DiBenedetto E. Degenerate Parabolic Equations
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376
XII.
Non-negative
solutions
in
Er.
The
case
I <p<2
13.
Bounded solutions. A counterexample
Let r
2:
1 satisfy
Ar
::=
N(p
-
2)
+
rp
>
O.
If U
o
E
L'oc(R
N
),
then
by
energy
estimates, the sequence of approximating solutions of
(8.1)n satisfies
{un} E
L'oA~T)
uniformly in n.
Therefore
by
Theorem
5.1
of Chap.
V,
{un} E
L~(ET)
uniformly in n. It follows
from
the regularity results
of
Chaps.
IV
and
IX
that
{un}, {u
n
.
x
;}
E
C;:'c(~T)'
j=l,
2,
...
,N,
uniformly
in
n,
for
some a E
(0,
1)
depending only upon N and p. This gives a regular solutions
to the Cauchy problem (1.1)-(1.2). A similar analysis can
be
carried if the initial
datum
is
a measure
JI.
and
Al
>
0,
i.e.,
(13.1)
2N
p>
N+1'
We
show next that if(l3.1)
is
violated, then initial data
in
Ltoc(RN) might produce
unbounded solutions.
13-(i). A
counterexample
Let a E (0,
1)
be
a positive constant and let Be denote the ball of radius a in
RN centered
at
the origin. Consider the functions
(a
2
-lxI2)2
z = + and v =
(1
- ht)+ z,
IxlNlln
Ix12113
(13.2)
where {j, h > 1 are to be chosen. One verifies that
Consider also the Cauchy problem
{
Ut
- div
IDulp-2
Du
=
0,
in
E1
::=RN
x (0,1),
(13.3)
u(·,O)=z.
1be p.d.e.
is
meant in the sense of
(2.1
)-(2.2)
and the initial datum
is
taken in the
sense of
Lloc(RN).
LEMMA
13.1. Assume that
N(p
-
2)
+ p =
O.
The constants a E (0,1) and
(j, h > 1 can be determined a priori so that v is a non-negative, weak subsolution
0/(13.3) in E
1
•
PROOF:
By
calculation
on
the set 0 <
Ixl
<
a,
z .
Dz
=
-lxl
2
Fx,
13.
Bounded
solutions.
A
counterexample
377
where
{
2/3
41X12}
F = N +
In
Ixl2
+
a2
_ I
x
l
2
.
We
choose a = e
-k
and
k>
1 so large that
F>
O.
Compute .
p-IFp-1
IDzlp-2 Dz
= z x
.
Ixl
p
zp-2 FP-l
zp-l
FP-l
div(lDzlp-2 Dz)
=
-(p
-
1)
Ixl
p
Dz . x
+ P
Ixl1>+1
Dlxl· X
Zp-l FP-l
zp-l
FP-2
-N
Ixl
p
-(P-I)
Ixl
p
DF·x.
Using the fonnulae
Dz
. x =
-zF,
Dlxl· x =
Ixl
DF
. x =
-4/3
+
81xl
2
+
81xl
4
In
2
1xl
2
(a
2
-lxI
2
)
(a
2
-
Ix12)2'
we obtain
We calculate the expression in braces on the right-hand side using the definition
of
F and the fact that
N(p
- 2) +
p=O,
to obtain
Consider the sets
Cil)
==
{~e-2k
~
Ixl
2
<
e-
2k
} ,
ci
2
)
==
{lxl2 <
~e-2k
},
k >
1.
One verifies that on Cil) we have
1t > 8
(2
_
!!.)
_
Nf3.
- k
2k
378
XII.
Non-negative
solutions
in
I:T.
The
case
I
<p<2
Therefore
'H.
~O
on
£1
if k
is
sufficiently
large.
On£~2)
we
have
F~
(N
-
f3/k)
>
1.
Therefore
div(IDzIP-2 Dz) > zp-1 'Y(N,p).
-
Ixl
P
In
Ixl
2
Finally
we
compute
in
{O<
Ixl
<a}
.c(v)
==
Vt
- div(IDvI
P
-
2
Dv)
=
-hz
-
(1
-
ht)~-l
div(IDzIP-2 Dz).
On
£(1)
.c(v) < 0
and
on
£(2)
k'
- k
[
zp-2 'Y(N,P)]
.c(v)
~
z
-h
-
Ixl
p
In
Ixl
2
.
By
calculation
on
£~2)
•
zp-2 'Y(N,p)
(a
2
_lxI2)2(p-2)
-
Ixl
p
In
Ixl
2
~
'Y(N,p)
lin
IxI21~(p-2)+l
'
where
we
have
used
the
fact
that
Al
==
N (p -
2)
+ p =
O.
We
select
f3
> 1 so
that
f3(P-
2)
+ 1 >
O.
This
gives
zp-2 'Y(N,p) •
-lxl
P
Inlxl
2
~'Y
(N,p,k).
Therefore
.c(w)
~
z(
-h
+ 'Y·(k».
Cltoosing
h = 'Y. ( k)
proves
that
(13.4)
.c(V)
~
0
on
{O<lxl<a}x(O,I).
To
prove
that
indeed v
is
a
weak
subsolution
in
the
whole
E
..
multiply
(13.4)
by
a
non-negative
function
x -
cp(x)
E
C~
(E
1
).
and
integrate over the cylindrical
domain
with
annular
cross
section
Q~
=={e<
Ixl
<a-e}X
(0,1).
We
obtain
II{
vtCP+IDvI
P
-
2
Dv.Dcp}
dxdr
1:1
= lim
jr
{{
vtcp+IDvlp-2
Dv.Dcp}
dxdr
~\,O
J
Q.
I
~
~~
I I
IDvl
p
-
2
Dv·
1:1
tpdtrdr
o{lzl=G-~}
I
-
lim
J"
{IDvIP-2
Dv
.
-ixi
tpdtrdr,
~
....
"'O
J I x
O{lzl=~}
14.
Bibliographical notes
319
where du denotes the surface measure
on
{Ixl
=e}
and
on
{Ixl
=e}.
The limits
on
the right hand side are zero.
In
particular
we
have
Vt/J
E x'oc
(Ed,
V( E
C~(RN),
(
~
0,
(13.5)
jj{vt(t/J
- v)++IDvI
P
-
2
Dv·D(t/J -
v)+}
dxdr
~
O.
1:1
One also verifies by direct calculation that v satisfies (5.1) and (5.2) and therefore
is a subsolution
of
(13.3) in the class
S·.
Next we return to (13.3). This problem has a unique solution U E
S·,
by
the construction
of
§§8-12
and the uniqueness theorem 7.1. By the comparison
principle
U
~
v and therefore U is not bounded.
The
comparison principle here
is applied as follows. By the definition
of
weak solution the truncated functions
Uk
==
mint
Ui
k} are, for all k > 0 distributional subsolutions
of
(13.3). Setting
w
==
v - U and
W(k)
==
v -
Uk
and using (13.5) we find
VO<s<t~T,
Vt/JeX,oc
(Ed,
V(EC~(RN),
(
~
0
+ [lDvlp-2 Dv
-IDuklp-2
DUk]·D
«t/J
- v)+()
}dxdr
~
O.
Observe that
w(t)-O
as t'\,O in Lloc(RN). Therefore we may proceed as in
the
proof
of
the uniqueness theorem and establish an analog
of
Proposition 6.1, i.e.
VO<t~T,
Vq>l,
Vp>O,
Vue
(0,
1)
t
j
(w+(t»)q
dx
~
_'Y_
11
(w+)P-2+
q
dxdr.
(up)p
Kp
OK(1+a)p
Proceeding as in the proof
of
Theorem 7.1 we find
w+
=
O.
Remark
13.1.
If
N(p-2)+p>0
then v satisfies (13.4) but it is not a subsolution
of
(13.3) in the whole E
1
•
In
particular it does not satisfy the requirement (5.2)
of
the class
S·.
If
N(P-2)+p<0
then v does not satisfy (13.4).
14.
Bibliographical notes
Equations
of
the type
of
(1.1) arise in modelling
of
non-newtonian fluids (see
Kalashnikov [57],
Martinson-Paplov
[74,75], Antonsev [5] and Joseph-Nield-
Papanicolau [56]). Questions
of
solvability, even though in a different context,
380
XU.
Non-negative
solutions in
ET.
The
case 1 <p<2
were
fmt
investigated by Btizis and Friedman
[18J.
The notion
of
weak solution
introduced in
§2
is
taken
from
[42J.
Benilan has infonned us
of
a more general
no-
tion of solution. introduced
in
[II
J.
that would include solutions
of
variable sign.
The
remainder
of
the chapter
is
essentially taken from
[42J.
It would
be
of
interest
to investigate questions
of
existence/Uniqueness for (1.1) in ET when
the
initial
datum
is
of
variable sign or
is
a measure. Singular equations are little understood.
mostly
if
p violates (13.1). Preliminary investigations seem
to
indicate questions
of
limiting Sobolev exponent (see [19]) and differential geometry.
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