
=
t
0
ˆ
b
s
ds,
where
ˆ
b
s
is an R
d
-valued locally integrable function on R. Here is the formula.
Theorem 4.3. Let f,u ∈ D(S, T ).Introduce
v
t
(x)=u
t
(x + x
t
)
and assume that (4.2) holds (in the sense of distributions).Then
∂
t
v
t
(x)=f
t
(x + x
t
)+
ˆ
b
i
t
D
i
v
t
(x),t∈ [S, T ] ∩ R
(in the sense of distributions).
Corollary 4.4. Under the assumptions of Theorem 4.3 for any η ∈ C
∞
0
we have
∂
t
[u
t
(x)η(x − x
t
)] = f
t
(x)η(x − x
t
) −u
t
(x)
ˆ
b
i
t
D
i
η(x − x
t
),t∈ [S, T ] ∩ R.
Indeed, what we claim is that for any φ ∈ C
∞
0
and finite s, t ∈ [S, T ]
((u
t
φ)(· + x
t
),η)=(u
s
φ, η)+
t
s
5
f
r
φ +
ˆ
b
i
r
D
i
(u
r
φ)
6
(·+ x
r
),η
dr.
However, to obtain this result it suffices to write down an obvious equation for
u
t
φ, then use Theorem 4.3 and, finally, use Definition 4.2 to interpret the result.
5. Proof of Theorem 3.1
Throughout this section we suppose that Assumptions 3.1, 3.2, and 3.3 are satisfied
(with a γ ∈ (0, 1]) and start with analyzing the integral in (2.3). Recall that q was
introduced before Assumption 3.2.
Divergence Equations with Growing Coefficients 399
Lemma 5.1. Let 1 ≤ r<pand
η := 1 +
d
p
−
d
r
≥ 0 (5.1)
with strict inequality if r =1. Then for any U ∈L
r
and ε>0 there exist V
j
∈L
p
,
j =0, 1,...,d, such that U = D
i
V
i
+ V
0
and
d
!
j=1
V
j
L
p
≤ N (d, p, r)ε
η/(1−η)
U
L
r
, V
0
L
p
≤ N (d, p, r)ε
−1
U
L
r
. (5.2)
In particular, for any w ∈ W
1
p
|(U, w)|≤N(d, p, r)U
L
r
w
W
1
p
.
Proof. Iftheresultistrueforε = 1, then for arbitrary ε>0 it is easily obtained
by scaling. Thus let ε = 1 and denote by R
0
(x)thekernelof(1− Δ)
−1
.For
i =1,...,d set R
i
= −D
i
R
0
. One knows (see, for instance, Theorem 12.7.1 of
[12]) that R
j
(x) decrease faster than |x|
−n
for any n>0as|x|→∞(actually,
exponentially fast) and (see, for instance, Theorem 12.7.4 of [12]) that for all x =0
|R
j
(x)|≤
N
|x|
d−1
,j=0, 1,...,d.
Define
V
j
= R
j
∗U, j =0, 1,...,d.
If r = 1, one obtains (5.2) from Young’s inequality since, owing to the strict
inequality in (5.1), we have p<d/(d − 1), so that R
j
∈L
p
.Ifr>1, then for ν
defined by
1
p
=
1
r
−
ν
d
we have ν ∈ (0, 1], so that
|R
j
(x)|≤
N
|x|
d−ν
,j=0, 1,...,d,
and we obtain (5.2) from the Sobolev-Hardy-Littlewood inequality (see, for in-
stance, Lemma 13.8.5 of [12]). After this it only remains to notice that in the
sense of generalized functions
D
i
V
i
+ V
0
= R
0
∗ U − ΔR
0
∗ U = U.
The lemma is proved.
Observe that by H¨older’s inequality for r = pq/(p + q)(∈ [1,p) due to q ≥ p
,
see (3.1)) we have
hv
L
r
≤h
L
q
v
L
p
.
Furthermore, if r =1,thenq = p
>d(see (3.1)), p<d/(d − 1), and η>0. In
this way we come to the following.
400 N.V. Krylov
Corollary 5.2. Let h ∈L
q
, v ∈L
p
,andw ∈ W
1
p
. Then for any ε>0 there exist
V
j
∈L
p
, j =0, 1,...,d, such that hv = D
i
V
i
+ V
0
and
d
!
j=1
V
j
L
p
≤ N (d, p)ε
(q−d)/d
h
L
q
v
L
p
,
V
0
L
p
≤ N (d, p)ε
−1
h
L
q
v
L
p
.
In particular,
|(hv, w)|≤N(d, p)h
L
q
v
L
p
w
W
1
p
. (5.3)
Lemma 5.3. Let h ∈L
q
and u ∈ W
1
p
. Then for any ε>0 we have
hu
L
p
≤ N(d, p)h
L
q
ε
(q−d)/d
Du
L
p
+ ε
−1
u
L
p
. (5.4)
Proof. As above it suffices to concentrate on ε =1.Incaseq>pobserve that by
H¨older’s inequality
hu
L
p
≤h
L
q
u
L
s
,
where s = pq/(q−p). After that it only remains to use embedding theorems (notice
that 1 − d/p ≥−d/s since q ≥ d). In the remaining case q = p, which happens
only if p>d(see (3.1)). In that case the above estimate remains true if we set
s = ∞. The lemma is proved.
Before we extract some consequences from the lemma we take a nonnegative
ξ ∈ C
∞
0
(B
ρ
1
) with unit integral and define
¯
b
s
(x)=
B
ρ
1
ξ(y)b
s
(x −y) dy,
¯
b
s
(x)=
B
ρ
1
ξ(y)b
s
(x − y) dy,
¯c
s
(x)=
B
ρ
1
ξ(y)c
s
(x − y) dy. (5.5)
We may assume that |ξ|≤N(d)ρ
−d
1
.
One obtains the first assertion of the following corollary from (5.3) by ob-
serving that
I
B
ρ
1
(x
t
)
(b
t
−
¯
b
t
(x
t
))
q
L
q
=
B
ρ
1
(x
t
)
|b
t
−
¯
b
t
(x
t
)|
q
dx
=
B
ρ
1
(x
t
)
"
"
"
"
"
B
ρ
1
(x
t
)
[b
t
(x) − b
t
(y)]ξ(x
t
−y) dy
"
"
"
"
"
q
dx
≤ N
B
ρ
1
(x
t
)
"
"
"
"
"
ρ
−d
1
B
ρ
1
(x
t
)
|b
t
(x) − b
t
(y)|dy
"
"
"
"
"
q
dx
≤ Nρ
−d
1
B
ρ
1
(x
t
)
B
ρ
1
(x
t
)
|b
t
(x) − b
t
(y)|
q
dy dx
≤ Nρ
−d
1
KI
q>d
+ Nγ. (5.6)
Divergence Equations with Growing Coefficients 401
The second assertion follows from estimates like (5.6) and (5.4) where one
chooses ε appropriately if q>d.
Corollary 5.4. Let u ∈ W
1
p
(S, T ),letx
s
be an R
d
-valued measurable function, and
let η ∈ C
∞
0
(B
ρ
1
).Setη
s
(x)=η(x − x
s
),
K
1
=sup|η| +sup|Dη|.
Then on (S, T )
(i) For any w ∈ W
1
p
and v ∈L
p
(|b
s
−
¯
b
s
(x
s
)|η
s
v, |w|) ≤ N (d, p, ρ
1
,K)η
s
v
L
p
w
W
1
p
;
(ii) We have
η
s
|b
s
−
¯
b
s
(x
s
)|u
s
L
p
+ η
s
|c
s
− ¯c
s
(x
s
)|u
s
L
p
≤ N(d, p)γ
1/q
η
s
Du
s
L
p
+ N(d, p, γ, ρ
1
,K,K
1
)I
B
ρ
1
(x
s
)
u
s
L
p
.
(iii) Almost everywhere on (S, T ) we have
(b
i
s
−
¯
b
i
s
(x
s
))η
s
D
i
u
s
= D
i
V
i
s
+ V
0
s
, (5.7)
d
!
j=1
V
j
L
p
≤ N (d, p)γ
1/q
η
s
Du
s
L
p
,
V
0
s
L
p
≤ N (d, p, γ, ρ
1
,K)η
s
Du
s
L
p
, (5.8)
where V
j
s
, j =0,...,d, are some measurable L
p
-valued functions on (S, T ).
To prove (iii) observe that one can find a Borel set A ⊂ (S, T )offullmeasure
such that I
A
D
i
u, i =1,...,d, are well defined as L
p
-valued Borel measurable func-
tions. Then (5.7) with I
A
D
i
u in place of D
i
u and (5.8) follow from (5.6), Corollary
5.2, and the fact that the way V
j
are constructed uses bounded hence continuous
operators and translates the measurability of the data into the measurability of the
result. Since we are interested in (5.7) and (5.8) holding only almost everywhere
on (S, T ), there is no actual need for the replacement.
Corollary 5.5. Let u ∈ W
1
p
(S, T ), R ∈ (0, ∞), φ ∈ C
∞
0
(B
R
), and let finite S
,T
∈
(S, T ) be such that S
<T
. Then there is a constant N independent of u and φ
such that
T
S
(|(b
i
s
D
i
u
s
,φ)| + |(b
i
s
u
s
,D
i
φ)| + |(c
s
u
s
,φ)|) ds ≤ Nu
W
1
p
(S,T )
φ
W
1
p
, (5.9)
so that requirement (i) in Definition 2.3 can be dropped.
Proof. By having in mind partitions of unity we convince ourselves that it suffices
to prove (5.9) under the assumption that φ has support in a ball B of radius ρ
1
.Let
402 N.V. Krylov
x
0
be the center of B and set x
s
≡ x
0
. Observe that the estimates from Corollary
5.4 imply that
|(b
i
s
u
s
,D
i
φ)|≤|(b
i
s
−
¯
b
i
s
(x
0
))u
s
,D
i
φ)| + |
¯
b
i
s
(x
0
)(u
s
,D
i
φ)|
≤ N u
s
W
1
p
φ
W
1
p
+ |
¯
b
s
(x
0
)|u
s
W
1
p
φ
W
1
p
.
By recalling Assumption 3.1 (iii) and H¨older’s inequality we get
T
S
|(b
i
s
u
s
,D
i
φ)|ds ≤ Nu
W
1
p
(S,T )
φ
W
1
p
.
Similarly the integrals of |(b
i
s
D
i
u
s
,φ)| and |(c
s
u
s
,φ)| are estimated and the
corollary is proved.
Since bounded linear operators are continuous we obtain the following.
Corollary 5.6. Let φ ∈ C
∞
0
, T ∈ (0, ∞). Then the operators
u
·
→
·
0
(b
i
t
D
i
u
t
,φ) dt, u
·
→
·
0
(b
i
t
u
t
,D
i
φ) dt, u
·
→
·
0
(c
t
u
t
,φ) dt
are continuous as operators from W
1
p
(∞) to L
p
([−T,T]).
This result will be used in Section 6.
Before we continue with the proof of Theorem 3.1, we notice that, if u ∈
W
1
p
(S, T ), then as we know (see, for instance, Theorem 2.1 of [14]), the function
u
t
is a continuous L
p
-valued function on [S, T ] ∩ R.
Now we are ready to prove Theorem 3.1 in a particular case.
Lemma 5.7. Let b
i
, b
i
,andc be independent of x and let S = −∞. Then the
assertion of Theorem 3.1 holds, naturally, with λ
0
= λ
0
(d, δ, p, ρ
0
)(independent of
ρ
1
and K).
Proof. First let c ≡ 0. We want to use Theorem 4.3 to get rid of the first-order
terms. Observe that (2.1) reads as
∂
t
u
t
= D
i
(a
ij
t
D
j
u
t
+[b
i
t
+ b
i
t
]u
t
+ f
i
t
)+f
0
t
− λu
t
,t≤ T. (5.10)
Recall that from the start (see Definition 2.3) it is assumed that u ∈ W
1
p
(T ).
Then one can find a Borel set A ⊂ (−∞,T) of full measure such that I
A
f
j
, j =
0, 1,...,d,andI
A
D
i
u, i =1,...,d, are well defined as L
p
-valued Borel functions
satisfying
T
−∞
I
A
⎛
⎝
d
!
j=0
f
j
t
p
L
p
+ Du
t
p
L
p
⎞
⎠
dt < ∞.
Replacing f
j
and D
i
u in (5.10) with I
A
f
j
and I
A
D
i
u, respectively, will not affect
(5.10). Similarly one can treat the term h
t
=(b
i
t
+ b
i
t
)u
t
for which
T
S
h
t
L
p
dt < ∞
Divergence Equations with Growing Coefficients 403
for each finite S
,T
∈ (−∞,T],owingtoAssumption3.1andthefactthatu ∈
L
p
(T ).
After these replacements all terms on the right in (5.10) will be of class
D
1
(−∞,T)sincea is bounded. This allows us to apply Theorem 4.3 and for
B
i
t
=
t
0
(b
i
s
+ b
i
s
) ds, ˆu
t
(x)=u
t
(x − B
t
)
obtain that
∂
t
ˆu
t
= D
i
(ˆa
ij
t
D
j
ˆu
t
) − λˆu
t
+ D
i
ˆ
f
i
t
+
ˆ
f
0
t
, (5.11)
where
(ˆa
ij
t
,
ˆ
f
j
t
)(x)=(a
ij
t
,f
j
t
)(x − B
t
).
Obviously, ˆu is in W
1
p
(T ) and its norm coincides with that of u.Equation
(5.11) shows that ˆu ∈W
1
p
(T ).
By Theorem 4.4 and Remark 2.4 of [11] there exist γ = γ(d, δ, p)andλ
0
=
λ
0
(d, δ, p, ρ
0
) such that if λ ≥ λ
0
,then
Dˆu
L
p
(T )
+ λ
1/2
ˆu
L
p
(T )
≤ N
3
d
!
i=1
ˆ
f
i
L
p
(T )
+ λ
−1/2
ˆ
f
0
L
p
(T )
4
. (5.12)
Actually, Theorem 4.4 of [11] is proved there only for T = ∞, but it is a standard
fact that such an estimate implies what we need for any T (cf. the proof of Theorem
6.4.1 of [12]). Since the norms in L
p
and W
1
p
are translation invariant, (5.12) implies
(3.3) and finishes the proof of the lemma in case c ≡ 0.
Our next step is to abandon the condition c ≡ 0 but assume that for an
S>−∞ we have u
t
= f
j
t
=0fort ≤ S. Observe that without loss of generality
we may assume that T<∞. In that case introduce
ξ
t
=exp
t
S
c
s
ds
.
Then we have v := ξu ∈ W
1
p
(T )and
∂
t
v
t
= D
i
(a
ij
t
D
j
v
t
+[b
i
t
+ b
i
t
]v
t
+ ξ
t
f
i
t
)+ξ
t
f
0
t
−λv
t
,t≤ T.
By the above result for all T
≤ T
T
−∞
ξ
p
t
Du
t
p
L
p
dt + λ
p/2
T
−∞
ξ
p
t
u
t
p
L
p
dt
≤ N
1
d
!
i=0
T
−∞
ξ
p
t
f
i
t
p
L
p
dt + N
1
λ
−p/2
T
−∞
ξ
p
t
f
0
t
p
L
p
dt. (5.13)
We multiply both part of (5.13) by pc
T
ξ
−p
T
and integrate with respect to T
over
(S, T ). We use integration by parts observing that both parts vanish at T
= S.
404 N.V. Krylov
Then we obtain
T
−∞
Du
t
p
L
p
dt + λ
p/2
T
−∞
u
t
p
L
p
dt
− ξ
−p
T
T
−∞
ξ
p
t
Du
t
p
L
p
dt −ξ
−p
T
λ
p/2
T
−∞
ξ
p
t
u
t
p
L
p
dt
≤ N
1
d
!
i=0
T
−∞
f
i
t
p
L
p
dt + N
1
λ
−p/2
T
−∞
f
0
t
p
L
p
dt
− ξ
−p
T
N
1
d
!
i=0
T
−∞
ξ
p
t
f
i
t
p
L
p
dt − ξ
−p
T
N
1
λ
−p/2
T
−∞
ξ
p
t
f
0
t
p
L
p
dt.
By adding up this inequality with (5.13) with T
= T multiplied by ξ
−p
T
we ob-
tain (3.3).
The last step is to avoid assuming that u
t
= 0 for large negative t.Inthat
case we find a sequence S
n
→−∞such that u
S
n
→ 0inW
1
p
and denote by v
n
t
the unique solution of class W
1
p
((0, 1) ×R
d
) of the heat equation ∂v
n
t
=Δv
n
t
with
initial condition u
S
n
. After that we modify u
t
and the coefficients of L
t
for t ≤ S
n
as in the proof of Theorem 3.2 by taking there v
n
t
and S
n
in place of v
t
and S,
respectively. Then by the above result we obtain
λu
2
L
p
(S
n
,T )
+ Du
2
L
p
(S
n
,T )
≤ N
3
d
!
i=1
˜
f
i
2
L
p
(T )
+ λ
−1
˜
f
0
2
L
p
(T )
4
,
≤ N
3
d
!
i=1
f
i
2
L
p
(T )
+ λ
−1
f
0
2
L
p
(T )
4
+ N(1 + λ
−1
)u
S
n
p
W
1
p
.
By letting n →∞we come to (3.3) and the lemma is proved.
Remark 5.1. In [11] the assumption corresponding to Assumption 3.3 is much
weaker since in the corresponding counterpart of (3.2) there is no supremum over
x ∈ R
d
. We need our stronger assumption because we need a
ij
t
(x − B
t
)tosatisfy
the assumption in [11] for any function B
t
.
To proceed further we need a construction. Recall that
¯
b and
¯
b are introduced
in (5.5). From Lemma 4.2 of [13] and Assumption 3.2 it follows that, for h
t
=
¯
b
t
,
¯
b
t
,
it holds that |D
n
h
t
|≤κ
n
,whereκ
n
= κ
n
(n, d, p, ρ
1
,K) ≥ 1andD
n
h
t
is any
derivative of h
t
of order n ≥ 1 with respect to x. By Corollary 4.3 of [13] we have
|h
t
(x)|≤K(t)(1 + |x|), where the function K(t) is locally integrable with respect
to t on R. Owing to these properties, for any (t
0
,x
0
) ∈ R
d+1
, the equation
x
t
= x
0
−
t
t
0
(
¯
b
s
+
¯
b
s
)(x
s
) ds, t ≥ t
0
,
has a unique solution x
t
= x
t
0
,x
0
,t
.
Divergence Equations with Growing Coefficients 405
Next, for i =1, 2setχ
(i)
(x) to be the indicator function of B
ρ
1
/i
and intro-
duce
χ
(i)
t
0
,x
0
,t
(x)=χ
(i)
(x − x
t
0
,x
0
,t
).
Here is a crucial estimate.
Lemma 5.8. Suppose that Assumptions 3.1, 3.2,and3.3 are satisfied with a γ ∈
(0,γ(d, p, δ)],whereγ(d, p, δ) is taken from Lemma 5.7.Take(t
0
,x
0
) ∈ R
d+1
and
assume that t
0
<T and that we are given a function u which is a solution of (2.1)
with S = t
0
, with zero initial condition, some f
j
∈ L
p
(t
0
,T),andλ ≥ λ
0
,where
λ
0
= λ
0
(d, δ, p, ρ
0
) is taken from Lemma 5.7.Then
λχ
(2)
t
0
,x
0
u
2
L
p
(t
0
,T )
+ χ
(2)
t
0
,x
0
Du
2
L
p
(t
0
,T )
≤ N
d
!
i=1
χ
(1)
t
0
,x
0
f
i
2
L
p
(t
0
,T )
+ Nλ
−1
χ
(1)
t
0
,x
0
f
0
2
L
p
(t
0
,T )
+ Nγ
2/q
χ
(1)
t
0
,x
0
Du
2
L
p
(t
0
,T )
+ N
∗
λ
−1
χ
(1)
t
0
,x
0
Du
2
L
p
(t
0
,T )
+ N
∗
χ
(1)
t
0
,x
0
u
2
L
p
(t
0
,T )
+ N
∗
λ
−1
d
!
i=1
χ
(1)
t
0
,x
0
f
i
2
L
p
(t
0
,T )
, (5.14)
where and below in the proof by N we denote generic constants depending only on
d, δ,andp and by N
∗
constants depending only on the same objects, γ, ρ
1
,andK.
Proof. Shifting the origin allows us to assume that t
0
=0andx
0
=0.Withthis
stipulations we will drop the subscripts t
0
,x
0
.
Fix a ζ ∈ C
∞
0
with support in B
ρ
1
and such that ζ =1onB
ρ
1
/2
and
0 ≤ ζ ≤ 1. Set x
t
= x
0,0,t
,
ˆ
b
t
=
¯
b
t
(x
t
),
ˆ
b
t
=
¯
b
t
(x
t
), ˆc
t
=¯c
t
(x
t
)
η
t
(x)=ζ(x − x
t
),v
t
(x)=u
t
(x)η
t
(x).
The most important property of η
t
is that
∂
t
η
t
=(
ˆ
b
i
t
+
ˆ
b
i
t
)D
i
η
t
.
Also observe for the later that we may assume that
χ
(2)
t
≤ η
t
≤ χ
(1)
t
, |Dη
t
|≤Nρ
−1
1
χ
(1)
t
, (5.15)
where χ
(i)
t
= χ
(i)
0,0,t
and N = N(d).
By Corollary 4.4 (also see the argument before (5.11)) we obtain that for
finite t ∈ [0,T]
∂
t
v
t
= D
i
(η
t
a
ij
t
D
j
u
t
+ b
i
t
v
t
) − (a
ij
t
D
j
u
t
+ b
i
t
u
t
)D
i
η
t
+ b
i
t
η
t
D
i
u
t
− (c
t
+ λ)v
t
+ D
i
(f
i
t
η
t
) −f
i
t
D
i
η
t
+ f
0
t
η
t
+(
ˆ
b
i
t
+
ˆ
b
i
t
)u
t
D
i
η
t
.
We transform this further by noticing that
η
t
a
ij
t
D
j
u
t
= a
ij
t
D
j
v
t
− a
ij
t
u
t
D
j
η
t
.
406 N.V. Krylov
To deal with the term b
i
t
η
t
D
i
u
t
we use Corollary 5.4 and find the correspond-
ing functions V
j
t
. Then simple arithmetics show that
∂
t
v
t
= D
i
a
ij
t
D
j
v
t
+
ˆ
b
i
t
v
t
−(ˆc
t
+ λ)v
t
+
ˆ
b
i
t
D
i
v
t
+ D
i
ˆ
f
i
t
+
ˆ
f
0
t
,
where
ˆ
f
0
t
= f
0
t
η
t
−f
i
t
D
i
η
t
−a
ij
t
(D
j
u
t
)D
i
η
t
+(
ˆ
b
i
t
− b
i
t
)u
t
D
i
η
t
+ V
0
t
+(ˆc
t
−c
t
)u
t
η
t
,
ˆ
f
i
t
= f
i
t
η
t
−a
ij
t
u
t
D
j
η
t
+(b
i
t
−
ˆ
b
i
t
)u
t
η
t
+ V
i
t
,i=1,...,d.
It we extend u
t
and f
j
t
as zero for t<0, then it will be seen from Lemma 5.7
that for λ ≥ λ
0
λv
2
L
p
(0,T )
+ Dv
2
L
p
(0,T )
≤ N
d
!
i=1
ˆ
f
i
2
L
p
(0,T )
+ Nλ
−1
ˆ
f
0
2
L
p
(0,T )
. (5.16)
Recall that here and below by N we denote generic constants depending only on
d, δ,andp.
Now we start estimating the right-hand side of (5.16). First we deal with
ˆ
f
i
t
.
Recall (5.15) and use Corollary 5.4 to get
(b
i
t
−
ˆ
b
i
t
)u
t
η
t
2
L
p
≤ Nγ
2/q
χ
(1)
t
Du
t
2
L
p
+ N
∗
χ
(1)
t
u
t
2
L
p
(5.17)
(we remind the reader that by N
∗
we denote generic constants depending only on
d, δ, p, γ, ρ
1
,andK). By adding that
a
ij
uD
j
η
2
L
p
(0,T )
≤ N
∗
χ
(1)
·
u
2
L
p
(0,T )
,
we derive from (5.8) and (5.17) that
d
!
i=1
ˆ
f
i
2
L
p
(0,T )
≤ N
d
!
i=1
χ
(1)
·
f
i
2
L
p
(0,T )
(5.18)
+ Nγ
2/q
χ
(1)
·
Du
2
L
p
(0,T )
+ N
∗
χ
(1)
·
u
2
L
p
(0,T )
.
While estimating
ˆ
f
0
we use (5.8) again and observe that we can deal with
(
ˆ
b
i
t
− b
i
t
)u
t
D
i
η
t
and (c
t
− ˆc
t
)u
t
η
t
as in (5.17) this time without paying too much
attention to the dependence of our constants on γ, ρ
1
,andK and obtain that
(
ˆ
b
i
−b
i
)uD
i
η
2
L
p
(0,T )
+ (c − ˆc)uη
2
L
p
(0,T )
≤ N
∗
(χ
(1)
·
Du
2
L
p
(0,T )
+ χ
(1)
·
u
2
L
p
(0,T )
).
By estimating also roughly the remaining terms in
ˆ
f
0
and combining this with
(5.18) and (5.16), we see that the left-hand side of (5.16) is less than the right-
hand side of (5.14). However,
|χ
(2)
t
Du
t
|≤|η
t
Du
t
|≤|Dv
t
|+ |u
t
Dη
t
|≤|Dv
t
|+ Nρ
−1
1
|u
t
χ
(1)
t
|
which easily leads to (5.14). The lemma is proved.
Next, from the result giving “local” in space estimates we derive global in
space estimates but for functions having, roughly speaking, small “past” support
Divergence Equations with Growing Coefficients 407
in the time variable. In the following lemma κ
1
is the number introduced before
Lemma 5.8.
Lemma 5.9. Suppose that Assumptions 3.1, 3.2,and3.3 are satisfied with a γ ∈
(0,γ(d, p, δ)],whereγ(d, p, δ) is taken from Lemma 5.7. Assume that u is a solution
of (2.1) with S = −∞,somef
j
∈ L
p
(T ),andλ ≥ λ
0
,whereλ
0
= λ
0
(d, δ, p, ρ
0
)
is taken from Lemma 5.7. Take a finite t
0
≤ T and assume that u
t
=0if t ≤ t
0
.
Then for I
t
0
:= I
(t
0
,T
)
,whereT
=(t
0
+ κ
−1
1
) ∧T , we have
λ
p/2
I
t
0
u
p
L
p
+ I
t
0
Du
p
L
p
≤ N
d
!
i=1
I
t
0
f
i
p
L
p
+ Nλ
−p/2
I
t
0
f
0
p
L
p
(5.19)
+ Nγ
p/q
I
t
0
Du
p
L
p
+ N
∗
λ
−p/2
I
t
0
Du
p
L
p
+ N
∗
I
t
0
u
p
L
p
+ N
∗
λ
−p/2
d
!
i=1
I
t
0
f
i
p
L
p
,
where and below in the proof by N we denote generic constants depending only on
d, δ,andp and by N
∗
constants depending only on the same objects, γ, ρ
1
,andK.
Proof. Take x
0
∈ R
d
and use the notation introduced before in Lemma 5.8. By
this lemma with T
in place of T we have
λ
p/2
I
t
0
χ
(2)
t
0
,x
0
u
p
L
p
+ I
t
0
χ
(2)
t
0
,x
0
Du
p
L
p
≤ N
d
!
i=1
I
t
0
χ
(1)
t
0
,x
0
f
i
p
L
p
+ Nλ
−p/2
I
t
0
χ
(1)
t
0
,x
0
f
0
p
L
p
+ Nγ
p/q
I
t
0
χ
(1)
t
0
,x
0
Du
p
L
p
+ N
∗
λ
−p/2
I
t
0
χ
(1)
t
0
,x
0
Du
p
L
p
+ N
∗
I
t
0
χ
(1)
t
0
,x
0
u
p
L
p
+ N
∗
λ
−p/2
d
!
i=1
I
t
0
χ
(1)
t
0
,x
0
f
i
p
L
p
. (5.20)
One knows that for each t ≥ t
0
, the mapping x
0
→ x
t
0
,x
0
,t
is a diffeomorphism
with Jacobian determinant given by
"
"
"
"
∂x
t
0
,x
0
,t
∂x
0
"
"
"
"
=exp
−
t
t
0
d
!
i=1
D
i
[
¯
b
i
s
+
¯
b
i
s
](x
t
0
,x
0
,s
) ds
.
By the way the constant κ
1
is introduced, we have
e
−Nκ
1
(t−t
0
)
≤
"
"
"
"
∂x
t
0
,x
0
,t
∂x
0
"
"
"
"
≤ e
Nκ
1
(t−t
0
)
,
where N depends only on d. Therefore, for any nonnegative Lebesgue measurable
function w(x) it holds that
e
−Nκ
1
(t−t
0
)
R
d
w(y) dy ≤
R
d
w(x
t
0
,x
0
,t
) dx
0
≤ e
Nκ
1
(t−t
0
)
R
d
w(y) dy.