SINGULAR SOLUTIONS FOR A CONVECTION DIFFUSION EQUATION WITH ABSORPTION

1995,15(4) :431-441

vd'atheThiacta!7cientia

l~tmJl~fIl SINGULAR SOLUTIONS FOR A CONVECTION DIFFUSION EQUATION WITH ABSORPTION· Zhao Junning(~1~~) Dept. of Math. , Jilin University, Changchun 130023, China.

Abstract

In this paper we discuss the existence and nonexistence of singular solutions for a

porous medium equations with convection and absorption terms.

Key words

1

Convection diffusion equation, singular solution, existence and nonexistence.

Introduction Consider the Cauchy problem u,

=

L1u

u(x,O)

where m

>

l,p

m

+ -ad-b;(u) a";

= 0

> 0, and b.Cs)

x

E C

1

-

u"

in ST = R N X (O,T),

(1.1)

E RN\{O},

(1.2)

(R ) .

Equation (1. 1) arises from many applications. We will not recall them here, since they can be found in many papers, for example in [IJ [2J. The case when the initial datum is a measure is also a model for physical phenomena (see [3 J [4 J). For the case b.Cu')

= 0, it was shown in [5J and [6J respectively, that if m > 1,0 < p < m

+ ~, the

problem (1. 1) (1. 2) has a solution which satisfies the initial condition u(x,0)=8(x),

where 8(x) denotes the Dirac mess. concentrated at the origin and that if p

(1.3)

>

m

+~,

the problem (1.1) (1.3) has no solution. In addition, it was shown in [7J and [8J that for b.Cu') = 0, max {l ,m}

< p < m < ~ O. 1) -

O. 2) has a very singular solution, i,

e. a solution w with the following properties w

E C(Sr\{ (O,O)})

w(x,O)

=

0

if

x

(1.4)

E RN\{O}

• Received Apr. 13,1992. Supported by the National Natural Science Fundation of China.

(1.5)

432

ACTA MATHEMATICA SCIENTIA

lim t-O+

I

w(x,t)dx

Ixl
=

+

00,

for every r

Vol. 15

> o.

(1.6)

=

t1cp(u) - u" some similar results have

For the general porous medium equation u,

been obtained in [9J by using a method different to that in [7J[8J. In this paper, we are interested in the effect of the convection term aCht on the exis~t

tence and nonexistence of singular solutions to Cauchy problem (1. 1) - (1. 2). For the case m'

=

l,b;(u)

=

uq ,

J. Aguirre, M. Escobeto[lO] and W. Liu[lO] proved that if 1 < p <

1

+ ~ and 1 < q < 1 + ~, O. 1) , O. 3) has a unique solution and that if p > 1 + ~ and

1

< q<

p

t

1, O. 1) O. 2) has no singular soluton and that if 1 < p

~ p ~ 1 , then

(1. 1)

O.

< 1 + ~, 1 < q

2) has a very singular solution. Here we use the method simi-

lar to that in [9J to improve and generalize the results in [10-IIJ. We assume that Ib~(s)

I<

Msq-l

if s

> o.

(1.7)

We shall prove the following theorems. Suppose that (1.7) holds and let 0 < p

Theorem 1

< m + N2

0
+ N2 .

Then (1. 1) (1. 3) has a solution. Theorem 2

Suppose that (1.7) holds and let p ~ m

+

2 N' 0

mN+ 1 nN + 2 P ·


Then (1. 1) (1. 3) has no solution. Theorem 3


Suppose that (1.7) holds and let m

2 N' 0 < q

< p.

Then

(1.1) (1.2) has a very singular solution. Suppose that (1.7) holds and let p

Theorem 4

2 > m + N' 0< q < p -

1

N. Then

(1.1) (1.2) has no very singular solution. Clearly, for the case m

= 1,

b, = u" , Theorem 1- Theorem 4 weakened the condi-

tions of index q in [10J [11J.

2

Proof of Theorem 1 Definition 2. 1

A solution u of (1. 1), (1. 3) is a nonnegative function defined in Sr

such that 1.

2.

u

E

It

T

3.

lim t-O+

Ll(Sr)

(1/.u

I

n LtXJ(R

+ /iTJum -

N

X (r,T))

n C(Sr\{(O,O)}), for every

t:

E

bi(u)1/Zi - TJuP)dxdt = 0, for every 1/ E Q(8T )

u(x,t)¢(x)dx

=

¢(O), for every ¢

RN

E

c: (R

N

(O,T), ,

) .

We first discuss the solutions of equation (1.1) with initial date

u(x,O)

= kNh(kx)

(2.1)

No.4

Zhao: SINGULAR SOLUTIONS FOR A CONVECTION DIFFUSION

where h(x) E' C';' (R N )

,

h(x)

>

I

0,

RN

= 1,

h(x)dx

433

> o.

k

Such solutions are defined as in Definition 2. 1. Without loss of generalities, in the following context, we use C to denote the constants independent of k , although they may change from line to line in the same proof. In this section we suppose that the hypotheses of Theorm 1 are satisfied. Denote BR(xo)

Lemma 2.1

=

{x E R N

I<

R}.

The problem (1.1) (2.1) have a nonnegative solution u, E LOCJ(ST)

C(ST) which satisfies

I

RN

Proof

Ix - Xo

:

Uk(x,t)dx

+

II

S,

ufdxdt

<

n

1.

We consider the approximte problem of (1. 1) (2.1) u,

= Llum + ~b;(u) aXt

-

uP

+ (l)P
in B; X (O,T),

(2.3)

on Ixl = n,

u(x,t) = (l)N+l n

(2.2)

(2.4) It is well known that (2. 2) - (2. 4) has a unique classical solution U,tn and (l)N+1 n

<

< B(k),

Ukn

where B(k) is a constant independent of n, By [12J, for every compact set K CST' the {Ukn} is ~quicontinuous on K . Thus we can select a subsequence from {u,tn} , which denotes again by {u,tn} , such that for any compact set K C ST U,tn -. u,

as n -.

(X)

in C(K).

It is easy to verify that u, is a solution of (1. 1) (2. 1). Set o., Vkn

+ 1J)2

----1

(vin

Notice that

=

Ukn -

(l )N+l, n

so that o.;

=

Ix I

0 on

n, We multiply (2. 2) by

. . over B n X (0 , T) . an d Integrate It

rJ =rJ

a':: bi(Ui.)

JOB.; JOB.

~~Jv"b~(s

CAL; 0

(vin

Vi.

+ 1J) 2: 1

dxdt

+ (l)N+l) n

(S2

s

1dsdxdt

+ 1J) 2

=o.

Using an analogous argument with Lemma 2. 1 in [9J, we can prove Lemma 2. 1. u,

>

Without loss of generalities, in the following lemmas, we assume u, E C 2 (ST) and 0. Otherwise we consider the approximate problem

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ACTA MATHEMATICA SCIENTIA

Lemma 2. 2

The solution UJ. of (1.1) (2.1) satisfies

[J o

where 0

< a<

Proof

Vol. 15

min {~, m

Let ~(x)

E

BR(x

(uJ.)m+~-adxdt <

+~-

q} and C(a) is a constant independent of

C~(B2R(XO))'

< ~<

0

1, ~ = 1 if

+ U~)-1~2 and integrate over ST to obtain

(1.1) by u~(1

sa 2 -1 adx~ dx RN o · S

JJ = - [J

+

UIt ( X . T }

o

R

N

(2.6)

C(a)

O)

+ (m

4ma

a )2

-

x

Xo

and k •

E BR(xo). We multiply

[J 0

ui a m-a 2 2 2 RN (1 + UJ.a)21 \1UJ. I ~ dxdt

-2u: +1 a\I UJ.m • \I ~~dxdt - 2JTJ N u, 0 R

J"'. b (s) 1 +sa I

i

0

-;ds~~x.dxdt

S

I

Using Schwarz's inequality, we get m-a ~ dxdt [JR +uiau~)21 \1ui21
o

N

2

(1

o

R

1

N

0

R

N

~ I \I ~ Iuidxdt,

where the hypotheses (1. 7) is used. Let UJ.l = max{uJ.,l}.

Then by Sobolev's imbedding inequality (see [13

[J o

R

N

2 ~2U!1+N-adxdt~
J

O
p74J)

2JTJ Uk1dx}N. 0

B ZR

R

N

m-a 1\I(~uiT)12dxdt.

Notice that

[LN¢frurl+-k-adxdt
{I + [LN¢fr-2Ui'I+adxdt + [LN¢2 r-l u:dxdt}

{[JRN¢2rU';1+N-adxdt ,-~
This implies

2

0

2

(2.7)

Zhao: SINGULAR SOLUTIONS FOR A CONVECTION DIFFUSION

No.4

435

and Lemma 2. 2 is proved. Lemma 2. 3

The solution (1.1) (2. 1) satisfies

(2.8)

where C·

(p

=

Proof

11)

p':) •

Since C* t" p2.} is a solution of (1. 1) by comparison principle we can obtain

(2.8).

For every ball B R (x o ) C R N \ {O} and 0 <

Lemma 2. 4

r

< 1 + ~ the solution u, of

(1. 1) (2. 1) satisfies

(2. 9) and

(2.10)

Proof

Let~(x)EC~(RN), O<~
tiply (1.1) by ~2U~(Y Y

>

0) and integrate over ST to obtain

+1 1f RNC;eau;r+l (


R

N

X

,t

)d

+ (m4mY + y)2 ftf RNC;e21 \l u,r+m 12d

x

0

u!+r-lUkx.~x.dxds •

2ftf 0

•

R

N

2

f"i rrb~(r)dr~~x.dxds 0

,

d

s

I

+ - l - f N ~2(kNh(kx))r+ldx Y+ 1 R • Since supp ~ C R N \ {O}, h E C~ (R N )

X

(2.11)

we have

~2kNh(kx)

=

0

if k is large enough. Hence, we obtain from (2.11) that supf ~2ur+l(x,t)dx+[fN ~21\lu:~mI2dxds O
Using the imbedding inequality, we have [of R N

~sur+m+2k+~dxdt 2


N~2ur+ldx + [ f 0

RN

I \l (~u:tr)

12dxdt}1+~ (2. 12)

for some s

>

O.

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ACTA MATHEMATICA SCIENTIA

For t: E (

~,

Vol. 15

1) , set

R = R( r + 1 2 1

1')

1

1

and let ~l be a cutoff function in B R1(xo) with ~l 2 Denote 1 N by K and choose r such that

=

=

0,1,2,···,

1 on B R1+1 •

+

r= r = N

= 2q

where d

mN

+q - 2 -

+K

d - q m - 1

+K

if q

1

if q

>m,

< m,

1.

We first dicuss the case q

> m,

From (2. 12), we get

I [ojf 1+1ut+KI+1dxdt< { (1 - C 2R2([f.J B

1

r)

R

Ud+K1dXdt+[f ur;-q+d+K1dxdt) }K. oj B R1 1

BR

Without loss of generality, we can assume

[J o

B

ut+K'dxdt

>

for any 1 > 1.

1

R1

and then we have

([L

R

1

K'

+1

1

ut+ + dxdt ) R'dxdt I

1

<

1 2[ f oj

C

(R(l -

Hence the Standard Moser's iteration yields

BrRSX~f.T)U! <

{(O - ;)R)N+2[L

r))

r

ut+ K1d xdt, BR

I

1

Rdxdt

for

t:

1

E (2' 1).

(2. 13)

From (2. 13), we get

Applying Schwarz's inequality, we get 1 sup Ul< 2

B-rRX (O.T)

1

sup u l+C(r){(O_l)R)N+2[f ut+rdxdt}r.

BRX (O.T)

r

0

J

BR

Hence by [14 p161 Lemma 3. IJ, we get

and Lemma 2. 4 when q

>m

is proved.

The proof of (2.9) is similar to the proof of (2. 10). Proof of Theorem 1

By Lemma 2. 2- Lemma 2. 4, q <

m

+ ~ and using an argu-

ment similar to the proof of Theorem 1 in [9J, we can prove Theorem 1.

No.4

3

Zhao: SINGULAR SOLUTIONS FOR A CONVECTION DIFFUSION

437

Proof of Theorem 2 The proof of Theorem 2 proceeds via two lemmas. Let p>m+

Lemma 3.1

~, o
t ~p,

whereqis the constant in 0.7).

Then any solution of (1. 1) (1. 3) satisfies

[J

u/ d xdt

{u~t

+ um~~ - b;(u)~x. - uP~}dxdt =

o BR

<

>

for any R

00,

0

(3.1)

and

[J o

R

N

0,

(3.2)

•

for any ~ E C~ (R N X (- T, T». Proof

The proof of (3.1) is similar to the proof of Lemma 3.2 in [9J, we omit it.

We now prove (3.2). Let
'b(

+

Ixl 2

where ~(x,t) E C;;" (R N X (- T ,T», Y = m -1 1](s)

= 1 if s > 2,

1](s)

Since for every k

= 0 if S < 1 and

>

1]k(S)

=

t~) ~(x,t)

+ ~ and TJ E C= (R) has the properties: 1](ks).

0,
follows from Definition 2.1 that

[LN(U¢'it

satisfies

u

+ UmLi¢'i -

b,(U)¢'h; - uP¢'i)dxdt = O.

Therefore, it is sufficient to verify that as k --.

00

[J «: ~1]k~dxdt --. 0, [J um~1]k - \J~dxdt--.O, [J b;(U)1]kx.~dxdt--.O. [J o

o

R

R

N

u1]kt~dxdt --. 0,

0

N

R

N

0

R

N

•

Set Then

[JRNUTJit~dxdt I
I

D

A:

I[LNumLiTJi~dxdtl < CkJL.umdxdt, I[JRNum\lTJi

0

\l~dxdtl < C

I[LNb,(U)TJiz,Uxdtl

<

C

./kHD.umdxdt,

./kJL

uidxdt;

i

Noting that

IDkl =

measure of D,

(mN+2)

= Ck--2-,

(3.3)

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ACTA MATHEMATICA SCIENTIA

Vol. 15

we obtain by means of Holder's inequality

C{JL.uPdxdt}.

k~JL.udxdt =

kJL.Umdxdt = C( JL.UPdxdt)~ . .fkJL.u

9dxdt

< C(JL.uPdxdt) '.

Since

(3. 3) is proved. Similarly to the proof of Lemma 3. 2 in [9J, we can prove the following lemma. Lemma 3. 2

Suppose that

[J o

BR

u

satisfies

uPdxdt

<

for every R

00

> 0,

and

[ ojr

R

N

(uet

+ um~e -

for any

~

E

b.(u)ex .

C~ (R

Then limr N u(x,t)X(x)dx

t-oJ

R

-

upe)dxdt

•

=

N

=

° (3.4)

X (- T ,T) ) .

° for any X E C~(RN).

By Lemma 3.1- Lemma 3.2 and using an argument similar to that in the proof of' Theorem 2 in [9], we can prove Theorem 2.

4 Proof of Theorem 3 and Theorem 4 We discuss the solution of equation (1.1) with initial data u(x,O)

where h Ca:') E

=

(4. 1)

k N + 1h(kx) ,

C;;'(R h(x) > 0, LNh(X)dx = 1. N

) ,

It is well-known that (1.1) (4.1) has a nonnegative solution u, E LrxJ(O,T;L 1(R N )

)

n C(5

T )

and \lu'; E L 2(B R X

os.r».

Similarly to proof of Lemma 2. 3, we can prove (4.2) Moreover, we have the following estimates. Lemma 4.1 isfies

Let BR(x O ) CRN\{O}. Then the solution of problem (1.1) (4.1) sat-

Zhao: SINGULAR SOLUTIONS FOR A CONVECTION DIFFUSION

No.4

439

(4.3) Let ~ E CC; (R N )

Proof

choose 1J

=

U:~2 (a

a

0 < ~ < 1, ~ (z)

,

> m) in Definition 2. 1 to

r

=

1 if x E B R(xo) , supp ~ C R N \ { 0 }. We

obtain

l l ui+l(x,T)e2dx+ rr uf+ae J JR + JR

=-

+

2dxdt+

N

rr J J ~u:V'~.

2

_1_

a

+

r

IJR N

JrrJR.

N

e2\l ui . \lui

0

rr J J Jr"ib;(s)sads~~x.dxdt

V'u';dxdt - 2

RN

0

N

0

0

R

N

0

r

(k N+ 1 h ( k x ) )a+l~2dx.

(4.4)

Notice that

and kN+1h(kx)

= 0 on supp

~

if k is large enough. From (4. 4) we obtain

rr

J0 JR Since p

N

~2ur+adxdt <

> max {q ,m },

Lemma 4.2

where 0 < r

crr I V'~12u!+adxdt + C + crr I V'~lu1+adxdt. J JR J JR 0

N

N

0

(4.5)

(4. 3) follows from (4. 5), as in the proof of Lemma 2. 2.

Let BR(xo) C RN\{O}. Then the solution u, of (1.1) (4.1) satisfies

< 1+

2 N.

The proof of Lemma 4. 2 is similar to that of Lemma 2. 4. By Lemma 4. 1, Lemma 4. 2, (4. 2) and using an argument similar to that in the proof of Theorem 3 of [9J, we can prove Theorem 3. To prove Theorem 4, we need the following lemma. Lemma 4.3

Let p

> m + ~, 0 < q < P - ~, where q is the constant in

Then there exist constants C 1 ,C2 ,C3 such that any solution u(x,t)
u

of (1.1) (1.2) satisfies

+ t)-P:"l + C ( lx 1 + t)-p:"m + C ( lx 1 + t)-2(/-q> , 2

2

3

where C 1 ,C2,C3 depend only on N ,m,p,q. Proof

Let Xo E R N \ { 0 }, 0

< to < T

and let 0 < R

0.7).

< IXo I.

2

(4.6)

440

Vol. 15

ACTA MATHEMATICA SCIENTIA

We consider in the set

G= {(x,t): Ix-xol
where r

= Ix

C(R 2 -

=

v(x,t)

r2

+ t)-p-m, 2

Xo I and C is a constant such that

-

=

Lv

~vm

vt -

CiJ;ax; (v)

-

+

> -

u"

0

. G tn.

(4.7)

In fact, noting that 2C- (R 2 -

vt = -

p-m

~Vm = 8m(m

r

+ t )__p-m

2

2_ _ 1

4C- (R 2 'VI. = -

p-m

c

+ P)C"(R 2 _

+ t) __2_p-m

+ t)-p:!mr 2 + p4mN cm(R 2 - m

r2

(p - m)2

r2

r2

1(

XOi )

x; -

+ t)-;~~

we have > -

Lv

-2C - (R 2 -

p - m

+ t) __2_p-m

p-m

4M m cq

-

p-m 2

-

r

_ 8(m

(p -

2

(R 2

r2

-

+

8m(m P)cm(R2 - r 2 (p - m)2

2C

2

--(R -

p-m

r

4mN cm(R2 _ r 2

P- m

m)2

4M mcq(R 2 - r 2

-

-

+ t) m-p-2q p-m-r + C P(R 2 -

-!:.L { + t)-p-m CP -

+ p) Cmr2 _

1

+ t)_-1L p-mr2

+ t)-~

4mN cm(R 2 _ r 2

-

_(R >

r2

p-m

2

r2

-2p + t)p-m

+ t') p+m-2 p-m

+ t')

q + t )p+m-2 p-m r }

where the hypothesis (1. 7) is used. To satisfy (4. 7), it is sufficient to take p+m-2q + to)p-m + C 2 + t o)(---p=-m+2:)p-q ,C2 ,C depend only on N, P .q, Since v = + on CG and u Cx, 0) = 0
C where C 1

= C1 (R 2 + to) (p-m)(p-l> p+m-2

+C

2(R

1

2

1

1

3(R

00

3

it follows from the comparison principle (see [15J) that u(xo,t o)

for every 0 < R


1(R

Z

+ to)-P-l + C 2(R + to)-p-m + C

+ t o)-2(P-q)

(4.8)

(0, T). Hence (4. 6) follows by letting R

= IXo I.

1

1

Z

< IXo I and every to E

Proof of Theorem 4 By Lemma 4.3, if p > m

3(R

+ ~, q <

f B u(x,t)dx < C(R) R

where C(R) does not depend on t. Thus limf u(x,t)dx 1-0

BR

<

C(R)

and this implies that (1. 1) (1. 2) has no very singular solution.

2

1

p - ~ we have

No.4

Zhao: SINGULAR SOLUTIONS FOR A CONVECTION DIFFUSION

441

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Peletier L A. The porous media equation, Application of Nonlinear Analysis in the Physical Sciences. Pitman, 1981.229-241.

2

W u Zhouqun. Degenerate quasilinear parabolic equation. China: Advance in Mathematics, 1987 , 16 (2) : 121-158.

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Kamin S. Source-type solutions for equations of nonstationary filtrations. J. Math. Analysis Applic , 1978,63.

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Zeldovich Ya B, Raizer Yu P. Physics of shock waves and high - temperature hydrodynamic phenomena, Vol.

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Brezis H, Friedmen A. Nonlinear parabolic equations involving measures as initial conditions. J.

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Kamin S, Peletier L A. Source - type solutions of degenerate diffusion equations with absorption.

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Brezis H, Peletier LA, Terman D. A very singular solution of the heat equation with absorption.

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Peletier LA, Terman D. A very singular solution of the porous media equation with absorption. J.

9

Zhao Junning. Source-type solutions of degenerate quasiliear parabolic equations. J. Differential E-

Math. Pures Appl. ,1983,62: 73-79. Israel Journal of Math. , 1985,50(3) Arch. Rational Anal. ,1986,95: 185-207 Diff.Equ. ,1986,65:396-410. quations, 92(2) :179-198 10

Aguirre J, Escobedo M. Source solutions for a convection diffusion problem: global existence and blow-up. preprint.

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Liu Wenxiong. Singular solutions for a convection diffusion equation with absorption. IMA, preprint Series 653, Univ. of Minnesota.

12

DiBenedetto E. Continuity of weak solutions to a general porous media equation. Indiana Univ. Math.J. ,1983,32:83-118

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Ladyzenskaja 0 A, Solonnikov V A, U ralceva N N. Linear and quasiliear equations of parabolic type. Trans. of Math. Mono. Providence, 1968.

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Giaquinta M. Multiple integrals in the calculus of variations and nonlinear elliptic systems. Princeton Univ. Press, 1983.

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Zhao Iunning, Uniqueness of generalized solution for the first boundary value problem of degenerate parabolic equation. Acta Mathematica Scientia . 1986,6(1):63-70.