A Porous Media Equation with Absorption. II. Uniqueness of the Very Singular Solution

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

223, 111]125 Ž1998.

AY985962

A Porous Media Equation with Absorption. II. Uniqueness of the Very Singular Solution Minkyu Kwak* Department of Mathematics, Chonnam National Uni¨ ersity, Kwangju, 500-757, Korea Submitted by Howard Le¨ ine Received June 13, 1997

We prove the uniqueness of the very singular solution for an equation of the form ut s D Ž um . y u p

in Q s R N = Ž 0, ` . ,

Ž1.

.q

where Ž1 y 2rN - m - 1 and 1 - p - m q 2rN. The solution we find is of the form u Ž x, t . s ty1 rŽ py1. f Ž h . ,

h s < x < tyŽ pym.rŽ2Ž py1.. ,

where f is the unique nontrivial solution of an ordinary differential equation

Ž f m .0 q

Ny1

h

Ž f m .9 q

pym 2 Ž p y 1.

f9 q

1 py1

h ) 0, Ž 2 .

f y f p s 0,

with conditions f G 0 on w0, `., f 9Ž0. s 0, and lim n ª `h 2r Ž pym. f Žh . s 0.

Q 1998

Academic Press

Key Words: a very singular solution; a porous media equation; a self-similar solution; uniqueness.

1. INTRODUCTION In this paper we consider a quasilinear degenerate diffusion equation with absorption ut s D Ž um . y u p ,

in Q s R N = Ž 0, ` . ,

Ž 1.1.

where m ) Ž1 y 2rN .q and max 1, m4 - p - m q 2rN. *Research supported in part by the Korea Science and Engineering Foundation through the GARC at Seoul National University, KOSEF-95-0701-01-01-3, and also by BSRI-97-1426. E-mail: [email protected] 111 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

112

MINKYU KWAK

A very singular solution W of Ž1.1. is a nonnegative continuous function in Q y Ž0, 0.4 such that Ži. W Ž x, 0. s 0 for x / 0; Žii. W satisfies Ž1.1. in the sense of distribution in Q; Žiii. HR N W Ž x, t . dx ª ` as t ª 0. Such a solution arises naturally when we study the long time behaviour of solutions of Ž1.1. with nonnegative initial data uŽ x, 0. s u 0 Ž x . satisfying lim < x < a u 0 Ž x . dx s 0,

as

< x <ª`

2 pym

Ž 1.2.

Žsee w3x, w6x, and w9x.. Brezis et al. w2x found in 1986 that the heat equation with absorption admits a unique very singular solution for the range corresponding to m s 1. For m ) 1, Peletier and Terman w8x have proved the existence of a very singular solution of Ž1.1. and Kamin and Veron w5x have proved its existence in a simpler way and also uniqueness. Later the existence proof was extended in w9x for the case Ž1 y 2rN .q- m - 1. We may construct a very singular solution simply by taking a monotone limit of nonnegative solutions of Ž1.1. with initial data u Ž x, 0 . s A < x
A ) 0;

see w6, Proposition 4.1x for details. The main purpose of this paper is to show its uniqueness. THEOREM A. Let Ž1 y 2rN .q- m - 1 and 1 - p - m q 2rN. Then there exists a unique ¨ ery singular solution W0 Ž x, t . for Ž1.1.. For the proof, we borrow some ideas from w4x and w5x and construct a minimal and maximal very singular solution. These solutions are invariant under a scaling transformation and become self-similar solutions. Hence these solutions will be of the form W Ž x, t . s ty1 rŽ py1. f Ž h . ,

h s < x < ty1 r b ,

where b s Ž p y 1. a s 2Ž p y 1.rŽ p y m. and f is the unique nontrivial solution of an ordinary differential equation

Ž f m.0 q

Ny1

h

Ž f m.9 q

1

b

f9 q

1 py1

f y f p s 0,

h ) 0, Ž 1.3.

with conditions fG0

on 0, ` . ,

f 9 Ž 0 . s 0, and

lim h a f Ž h . s 0. Ž 1.4.

nª`

113

POROUS MEDIA EQUATION II

The uniqueness proof is then reduced to showing that the above O.D.E. problem has a unique solution. When m ) 1, f has a compact support and the support of the scaled function flŽh . s l f Ž ldh ., d s Ž1 y m.r2, covers the support of f for l ) 1. These facts were essential in the proof of uniqueness Žsee w5x for details.. On the other hand when Ž1 y 2rN .qm F 1, the support of f becomes the whole R N and the argument for the case m ) 1 cannot be applied directly. The case m s 1 has been treated in a different way in w2x Žsee w2, p. 206x. and the proof cannot be applied to the case m - 1, either. Here we investigate the exact asymptotic decay rate of solutions of Ž1.4. and prove Theorem A in a rather simple way by adapting some ideas from w4x. Concerning the asymptotic behaviour of solutions, we in particular prove the following. THEOREM B. Let uŽ x, t . be a solution of Ž1.1. with nontri¨ ial and nonnegati¨ e initial data uŽ x, 0. s u 0 Ž x . satisfying Ž1.2.. Then lim < x < gu Ž x, t . s K#t Ž1r b .Ž gy a .

< x <ª`

for e¨ ery t ) 0, where

gs

2 1ym

K# s

,

ž

m bg

1r Ž1ym .

1 y Ž a y N . rŽ g y N .

/

.

2. A PRIORI ESTIMATES Throughout this paper we assume

ž

1y

2 N

q

- m - 1 and 1 - p - m q

/

2 N

,

Ž 2.1.

and we denote for notational simplicity

as

2 pym

,

bs

2 Ž p y 1. pym

Then the assumption implies that N- a - g.

,

gs

2 1ym

.

114

MINKYU KWAK

We now consider an ordinary differential equation

Ž um . 0 q

Ny1 x

Ž um . 9 q

x

b

1

u9 q

py1

u y u p s 0,

x ) 0, Ž 2.2.

with conditions uG0

on 0, ` . ,

u9 Ž 0 . s 0, and

lim x a u Ž x . s 0. Ž 2.3.

xª`

Let u be a nontrivial solution of Ž2.2. ] Ž2.3.. Then the main purpose of this section is to derive the exact decay rate of solution uŽ x .. We first show that such a solution is positive for all x G 0. LEMMA 2.1. 0 - uŽ x . - C* and u9Ž x . - 0 for all x ) 0, where C* s Ž p y 1.y1 rŽ py1.. Proof. We see from w6, Lemma 2.3x that 0 F uŽ x . - C* s Ž p y 1.y1 rŽ py1. for all x G 0 and uŽ x . is nonincreasing on w0, `.. Thus 1rŽ p y 1. u y u p G 0 and u9 F 0. Let ¨ s u m . Then Ž2.2. is rewritten as ¨0 q

Ny1

¨9 q

x

x

b

Ž ¨ 1r m . 9 F 0

Ž 2.4.

or ¨ 0 q a Ž x . ¨ 9 F 0,

Ž 2.5.

where aŽ x . s

Ny1 x

q

x mb

¨ Ž1ym.r m .

Notice that aŽ x . is continuous and bounded for x away from zero. Now suppose that ¨ 9Ž R . s 0 for some R ) 0. By multiplying an integrating factor to Ž2.5., one has

Ž expŽyH

R x aŽ r .

dr.

¨ 9 . 9 F 0.

Ž 2.6.

An integration yields R

exp ŽyHx aŽ r . d r . ¨ 9 Ž x . G ¨ 9 Ž R . s 0, which implies ¨ 9Ž x . s 0 for every 0 - x F R. This is impossible unless ¨ Ž x . is identically zero. Hence ¨ 9Ž x . - 0 and ¨ Ž x . ) 0 for every x ) 0. Then the lemma follows immediately.

115

POROUS MEDIA EQUATION II

LEMMA 2.2. lim x Ny 1 Ž u m . 9 s 0.

xª`

Proof. Equation Ž2.2. is rewritten as 1

Ž x Ny 1 Ž u m . 9 . 9 q

x N u9 q

b

1 py1

x Ny1 u y x Ny1 u p s 0.

Ž 2.7.

An integration of Ž2.7. over Ž0, x . yields x

Ny 1

Ž u . 9Ž x . q m

xN

b

ayN

uŽ x . s y

b

q

x

H0 x

x

H0 s

Ny 1

Ny1

u Ž x . ds

u p Ž s . ds.

Ž 2.8.

Assumption Ž2.1. implies a ) N and the decay condition in Ž2.3. implies that integrals on the right-hand side of Ž2.8. have limits, taking x ª `. Hence a limit lim x ª` x Ny 1 Ž u m .9 s yl, l G 0, exists. Suppose l ) 0. There exists R 0 ) 0 such that x Ny 1 Ž u m . 9 Ž x . - y

l 2

for x ) R 0 ,

,

or l

Ž u m . 9 Ž x . - y xyŽ Ny1. , 2

for x ) R 0 .

Ž 2.9.

Choose R ) x ) R 0 , and integrate Ž2.9. over Ž x, R . to obtain um Ž R. y um Ž x . - y

l 2

¡y

Hx

R yŽ Ny1.

x

ds

1

Ž R 2y N y x 2yN . , 2Ž 2 y N . s l y Ž ln R y ln x . , 2

~

¢

N/2 N s 2.

When N s 1 or N s 2, one has uŽ x . s ` by taking R ª `, which is impossible. On the other hand, when N ) 2, one has Žby taking a limit as R ª `. um Ž x . G

l 2 Ž N y 2.

x 2yN ,

for x ) R 0 .

116

MINKYU KWAK

This is incompatible with the decay condition lim x ª` x a uŽ x . s 0 since a ) N and N ) Ž N y 2.rm. This completes the proof. We now integrate Ž2.7. over Ž x, `. to obtain yx Ny 1 Ž u m . 9 Ž x . q s

xN

b

uq

`

Hx

ayN

`

Hx

b

x Ny1 u Ž x . ds

s Ny 1 u p Ž s . ds.

Ž 2.10.

Dividing by x N u, we have y

Ž um . 9 xu

q

a y N Hx`s Ny 1 u Ž s . ds b

s

x Nu

1

q

b

Hx`s Ny 1 u p Ž s . ds x Nu

. Ž 2.11.

Then we prove PROPOSITION 2.3. x < u9 <

lim

u

xª`

lim

Ž um . 9

xª`

xu

sy

1

b

sg,

ž

1y

Ž 2.12. ayN gyN

/

,

Ž 2.13.

lim x g u Ž x . s K#.

Ž 2.14.

xª`

Here

gs

2 1ym

and K# s

ž

m bg

1r Ž1ym .

1 y Ž a y N . rŽ g y N .

/

Proof. Assume that lim x ª`Ž x < u9
F

Nqa 2

,

for x G R.

A calculation yields uŽ x . G uŽ R .

R

ž / x

Ž Nq a .r2

,

which conflicts with the decay condition Ž2.3..

x G R,

Ž 2.15.

117

POROUS MEDIA EQUATION II

We now have, by l’Hopital’s rule, ˆ lim

Hx`s Ny 1 u Ž s .

s lim

N

x u

xª`

xª`

yx Ny 1 u Ž x . x u9 q Nx N

Ny1

u

1

s lim

x < u9
xª`

s

1 lyN

Ž 2.16. and lim

Hx`s Ny 1 u p Ž s . x Nu

xª`

s lim

xª`

u py 1 Ž x .

s 0.

x < u9
Ž 2.17.

Using these in Ž2.11. we obtain lim

Ž um . 9

xª`

xu

sy

1

b

q

1

b

?

ayN lyN

.

Thus l G a . Moreover, since we assume that lim x ª` x a u s 0, we have lim x aq1 u9 s 0.

Ž 2.18.

xª`

If we define f Ž x . s Hx`s Ny1 uŽ s . ds, then Ž2.10. is rewritten as

f 9Ž x . q

ayN x

f Ž x . s b x Ny 2 Ž u m . 9 q

b

`

H x x

s Ny1 u p Ž s . ds. Ž 2.19.

Multiplying Ž2.19. by an integrating factor x ayN , we get `

Ž x ayNf Ž x . . 9 s b x ay2 Ž u m . 9 q b x ayNy1 H

s Ny1 u p Ž s . ds. Ž 2.20.

x

Now, x a pyNf Ž x . s x ayNf Ž x .rx a Ž1yp. and, by l’Hopital’s rule, ˆ lim x a pyNf Ž x .

xª`

s lim

xª`

s

Ž x ayNf Ž x . . 9 a Ž 1 y p . x ay a py1 b

a Ž1 y p

. ž

lim x a py1 Ž u m . 9 q lim x a pyN

xª`

xª`

`

Hx

x Ny1 u p Ž s . ds ,

/

118

MINKYU KWAK

which becomes 0 from Ž2.18. and another application of l’Hopital’s rule. ˆ Notice that xu9

x a py1 Ž u m . 9 s mx a py2 u m

s mŽ x a u.

u

m

xu9 u

,

which tends to 0 as x tends to `. This fact and Ž2.16. imply that lim x ª` x a p uŽ x . s 0, which in turn implies l ) a . Writing

Ž um . 9 xu

sy

x < u9 < u

m

?

2

x u1ym

,

we have lim x ª ` x 2 u 1y m s lm b r Ž 1 y Ž a y N . r Ž l y N .. and lim x ª` x 2rŽ1ym. uŽ x . / 0. Hence one must have l s 2rŽ1 y m. and it suffices to prove that lim x ª` x < u9
Ž um . 9 xu

Fy

1 2b

or

Ž u my 1 . 9 G

1ym 2 mb

x

for all x G R. An integration yields u my 1 Ž x . y u my1 Ž R . G

1ym 4 mb

Ž x 2 y R2 . ,

for x G R ,

which implies that lim sup x ª` x 2rŽ1ym. uŽ x . - `. We next observe that for every R ) 0 there exists x ) R such that x < u9
ž / u

9s

< u9 < u

y

xu0 u

q

x < u9 < 2 u2

.

Ž 2.21.

119

POROUS MEDIA EQUATION II

We multiply Ž2.21. by mu m rx and use Ž2.2. and the fact that Ž u m .0 s mŽ m y 1. u my 2 u9 2 q mu my1 u0 to obtain mu m x

x < u9 <

ž / u

mu my 1

9s

< u9 < q mu my 2 u9 2 y mu my1 u0

x mu my 1

s

< u9 < q mu my2 u9 2 q m Ž m y 1 . u my2 u9 2

x q

Ny1 x

Ž um . 9 q

x

b

u9 q

a b

u y u p.

Ž 2.22.

By using the fact Ž u m .9 s ymu my 1 < u9 <, one has mu my 1 x

x < u9 <

ž / u

9s

yŽ um . 9 xu y

ž

1 x < u9 <

b

u

m

x < u9 <

q

u

a

q2yN

/

y u py 1 .

b

Ž 2.23.

When x < u9
ž

ma q 2 y N q

xu p

/

Ž um . 9

,

positive, and x < u9
Ž u .9 m

s

xu p mu

my1

s

u9

u mx < u9 <

Ž x a u.

py m

,

which tends to 0 as x ª `. This reveals that lim inf x ª` x < u9
x < u9 <

ž / u

9s

x < u9 < u q

ž

m

x < u9 <

x2 mu

my 1

u

ž

y

q2yN 1 x < u9 <

b

u

/

q

a b

y u py 1 .

/

Ž 2.24.

120

MINKYU KWAK

Let us assume that Ž x < u9
x < u9 <

x2

0s

ž / ž u

mu

y

9 y

/ž

my 1

py1

1 x < u9 <

b

q

u

a

y u py 1

b

/

x 2 u pymy1 u9.

m

Ž 2.25.

One now has x2

sy

mu my 1

x < u9 <

Ž um . 9

u

xu

and, at x s x 0 , x2

ž

mu my 1

/

x < u9
9s

Ž Ž u m . 9r Ž xu . .

Ž um . 9

ž

2

xu

/

9.

With the use of Ž2.7., one also has x

ž

Ž um . 9 xu

x Ny 1 Ž u m . 9

9sx

/ ž s s

/

x Nu

Ž x Ny 1 Ž u m . 9 . 9 x Ny 1 u 1 x < u9 <

b

u

y

a b

9 Nx Ny 1 u q x N u9

y x Ž u .9 N

m

q u py 1 q

x 2 N u2

Ž u m . 9 x < u9 <

ž

xu

y N . Ž 2.26.

/

u

When Ž x < u9
Ž um . 9 xu

ž

m

x < u9 <

q2yN sy

/

u

1 x < u9 <

b

u

q

a b

y u py 1 .

Using this in Ž2.26., one obtains x

ž

Ž um . 9 xu

/

9sy

Ž um . 9 xu

ž

Ž m y 1.

x < u9 < u

q2

/

and thus

ž

x2 mu my1

/

9s s

at x s x 0 .

x < u9
ž

ž

Ž m y 1.

Ž m y 1.

x < u9 < u

x < u9 <

q2

u

/

q2

/

121

POROUS MEDIA EQUATION II

Now Ž2.25. is rewritten as

x2

x < u9 <

ž / u

0s

x2 mu

my 1

ž

q Ž p y 1. s

x2 mu

my 1

x < u9 <

Ž 1 y m.

žž

u

y2

x2

x < u9 <

mu my 1

u

Ž 1 y m.

x < u9 < u

/ž

1 x < u9 <

b

u

y2

1 x < u9 <

/ž

b

u

ž

1ym

b

x2 mu

my 1

ž

x < u9 < u

b

q u py1

/

u py 1

qu py 1 Ž p y m . s

a

y

ya

/ž

x < u9 < u

y

x < u9 < u

a b

y2

ygq

/ //

b Ž r y m. 1ym

u py 1 .

/

Ž 2.27. We are now ready to complete the proof. Let us denote S s lim sup xª`

x < u9 < u

,

I s lim inf xª`

x < u9 < u

and suppose to the contrary the lim x ª` x < u9
3. UNIQUENESS We now turn to the proof of the uniqueness of the very singular solution. A very singular solution may be found as a monotone limit of singular solutions and we have also seen that a monotone limit of WAŽ x, t . yields also a very singular solution, where WAŽ x, t . is a unique solution of Ž1.1. with initial data uŽ x, 0. s A < x
122

MINKYU KWAK

As we have seen in w3x and w5x, we can show that there exist a minimal and a maximal very singular solution. Remarked earlier, such a solution has to be invariant under a scaling transformation Tl which associates with any solution of Ž1.1. another solution Tl u defined by

Ž Tl u . Ž x, t . s l a u Ž l x, l b t .

Ž 3.1.

and becomes a spherically symmetric self-similar solution. Hence such a solution must be of the form u Ž x, t . s ty1 rŽ py1. f Ž r . ,

r s < x < ty1 r b .

Moreover f satisfies an ordinary differential equation

Ž f m.0 q

Ny1 r

Ž f m.9 q

1

b

f9 q

1 py1

f y < f < py 1 f s 0,

r ) 0,

Ž 3.2. and additional conditions f)0

0, ` . ,

f 9 Ž 0 . s 0, and

lim r a f Ž r . s 0.

rª`

Ž 3.3.

The uniqueness proof is then reduced to showing that the above O.D.E. problem has a unique solution. Let F and f be any two solutions of Ž3.2. and Ž3.3.. Following w3x, we define 1 1ym f k Ž r . s kf Ž k d r . , ds s Ž 3.4. g 2 and then f k will be larger than F on w0, `. for sufficiently large k. In fact, we first observe that when uŽ x, t . s ty1 rŽ py1. f Ž r ., r s < x < ty1 r b , is a solution of Ž1.1., u k Ž x, t . s ty1 rŽ py1. f k Ž r . satisfies

Ž u k . t y D u km q u kp s k Ž k py 1 y 1 . u p Ž k d x, t . G 0 and is a supersolution of Ž1.1.. By Proposition 2.3, lim r g f Ž r . s lim r g F Ž r . s l

rª`

rª`

for some l ) 0 and lim < x < gu k Ž x, 2 . s lim < x < g 2y1rŽ py1. kf k d

< x <ª`

ž

< x <ª`

s lim 2

1r b Ž g y a .

< x <ª`

s 2 1r b Ž gy a . l ) l

kd < x<

< x< 2 1r b

g

f kd

ž / ž 2

1r b

/ < x< 2 1r b

/ Ž 3.5.

123

POROUS MEDIA EQUATION II

uniformly for k G 1. Thus we may find k large enough so that u k Ž x, 2. G F Ž< x <. for all x g R N . Using the maximum principle w1, 7x we obtain that u k Ž x, t q 1 . G ty1 rŽ py1. F Ž < x < ty1 r b .

Ž 3.6.

for every x g R N and t G 1. Put r s < x < ty1 r b . Then k

ž

1r Ž py1 .

t

/

tq1

f

1r b

t

kdr G F Ž r . ,

for t G 1.

žž / / tq1

We now let t ª ` to get f k Ž r . G F Ž r . for all r G 0 as we claimed. We now define

m s min  k G 1: f k Ž r . G F Ž r . , 0 F r - ` 4 .

Ž 3.7.

The uniqueness proof is now reduced to showing that m is not greater than 1 by reversing the role of f and F. Suppose m ) 1, to the contrary. We will show that there exists « ) 0 such that fmy e Ž r . G F Ž r . for every r G 0. We first have

typ rŽ py1. f

žt

t

d

s

H1

s

H1 s

t

ds

r dq1r b

ž

/

y fŽ r.

syp rŽ py1. f

yp rŽ py1.y1

ž

r s

//

dq1r b

fŽf. y

ž

ds

p py1

q dq

ž

1

b

/

f < f 9Ž f . < fŽf.

/

ds,

where f s rrs dq1r b . We see from Proposition 2.3 that the last line is positive for large r and that there exists R ) 0 such that

typ rŽ py1. f

žt

r dq1r b

/

G fŽ r.

Ž 3.8.

for all r G R and 1 F t F 2. For any e G 0, we now let t s mrŽ m y e .. Then d umy e Ž x, t . s ty1rŽ py1. Ž m y e . f Ž Ž m y e . < x
s mtyp rŽ py1. f Ž m d < x
Ž 3.9.

124

MINKYU KWAK

We also note that fmŽ r . does not touch F Ž r . in a compact subset of w0, `.. In fact fmŽ r . solves

Ž fmm . 0 q

Ny1 r

r

Ž fmm . 9 q b fmX q

1 py1

fm y fmp s Ž m y m p . f p . Ž 3.10.

If fm touches F at r 0 ) 0, then fmX Ž r 0 . s F9Ž r 0 . / 0 and

Ž fmm . 0 Ž r 0 . - Ž F m . 0 Ž r 0 . . But fmŽ r . G F Ž r . near r s r 0 , which obviously violates the strong maximum principle w1, 7x. The other possibility to be checked is the case when fm and F touch at the origin. We have from w6, Ž2.15.x that

Ž fmm . 0 Ž 0. s m Ž f m . 0 Ž 0. s -

m N 1 N

1

ž

f p Ž 0. y

ž

m p f p Ž 0. y

py1

f Ž 0.

m py1

/

f Ž 0. s Ž F m . 0 Ž 0. ,

/

which leads to a contradiction since fmŽ r . G F Ž r . near the origin. Hence we may find e ) 0 so that umy e Ž x, t . G F Ž < x < .

Ž 3.11.

for < x < F R. Here t s mrŽ m y e .. By Ž3.10., Ž3.11., and the comparison argument as above, we obtain that fmy e Ž r . G F Ž r . ,

for r G 0,

which means that we can slightly reduce the factor m. Hence we may conclude that m s 1 and f Ž r . s F Ž r ., which proves the uniqueness of the very singular solution.

4. ASYMPTOTIC BEHAVIOUR Let uŽ x, t . be a solution of Ž1.1. with nonnegative and nontrivial initial data u 0 Ž x . satisfying Ž1.2.. We have already seen that the asymptotic behaviour as t ª ` is deduced from the limiting behaviour of a family of scaled functions ulŽ x, t . s l a uŽ l x, l b t . and determined by the very singu-

125

POROUS MEDIA EQUATION II

lar solution W0 Ž x, t . Žsee w9, Theorem 6.1x.. In fact we see that lim t 1rŽ py1. < u Ž x, t . y W0 Ž x, t . < s 0

tª`

uniformly on the set  x g R N : < x < F g t 1r b 4 for every g ) 0. The main purpose of this section is to derive the exact decay property in Theorem B. For fixed t ) 0, we put l s < x < and s s tr< x < b. Then l ª ` and s ª 0 as < x < ª `. Thus lim < x < gu Ž x, t . s lim lim

< x <ª`

t

sª0 l ª`

s lim

sª0

t

1r b Ž g y a .

la u l

ž/

ž

s

1r b Ž g y a .

ž/

W0

s

x

x < x<

, l bs

/

ž / < x<

,s

g

s t 1r b Ž gy a . lim Ž sy1r b . f Ž xy1r b . sª0

s K#t

1r b Ž g y a .

Ž by Proposition 2.3. .

REFERENCES 1. D. Aronson, M. G. Crandal, and L. A. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Anal. 6 Ž1982., 1001]1022. 2. H. Brezis, L. A. Peletier, and D. Terman, A very singular solution of the heat equation with absorption, Arch. Rational Mech. Anal. 95 Ž1986., 185]209. 3. S. Kamin and L. A. Peletier, Large time behaviour of solutions of the porous media equation with absorption, Israel J. Math. 55 Ž1986., 129]146. 4. S. Kamin and J. L. Vazquez, Singular solutions of some nonlinear parabolic equations, IMA Preprint 834, 1991. 5. S. Kamin and L. Veron, Existence and uniqueness of the very singular solution of the porous media equation with absorption, J. Anal. Math. 51 Ž1988., 245]258. 6. M. Kwak, A porous media equation with absorption. I. Long time behaviour, J. Math. Anal. Appl. Ž1998., 000]000. 7. P. de Mottoni, A. Schiaffino, and A. Tesei, Attractivity properties of nonnegative solutions for a class of nonlinear degenerate parabolic problem, Ann. Mat. Pura Appl. 136 Ž1984., 35]48. 8. L. A. Peletier and D. Terman, A very singular solution of the porous media equation with absorption, J. Differential Equations 65 Ž1986., 396]410. 9. L. A. Peletier and J. Zhao, Large time behaviour of solutions of the porous media equation with absorption: The fast diffusion case, Nonlinear Anal. 17 Ž1991., 991]1009.