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Source-type solutions of the porous media equation with absorption: The fast diffusion case
hronlineor Anulpis. Theory. Printed in Great Britain.
h4crhod.c
&
Apphmrions. Vol.
14.
No.
2. pp.
107-121.
0362-%6X/90 S3.00+ .OO @ 1990 Pergmlon Press PlC
1990.
SOURCE-TYPE SOLUTIONS OF THE POROUS MEDIA EQUATION WITH ABSORPTION: THE FAST DIFFUSION CASE L.
A.
PELETIER
Mathematical Institute, Leiden University,
The Netherlands
and ZHAO JUNNING Departmentof Mathematics,Jilin University,China (Received 2 January 1989; received for publication 20 February 1989) Key words and phrases: Degenerate diffusion, singular solutions.
1. INTRODUCTION
IN THIS PAPER
we
consider the Cauchy Problem u, = A(u”) - up
on S, = R” x (0, T)
u(x, 0) = 0
x E R”\(O],
(1.1) (1.2)
where n L 1,p > 1 and 0 < m < 1. For the cases of regular diffusion (m = 1) and slow diffusion (m > 1) it was shown in respectively [l] and [2] that the problem (1. l), (1.2) has a solution which satisfies the initial condition 0,
0) = 6(x),
(1.3)
where 6 denotes the Dirac mass centered at the origin if and only if 1 < p c m + (2/n). In addition it was shown in [3] and [4] that the same condition was necessary and sufficient for the existence of a very singular solution of (1 .l), (1.2), i.e. a solution W with the properties w E C(%WO* 0)l); W(x, 0) = 0
(1.4)
if x E R”\(O);
(1.5)
,n
W(x, t)dx lim ‘10 ,I lx/
= +aJ
for any r > 0.
(1.6)
The object of this paper is to extend these results to the case when (1.7) where I. II. III.
(s)+ = max(0, s]. Specifically, we shall prove Ifp L m + (2/n), then (l.l), (1.3) has no solution. If 1 < p < m + (2/n), then (l.l), (1.3) has a solution. Let 1 < p < m + (2/n). Then there exists a solution u(x, t) of (1 .I), (1.3) such that t’ql(x,
t) - E(x, 1)j + 0 107
ast+O
L. A. PELETIERand Z. JUNNING
108
uniformly on R”. Here
and E(x,
I)
is the Barenblatt-Pattle
=
1-1’6
I
a2
+
l-
-
m - 1x1’
-1/(1-m)
2mnj3 t2’On1
’
fZER+
(1.8)
solution of the Cauchy Problem II, = A@“),
U(X,0) = 6(x).
IV. If 1 < p < m + (2/n), then (1.1) has a very singular solution. The ideas of the proofs of I, II and III are similar to those given in [I] and [2]; for the proof of IV we use ideas of (51. However, the fact that when m < 1, u’“-~ is unbounded, and urn TVL’(R” x (T, 7)) in general, requires a number of changes in the arguments. 2. PRELIMINARIES
Let BR = [x E R”: 1x1< R] and aBR = (x E R”: 1x1 = R). Definition 2.1. A solution u of (1. l), (1.3) is a nonnegative
function defined in ST such that (a) u E C(0, 7; L’(R”)) n C(S,) rl L”(R” x (T, T)) for every 7 E (0, T); (b) for any R > 0 and any 5 E C2(B, x [0, T]) such that [ vanishes on aBR x [0, T] and near BR x IO] and B, x (T], u satisfies ‘T’
T (UC,
+
u”‘AC - u”T) -
I I .O,BR
U”(Vl;, v) = 0, CT vaB~ “0
where v denotes the outward pointing normal on i3BR; (c) for any x E C’,“(R”), n lim
rio ,\ R”
w,
f)Xce dx = x(O).
In the next three theorems we discuss solutions of equation (1.1) in R” with bounded, nonnegative initial data: 0,
0) = 6(.x),
410
in R”.
(2.1)
Such solutions are defined as in definition 2.1. THEOREM 2.1. Let 4 E L’(R”) n L”(R”). Then L-(0, c L’(R”)) n L’(&) n C(&).
Proof. We consider the approximate
problem
(1 .l),
(2.1) has a solution
uE
problem
II, = A(u”) - up +- E”-’
in B1,, x (0, T)
u(x, f)l,,,=,-1 = &“+’ u(x, 0) = #Q(x)
x E &,,
(2.2) (2.3) (2.4)
109
Source-type solutions of the porous media equation
where 0 < E < 1 and 4, E C”(R”) has the properties (i) 4, r P+‘, JBl,j& - $1 -+ 0 as ~10; (ii) I#I,(x) = .9+l near 1x1 = l/E. It is well-known that (2.2)-(2.4) has a unique classical solution uE, and that &“+I I u, I c,
(2.5)
where C is a constant which does not depend on E. By [6], for any compact set K C S,, the family (u,] is equicontinuous in K. Thus we can select a sequence from (u,], which we denote again by [u,] and a function u E C(S,) such that for every compact set K C ST, in C(K). u, + u as&l0 (2.6) For u we have the following estimates
Il~llrw, 9 lluPll L’(ST)5 Il4llrqRn,.
(2.7)
IlullL”(0.C’;L’(R”)) 5
multiply (2.2) by u,/(u, + q) (V > 0) and
Indeed, set U, = u, - E”+‘, so that v, = 0 on M,,,, integrate over B,,, x [0, T]. Then one obtains
_T'
-
JI
upA+
0 . Bl/r
v, +
'T"
rl
v-v
3I 0
L
&
n+l
-.
0,
v, +
BI/E
rl
Letting q ---)0 and E -) 0 in turn we obtain
ci R”
utx, TJk
+‘{;j)
5 JR$#l
from which (2.7) and (2.8) follow. From these bounds we can readily conclude that u is a solution of (1 .l), (2.1). THEOREM 2.2. Let ulr u2 E L”(S,) rl L’(S,) be solutions 4~~)I#J*E L”(R”) n L’(R”), and suppose ‘?I ’
1 1+ 2 - n ’
( >
p>
of (l.l),
(2.1) with initial values
I.
Then O1 s 4* on R” implies that u1 s u2 on S,. COROLLARY
2.1. Let ~5E L”(R”) n L’(R”). Then (1 .l), (2.1) has a unique solution.
Proof of theorer?? 2.2. Let u1 and u2 be solutions of (1. I) with initial values O1 and & , respectively. Then, by the definition of a solution, for any R > 0 and [ E C*(Sr) such that <(x, t) = 0
L. A. PELETIER and Z.
110
for x E dB,,
JUNNING
u1 and ZQsatisfy
s BR
@r(T) - @%I~)
-
s BR
(d, - &)5(O)
T =
((~1 ss0
-
u2K
+
(6
-
4’)
Al
-
d’)Cl
BR T
ss0
U,m)P~~
(2.9)
VI*
aBR
Let E > 0, p > 0, and define the functions u’: - uz” &(U;” - t(F) *
a, = U’
-
u2 +
aw= a,*
Jp,
where Jp E C=(R”+‘) is a mollifier with the properties supp Jp C ((x, 1): 1x1< p, It] < p) and jR”+lJp = 1. Clearly as, acp 2 co > 0 where co is a constant independent of E and p. We consider the Dirichlet problem in BR x (0, T)
[f + a+,Ac - Ac = 0 <=o
(2.10) (2.11)
on i3BR x (0, T) in BR,
[(x, T) = x(x)e-fi’“’
(2.12)
where 1. > 0 and x E C,“(B,), 0 I x I I. Problem (2.10)-(2.12) has a unique nonnegative smooth solution [ which is uniformly bounded with respect to R, E and p. Multiplying (2.10) by AC and then integrating the result over BR x (0, T) we obtain T
(2.13) b’d2 5 M, a&W2 5 M, s BR is0 BR where M is a constant, which does not depend on R, E and p. In the next lemma we give two further estimates. LEMMA2.1. There exists a constant M, which does not depend on R, E and p such that [(x, 1) 5 MemmXi IV[(x, r)l 5 MeeRG Accepting this lemma temporarily, in (2.9). This yields [u,(T)
i
-
u2(T)lxe-Jzlx’
in 8, x [0, T]
(2.14)
On a& X [0, T].
(2.15)
we use the function 4’constructed above as a test function
-
(4,
-
42K(O)
BR
BR
=
wi”
-
~3
-
(u,
-
@WC,
-
u2)aepl N
+
A--
u:
-
u1 -
uz” u2
(u,
-
u2)C
T
-
11
,o
=x,+x,+x,.
@I” aBR
v)
(2.16)
111
Source-type solutions of the porous media equation
Write
(Ss r
I
0
BR(UI
E2(U;” -
-
u2
+
U24u"m &(UI
u~,,IA[[~)~”
2y'2(j:lBR
-
u2
))
(2.17) where we have used the fact that ucP is bounded away from zero. Thus: (i) if m 1 +
(ii) if m c *
x,
I
,JBR u,r+(1:1.,u2)J+
1~2Rn-~2[ ( cc?
&,
where we have chosen p = P(E, R) so small that T
L
ii0
(4 - acJ2 c
e2.
5
BR
Choosing uf A > sup ST I u1-"2
4
,
I
which is possible because p > 1, we ensure that '-7
x2 5
c
Ii
@I
-
u2)+.
I 0 ,BR
Finally, using lemma 2.1 we obtain X3 5 CR”-‘e-RJZ. Substituting the bounds for X, and X3 into (2.16), we obtain
ii T
(u,(T) i
c BR
- u2(T)]e-‘X’G~ 5 X, + C
(4, BR
-
42)+
+
,o
,BR
(u,
-
u2)+
+
.
L. A. PELETIERand Z. JIJNNING
112
Now let R = E-(~‘~)-~, where cc > 0 and 2 - n + 2mn ’ < 2n(l - am)
if m < +.
fss
Then, if we let E + 0, X1 -* 0 and we obtain, since +i S 62, (u,u-)
-
U2(n)+
5
c
b,
-
R”
0
u21+.
This implies that u1 I u2 and theorem 2.2 is proved. In analogous fashion we can prove the following comparison theorem. THEOREM2.3. Let ul, u2 E C(0, Z L’(R”)) 17L“‘(S,), and let
s
ss I
(u,(t) - u,(t))tv) 5
BR
lb,
0
-
u2K
+
-
4”)
AC
-
04
-
GXl
BR
“T
-
1s
0
-
GWY,
VI
aBR
for any c E C2(BR x [0, T)), R > 0 such that [ L 0 and c = 0 on aBR x [0, T]. Then U, 5 u2, It remains to prove lemma 2.1. Proof of lemma 2. I. Define for rl, r2 > 0, r, > r2 the cylinders
Qr,.rl = @,,W,) x 10, Tl. In QR, r, we consider the auxiliary function w(x, t) = -c(x, t) + ke-‘X’G+acT-r). Clearly, w(x, 1) L 0
if 1x1= 1
and
.
1x1 = R
provided k is chosen large enough. Moreover, we have w(x, T) = -xe-IxI*
+ ke-l”lJ” 1 0
if k is large enough. Finally, since C~,Eis uniformly bounded above in E and p, we can choose /3 so that VW= w, + acPAw - 13~ =
ke-i”‘&+“‘T-0
o,,E
_
4/Z
+p-l
co. >
Thus, by the maximum principle, (2.14) follows. To prove (2.15) we use a barrier function argument, and we consider in QR,R_l the auxiliary function 2(x, t) = 5(x, 1) + K,e-R~(eKz(‘xi-R) - 1). Clearly, z(x, t) = 0 if 1x1 = R. Since suppx C B, we also have z(x, T) = *(x)e-‘“iG + Kle-RG(eXZ(‘X’-Rf - 1) < 0
113
Source-type solutions of the porous media equation
if K, is large enough. Moreover by (2.14) we can choose K, and K2 so large that if 1x1 = R - 1 2(x, t) = ((x, t) + K,e-RG(e-K2 5 Me-(R-r)fi
- 1)
+ K,e-RJr(e-x2
- 1)
+ -n-l - A + AK,emRz> 1x1 K2 >
0.
z s 0 in QR,R_r and if 1x1 = R,
az > 0 zwhich implies that ai,_~~~-Rfi av12
if 1x1= R.
Observing that ac/av I 0 on (xl = R, (2.18) implies (2.15). We conclude this section with an upper bound for the solution of (l.l), THEOREM
(2.18)
(1.3).
2.4. Let p > 1, and let u be a solution of (].I), (1.3). Then U(X,1) 5 c*t-l’@-‘)
where c* = (l/(p
in ST,
- l))““‘-‘).
Proof. It is readily verified that the function c*(l - I~)-~‘~-‘) satisfies (1 .I) in R” x (to, co), to E R. For any 6 > 0 we choose 6r E (0,6) so that
Ilu(*, B)IJL”(R”)_( c*(t - 6,)-“@-? Let U, = u and u2 = c*(t - 6,)-1’(p-1). Then we have 'T" [u,(T)
- u2(T))xe-IX’ s ?6
BR
I
[UT -
uy -
(u, - u2)a,l
Al
, BR
-H T
c
6
L
aBR
in which < is the solution of the problem c, + acPAc - ,Ic = 0 [=O
xEBR,
tE (0, T)
if 1x1 = R, t E (0, T)
(lx, T) = X(x)e-‘“I,
XEBR,
where x E C,“(B,), 0 5 x 5 1, and atp is the function defined in the proof of theorem 2.2. Clearly (2.13) holds again.
114
L. A. PELET~R and
Z. JUNNMC
Because I.+ is bounded away from zero on S,, there exists a constant M, which does not depend on E, p or R such that sup[a,,(x, 1): 0 < 1x1< R) I M. Also, as in lemma 2.1, we have I[(x, t)l I Me-‘“’
in BR x [0, T]
and IV[(x, t)l 5 MemR
on aBR
[0, T],
X
where M is again a generic constant which does not depend on p, E and R. Using these estimates for [ we obtain, as in the proof of theorem 2.2. (u,(T)
- u,(T)je-‘*IX
I
t
BR
Choose
R =
~-l’(“+l)
ss 7
M R"-'emR + .c"*R"'*
+
6
(ul - u2)+e-‘x’ . BR
and let E -+ 0. Then we arrive at the bound T (u,(T) - u2(r))e-'"'X 5
6 R”(l(l - zf2)+e-1”1.
M
.Ts
s R" Setting x(x) =
1 o
if ur(x, T) > u,(x, T) if ut(x, T) 5 u2(x, T),
we conclude that (u,(7) - u2(T))+emiXi 5
i
(u, - u2)+e-lxi,
M
L R”
which implies that u1 I u2, i.e. u(x, t) I
c*(t
-
8J”(J-
t > 6.
The proof is completed by letting 6 + 0. 3. EXISTENCE
THEOREM3.1. Suppose that
( > l-f
+
l
Then equation (1 .l) has a solution u E C(ST\((O, 0))) which satisfies the initial condition (1.3). In addition we have u(x, t) 5 E(x, t)
in ST,
where E is defined in (1.8). Proof.
We consider the Cauchy Problem u, = A&“) - up u(x, 0) = E
in Sf
(3.1)
on R”,
(3.2)
115
Source-type solutions of the porous media equation
where P is a positive number. Clearly, for every 0 > 0, E(*, l/k’) E L’(R”) fl P(R”). theorems 2.1 and 2.2, (3.1), (3.2) has a solution uf and
Hence, by
u~(x, t) I E(x, t + (IA’)).
(3.3)
From (3.3) we can deduce that for 1 < p c m + (2/n) 7 up”x 5 c, 1s0 R” where x E C,“(R”) and C is a constant independent of P. Moreover, by [6] the family (~0 is equicontinuous in every compact subset K of ST. Thus, there exists a subsequence {ulj] of lur] and a function u E C(S,) such that for every compact K C S7 as Pi + 03
4, -+ u
in C(K).
Plainly, by (3.3), u(x, t) I E(x, 2)
in ST,
(3.4)
and u E C(0, Z L’(R”)) fl L”(R” x (T, T)) for every 7 E (0, T). Because u 2 0, (3.4) implies that u E C(&\((O, 0))) and that u(x, 0) = 0 if x # 0. As in [2] we can now prove that u is a solution of (1. I), (1.3). THEOREM 3.2.
+ ( >
Suppose that
l-1
lcp
cm-cl,
n
and u is the solution of (1. l), (1.3) which was constructed t”Q(x,
I) - E(x, f)) + 0
in theorem 3.1. Then as t + 0,
where /3 = m - 1 + (2/n), uniformly in R”. .
We first prove an auxiliary lemma. LEMMA 3.1.
Suppose that u, and u2 are two solutions of the Cauchy Problem u, = A@‘“)
in ST
u(x,O) = 6(x)
(3.5)
in R”.
(3.6)
Then u1 L u2 in ST implies u1 = u2. Proof. C’@(R) n \
,BR
From the definition of a solution, we deduce that for any R > 0 and x [0, T]) such that < 1 0 and 5 = 0 on aB(R) x [0, T], u1 and u2 satisfy f lu,(x, 2) -
u,(x,
t)K(x,
1)
dJ,
=
c
lu,(x
s)
-
u2(x,
GMX,
9
ds
+
(u,
rS
c BR
+
u s
CUT
-
GY
A5
-
CUT
-
u,K,
BR
-
4wL
VI.
[ E
116
L. A. PELETER and Z. JUNNWG
Following the method used to prove theorem 2.2, we can show that for any x E C,“(R”), x B 0, u, and u2 satisfy
Letting s + 0 we obtain
s R”
lu,(x, 1) - GG m(x) dx 5 0
which implies that tlr I t12. Since we assumed that U, 2 u2, it follows that U, = u2. Proof of theorem 3.2. Let u be the solution of (1 .l), (1.3) constructed
in theorem 3.1. Then
for any k > 0 u,(x, t) dAfk”u(kx , k‘?)
is a solution of the Cauchy Problem u, = A@“) - kW
U(X,0) = 6(x)
in ST
on R”,
where p = n(m + (2/n) - p). Since E(x, t) = k”E(kx, k%)
k>O
it follows from theorem 3.1 that Uk(X, 1) 5 Ek(X, I) = E(x, I).
Clearly, the continuity of u implies that the family (u,: 0 < k < 1) is equicontinuous, and so there exists a subsequence (uk,), kj -, 0 asj -, co, and a function U E C(S,) such that for every compact subset K of R” x (0, co), uk,
+
u
in C(K).
asj+a,
It is easy to show that U E C(0, 7; L’(R”)) fl L”(R” x (5, T)) for every r E (0, T), that U I E, and that U is a solution of (3.9, (3.6). Hence, by lemma 3.1, U = E and the entire sequence (uk) converges to E as k + 0. Therefore k”u(kx, kBn) = tlk(x, 1) + E(x, 1) = k”E(kx, k”‘)
Since E(x, 1) + 0 as 1x1+ oo the convergence kx = y, we conclude that t’%(~, uniformly in R”.
as k + 0.
is uniform in R”. Thus, writing k5” = t and
t) - E(Y, t)l + 0
as t + 0
Source-type solutions of the
117
porous media equation
4. NONEXISTENCE
THEOREM4.1. Suppose that
+ ( >
Then problem (l.l),
2
pzm+-.
1-i
n
(1.3) has no solution.
The proof proceeds via two lemmas. 4.1. Let p z m + (2/n), and suppose u is a solution of (l.l), (i) for any R > 0,
(1.3). Then
LEMMA
T
up
ss0
c
00;
1x1
(ii) u satisfies
for any 5 E C,“(R” x (-T, Proof.
T)).
(i) Because u E C(0, T; L’(R”))
and
lim , o R w, -
.,c n
Mx)
dx = K(O)
for every x E C,“(R”), it follows from the uniform boundedness principle that L”(0, r; L’(R”)). From the definition of a solution, we deduce that for any x E C,“(R”) and E E (0, T),
s R”
W->TMx~~
+
[tTs.nupx
=
jcTjRnu-Ax
5
([:S.!Aq--(
+
u E
jRnWx(X)~
+ I;:s.”
uiA,i>
Now choose x(x) = 1 if 1x1 < R and let E -+ 0. Then we obtain
s
U(& &t(X) dr.
(4.1)
R”
T up
ji0
<
00.
Ixl
(ii) Let 4’k(X,1) = rlk(lX12+ r2’9i(x, where c(x, t) E C,“(R” q(s) = 0 of s s 1, q(s) Since for every k > from the definition of
x (-7,
r),
T)), /I = m - 1 + (2/n) and q E C”(R) has the properties: = 1 ifs 1 2 and q&) = q(ks). 0, &(x, 1) vanishes in a neighbourhood of the origin (0, 0), it follows a solution that u satisfies
118
L. A. PELETIERand Z. JUNNXNG
r JJ
It is therefore sufficient to verify that as k + a0
0
Set
JJ T
Uh)rC
+
0,
R"
0
JJ T
umAqkc + 0
R" .
0
U”(Vrlk, v43 + 0.
(4.2)
R"
Dk = ((x, t): t > 0, k-’ < 1x1’ + I*‘@”< 2k-‘). Then / JoTJRnuctl,),il
5 cknB/j’JDk
(4.3)
u
(4.4)
(4.5)
Noting that we obtain by means of HCilder’s inequality
where
Similarly, since p > m .
k
Since y c 0 because p 2 m + (2/n) and n
11up+
. ,Dk
0
ask+oo,
by part (i), (4.2) is proved. LEMMA 4.2.
Suppose u E C(0, Z L’(R”)), u 2 0 and T
up< a0
for any R > 0.
(4.6)
Source-type solutions of the porous media equation
If for any c E C,“(R” x (-T,
119
T)),
ss T
(UC, +
0
then
urnAC -
up0 = 0,
(4.7)
R"
U(X,f)c#J(X)dw = 0 lim f-0. cR” for every $ E C,“(R”). Proof. Let j E C”(R) have the properties: j Z 0, j(s) = 0 if IsI L 1 and jr&s) ds = 1. For h > 0 we define j,,(s) = h-‘j(s/h) and r-r-Zh &l(f)
=
1
_oD
-
i&l
c-b
s T E (0, T) is some fixed number. Clearly, qh E C”(R), qh(t) = 1 if 2 < T + h, 0 I qh I 1 and limh_o q,,(f) = 0 if 1 > 7. For 4 E C,“(R”) we set ((x, 2) = +(X)qh(t) in (4.7) to obtain
where
ss is T
-
Rnjh(t
-
‘5 -
2h)@
+
(u”qh
0
If we now let h * 0 we obtain
Aq6 - u’q,,$‘) = 0.
R"
7
i ,R”
(u” A4 - ~“4)
u(x, +#4e CL%=
,O R"
and this implies, in view of (4.6), that lim u(x, T)4(X) dX = 0. 2-o ,R” i Proof of theorem 4.1. Suppose to the contrary that (1. I), (1.3) has a solution. Then by lemma
4.1 and 4.2 we have
ID lim
f-0 .R” I
for every 4 E C,“(R”). This contradicts 5. A VERY
u(x, ()4(X) du = 0
(1.3). SINGULAR
SOLUTION
Definition 5.1. A nonnegative
solution Wof (1.1) is called a very singular solution if it satisfies (1.4)-(1.6) and (1.1) in the sense of distributions.
THEOREM
+ ( >
5.1. Suppose that
l-2
n
Then (1.1) has a very singular solution.
l
L. A. PELETIERand Z. JIJNNING
120
We consider the Cauchy Problem
Proof.
u, = A@“) - up
uro
in ST
in ST in R”,
u(x, 0) = c&x)
where c > 0 and construct as in theorem 3.1 a solution uc(x, t), Plainly,
r) 5
r&k
.in Sr,
&(x, t)
(5.1)
where E,(x, 1) = t-‘/O
a,’ + -
-
(5.2)
’
is the fundamental solution of the Porous Media Equation with u(x, 0) = c&x). By theorem 2.2, c, 5 c, * u,, 5 u,, in Sr. Indeed, if c, I c2, then E,, s E,, and in particular E,,(x, l/Q 5 EJx, the approximate solutions ur; and I+; satisfy UC; 5 u=; for every a > 0, from which (5.3) follows. Thus the set fu,: c > 0) is nondecreasing (5.2) uc(x, t)
5
(5.3)
l/P), t’ > 0. Therefore
in ST
with respect to c. On the other hand, by (5.1) and
t-“8
!$)-‘(‘-m)
($
(5.4)
for every c > 0 and by theorem 2.3 applied to (3.1) and (3.2) z&(x,
t) I
1
l/@- 1) t - l/(p- 1)
( > -
P-l
(5.5)
Therefore the set (u,: c > 0) is uniformly bounded on any compact set K c Sr\((O, 0)). Since the modulus of continuity of u, only depends on the upper bound of uc, the set IU,: c > Oj is equicontinuous on any compact set K C Sr\((O, 0)), lim U, = W,
C-CC
where WE C(S,). From (5.1) we conclude that WE C&\((O, 0))) and that W(x, 0) = 0 if x # 0. It is easily verified that W satisfies (1 .l) in the sense of distributions and satisfies (1.4)-(1.6). REFERENCES 1. BR~ZIS H. & FRIEDMAN A., Nonlinear parabolic equations involving measures as initial conditions, J. Marh. Pure er Appliqudes 62, 13-97 (1983). 2. KAMW S. & PELETIER 1. A., Source-type solutions of degenerate diffusion equations with absorption, Isruei J. Math. 50, 219-230 (1985). 3. BR~ZISH., PELETIER L. A. & TERMAND., A very singular solution of the heat equation with absorption, Arch. Ration.
Mech.
Anal.
95, 185-207
(1986).
Source-type solutions of the porous media equation
121
4. PELET~ER L. A. & TEW D., A very singular solution of the porous media equation with absorption, J. o’$J Eqns 65, 396-410 (1986). 5. KMQN S., PELET~ER L. A. & VA~QUEZJ. L., Classification of singular solutions of a nonlinear heat equation, Duke Murh. JI 58, 601-615 (1989). 6. SACHSP. E., Continuity of solutions of a singular parabolic equation, Nonlineur Anelysis 7, 387-409 (1983).
h4crhod.c
&
Apphmrions. Vol.
14.
No.
2. pp.
107-121.
0362-%6X/90 S3.00+ .OO @ 1990 Pergmlon Press PlC
1990.
SOURCE-TYPE SOLUTIONS OF THE POROUS MEDIA EQUATION WITH ABSORPTION: THE FAST DIFFUSION CASE L.
A.
PELETIER
Mathematical Institute, Leiden University,
The Netherlands
and ZHAO JUNNING Departmentof Mathematics,Jilin University,China (Received 2 January 1989; received for publication 20 February 1989) Key words and phrases: Degenerate diffusion, singular solutions.
1. INTRODUCTION
IN THIS PAPER
we
consider the Cauchy Problem u, = A(u”) - up
on S, = R” x (0, T)
u(x, 0) = 0
x E R”\(O],
(1.1) (1.2)
where n L 1,p > 1 and 0 < m < 1. For the cases of regular diffusion (m = 1) and slow diffusion (m > 1) it was shown in respectively [l] and [2] that the problem (1. l), (1.2) has a solution which satisfies the initial condition 0,
0) = 6(x),
(1.3)
where 6 denotes the Dirac mass centered at the origin if and only if 1 < p c m + (2/n). In addition it was shown in [3] and [4] that the same condition was necessary and sufficient for the existence of a very singular solution of (1 .l), (1.2), i.e. a solution W with the properties w E C(%WO* 0)l); W(x, 0) = 0
(1.4)
if x E R”\(O);
(1.5)
,n
W(x, t)dx lim ‘10 ,I lx/
= +aJ
for any r > 0.
(1.6)
The object of this paper is to extend these results to the case when (1.7) where I. II. III.
(s)+ = max(0, s]. Specifically, we shall prove Ifp L m + (2/n), then (l.l), (1.3) has no solution. If 1 < p < m + (2/n), then (l.l), (1.3) has a solution. Let 1 < p < m + (2/n). Then there exists a solution u(x, t) of (1 .I), (1.3) such that t’ql(x,
t) - E(x, 1)j + 0 107
ast+O
L. A. PELETIERand Z. JUNNING
108
uniformly on R”. Here
and E(x,
I)
is the Barenblatt-Pattle
=
1-1’6
I
a2
+
l-
-
m - 1x1’
-1/(1-m)
2mnj3 t2’On1
’
fZER+
(1.8)
solution of the Cauchy Problem II, = A@“),
U(X,0) = 6(x).
IV. If 1 < p < m + (2/n), then (1.1) has a very singular solution. The ideas of the proofs of I, II and III are similar to those given in [I] and [2]; for the proof of IV we use ideas of (51. However, the fact that when m < 1, u’“-~ is unbounded, and urn TVL’(R” x (T, 7)) in general, requires a number of changes in the arguments. 2. PRELIMINARIES
Let BR = [x E R”: 1x1< R] and aBR = (x E R”: 1x1 = R). Definition 2.1. A solution u of (1. l), (1.3) is a nonnegative
function defined in ST such that (a) u E C(0, 7; L’(R”)) n C(S,) rl L”(R” x (T, T)) for every 7 E (0, T); (b) for any R > 0 and any 5 E C2(B, x [0, T]) such that [ vanishes on aBR x [0, T] and near BR x IO] and B, x (T], u satisfies ‘T’
T (UC,
+
u”‘AC - u”T) -
I I .O,BR
U”(Vl;, v) = 0, CT vaB~ “0
where v denotes the outward pointing normal on i3BR; (c) for any x E C’,“(R”), n lim
rio ,\ R”
w,
f)Xce dx = x(O).
In the next three theorems we discuss solutions of equation (1.1) in R” with bounded, nonnegative initial data: 0,
0) = 6(.x),
410
in R”.
(2.1)
Such solutions are defined as in definition 2.1. THEOREM 2.1. Let 4 E L’(R”) n L”(R”). Then L-(0, c L’(R”)) n L’(&) n C(&).
Proof. We consider the approximate
problem
(1 .l),
(2.1) has a solution
uE
problem
II, = A(u”) - up +- E”-’
in B1,, x (0, T)
u(x, f)l,,,=,-1 = &“+’ u(x, 0) = #Q(x)
x E &,,
(2.2) (2.3) (2.4)
109
Source-type solutions of the porous media equation
where 0 < E < 1 and 4, E C”(R”) has the properties (i) 4, r P+‘, JBl,j& - $1 -+ 0 as ~10; (ii) I#I,(x) = .9+l near 1x1 = l/E. It is well-known that (2.2)-(2.4) has a unique classical solution uE, and that &“+I I u, I c,
(2.5)
where C is a constant which does not depend on E. By [6], for any compact set K C S,, the family (u,] is equicontinuous in K. Thus we can select a sequence from (u,], which we denote again by [u,] and a function u E C(S,) such that for every compact set K C ST, in C(K). u, + u as&l0 (2.6) For u we have the following estimates
Il~llrw, 9 lluPll L’(ST)5 Il4llrqRn,.
(2.7)
IlullL”(0.C’;L’(R”)) 5
multiply (2.2) by u,/(u, + q) (V > 0) and
Indeed, set U, = u, - E”+‘, so that v, = 0 on M,,,, integrate over B,,, x [0, T]. Then one obtains
_T'
-
JI
upA+
0 . Bl/r
v, +
'T"
rl
v-v
3I 0
L
&
n+l
-.
0,
v, +
BI/E
rl
Letting q ---)0 and E -) 0 in turn we obtain
ci R”
utx, TJk
+‘{;j)
5 JR$#l
from which (2.7) and (2.8) follow. From these bounds we can readily conclude that u is a solution of (1 .l), (2.1). THEOREM 2.2. Let ulr u2 E L”(S,) rl L’(S,) be solutions 4~~)I#J*E L”(R”) n L’(R”), and suppose ‘?I ’
1 1+ 2 - n ’
( >
p>
of (l.l),
(2.1) with initial values
I.
Then O1 s 4* on R” implies that u1 s u2 on S,. COROLLARY
2.1. Let ~5E L”(R”) n L’(R”). Then (1 .l), (2.1) has a unique solution.
Proof of theorer?? 2.2. Let u1 and u2 be solutions of (1. I) with initial values O1 and & , respectively. Then, by the definition of a solution, for any R > 0 and [ E C*(Sr) such that <(x, t) = 0
L. A. PELETIER and Z.
110
for x E dB,,
JUNNING
u1 and ZQsatisfy
s BR
@r(T) - @%I~)
-
s BR
(d, - &)5(O)
T =
((~1 ss0
-
u2K
+
(6
-
4’)
Al
-
d’)Cl
BR T
ss0
U,m)P~~
(2.9)
VI*
aBR
Let E > 0, p > 0, and define the functions u’: - uz” &(U;” - t(F) *
a, = U’
-
u2 +
aw= a,*
Jp,
where Jp E C=(R”+‘) is a mollifier with the properties supp Jp C ((x, 1): 1x1< p, It] < p) and jR”+lJp = 1. Clearly as, acp 2 co > 0 where co is a constant independent of E and p. We consider the Dirichlet problem in BR x (0, T)
[f + a+,Ac - Ac = 0 <=o
(2.10) (2.11)
on i3BR x (0, T) in BR,
[(x, T) = x(x)e-fi’“’
(2.12)
where 1. > 0 and x E C,“(B,), 0 I x I I. Problem (2.10)-(2.12) has a unique nonnegative smooth solution [ which is uniformly bounded with respect to R, E and p. Multiplying (2.10) by AC and then integrating the result over BR x (0, T) we obtain T
(2.13) b’d2 5 M, a&W2 5 M, s BR is0 BR where M is a constant, which does not depend on R, E and p. In the next lemma we give two further estimates. LEMMA2.1. There exists a constant M, which does not depend on R, E and p such that [(x, 1) 5 MemmXi IV[(x, r)l 5 MeeRG Accepting this lemma temporarily, in (2.9). This yields [u,(T)
i
-
u2(T)lxe-Jzlx’
in 8, x [0, T]
(2.14)
On a& X [0, T].
(2.15)
we use the function 4’constructed above as a test function
-
(4,
-
42K(O)
BR
BR
=
wi”
-
~3
-
(u,
-
@WC,
-
u2)aepl N
+
A--
u:
-
u1 -
uz” u2
(u,
-
u2)C
T
-
11
,o
=x,+x,+x,.
@I” aBR
v)
(2.16)
111
Source-type solutions of the porous media equation
Write
(Ss r
I
0
BR(UI
E2(U;” -
-
u2
+
U24u"m &(UI
u~,,IA[[~)~”
2y'2(j:lBR
-
u2
))
(2.17) where we have used the fact that ucP is bounded away from zero. Thus: (i) if m 1 +
(ii) if m c *
x,
I
,JBR u,r+(1:1.,u2)J+
1~2Rn-~2[ ( cc?
&,
where we have chosen p = P(E, R) so small that T
L
ii0
(4 - acJ2 c
e2.
5
BR
Choosing uf A > sup ST I u1-"2
4
,
I
which is possible because p > 1, we ensure that '-7
x2 5
c
Ii
@I
-
u2)+.
I 0 ,BR
Finally, using lemma 2.1 we obtain X3 5 CR”-‘e-RJZ. Substituting the bounds for X, and X3 into (2.16), we obtain
ii T
(u,(T) i
c BR
- u2(T)]e-‘X’G~ 5 X, + C
(4, BR
-
42)+
+
,o
,BR
(u,
-
u2)+
+
.
L. A. PELETIERand Z. JIJNNING
112
Now let R = E-(~‘~)-~, where cc > 0 and 2 - n + 2mn ’ < 2n(l - am)
if m < +.
fss
Then, if we let E + 0, X1 -* 0 and we obtain, since +i S 62, (u,u-)
-
U2(n)+
5
c
b,
-
R”
0
u21+.
This implies that u1 I u2 and theorem 2.2 is proved. In analogous fashion we can prove the following comparison theorem. THEOREM2.3. Let ul, u2 E C(0, Z L’(R”)) 17L“‘(S,), and let
s
ss I
(u,(t) - u,(t))tv) 5
BR
lb,
0
-
u2K
+
-
4”)
AC
-
04
-
GXl
BR
“T
-
1s
0
-
GWY,
VI
aBR
for any c E C2(BR x [0, T)), R > 0 such that [ L 0 and c = 0 on aBR x [0, T]. Then U, 5 u2, It remains to prove lemma 2.1. Proof of lemma 2. I. Define for rl, r2 > 0, r, > r2 the cylinders
Qr,.rl = @,,W,) x 10, Tl. In QR, r, we consider the auxiliary function w(x, t) = -c(x, t) + ke-‘X’G+acT-r). Clearly, w(x, 1) L 0
if 1x1= 1
and
.
1x1 = R
provided k is chosen large enough. Moreover, we have w(x, T) = -xe-IxI*
+ ke-l”lJ” 1 0
if k is large enough. Finally, since C~,Eis uniformly bounded above in E and p, we can choose /3 so that VW= w, + acPAw - 13~ =
ke-i”‘&+“‘T-0
o,,E
_
4/Z
+p-l
co. >
Thus, by the maximum principle, (2.14) follows. To prove (2.15) we use a barrier function argument, and we consider in QR,R_l the auxiliary function 2(x, t) = 5(x, 1) + K,e-R~(eKz(‘xi-R) - 1). Clearly, z(x, t) = 0 if 1x1 = R. Since suppx C B, we also have z(x, T) = *(x)e-‘“iG + Kle-RG(eXZ(‘X’-Rf - 1) < 0
113
Source-type solutions of the porous media equation
if K, is large enough. Moreover by (2.14) we can choose K, and K2 so large that if 1x1 = R - 1 2(x, t) = ((x, t) + K,e-RG(e-K2 5 Me-(R-r)fi
- 1)
+ K,e-RJr(e-x2
- 1)
+ -n-l - A + AK,emRz> 1x1 K2 >
0.
z s 0 in QR,R_r and if 1x1 = R,
az > 0 zwhich implies that ai,_~~~-Rfi av12
if 1x1= R.
Observing that ac/av I 0 on (xl = R, (2.18) implies (2.15). We conclude this section with an upper bound for the solution of (l.l), THEOREM
(2.18)
(1.3).
2.4. Let p > 1, and let u be a solution of (].I), (1.3). Then U(X,1) 5 c*t-l’@-‘)
where c* = (l/(p
in ST,
- l))““‘-‘).
Proof. It is readily verified that the function c*(l - I~)-~‘~-‘) satisfies (1 .I) in R” x (to, co), to E R. For any 6 > 0 we choose 6r E (0,6) so that
Ilu(*, B)IJL”(R”)_( c*(t - 6,)-“@-? Let U, = u and u2 = c*(t - 6,)-1’(p-1). Then we have 'T" [u,(T)
- u2(T))xe-IX’ s ?6
BR
I
[UT -
uy -
(u, - u2)a,l
Al
, BR
-H T
c
6
L
aBR
in which < is the solution of the problem c, + acPAc - ,Ic = 0 [=O
xEBR,
tE (0, T)
if 1x1 = R, t E (0, T)
(lx, T) = X(x)e-‘“I,
XEBR,
where x E C,“(B,), 0 5 x 5 1, and atp is the function defined in the proof of theorem 2.2. Clearly (2.13) holds again.
114
L. A. PELET~R and
Z. JUNNMC
Because I.+ is bounded away from zero on S,, there exists a constant M, which does not depend on E, p or R such that sup[a,,(x, 1): 0 < 1x1< R) I M. Also, as in lemma 2.1, we have I[(x, t)l I Me-‘“’
in BR x [0, T]
and IV[(x, t)l 5 MemR
on aBR
[0, T],
X
where M is again a generic constant which does not depend on p, E and R. Using these estimates for [ we obtain, as in the proof of theorem 2.2. (u,(T)
- u,(T)je-‘*IX
I
t
BR
Choose
R =
~-l’(“+l)
ss 7
M R"-'emR + .c"*R"'*
+
6
(ul - u2)+e-‘x’ . BR
and let E -+ 0. Then we arrive at the bound T (u,(T) - u2(r))e-'"'X 5
6 R”(l(l - zf2)+e-1”1.
M
.Ts
s R" Setting x(x) =
1 o
if ur(x, T) > u,(x, T) if ut(x, T) 5 u2(x, T),
we conclude that (u,(7) - u2(T))+emiXi 5
i
(u, - u2)+e-lxi,
M
L R”
which implies that u1 I u2, i.e. u(x, t) I
c*(t
-
8J”(J-
t > 6.
The proof is completed by letting 6 + 0. 3. EXISTENCE
THEOREM3.1. Suppose that
( > l-f
+
l
Then equation (1 .l) has a solution u E C(ST\((O, 0))) which satisfies the initial condition (1.3). In addition we have u(x, t) 5 E(x, t)
in ST,
where E is defined in (1.8). Proof.
We consider the Cauchy Problem u, = A&“) - up u(x, 0) = E
in Sf
(3.1)
on R”,
(3.2)
115
Source-type solutions of the porous media equation
where P is a positive number. Clearly, for every 0 > 0, E(*, l/k’) E L’(R”) fl P(R”). theorems 2.1 and 2.2, (3.1), (3.2) has a solution uf and
Hence, by
u~(x, t) I E(x, t + (IA’)).
(3.3)
From (3.3) we can deduce that for 1 < p c m + (2/n) 7 up”x 5 c, 1s0 R” where x E C,“(R”) and C is a constant independent of P. Moreover, by [6] the family (~0 is equicontinuous in every compact subset K of ST. Thus, there exists a subsequence {ulj] of lur] and a function u E C(S,) such that for every compact K C S7 as Pi + 03
4, -+ u
in C(K).
Plainly, by (3.3), u(x, t) I E(x, 2)
in ST,
(3.4)
and u E C(0, Z L’(R”)) fl L”(R” x (T, T)) for every 7 E (0, T). Because u 2 0, (3.4) implies that u E C(&\((O, 0))) and that u(x, 0) = 0 if x # 0. As in [2] we can now prove that u is a solution of (1. I), (1.3). THEOREM 3.2.
+ ( >
Suppose that
l-1
lcp
cm-cl,
n
and u is the solution of (1. l), (1.3) which was constructed t”Q(x,
I) - E(x, f)) + 0
in theorem 3.1. Then as t + 0,
where /3 = m - 1 + (2/n), uniformly in R”. .
We first prove an auxiliary lemma. LEMMA 3.1.
Suppose that u, and u2 are two solutions of the Cauchy Problem u, = A@‘“)
in ST
u(x,O) = 6(x)
(3.5)
in R”.
(3.6)
Then u1 L u2 in ST implies u1 = u2. Proof. C’@(R) n \
,BR
From the definition of a solution, we deduce that for any R > 0 and x [0, T]) such that < 1 0 and 5 = 0 on aB(R) x [0, T], u1 and u2 satisfy f lu,(x, 2) -
u,(x,
t)K(x,
1)
dJ,
=
c
lu,(x
s)
-
u2(x,
GMX,
9
ds
+
(u,
rS
c BR
+
u s
CUT
-
GY
A5
-
CUT
-
u,K,
BR
-
4wL
VI.
[ E
116
L. A. PELETER and Z. JUNNWG
Following the method used to prove theorem 2.2, we can show that for any x E C,“(R”), x B 0, u, and u2 satisfy
Letting s + 0 we obtain
s R”
lu,(x, 1) - GG m(x) dx 5 0
which implies that tlr I t12. Since we assumed that U, 2 u2, it follows that U, = u2. Proof of theorem 3.2. Let u be the solution of (1 .l), (1.3) constructed
in theorem 3.1. Then
for any k > 0 u,(x, t) dAfk”u(kx , k‘?)
is a solution of the Cauchy Problem u, = A@“) - kW
U(X,0) = 6(x)
in ST
on R”,
where p = n(m + (2/n) - p). Since E(x, t) = k”E(kx, k%)
k>O
it follows from theorem 3.1 that Uk(X, 1) 5 Ek(X, I) = E(x, I).
Clearly, the continuity of u implies that the family (u,: 0 < k < 1) is equicontinuous, and so there exists a subsequence (uk,), kj -, 0 asj -, co, and a function U E C(S,) such that for every compact subset K of R” x (0, co), uk,
+
u
in C(K).
asj+a,
It is easy to show that U E C(0, 7; L’(R”)) fl L”(R” x (5, T)) for every r E (0, T), that U I E, and that U is a solution of (3.9, (3.6). Hence, by lemma 3.1, U = E and the entire sequence (uk) converges to E as k + 0. Therefore k”u(kx, kBn) = tlk(x, 1) + E(x, 1) = k”E(kx, k”‘)
Since E(x, 1) + 0 as 1x1+ oo the convergence kx = y, we conclude that t’%(~, uniformly in R”.
as k + 0.
is uniform in R”. Thus, writing k5” = t and
t) - E(Y, t)l + 0
as t + 0
Source-type solutions of the
117
porous media equation
4. NONEXISTENCE
THEOREM4.1. Suppose that
+ ( >
Then problem (l.l),
2
pzm+-.
1-i
n
(1.3) has no solution.
The proof proceeds via two lemmas. 4.1. Let p z m + (2/n), and suppose u is a solution of (l.l), (i) for any R > 0,
(1.3). Then
LEMMA
T
up
ss0
c
00;
1x1
(ii) u satisfies
for any 5 E C,“(R” x (-T, Proof.
T)).
(i) Because u E C(0, T; L’(R”))
and
lim , o R w, -
.,c n
Mx)
dx = K(O)
for every x E C,“(R”), it follows from the uniform boundedness principle that L”(0, r; L’(R”)). From the definition of a solution, we deduce that for any x E C,“(R”) and E E (0, T),
s R”
W->TMx~~
+
[tTs.nupx
=
jcTjRnu-Ax
5
([:S.!Aq--(
+
u E
jRnWx(X)~
+ I;:s.”
uiA,i>
Now choose x(x) = 1 if 1x1 < R and let E -+ 0. Then we obtain
s
U(& &t(X) dr.
(4.1)
R”
T up
ji0
<
00.
Ixl
(ii) Let 4’k(X,1) = rlk(lX12+ r2’9i(x, where c(x, t) E C,“(R” q(s) = 0 of s s 1, q(s) Since for every k > from the definition of
x (-7,
r),
T)), /I = m - 1 + (2/n) and q E C”(R) has the properties: = 1 ifs 1 2 and q&) = q(ks). 0, &(x, 1) vanishes in a neighbourhood of the origin (0, 0), it follows a solution that u satisfies
118
L. A. PELETIERand Z. JUNNXNG
r JJ
It is therefore sufficient to verify that as k + a0
0
Set
JJ T
Uh)rC
+
0,
R"
0
JJ T
umAqkc + 0
R" .
0
U”(Vrlk, v43 + 0.
(4.2)
R"
Dk = ((x, t): t > 0, k-’ < 1x1’ + I*‘@”< 2k-‘). Then / JoTJRnuctl,),il
5 cknB/j’JDk
(4.3)
u
(4.4)
(4.5)
Noting that we obtain by means of HCilder’s inequality
where
Similarly, since p > m .
k
Since y c 0 because p 2 m + (2/n) and n
11up+
. ,Dk
0
ask+oo,
by part (i), (4.2) is proved. LEMMA 4.2.
Suppose u E C(0, Z L’(R”)), u 2 0 and T
up< a0
for any R > 0.
(4.6)
Source-type solutions of the porous media equation
If for any c E C,“(R” x (-T,
119
T)),
ss T
(UC, +
0
then
urnAC -
up0 = 0,
(4.7)
R"
U(X,f)c#J(X)dw = 0 lim f-0. cR” for every $ E C,“(R”). Proof. Let j E C”(R) have the properties: j Z 0, j(s) = 0 if IsI L 1 and jr&s) ds = 1. For h > 0 we define j,,(s) = h-‘j(s/h) and r-r-Zh &l(f)
=
1
_oD
-
i&l
c-b
s T E (0, T) is some fixed number. Clearly, qh E C”(R), qh(t) = 1 if 2 < T + h, 0 I qh I 1 and limh_o q,,(f) = 0 if 1 > 7. For 4 E C,“(R”) we set ((x, 2) = +(X)qh(t) in (4.7) to obtain
where
ss is T
-
Rnjh(t
-
‘5 -
2h)@
+
(u”qh
0
If we now let h * 0 we obtain
Aq6 - u’q,,$‘) = 0.
R"
7
i ,R”
(u” A4 - ~“4)
u(x, +#4e CL%=
,O R"
and this implies, in view of (4.6), that lim u(x, T)4(X) dX = 0. 2-o ,R” i Proof of theorem 4.1. Suppose to the contrary that (1. I), (1.3) has a solution. Then by lemma
4.1 and 4.2 we have
ID lim
f-0 .R” I
for every 4 E C,“(R”). This contradicts 5. A VERY
u(x, ()4(X) du = 0
(1.3). SINGULAR
SOLUTION
Definition 5.1. A nonnegative
solution Wof (1.1) is called a very singular solution if it satisfies (1.4)-(1.6) and (1.1) in the sense of distributions.
THEOREM
+ ( >
5.1. Suppose that
l-2
n
Then (1.1) has a very singular solution.
l
L. A. PELETIERand Z. JIJNNING
120
We consider the Cauchy Problem
Proof.
u, = A@“) - up
uro
in ST
in ST in R”,
u(x, 0) = c&x)
where c > 0 and construct as in theorem 3.1 a solution uc(x, t), Plainly,
r) 5
r&k
.in Sr,
&(x, t)
(5.1)
where E,(x, 1) = t-‘/O
a,’ + -
-
(5.2)
’
is the fundamental solution of the Porous Media Equation with u(x, 0) = c&x). By theorem 2.2, c, 5 c, * u,, 5 u,, in Sr. Indeed, if c, I c2, then E,, s E,, and in particular E,,(x, l/Q 5 EJx, the approximate solutions ur; and I+; satisfy UC; 5 u=; for every a > 0, from which (5.3) follows. Thus the set fu,: c > 0) is nondecreasing (5.2) uc(x, t)
5
(5.3)
l/P), t’ > 0. Therefore
in ST
with respect to c. On the other hand, by (5.1) and
t-“8
!$)-‘(‘-m)
($
(5.4)
for every c > 0 and by theorem 2.3 applied to (3.1) and (3.2) z&(x,
t) I
1
l/@- 1) t - l/(p- 1)
( > -
P-l
(5.5)
Therefore the set (u,: c > 0) is uniformly bounded on any compact set K c Sr\((O, 0)). Since the modulus of continuity of u, only depends on the upper bound of uc, the set IU,: c > Oj is equicontinuous on any compact set K C Sr\((O, 0)), lim U, = W,
C-CC
where WE C(S,). From (5.1) we conclude that WE C&\((O, 0))) and that W(x, 0) = 0 if x # 0. It is easily verified that W satisfies (1 .l) in the sense of distributions and satisfies (1.4)-(1.6). REFERENCES 1. BR~ZIS H. & FRIEDMAN A., Nonlinear parabolic equations involving measures as initial conditions, J. Marh. Pure er Appliqudes 62, 13-97 (1983). 2. KAMW S. & PELETIER 1. A., Source-type solutions of degenerate diffusion equations with absorption, Isruei J. Math. 50, 219-230 (1985). 3. BR~ZISH., PELETIER L. A. & TERMAND., A very singular solution of the heat equation with absorption, Arch. Ration.
Mech.
Anal.
95, 185-207
(1986).
Source-type solutions of the porous media equation
121
4. PELET~ER L. A. & TEW D., A very singular solution of the porous media equation with absorption, J. o’$J Eqns 65, 396-410 (1986). 5. KMQN S., PELET~ER L. A. & VA~QUEZJ. L., Classification of singular solutions of a nonlinear heat equation, Duke Murh. JI 58, 601-615 (1989). 6. SACHSP. E., Continuity of solutions of a singular parabolic equation, Nonlineur Anelysis 7, 387-409 (1983).