DiBenedetto E. Degenerate Parabolic Equations
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116
IV.
HOlder
continuity
of
solutions
of
singular parabolic equations
15.
Bibliographical notes
Theorems
1.1-1.3
were established
in
[26J
for the case
when
the
principal part of
the
operator
is
independent of
t.
This restriction
has
been removed in
[27J
and
entails
the
new
iteration technique presented
in
§6-10.
This technique differs
sub-
stantially
from
the
classical iteration of Moser
[81.82.83J
or DeGiorgi
[33J.
It
ex-
tracts
the
'almost elliptic' nature of
the
singular p.d.e.
as
follows
from
the remarks
in
§14-(i).
We
will
further discuss this point
in
Chap.
VII in
the context of
Har-
nack estimates.
The
method
is
rather flexible and adapts to a variety of singular
parabolic equations. For example
it
implies the
HOlder
continuity of solutions of
singular equations of porous
medium
type.
To
be specific. consider
the
p.d.e.
Ut
- div a(x,
t,
u, Du) + b(x,
t,
u,
Du)
= 0,
in
{h,
with
the
structure conditions
(At>
a(x, t, u,
Du)·Du
~
Co
lul
m
-
1
lDul
2
-
'Po(x,
t),
mE
(0,1),
(A
2
)
la(x,
t,
u,
Du)1
~
C
1
lul
m
-
1
IDul +
'PI
(x, t),
(A
3
)
Ib(x,
t,
u,
Du)1
~
C
2
1 D lul
m
12
+
'P2(X,
t).
We
require that
u E
L::
(0,
Tj
L~oc(
ll») and lul
m
E
L~oc
(
0,
Tj
W,!:(
ll»)
.
The non-negative functions 'Pi, i = 0,1,2. satisfy (A4) - (As) of §I of
Chap.
I
with
P =
2.
Further generalisations can be obtained
by
replacing
sm-l,
S > 0
with
a function
cp(
s) that blows
up
like a power
when
s - 0 and
is
regular otherwise.
Results concerning doubly non-linear equations bearing singularity and/or degen-
eracy
are
due
to
Ivanov
[52.53.54J
and
Vespri
[l02J. A complete theory of doubly
singular equations. however.
is
still lacking.
v
Boundedness
of
weak solutions
1.
Introduction
Let u be a weak solution
of
equations
of
the type
of
(1.1)
of
Chap. II in
aT.
We
will establish local and global bounds for u in
flT.
Global bounds depend
on
the
data prescribed
on
the parabolic boundary
of
flT.
Local bounds
are
given in tenns
of
local integral nonns
of
u.
Consider the cubes Kp C K
2p
. After a translation we
may assume they are contained in
fl
provided p is sufficiently small. For 0
~
tl
<
to
< t
~
T consider the cylindrical domains
The local estimates are
of
the type.
(1.1)
lIulloo,Q.
,;
~
(
1
+
If
lUI'.)
..
,
where the numbers qi, i =
0,
1,
are
detennined a priori in tenns
of
p and N and the
constant
'Y
is detennined a priori in terms
of
the structure conditions
of
the p.d.e.
and
Ql.
Unlike the elliptic theory, the estimate (1.1) discriminates between the degen-
erate case
p > 2 and the singular case 1 < p <
2.
To illustrate this point, consider
local weak solutions
of
the elliptic equation
{
u
E w:l'''(fl)
loe
,
div
IDul,,-2
Du
=
0,
p>l
in
fl.
118
V.
Boundedness
of
weak
solutions
These solutions satisfy the following estimate for any p >
1.
For every e > ° there
exists a constant
'Y
=
'Y
(N,p,e), such that
Consider now the corresponding parabolic equation
{
u
E
Gloe
(0,
Tj
L~oc(D))
nLfoe
(0,
Tj
WI!;: (D)) , p >
1,
(1.2)
Ut
- div IDulp-2
Du
=
0,
in
DT,
and the cylindrical domain
Q~p
==
K
2p
X (0,
tl.
Assume
fmt
that p > 2. Then for
all
eE
(0,
21
there exists a constant 'Y='Y (N,p,
e)
such that for all
tt~s~t
For the singular case 1 < p < 2 a local sup-estimate can be derived only
if
u is
$llfficiently integrable. Introduce the numbers
(1.4)
A,.
==
N(P-
2)
+rp,
r
~
1,
and assume that u E
Lfoe(DT)
for some r
~
1 such that
A,.
> 0. Then there exist a
constant
'Y
=
'Y(
N,
p,
r) such that for all t t < 8 = t there holds
When 1
< p < 2 such an order
of
local integrability is not implicit in the notion
of
weak solution and it must be imposed. The counterexample
of
§ 12
of
Chap. XII
shows that it is sharp.
2.
Quasilinear parabolic equations
Consider quasilinear evolution equations
of
the type
(2.1)
Ut
-diva(x,t,u,Du)
=
b(x,t,u,Du)
in'D'(D
T
).
The functions
a:
nT
x R N
+1
-+
R N and
b:
nT
x
RN
+1
-+
R,
are measurable and
satisfy
2.
Quasilinear parabolic
equations
119
(B
l
) a(x,
t,
u,
Du) . Du
~
GoIDul"
-
Colul
6
-
rpo(x,
t),
(B
2
) la(x,
t,
u,
Du)1
:$
GlIDul,,-l +
cllul6~
+
rpl(X,
t),
(B3)
Ib(x,
t, u,
Du)1
:$
G2IDul"¥
+
c2lul
6
-
l
+
rp2(X,
t)
for p > 1 and a.e. (x, t) E
il
T
.
Here G
i
,
C;,
i =
0,1,2.
are
positive constants and 6
is
in
the range
N+2
p:$6<p~.
The non-negative functions
rpi,
i=O,
1,2.
are
defined in ilT and satisfy
(Bs)
where
1 p
- =
(I-It
)--
q 0
N+p'
Ito
E (0,1].
For
fELl
(ilT) and h E (0, T) we let
!h
denote the Steklov average
of
f. A
function
u is a
local
weak
sub(super)-solution
of
(2.1)
in
ilT
if
(2.2) u E
Gloc
(0,
Tj
L~oc{il»
nLfoc
(0,
Tj WI!:(il») ,
and for every compact subset
IC
of
il.
(2.3)
f
{!
uhrp+[a(x,
T,
u,
DU)]h
·Drp-
Ibex,
T,
u,
DU)]h
rp}
dx:$
(?)o
K:x{t}
for all ° < t
:$
T - h and all testing functions
(2.4)
rp
E
Gloc
(O,TjL2(1C»)nLfoc
(O,TjWJ'''(IC» ,
rp
~
0.
The statement that a constant
'Y
=
'Y
(data) depends only upon the data means
that it can be determined a priori only in terms
of
the numbers N, p,
q,
6,
Ito.
the
constants
G
i
,
C;,
i=O,
1,2. and the norms
~
8-1
IIrpo,
rpr
,rp;r
114,UT'
2-(i). The Dirichlet problem
Consider the boundary value problem
{
Ut
-
div
a(x, t, u, Du) = b(x, t, u, Du),
in
ilT,
u(·,
t)18U
= g(., t), a.e.
t.E
(0, T),
u(·,O) = u
o
,
(2.S)
120
V.
Boundedness
of
weak
solutions
We
retain the structure conditions (Bd-(B6). and on the Dirichlet data 9 and 1£0
we assume
(2.6)
(2.7)
9 E L
oo
(BT),
1£0 E
L2(n).
The notion
of
weak solution is in (2.5)
of
Chap. II.
Remark
2.1. Unlike the assumption
(U
o
)
in Chap. II. we do not assume here that
1£0 E Loo(n). Accordingly. our estimates
of
the norms 111£(', t)lIoo.n deteriorate as
t'\,O.
2-(ii).
Homogeneous
structures
Local and global sup-bounds. take an elegant form for solutions
of
equations
of
the type
(2.8)
(2.9)
Ut
-
diva(x,t,u,Du)
= 0, in
nT,
P>
1
{
a{x,t,u,Du)·
Du
~
CoIDuI
P
,
la{x,
t,
u,
Du)1
~
CtIDul
p
-
t
,
for two given constants 0
< Co
~
Ct.
The lower order terms are zero and the
principal part has the same structure as
(2.10)
Ut
-
div
IDul
p
-
2
Du
= 0
or
Ut
-
(IU:r:iIP-2u:r:;),c;
=
O.
Because
of
the structural analogy with (2.10) we will refer to (2.8)-(2.9) as equa-
tions with homogeneous structure.
3.
Sup-hounds
We
let u be a non-negative weak subsolution
of
(2.1) and will state several upper
bounds for it. The assumption that u is non-negative is not essential and is used
here only to deduce that u is locally or globally bounded.
If
u is a subsolution.
not necessarily bounded below. our results supply a priori bounds
above
for u.
Analogous statements hold for non-positive local supersolutions and in particular
for solutions.
The estimates
of
this section hold for P in the range
(3.1)
p>max{l;
::2}'
i.e
.•
~2=N(p-2)+2p>0
The case \ < p
~
max { 1;
;~2}
will be discussed in §5. Let 6 and
"'0
be the
numbers appearing in the structure conditions
(Bl)-(B6) and set
(3.2)
N+2
q=p--,
N
The range
of
6 in
(B4)
is
p:$
6 <
q.
We will assume that
(3.3)
max{pj
2}
:$
6 <
q.
3. Sup-bounds
121
This is
no
loss
of
generality
by
possibly modifying the constants c; and the func-
tions
'Pi, i = 0, 1, 2. We also observe that owing to (3.1), the range (3.3)
of
6 is
non-empty. In the theorems below
we
will establish local
or
global bounds for
s0-
lutions
of
(2.1). However precise quantitative estimates will
be
given only for the
case
(3.4)
'Pi E LOO(fh),
i =
0,1,2.
In this case we may take
Ko
= 1 in (B6) and
K=p/N.
3-(i). Local estimates
THEOREM
3.1.
Let (3.1) hold. Every non-negative. local weak subsolution U
0/
(2.1) in
DT
is locally bounded in D
T
. Moreover.
if
'Pi E
LOO(DT)'
i =
0,1,2.
there exists a constant
'Y
=
'Y
(data) such that V
[(xo,
to)
+ Q
(PP,
p)]
CDT
and
VUE(O,I).
(3.5)
sup
U
[(zo.to)+Q(upP
,up»)
:$'Y((I-u)-(N+P)+IQ(PP,p)I)~
(
ffU6dXdT)~
1\1.
[(zo,to)+Q(pp
,p»)
3-(ii). Global estimates: Dirichlet
data
THEOREM
3.2. Let u
be
a non-negative weak subsolution
o/the
Dirichlet prob-
lem
(2.5)
and
let (2.6) hold. Then u is bounded in
Dx(e,
T),
Ve
E (0, T). Moreover,
if'Pi E
LOO(D
T
),
i=O,
1,2.
there exists a constant
'Y
=
'Y
(data). such
that/or
all
O<t:$T.
(3.6)
~pu(.,
t)
$0:,"
9 + 1
(ti
+
D,!.
(If
U'dxdT)
i"J
AI
/fin
addition the initial datum U
o
is bounded above. then
(3.6)
~
u(·,
t)
"
max
{s:,"
g;
~P'"
} + 1
(l!
u'
dxdr
) i"J A
1.
122
V.
Boundedness
of
weak
solutions
THEOREM
3.3
(THE
WEAK MAXIMUM
PRINCIPLE).
Let u be a non-negative
weak subsolution
of
the Dirichlet problem (2.5) for equations with homogeneous
structure as
(2.8)-(2.9).
Then
(3.7)
sup
u
:5
max
{ess
sup
9 i ess
sup
u
o
} .
aT
ST a
Remark 3.1. The weak maximum principle holds for equations with homoge-
neous structure for all
p>
1.
As a particular case, Theorems 3.1-3.3 give a priori sup-estimates for non-
negative weak solutions
of
(3.8)
Ut
-
div
IDuI
P
-
2
Du
=
al
U
6
-
l
+
a2!p,
at
E
R,
i =
1,2,
where
and
N+2
1<6<p--
- N
1 p
"4
= N + p
(1
-
Ito),
Ito
E (0,1].
These conditions on the lower order terms are optimal for a sup-bound to hold as
it can be seen from
the 'linear' case p = 2. Set p = 2 and
a2
= 0
in
(3.8). For a
local weak solution
uE
V,~(flT)
to be locally bounded, 6 must not exceed
2N&2.
Likewise
if
al
=0.
the forcing term II' must satisfy
A
N+2
where q >
-2-.
These are classical and optimal results for the linear case
p=
2 (see [67]).
4.
Homogeneous
structures.
The
degenerate case p > 2
Here we consider non-negative local or global subsolutions
of
equations with the
homogeneous structure (2.8)-(2.9). These structures reveal the basic difference
between the degenerate case
p>
2 and the singular case 1 < p < 2.
4-(i). Local estimates
THEOREM
4.1.
Every non-negative, local weak subsolution u of(2.8)-(2.9)
in
flT
is
locally bounded
in
fl
T
. Moreover for all E E (0,2]
there
exists a constant
'Y
depending only upon
the
data
and
E,
such that V
[(zo,
to)
+ Q
(8,
p)]cfl
T
and
VuE (0, 1),
4.
Homogeneous structures. The degenerate case
p>
2
123
(4.1)
Remark
4.1.
If
9 =
p",
(4.1) is dimensionless but it is not homogeneous in
1.£.
In the linear case p = 2, (4.1) holds for any positive number
e.
In our case, e is
restricted in the range (0,2].
It is
of
interest
to
have sup-estimates that involve
'low'
integral norms
of
the
solution. The next theorem is a result in this direction. Even though it is
of
local
nature, it will be crucial in characterising the class
of
non-negative solutions in the
strip
RN
x (0, T). (1)
THEOREM
4.2.
Let u be a
non-negative,
local
subsolution
u
of
(2.8)-(2.9)
in
n
T
.
There
exists a constant
'Y
=
'Y
(data),
such
that V
[(xo,
to)
+ Q
(9,
p)]
c n
T
and
Vue
(0, 1),
(4.2)
(
)
P/2
'Y";9/PP
sup
1.£
:5
N(p+ll+p
sup
1.£(x,
r)dx
[(zo,t
o
)+Q(0'9,O'p»)
(1
_ u) 2
to-9<T<to
f
[zo+Kp)
(
p")~
1\ 9 .
4-(ii). Global estimates for solutions
of
the Dirichlet problem
Consider a non-negative weak subsolution
of
the Dirichlet problem (2.5) for equa-
tions with homogeneous structure and let (2.6) hold.
If
the initial datum
1.£0
is also
bounded above, then the weak maximum principle estimate
(3.7) holds true.
If
however
1.£t
is not bounded, it is
of
interest to investigate how the supremum
of
1.£
behaves when t
-+
0.
THEOREM
4.3. Let
1.£
be a non-negative
weak
subsolution
of
the
Dirichlet
prob-
lem
(2.5) and
let
(2.6)
hold.
There
exists a constant
'Y
=
'Y
(data),
such
that
Vte(O,T),
(4.3)
~
= N(P - 2) + p.
(1)
See §7
flf
Chap
VI
and
§2
of
Chap. XI.
124
V.
Boundedness
of weak solutions
Results of this kind could
be
used
to
construct solutions of
the
Dirichlet prob-
lem
with
initial data
in
L
1
(n)
or even finite measures. Indeed the regularity results
of Chap. III supply the necessary compactness
to pass
to
the
limit in a sequence of
approximating problems.
Consider a non-negative
weak
subsolution
1£
of (2.8)
in
the
whole
strip
ET'
By
this
we
mean that
1£
is
a local weak subsolution
of
(2.8)
in
nT
for every bounded
domain n c RN.
To
derive global sup-estimates,
we
must impose
some
control
on the behaviour
of
1£
as
Ixl-.oo.
We
assume that
the
quantity
(4.4)
- . f 1£(x,r)
1I1£II{r,t}
=
sup sup
>'/(
-2)
dx,
O<T<t
p~r
P P
Kp
~
=
N(P-
2)
+p,
is
finite for
some
r > 0
and
for
all t E
(0,
T). The subsolution
1£
at
hand
is
not
necessarily bounded. However it
is
locally bounded
and
and
as
lxi-
00
it grows
no
faster than
Ixl
~
. This
is
the
content of the next theorem.
THEOREM
4.4. Let
1£
be
a non-negative subsolution 0/(2.8)
in
ET, and assume
(4.4)
holds.
There
exist a constant
'Y
=
'Y
(data),
such
that/or all
tE
(0,
T),
(4.5)
sup
111£(',
t)lIoo,K
p
~
'YVt
1I1£II{~~}
A
C~.
p~r
ppI(p-2)
PROOF:
Apply
Theorem 4.2
with
the choices
(x
o
,
to)
==
(0,
t), 9 = t,
(1
= 1/2, P
~
r,
and
p replaced
by
2p.
It
gives
(
)
P/2
111£("
t)lIoo,K
p
1£(x,
r)
_~
I(
-2)
~
'YVt
sup
sup
f
p>./(
-2)
A t
p-
•
pp
P O<T<t
p~r
P
Kp
Remark
4.2. The assumption (4.4)
is
not restrictive.
We
will
show
in
Chap.
XI
that it
is
necessary and sufficient
for
a non-negative solution of (2.8)
to
exist
in
E
T
•
The right hand
side
of (4.5) blows
up
as
t
'\,
0
at
the
rate of
at
least t -
~
. Such
a rate
is
not
optimal. However
the
advantage of Theorem 4.4
is
that it
does
hold
for
all t E
(0,
T). The purpose of the next theorem
is
two-fold.
It
gives
an
optimal
estimate of
how
the local sup-bound for
1£
may
deteriorate
as
either
Ixl
-
00
or
t'\,O.
5. Homogeneous structures. TIle singular case 1 < p < 2
125
THEOREM
4.5. Let
1£
be
a non-negative subsolution 0/(2.8) in
ET
and assume
(4.4) holds. There exists constants
'Y.
and
'Y
depending only upon
N,p
and the
constants
C
i
,
i=O,
1 in the structure condition (2.9), such that
(4.6) for all
° < t <
'Y.llull~;':}
and/or
all p
~
r
pp/(,,-2)
pI>'
111£(·,
t)lIoo,K
p
~
'Y
tNI>'
Ilull{r,t}'
A = N(p -
2)
+ p.
Information of this kind are of interest
in
investigating the behaviour of
the
solutions for t near zero
and
in
studying the structure of
the
non-negative solutions
in
ET.
(
1
)
The functional dependence
in
(4.6)
is
sharp
as
it can
be
verified
from
the
explicit Barenblatt solution
(4.7)
B(x,')
= d
{1-~.
(.l~~
t}
~,
t>o
-L.
'Y"
=
(~)
,,-1
p;
2,
p>2.
The
function 8 solves the Cauchy problem
(4.8)
{
Ut
- div
IDul,,-2
Du =
0,
in
RN x
(0,
00),
8(·,0) =
M6
0
,
where
6
0
is
the
Dirac mass concentrated
at
the origin,
and
M
==
118(·,
t)1I1,RN,
'TIt>
0.
The
i~itial
datum
is
taken
in the
sense of
the
measures, i.e., for every
cp
E Co(RN)
f
8(x,
t)cpdx
--+
M
cp(O),
as
t
'\.
0.
RN
For t > ° and for every
p>
°
we
have
5.
Homogeneous structures.
The
singular case 1 < p < 2
The
estimates of
§3
are
valid for solutions
1£
E
L1oc(
D
T
)
as
long
as
p > max {
1;
::
2 } .
(1) See
Chap.
XI.