Nonlinear Diffusion Equations

(1.3)

where \f is the hydrostatic potential due to capillary suction and z the gravitational potential. Here we choose the (x,y,z) coordinate system in such a way that the z-coordinate is vertical and pointing upwards. l

2

Newtonian

Filtration

Equations

Combining (1.1), (1.2) and (1.3) we obtain

g=div W )V*) + ™

,,.4,

Between the variables 9 and 9 there exists an empirical relationship which might be quite complicated because of hysterical effects. d^> For many porous media, ^ is a function of 9 and K(9) —— and K(9) can be expressed reasonably as follows K(0)^

= Doem-1,

K(9) =

K0en,

where Do, KQ, m, n are positive constants and 1 < m < n. In this case, (1.4) becomes

after necessary change of variables (cf.[BEA], [BRC], [IR], [SWA]). Thus we have, for horizontal flow, I

(1.6)

! - " •

1 for vertical flow,

W dt

d29m dz2

+

(1.7)

dz

In another case, 9 is a function of 9, 9 = 0(9), and 0 < 9(9) < 0, with 9 increasing strictly for 9 < 0 and 6(9) = 8 for 9 > 0. The equation (1.4) now becomes

°JSl = dHDi*m) where D(9) = K(9(9)).

+

°J&l,

(1.8)

In the saturated region, (1.4) reduces to

Thus, the equation (1.8) is of elliptic type in the saturated region and of parabolic type in the unsaturated region.

Introduction

If K(9(^)) sional flow

3

= 1, then we obtain the following equation for one-dimen-

^ - S -

<•••>

There are a large number of other diffusion problems from which quasilinear degenerate parabolic equations are arisen. A typical example in dynamics of biological groups is the model for the space-diffusion of groups ^

= AA(p) + a(p),

(1.10)

where p denotes the density of distribution of the species, a(p) the growth rate and A'(p) the diffusion coefficient which is positive for p > 0 and vanishes for p = 0. Evolution of two classes of biological groups, for example, species with different levels of age, can be governed by the system du ,. . _ , .. — = div(uV{u + v)), dv — = kaiv{vV{u + vj), where u and v denote the densities of the two different groups, k > 0 is a constant. In the extreme case, k = 0, v does not vary with time t and the equation for u turns out to be ^

= A(±uA+dw(uVv).

(1.11)

Equation of the form (1.6) also appears in plasma physics, however, in which 0 < m < 1, corresponding to the fast diffusion. Such kind of equation also arises in the study of phenomena occurring at the beginning of a nuclear explosion. At the very first stage, immediately following energy release, thermal waves are propagated in the as yet stationary gas. Heat conduction is then determined mainly by radiation, and the thermal conductivity is a function of temperature. Thus the equation to be considered is du

d2A(u)

at = -a?1'

(L12)

where u denotes the temperature and A(u) satisfies the condition: A'(u) > 0 for u > 0, A'(0) > 0. The corresponding initial condition is u{x,0) = ES(x),

Newtonian

4

Filtration

Equations

where E > 0 is a constant and S(x) the Dirac measure. The initial condition describes instant energy release at t = 0. Such kind of physical process motivates the interest in consideration of the problem with unbounded and measure initial data. (cf.[ZR]) Finally it should be pointed out that a lot of diffusion problems can be described by filtration equations with absorption term and convection term Oil

— = AA(U)+C-VB(U)

+
(1.13)

Equations of the form (1.5)-(1.13) are referred to as porous medium equations or Newtonian filtration equations. They are called briefly filtration equations very often. 1.1.2

Definitions

of generalized

solutions

Consider equations of the form ^

= AA(u),

(1.14)

N d2 where A = Y J —-^ and A(u) £ C 1 ^ , oo) satisfies dx2

A'(Q)=0,A'(u)>0,

foru>0.

(1.15)

The equation (1.14) is of parabolic type when u > 0. However, it degenerates when u = 0. If we do not restrict ourselves to the study of nonnegative solutions, then it should be assumed that A'(u) £ Cl (—00,00) and the condition (1.15) should be replaced by A'(0) = 0, A'(u) > 0,

for u + 0.

(1.16)

We will pay more attention to the special case

with m > 1, corresponding to the slow diffusion. If we do not restrict ourselves to the study of nonnegative solutions, then (1.17) should be written as ^

= A(i«r-i«).

(1.18)

Introduction

5

Basically no attention will be paid to the fast diffusion case, although it will be mentioned somewhere. We will mainly consider the Cauchy problem with initial condition xGRN,

u(x,Q) = u0(x),

(1.19)

where uo(x) is a nonnegative and locally integrable function. Since quasilinear parabolic equations with degeneracy do not have classical solutions in general, it is necessary to generalize the notion of solutions. Let QT = ^N x (0, T) and G be a subdomain of QTDefinition 1.1.1 A nonnegative function u is called a generalized solution of the equation (1.14) on G, if u, A(u) € L}oc{G) and u satisfies

G\

u-^ +A(u)A

(1.20)

for any

(1.21)

Definition 1.1.2 A nonnegative function u is called a generalized solution of the Cauchy problem (1.14), (1.19) on QT, if w is a generalized solution of the equation (1.14) on QT and u satisfies lim /

u(x,t)h(x)dx

= /

*->0+ JRN

uQ(x)h(x)dx

(1-22)

JRN

heC^°(RN).

for any

If u, A(u) G L\oc{QT) (we mean u, A(u) G LX{G) for any bounded subdomain G of QT-), then Definition 1.1.2 is equivalent to the following Definition 1.1.3 A nonnegative function u is called a generalized solution of the Cauchy problem (1.14), (1.19) on QT, iiu,A(u) G Ljoc(QT) and u satisfies //

(u-£

+ A(u)Aip)

dxdt + /

uo(x)
(1.23)

6

Newtonian

Filtration

Equations

for any ip G C°° (QT),which vanishes for large \x\ and t — T. The function

[u-£+A(u)Aip)r]edxdt+

wprfedxdt = 0.

(1.24)

The first integral tends to the first term of (1.23) a s E ^ O . Moreover //

mprj'edxdt — /

Jm.N

JJQT

=

/

Uo(x)
utpn'edxdt

+ I V'e(t) ( I

<X, t)(lfi(x, t) - ip{x, 0)) ) dt

Jo \JMN + / r]'£(t) (u(x,t) — Jo

J uo{x))(f(x,0)dx)dt

= h + h + hSince it is clear that lim I2 = 0,
t G [0,t 0 - e], Ve(t) = 0 for t G [i0 + e,T\, \rj'£(t)\ < -

for t G [0,T] and

some constants C and £ G (0,£o)- Then we have pto+E

p

/

/

itn-E

i»"

rto+£

uhr)'Edxdt+ I JO

p

/ ^R w

p

A(u)Ahr]edxdt

+ / VEN

uo(x)h(x)dx

= 0,

7

Introduction

namely r-t0+£

/

Ve(t) ( /

u(x,t)h(x)dx

— /

\JRN

Jt0-E /•

rto+£

— I

u(x,to)h(x)dx

+ /

JK W

dt )

/

A(u)AhTjEdxdt

JRN

JO

+ f u0{x)h(x)dx Jw"

u(x,to)h(x)dx)

JRN r

= 0.

(1.25)

The absolute value of the first term of (1-25) is dominated by I Jto-e

/ u(x,t)h(x)dx JM.N

— I JUN

u(x,to)h(x)dx dt

which tends to zero as e —• 0 for almost all io- In addition lim / £_!,

°7o

/ JMN

A(u)Ahr]edxdt

= /

/

A(u)Ahdxdt.

JO

JRN

u(x,to)h(x)dx

+ /

Thus we may conclude from (1.25) that / Jo

/

JWLN

A(u)Ahdxdt

—/

JmN

JmN

uo(x)h(x)dx

=0

for almost all io G (0, T) and hence for all io G (0, T), which implies (1.22). This means that u is a generalized solution in Definition 1.1.2. v

If

' G L}oc{QT) (i = 1, • • • , AT), then (1.23) can be transformed to

the form // JJQT

(u-£-VA(u)-V
d t

J

uo(x)(p(x,0)dx JRN

= 0.

(1.26) g

Now we define generalized solutions of the boundary value problem. Let O C M.N be a domain with appropriately smooth boundary dfl. Denote fiT = fi x (0, T), T = 9 0 x (0, T). Consider the first boundary value problem for the equation (1.14) on fiy, with boundary value condition u(x,t)\r

= g(x,t)

tor{x,t)er

(1.27)

u{x,0)=uo(x),

i£fi.

(1.28)

and initial value condition

8

Newtonian

Filtration

Equations

Definition 1.1.4 A nonnegative function u is called a generalized solution of the boundary value problem (1.14), (1.27), (1.28) o n f l r , if u, A(u) G L 1 ( Q T ) and u satisfies

//

\UlH

+ A(u)Af)dxdt

+ / u0(x)
=0

(1.29) for any if G C°°(QT) which vanishes for (x,t) G T, t = T and large \x\, where n denotes the outward normal to T. If

dA(u) K ' G Lx(nT)

i = 1,2, • • • , N, then Definition 1.1.4 is equivalent

OX{

to the following Definition 1.1.5 A nonnegative function u is called a generalized solution of the boundary value problem (1.14), (1-27), (1.28) on Q,T, if u,A{u) G L1(Q,T) and u satisfies //

(u~-

+ A(u)A
(1.30)

for any (p G C°°(Q,T) which vanishes when \x\ is large enough, t — T and in a neighborhood of T, and u has boundary value w|r, namely, the limit \\mu{y,t) exists for almost all (x,t) G T, and y—>x

yen hmu(y,t)=g(x,t).

(1.31)

y —n

yen It is not difficult to check that if

v ;

G i ^ f i r ) (i = 1,2,• • • , AT),

then Definition 1.1.4 is equivalent to Definition 1.1.5. To prove, first let u be a generalized solution of (1.14), (1-27), (1-28) r)A(u) in Definition 1.1.4. Then clearly (1.30) holds. Since , / ; G L1^) OXi

(i = 1,2.-,N) implies the existence of the boundary value of A(u), hence of u itself on T and A(u) G i 1 ( r ) . To prove (1.31), let

and a test = V^e, < 1,

£ e (z) = 1 for a; G fie = {a: G ft;dist(z, dfi) > e} (e > 0), |V&| < — for

9

Introduction

x £ ft. Then from (1.29) we obtain

if ( « ^ p + A(u)A(p! J dxdt + ff

(u^+A{u)Aip2jdxdt+

f u0(x)^ds

= 0.(1.32)

Using (1-29) for

//

u—^—dxdt at

(1 -

Ze^dxdt

0

(e -> 0),

JJCIT

JJQ.J

A{u) LA{u)

ff A(u)A(f2dxdt

L

d
dn r

ds - / / VA(u) • Wfedxdt JJnT

^ds an

- ff VA(«)(1 - ie)V
+ / / VA(u)(fi • W^edxdt JJnT

IMX

ds

(e - t 0).

Here we have used the properties of £e and the fact y?|r = 0. Thus, from (1.32) we derive

f{A{u)\T-A{gy)^ds

=Q

which implies (1.31). Conversely, let u b e a generalized solution of (1.14), (1-27), (1-28) in Definition 1.1.5 and (p be a test function in Definition 1.1.4 Then

II \^eln

+ A A e dxdt+

^ ^n

IMxMx,o)^(x)dx

= o. (1.33)

Since ip\r = 0, we have

\lf

\JJnT

VVA{u)- V£sdxdt <

C

fl

Jo

0 Jn\nc

(e->0).

10

Newtonian

Filtration

Equations

Hence //


V(f££)dxdt

JJQ.T

=

- //

£,eVA{u)Vtpdxdt - [J

—> - If

fVA(u)

VA{u) • Vfdxdt

• V£edxdt

(e -4 0)

and from (1.33), //

(u-£

- VA(u)Vp)

dxdt+

/ uo{x)
which implies (1-29) by integrating by parts. The following assertion is valid obviously, which will be used in the sequel. Proposition 1.1.1

Let u be a generalized solution of the Cauchy problem

(1.14), (1.19) and - ^ _ Z G L}oc(QT),

(i = 1,2, • • • ,N).

Then for any

smooth domain 17 C R^, u is a generalized solution of the boundary value problem for the equation (1.14) on &T = fi x (0,T) with boundary value g{x,t)

=u(x,t)\r

and initial value uo(x). Remark 1.1.1 Sometimes we need to use the notion of generalized super (sub)-solutions. To define the generalized super-solutions(sub-solutions), it suffices to replace " = " in (1.20) by " < " (">") and require the test function if to be nonnegative. To define the generalized super-solutions(subsolutions) of the Cauchy problem (1.14), (1.19), it suffices to replace " = " in (1.23) by " < " ( " > " ) and require f to be nonnegative, or replace " = " in (1.20) and (1.22) by " < " ( " > " ) and require f>0,h> 0. Similarly, to define the generalized super-solutions(sub-solutions) of the boundary value problem (1.14), (1.27), (1.28), it suffices to replace " = " in (1.29) by " < " ( " > " ) and require tp > 0 , or replace " = " in (1.30) and (1.31) by " < " ( " > " ) and " > " ( " < " ) respectively and require f > 0. Remark 1.1.2

If u is a generalized solution of the Cauchy problem

11

Introduction

(1.14), (1.19) in Definition 1.1.3, then for any r <E (0,T) there holds

//

[u—+

JJQT

\

A(u)A
at

+ /

/

J

u(x,T)ip(x,T)dx

JRN

^

34^

uo(x)ip(x,Q)dx = 0

for any

'dt

JJQ-r

+ /

A(u)Aip I dxdt +

u

\L ^

dxdt (1.35)

uo(x)
Letting e —> 0 and noticing that

//

uiprjedxdt— \

JmN

JjQr

Ve(t) (u(x>t)tp(x>t)

// n

<

pr+e

—/ £ Jr-e

u(x,r)(p(x.T)dx —

U(X,T)IP(X,T))

p

/

JRN

dxdt

/.

u(x,t)ip(x,t)dxdt—

I

u(x,T)ip(x,r)dx dt

JUL"

(e->0)

from (1.35) we obtain (1.34). It is clear that if for any r € (0,T) and any ip G C°°(QT) vanishing when |a;| is large enough, (1.34) holds, then u is a generalized solution of the Cauchy problem (1.14), (1.19) in Definition 1.1.3. This means that we may use (1.34) to give another equivalent definition of generalized solutions of the Cauchy problem (1.14), (1.19). Similarly it is easy to see that if u is a generalized solution of the boundary value problem (1.14), (1.27), (1.28) in Definition 1.1.4, then for any

12

Newtonian

Filtration

Equations

r e (0,T), there holds ff

(u^+A(u)A,)dxdt-[u{x,rMX,r)dx

JJn v

dt

-

+ / uo(x)ip(X,0)dx Jo,

( L 3 6 )

d r - / A(g)-^-dS Jrr vn

=0

for any (p € C°°(QT) which vanishes when |a;| is large enough, t •= T and (a;, t) G T, where fiT = Q x (0, r ) , T r = 9 0 x (0, r ) . 1.1.3

Special

solutions

We will give some special solutions of the equation (1.17) to conclude this section. 1. Barenblatt solution

«•(«•')=«^(0-^ 1 J^) t ) 1 / < "" > wherefc= I m — 1 + — I

<'•*>

, u+ = max(tt, 0), AT is the dimension of RN.

Its initial value is the Dirac measure. 2. Pressure solution of degree two /Tnbl2\1/(m-1)

u{x t) =

'

Wh6re T =

°

2m(2 + N{m-l)Y

[n^t)

(L38)

'

** ^

^

iS

W^™"1'-

3. Pressure solution of degree one (N = 1) fm-1 u(x,t)=l

\V(™-i) c(ct±X)+\

,

(1.39)

where c > 0 is an arbitrary constant. Its initial value is !

, l/(m-l)

x m — -c(±ca;) 1 +

In fact, the function (1.38) is a classical solution of (1.17) on QT0\ this can be checked immediately.

Existence

and Uniqueness of Solutions:

One Dimensional

Case

13

For any fixed 0 < t < T, the function (1.37) has a compact support \x\2<

2mNt2k/N k(m - 1)

To check that the function (1.37) is a generalized solution of the equation (1.17), it suffices to observe that Bm(x,t) is a classical solution of (1.17) in the domain {(x,t);Bm(x,t)

> 0} and both Bm(x,t)

and ——B™(x,t) ux%

(i = 1, 2, • • • , N) vanish at the lateral boundary of the domain: 2 ]X]

_ 2mNt2k/N ~ jfe(m-l) '

Similarly, we can check that the function (1.39) is a generalized solution of the equation (1.17) in one dimension. 1.2

Existence and Uniqueness of Solutions: One Dimensional Case

In this section, we study the Cauchy problem for the filtration equation (1.14) in one dimension: du _ dt ~

d2A(u) dx* '

[Z l)

-

u(x,0) = u0(x),

(2.2)

where uQ(x) > 0 is a locally integrable function on R and A(s) £ C 1 [0, oo) satisfies the conditions A(s) > 0, A'(s) > 0, for s > 0, A(0) = A'(Q) = 0.

(2.3)

Denote Q T = R x ( 0 , T ) . 1.2.1

Uniqueness

of

solutions

We first discuss the uniqueness of generalized solutions of the Cauchy problem. Theorem 1.2.1

The Cauchy problem (2.1), (2.2) has at most one gendA(u) eralized solution u bounded together with the weak derivative —-^—^. ox

14

Newtonian

Filtration

Equations

dA(u) First notice that if both u and — - — are bounded, then by apax proximation, the integral identity in the definition of generalized solutions (Definition 1.1.3) holds for any ip G W1'°°(QT) vanishing when |a;| is large enough and t = T. Such functions will be also called test functions. Now let ui, u2 be generalized solutions of (2.1), (2.2). Then ui and u2 satisfy (1.26). Hence we have Proof.

//S>-»WU(^-^)- ™ for any test function ip. If
T)))dT

(2.5)

could be chosen as a test function, then from (2.4) we would have

I

(A(ui) — A(u 2 ))(«i — u2)dxdt

Q^

JJQT

fdA{Ul{x,r)) \ dx fdA{Ul{x,t)) \ dx JT T

_ dA(u2(x,T))\ ^ dx J dA{u2{x, t))\dxdt dx

and hence u\ = u2 a.e. on R due to the condition (2.3). However the function

/ (A(m) - A{u2))dT

Existence and Uniqueness of Solutions:

One Dimensional

Case

15

where an(x) is a smooth function such that a„(x) — 1,

when |a;| < n — 1;

an(x) = 0,

when|:r|>n;

0 < an(x) < 1,

whenn — 1 < \x\ < n;

\a'n(x)\ is bounded uniformly. Substituting

— A{u2)){ui — u2)dxdt

JJQ-i

dA(m)

+ [[ <*n(*)( [ JjQ n n >Q

dA{u2)\

(A(Ul)-A(U2))dS

\JT

JJ

(2,6)

fo_AM_^M)dxdt \ <

ox

[[

ox

J

a'n{x)lf\A{ux)-A{u2))dT

JjQn

\JT

\

ox

ox

J

where Qn = {(x,t);n1 < \x\ < n,0 < t < T}. dA(u) Since Ui and — (i = 1,2) are bounded and an(x) is bounded uniformly in n, the right hand side of (2.6), denoted by I2n, is bounded in n, so is the left hand side, denoted by I\n. Since an(x) increases with n, so does I\n. Thus lim I\n exists and we have n—>oo

lim Iln = / / "^•°°

{A(ui) - A(u2))(ui

-

u2)dxdt.

JJQT

Furthermore we can prove lim I2n = 0.

(2.7)

16

Newtonian

Filtration

Equations

dAtuA In fact, from the boundedness of ui and — and the uniform bounded edness of ct'n(x), we have \L2n|

< C (J

\A{Ul) -

A{u2)\dxdt 1/2


(Mui) -

A{u2)fdxdt\


{A(Ul) - A(u2))(Ul

1/2

- u2)dxdtj

,

where C is a constant independent of n. The finiteness of the integral //

{A(u\) — A{u2))(u\ — u2)dxdt implies that the right hand side of (2.7)

JJQT

tends to zero as n —> oo. Letting n ~>-ooin(2.6) yields //

{A(ui) - A(u2))(ui

- u2)dxdt = 0,

JJQT

and hence u\ = u2 a.e. on QT- This completes the proof of our theorem. 1.2.2

Existence

of

solutions

Next we discuss the existence of generalized solutions of the Cauchy problem (2.1), (2.2). • Theorem 1.2.2 Assume thatuo is a nonnegative, continuous and bounded function on K, A(uo) satisfies the Lipschitz condition, A(s) is appropriately smooth and A(s) —> +oo as s —>• +oo. Then for any T > 0 the Cauchy problem (2.1), (2.2) admits a continuous and bounded generalized solution dA(u) u on QT such that — - — is bounded. Moreover, the solution u is classical ox in the domain {(x,t) G Qr,u(x,t) > 0}. Proof. {VQ(X)}

Denote Vo = A(uo) and choose a sequence of smooth functions such that {VQ (X)} uniformly converges to VQ(X) as n —> oo and

!*<*>


0
(n=l,2,.

Existence

and Uniqueness of Solutions:

One Dimensional

Case

17

with some constants K0 and M. Construct a sequence of smooth functions {wn(x)} such that wn(x) = v${x),

\x\

wn(x) = M,

\x\ > n — 1,


(2.8)

0 < wn(x) < M d wn(x) dx


Now denote v = A(u), $(v) = A

(n=l,2,

(w). Then (2.1) becomes

_.,. .dv

d2v

(2.9)

Consider the initial-boundary value problem for (2.9) with conditions v{x,Q)=wn{x),

v(±n,t)

= M,

(2.10)

which admits a classical solution vn(x, t) on Gn — (—n, n) x (0, T) by virtue of the standard theory for parabolic equations. From the maximum principle for classical solutions, we have 0 < infiy„(a;) < vn{x,t)

< M.

(2-11)

We will further prove that dvn dx

< N

(2.12)

on Gn.

dvn For this purpose, let Pn = —— which satisfies ox dPn d2Pr 1 -*'(v &(Vn) dx2 $'(«„) 8x { dt

n>

) ^ dx '

The maximum principle shows that dvn max dx

dvn < max dx rn dvn dx t=o < N.

where Tn = dpGn denotes the parabolic boundary of Gn. Since 1^4(a) I < N, to prove (2.12) it suffices to prove

dvn dx

x=±n

Newtonian

18

Filtration

Equations

Notice that vn achieves its maximum M on the lateral boundary x = n, 0 < t < T. Hence dvn dx

(2.13)

>0.

Consider the auxiliary function zn=vn-

M(x

-n+1),

which satisfies dz d2z $ ' K ) ^ = - ^ ,

onfl„ =

{n-l
z„(x, 0) = wn(x) - M(x - n + 1 ) — M(n - x) > 0 on [n - 1, n] z„(n,i) = 0,

zn(n-l,t)=vn(n-l,t)>0

on (0,T].

z„ achieves its minimum min^ z„ on a; = n, 0 < t < T. Hence

9a;

9^n 9a:

which combining with (2.13) yields 0 <

-M<0, dv„ dx

<M


dv, <M. dx The estimate (2.12) implies the uniform Lipschitz continuity of vn in x: for any (x,t), (y,t) e QT) and n large enough such that (x,t), (y,t) G G„), Similarly we can prove

|u„(a;,t) — fn(2/,*)| <

N\x-y\.

(2.14)

Denote un = A~x(vn). Note that the assumption A(s) - > o o a s s - > o o and A'(s) > 0 imply the existence of the inverse function A _1 (w„). Prom (2.9) we have dun _ dt

d2vn dx 2 "

(2.15)

Existence and Uniqueness of Solutions:

One Dimensional

Case

19

For any (x,t), (y,t) G QT), choose n large enough such that (x,t), (y,t) G Integrating (2.15) yields Gn), x + \At\ll2 G \-n,n\ with At = t-s. x+\At\1/2

L

un(z,t)

t

=

rx+\At\

-un{z,s))dz

1/2

^dydT

I. L

Hence from (2.12) we obtain rx+\At\^2

/

(un(z,i)-un(z,s))dz <2N\At\.

JX

Using the mean value theorem for integrals, we see that there exists x* G [x,x+ | Ail 1 / 2 ] such that / Jx Thus

(un(z,t)

-un(z,s))dz

(un(x*,t)-un(x*,s))\At\1/2.

=

<2N\At\1'2

\un(x*,t) - un(x*,s)\ and hence for some constant C, \vn(x*,t)-vn(x*,s)\

= \A(un(x*,t))

- A(un(x*,

s))\ (2.16)

=

\A'(£n)\\un(x*,t)

-un(x*,s)\


Combining (2.14) with (2.16) deduces \vn(x,t) <

-vn(y,s)\

\vn(x,t) -vn(x*,t)\ +\vn(x*,s)

<

N(\x-x*\

<

C(\x-y\

+ \vn(x*,t) -vn(y,s)\

+ \At\^2 + +

\At\W)

\x*-y\\

-vn(x*,s)\

20

Newtonian

Filtration

Equations

for some constant C and proves that {vn} is uniformly Holder continuous on Gn with Holder exponent {1, - } . This together with (2.11), (2.12) implies that there exists a subsequence of {vn}, supposed to be {vn} itself, such that {vn} converges uniformly to a certain function v on any compact subset of QT and < —— > weak star converges to — on any bounded domain of QTFurthermore it is easy to see that {un} (un = A _ 1 (v„)) converges uniformly to u = A_1(v) on any compact subset of QT. dA(u) Obviously u and — - — are bounded. Given any test function ip, i.e. _ ox (f £ C°°(QT) such that

C Gn. Multiplying (2.15) by
IL (-£ - ^ s )

dxit+

£ »<*• »)^'(-w>^ -»

from which it follows by letting n -> oo and noticing that for large n, wn(x) = VQ(X) and wn(x) converges uniformly to VQ = A(UQ) on any finite interval, that u satisfies (1-24), i.e u is a generalized solution of (2.1), (2.2). Finally, we prove that the solution u is classical in {(x, t) € QT, U(X, t) > 0}. Let (x0, t0) £ QT, U(XQ, to) > 0. Then there is a neighborhood U C QT of (xo,io) and constant ao > 0 such that u(x,t) > ao > 0,

for (x,t) G U.

Hence un(x, t) > — > 0

for (x, t) E.U and large n.

This means that for large n the solution un satisfies dun

d (

dun

with a(x,t) = A'(un) being uniformly parabolic on U. From the standard theory for parabolic equations, it follows that for large n, un is uniformly bounded and equi-continuous in C2(U). Thus u eC2(U) and satisfies (2.1) in the classical sense. The proof of Theorem 1.2.2 is complete. •

Existence and Uniqueness of Solutions:

1.2.3

Comparison

One Dimensional

Case

21

theorems

First we have Theorem 1.2.3 Assume the conditions in Theorem 1.2.1 and Theorem 1.2.2. Let Ui and u2 be bounded generalized solutions of the equation (2.1) dA(ui) with bounded weak derivatives — and initial data uoi(x) (i = 1,2). / / ox u oi( ; c ) < UQ2(X) a.e. on K, then u\ < u2 a.e. on QTProof. As in the proof Theorem 1.2.2, we construct a sequence of approximate smooth functions {u^t{x)} (i = 1,2) of {uot(x)} (i = 1,2). However, an additional condition UQ^X) < UQ2(X) is required. Then, by the comparison theorem for classical solutions, we get u™(x,t) < u2(x,t) on Gn. Letting n —> oo and using the uniqueness of generalized solutions yield iii < u2 a.e on QT and complete the proof of our theorem. • Proving comparison theorem in this way seems to be too roundabout. Many approaches applied to prove uniqueness theorem are also adapted in establishing comparison theorem. The proof of Theorem 1.3.1 is an example in this respect. As we will point out in Remark 1.3.1, we can prove Theorem 1.2.4 Let ut G L1(QT) D L°°(Q r ) (i = 1,2) be generalized solutions of (2.1) with initial data Uoi, (i = 1, 2). Ifuoi(x) < uo2{x) a.e on R, then u\
Ui((3,t) = hi(t),

(i = 1,2).

} U i ' e i V f i r ) (i = 1,2). / / ox uoi(x) < u02{x) a.e on (a,/?), gi(t) < g2{t), hi(t) < h2{t) a.e. on (0,T), then u\
The basic idea of the proof is just the same as in Theorem 1.3.1. We leave it as an exercise and suggest that the readers do it after reading the proof of Theorem 1.3.1.

22

Newtonian

Filtration

Equations

Remark 1.2.1 Comparison theorem is still valid if (a,/3) is replaced by (a, oo) or (—oo,/?). In this case we need to require the generalized solutions considered to be bounded and integrable on (a, oo) x (0,T) or (-oo,/3) x (0,T). The proof is similar to §1.1.3 Theorem 1.3.1, but appropriate adaptations should be made. Checking the proof of Theorem 1.2.4 and Theorem 1.2.5 we see that in fact we have the following more general results. Theorem 1.2.6 Let u{ G L°°(QT) n Ll{QT) (i = 1,2) and m (u2) be a generalized subsolution (supersolution) of (2.1) with initial data UQI{X) (UQ2(X)) respectively. Ifuoi(x) < Uo2(x) a.e onR, then ui < u2 a.e on QTTheorem 1.2.7 Let u\ and u2 be a generalized subsolution and a generalized supersolution of (2.1) on D,T — (a, P) x (0, T) with initial data uoi(x) (i = 1,2) and boundary data Ui{a,t)

= 9i(t)

respectively. Assume that

Ui

Ui(f3,t)

€

= hi{t)

LM(QT),

dA(u) ^ ^

(t = l,2) G L^flr)

(i = 1 , 2 ) . / /

uoi(x) < uo2(x) a.e on (a,/3), gi(t) < g2(i), h\(t) < h^it) a.e on (0,T), then u\ < U2 a.e on Q. Remark 1.2.2 Since Proposition 1.1.1 implies that generalized solutions of the Cauchy problem on QT are generalized solutions of the boundary value problem on QT, we can compare generalized solutions of the Cauchy problem on any domain of the form Q,T- This fact will be used in the sequel. 1.2.4

Some

extensions

The results presented in Theorem 1.2.1 and Theorem 1.2.2 were first obtained by Oleinik, Kalashnikov and Zhou. In their creative work [OKZ] they studied also the first boundary value problem. What they considered are equations of the form du dt ~

d2A(x,t,u) Ox*

where A(x, t, u) satisfies A(x,t,0)

dA = —(x,t,0)

= 0,

[

V

Existence

and Uniqueness of Solutions:

One Dimensional

Case

23

M

A(x,t,u)>0,

^'*'") > 0 , f o r u >0. ou Following [OKZ], the existence and uniqueness theory for quasilinear degenerate parabolic equations in one dimension has been developed in several directions. 1. Extension to unbounded initial value. Kalashnikov studied the Cauchy problem for (2.17) with unbounded initial value UQ(X) in [KA3]. He discussed the existence and uniqueness of generalized solutions in a class of functions which are continuous and nonnegative but need not to be bounded. The uniqueness of generalized solutions of the Cauchy problem is established in a class of functions deT-,

.

i

-

i

dA(x,t,u)

noted by E, under the assumption that x „, A(x,t,u)<M1i

.

.

increases with u and ou fdA(x,t,u)^p V^'

for some positive constants M\ and p. By E he meant the class of all functions u(x, t) satisfying the condition dA(x,t, u(x,t)) —v ' v n <M2{l ou

,, + x2),

, 2.18

where Mi > 0 is a constant. The result in [BAl], [BA2] shows that for the uniqueness of generalized solutions to hold, the condition (2.18) defining the class E can not be replaced by dA(x,t,u) , , .„ , . u , —V ' <M1 {l + x2 1 + £ , ou

0<e
no matter how small e is. As for the existence, the following result is presented in [KA3]: the Cauchy problem for (2.17) admits a generalized solution in the class Ee defined by dA(x,t,u(x,t)) — L

,, . ,., , , ^M^l+z2)1 (e>0)

under the assumptions A(x,t, u) =

A(x,u),

Newtonian

24

m \

ou

Filtration

J

Equations

\

9A{X U 0{x)) <M2(l d u

+ X*y-*,

du (£>0)

(2.19)

and other appropriate conditions, where m, M, p are positive constants. It is indicated in [KA3] that if the condition (2.19) is replaced by dA{x,u0{x)) du

^ A/f /n t 2J < M i ( l + z ),

(2.20)

then one can not obtain the global solution in general. The pressure solution of degree two given in §1.1.1 also shows this matter. In fact, in case A(x,t,u) = um (m > 1), the condition (2.20) becomes / A/f \

1

/(m_1)

(l + z 2 ) 1 / ( m - 1 } .

«o(s)<(^)

The initial data of the solution (1.33) satisfy this condition, however (1.38) does exist locally. For the further work in this direction, we mention [KAM] in which the author studied the Cauchy problem for (2.1) with measure initial value u(x, 0) = ES(x)

(E>0)

and proved the existence and uniqueness of the so-called source-type solutions under appropriate conditions on A(u). 2.Extension to equations of other form. Kalashnikov [KA2] studied equations with absorption term du

d2A(u)

,, ,

where A(u) and ip(u) satisfy A'(u) > 0, V>'(«) > 0, A'(0)>0,ij(u)>0,

for u > 0, toiu>0.

He proved the existence and uniqueness of generalized solutions of the Cauchy problem under some additional conditions on A(u) and I/J(U), the solution obtained is Holder continuous.

Existence and Uniqueness of Solutions:

One Dimensional

Case

25

Kershner [KE1] has studied this kind of equations too. His method can be applied to more general equations d2A(u) dx2

du dt~

Wju) dx

+

+1{U)

-

In another paper, published a little early [KE2], he studied the first boundary value problem for equations of the special form du _ d2um dt dx2

dun _ dx

j

with some positive constants m, n, I. Gilding and Peletier [GP2] considered the Cauchy problem for the equation d2um

8u

m=-dx^

+

dun

.

.,

n

^x- (™>L»>o)

and proved that it admits at most one generalized solution whenever

n > i ( m + l) and it admits a generalized solution if UQ > 0 is bounded and continuous with uam Lipschitz continuous. Soon afterwards these results were extended by Gilding [GI] to more general equations du

m

=

d ( . ,du\ aiu)

dx-{

dx-)

,.

.du

+b{u)

dx-'

where a(u), b(u) are continuous and a(u) > 0 for u > 0, a(0) = 0. He proved the uniqueness for the Cauchy problem under the condition b2(u) = 0(a(u))

(u -»• 0+)

(2.21)

and proved the existence under the conditions that a'(u) and b'(u) are locally integrable , ua'{u),ub'{u) € Lx(0,1) and (x) ruu0ayx)

A(u0(x)) = / Jo

/

\

1

a{r)d

is Lipschitz continuous. It should be pointed out that as a condition to ensure the uniqueness of generalized solutions, (2.21) is unnatural. Wu Dequan [WD] and Dong Guangchang, Ye Qixiao [DY] have ever made efforts to improve this condition. Chen Yazhe [CHI] removed any kind of conditions in which b(u)

26

Newtonian

Filtration

Equations

is controlled by a(u) and established the uniqueness by assuming a(u) to satisfy the following condition: there exist constants 6, m > 1 such that 1 fm\

a(ui) < —,—r < m a.[u2)

m \u2J

for any 0 < u\ < u2 < 6. A substantial progress in the study of uniqueness was made by Zhao Junning [ZHl] who did not require a{u) and b{u) to have any relation and only assumed a(u) > 0 to satisfy the condition that the set E = {s,a{s) = 0} does not include any interior point; uniqueness was proved in L°°(QT)The uniqueness of solutions for equations without any additional assumption except a(u) > 0 is much more difficult to prove. Vol'pert and Hudjaev [VH1] tried to do that in the class of BV functions; however, as pointed out in [WZQl], their proof is incorrect. Based on a deep investigation of functions in BV and in a more general class BVX, Wu Zhuoqun and Yin Jingxue [WYl] completed the proof of uniqueness of solutions in BV. (For details see Chapter 3).

1.3

Existence and Uniqueness of Solutions: Higher Dimensional Case

We are ready to turn to the filtration equation in higher dimension. We will concentrate our attention on its typical case, i.e, equation of the form du

Aum

dt N

with constant m > 1, where A = ^

(3.1)

d2

- ^ , and only discuss the Cauchy

i=i

problem, whose initial value condition is u(a;)0) = uo(a:). 1.3.1

Comparison

theorem

and uniqueness

(3-2) of

solutions

Denote QT = RN x (0,T). Theorem 1.3.1 Let u{ G L1(QT) n L°°{QT) (i = 1,2) be generalized solutions of the Cauchy problem for (3.1) with initial data uoi(x), (i = 1,2). 7/0
Existence and Uniqueness of Solutions:

Proof.

Higher Dimensional

Case

27

Prom Remark 1.1.2 we have /

Ui(x, T)ip(x,r)dx — /

=

ft

Uio(x)ip(x,0)dx

JM.N

JRH

(ui^+u^Aip\dxdt

(t = l,2)

for any r G (0, T) and (p G C°°(QT) which vanishes when \x\ is large enough. Let z = u\ — U2, ZQ — U01 — W02. Then /

z(x,T)ip(x,r)dx

JM^

=

=

jj

(z^

— / JmN

zo(x)(p(x,0)dx.

+ iuT-u^A^dxdt.

(3.3)

Lz&+aA*)dxdt>

where uVl(x,t)-vZl(x,t) a(a;,i) = ^ «i(a;,i) - u 2 (x,i) mu™ _1 (:E,i),

„

,

. ,

.

,

ifu].(a;,i) = U2{x,t).

If for any nonnegative function g G CQ°(R ), the problem -£- + aA

(3.4)

had a solution ip G C°°(Qr) which vanishes when |ar| is large enough, then from (3.3) we would obtain /

z(x,t)g(x)dx<0.

(3.5)

JRN

Here we have used the assumption ZQ(X) < 0 and the fact 0 which follows from the maximum principle. Therefore by the arbitrariness of g(x) we get Z(X,T) < 0 or UI(X,T) < U2(X,T) a.e for x G RN and this is what we want to prove. However since the coefficient a in (3.4) is merely a nonnegative and locally integrable function, (3.4) does not admit any smooth solution in

28

Newtonian

Filtration

Equations

general and even if (3.4) does admit, the solution can not have compact support in x in general. In view of this point, we replace a by 1 an = Pn * a + -, n where pn is a mollifier on QT and consider the boundary value problem -T^ + a„A = 0

for|x|<.R,

ip = 0

for \x\ = R, 0 < t <

.
for |a;| <

0 < t < r, T,

(3-6)

R,

where R > Ro + 1 such that supp#(z) C BRo = {x € RN; \x\ < RQ}. We choose pn such that / / (a - pn * a)2 dxdt < \ . (3.7) Jo JBR ™2 Let ifin be a solution of (3.6). Extend the definition of ifn to the whole QT by setting y>n = 0 outside the domain \x\ < R, 0 < t < T. Since the extended function tpn may not necessarily be a sufficiently smooth function on QT, we use a function £R S C0X(M.N) with the following properties to "cut-off' (pn: ( 0 < £R(x) < 1, fr(a:) = l,

for | r r | < i ? - l , (3.8)

6i(aO = 0, {

ioi\x\>R-±,

\VZR(x)\,\AtR(x)\
Here and below, we always use C to denote a universal constant independent of R, n, which may take different values on different occasions. Choosing

z{x,T)g{x)£,R{x)dx

I

- \

z0(x)£R(x)
JM.N

J&N

« +

- i # ) ( 2 V ^ V ^ n + fnA^R)

dxdt.

z^R(a - a„)Aipndxdt = In + Jn.

(3.9)

Existence and Uniqueness of Solutions:

Higher Dimensional

Case

29

Now we are ready to estimate J„ and Jn. Multiplying the equation in (3.6) by Aipn, integrating over BR x (i,r) and applying Green's formula, we obtain \Vvn{x,t)\2dx+

\ j Z

JBR

I

I

Jt

J BR

an{Aipn)2dxds

\Vg\2dx,

= Z

JBR

from which it follows in particular /

\Vipn\2dxds
f

Jt

(3.10)

JBR

f Jt

an(Aipn)2dxds

f

< C.

(3.11)

JBR

Using (3.8), (3.10) and noting that ut € L°°{QT) uniformly bounded yield

\In\ < C f

[

Jo

(u? + v?)(\V
JBRXBR.!


(3

JBR\BR^1

Using (3.11) and noting that ut G L°°(QT)

12)

(ui + u2)dxdt. Jo

|Jn|

{i = 1,2) and
{a an)2

f

~

dxdt)

a

n

JBR

J

(i = 1,2) yield ( fT \Jo

f JBR

an(A

a \Jo JBR n J By virtue of (3.7) we further obtain

\Jn\
/

(a-pn*a--\

dxdt\

<—=.

(3.13)

Combining (3.12), (3.13) with (3.9) and noting that ZQ(X) < 0, 0, £R > 0, we finally arrive at /
z(x,T)g(x)£R(x)dx zo(x)pn(x,0)£R(x)dx+

\In\ + \Jn

30

Newtonian

<\In\ + |J„| < C /

Filtration

/

JO

Equations

(Ui + U2)dxdt JBRXBR-!

Letting n —> oo and then R —> oo and noting that m G L1(QT) (i = 1,2), we see that the right hand side tends to zero and hence (3.5) holds. The proof is complete. • R e m a r k 1.3.1 Checking the proof of Theorem 1.3.1, we see that the method applied is adapted to more general equations of the form % = AA(tt)

(3.14)

with A(u) G Cl[0, oo) and A(s) > 0, A'(s) > 0 for s > 0, A(0) = A'(0) = 0. . In other words, using the same method we can prove that for generalized solutions of (3.14) in ^(QT) fl L°°(QT), the comparison theorem is valid. R e m a r k 1.3.2 It should be pointed out that requiring a generalized solution u to belong to L 1 (Qr) (~1 L°°(QT) is too restrictive, which means that u must be "small" at infinity and excludes even the nonzero constant solution. Fortunately those generalized solutions determined by the initial data with compact support satisfy such condition. As an immediate corollary of Theorem 1.3.1, we have T h e o r e m 1.3.2 Suppose 0 < u0 G L1(RN)f)L00(RN). Then the Cauchy problem (3.1), (3.2) admits at most one generalized solution in i 1 ( Q ^ ) D L°°(QT). Similar to the proof of Theorem 1.3.1, we may obtain the following comparison theorem for the boundary value problem. T h e o r e m 1.3.3 Let ut G i 1 ( H r ) l~l L°°(Q.T) (i = 1,2) be generalized solutions of (3.1) satisfying the initial value condition and boundary value condition ui(x,0) = Uio(x),

for x G fl ( i = l , 2 )

and Ui(x,t)\r

= 9i{x,t)

for(x,t)€T

(i = 1,2),

Existence and Uniqueness of Solutions: Higher Dimensional

Case

31

where £lcRN is a smooth domain, T = <9fi x (0,T), fiT = fi x (0,T). If ui(x) < U2(x) a.e on fi and g\{x,t) < g2(x,t) a.e on T, then ui(x,t) < U2(x,t) a.e on QTRemark 1.3.3

Since the generalized solution u of the Cauchy problem dA(u) for (3.1) on QT with locally integrable derivatives — (i = 1,2, • • • ,N) OXi

is also a generalized solution of the boundary value problem for (3.1) on any domain of the form fi^ = fi x (0,T), we can use the comparison theorem on any fix for generalized solutions of the Cauchy problem. 1.3.2

Existence

of

solutions

Now we discuss the existence of generalized solutions of the Cauchy problem (3.1), (3.2). First we prove Theorem 1.3.4 Assume that u0 e L 1 ^ ) n L°°(RN) and u0(x) > 0. Then the Cauchy problem (3.1), (3.2) admits a generalized solution u G

L 1 (Q T )ni 0 0 (Q T ). Proof. The basic idea of the proof is to regularize the initial value and then to establish some estimates on the approximate solutions to obtain the desired compactness. Choose a sequence of positive numbers i?„ and r]n such that as n -> oo, and then construct Uon € Co°(BRn)

(3.15)

such that

l|wOn||i~(K") < |I«O|IL~(E' V )> ||«0n - U 0 ||LI(RW)-)• 0 as n-J-oo.

(3.16) (3.17)

To this purpose, it suffices to define von = wo f° r \x\ < -B.Rn_i, von = 0 elsewhere and then mollify it. Now consider the boundary value problem < du -£• = Aw™

for (x, t) e BRn x (0, T)

u = r)n

for (x, t) e dBRn x (0, T)

U = UQn + T]n

for

Xe

BRn.

(3-18)

32

Newtonian

Filtration

Equations

According to the classical theory, (3.18) admits a smooth solution un. By maximum principle and using (3.16), we obtain Vn
Multiplying the equation in (3.18) by pw£ _1 (1 < p < oo) we derive dvP - ^ = pur1 A < = pdiv (K^Vu™) =

- pVu?-1

•V <

p 3

- mp{p- l)u™+ - |Vu„| 2 .

pdiviuP^Vu^)

Integrating both sides of the above equality over BTn x (0, T) and noting du that — - < 0 on dBnn x (0,T), where v denotes the outward normal to dBRn

x ( 0 , r ) , we have upn{x,t)dx-

/ =

pf

f

(u0n(x)+rjn)pdx

ul^^-dadr-mpip-l)

JO JdBRn

<

I

-mp{p-l)

f

OV

I Jo

JO

u™+p-3\Vun\2dxdT

f

u™+p-3\Vun\2dxdT

[ JBRn

JBR„

or upn(x,t)dx + mp{p-l)

/

f

JBRn <

/ (UQn(x) •>BRn

Jo

f

u™+p-3\Vun\2dxdT

JBRn

+T)n)Pdx.

Using (3.15) and (3.17) we see that the right hand side of the above inequality tends to /

u^dx as n —> oo, which implies that for any fixed

p £ [1, oo) the left hand side is bounded. In particular we have / Un(x,t)dx
(3.19)

and /

f

•'O

JBRn

\Vu™\dxdt = m2 [ JO

f JBRn

u2J;m-V\Vun\2dxdt
(3.20)

Existence

and Uniqueness of Solutions:

Higher Dimensional

Case

33

Integrating over BRn x (0,T) and noting that — - = 0 on dBRn

x (0,T)

Multiplying the equation in (3.18) by m u ™ " 1 - ^ gives

/a^-)/n2

4m

(m + 1 ) 2 \

dt

= d i v

I

^

v

^

IS

n

\ dt

J

2 dt

OIL

we obtain

=

\Vu™(x,T)\2dx+l

- \ l

\Vu™(x,t)\2dx

[

rp

I \Vu™(x,t)\2dx.

< \ I z

JO

JBR„

Integrating with respect to t G (0, T) we further obtain .

4m

rTT rrTT rr

.„ ,

.

(m + l1)) 2 JJO 0

hJt

fT

f

4m

m

+1)/2\2

.

n

—

\

0t

/»„.( (duZ

.

JJE BR„

2'

dxdrdt I

(du{r+1)/^ t I —^— I dxdt

—2 f I -.

(m+i) 7o iBfi„ V2 9 t rT < \ l I \^<{x,t)\ dxdt z

Vo

JBR„ 'BR„

Using (3.20) and the uniform boundedness of un, we finally derive T

fdum\2

'

/ / JO

JET

2

i-

/

rT

/o

r

(m + l) 2 7 0 JBJ

\ 2

a

(m+l)/2\2

1 1

tit"n " I ^2-

dt

I dxdt < C.

The estimates (3.19), (3.20), and the Kolmogrov theorem imply that for any 5 G (0,T), fl > 0, {<*} is strongly compact in L2(BR x (<5,T) and hence there exits a subsequence of {u™}, supposed to be {u™} itself, which

34

Newtonian

Filtration

Equations

converges almost everywhere to a certain function v on QTu™ —• v

as n —> oo a.e on

QT,

namely un -> u = i;1'™

as n —> oo a.e on

QT-

Obviously u G L°°(QT) which together with (3.19) implies w G ^(QT). Given any
(un-^+Un&
(•uOn(x)+rin)
= 0.

Here we regard un as zero outside {(a;,t);|a;| < Rn,t € [0, T]}. Letting n - > o o w e see that u is a generalized solution of (3.1), (3.2). The proof is complete. • Remark 1.3.4 Different from one dimensional case, the above existence theorem does not provide a continuous solution. We will see in §1.5 that to prove the continuity of generalized solutions of the equation (3.1) in higher dimension is quite difficult. 1.3.3

Some

extensions

We have discussed the existence and uniqueness of nonnegative generalized solutions of the Cauchy problem for the equation (3.1) under the assumption UQ > 0 and UQ G L1(M.N) n L°°(WLN). The existence and uniqueness theory has been extended in several directions. 1. It is natural to consider the nonnegative solutions from the viewpoint of physics. However in mathematics we need not to restrict ourselves to this consideration. For signed solutions, the equation (3.1) should be changed into the form

^ = A(|ur-V

(3.21)

The argument stated in §1.3.1 and §1.3.2 is also available for (3.21) with signed initial data u0 G L 1 ^ ) fl L°°(RN). 2. All methods stated in §1.1.1 and §1.1.2 can be used to treat the first initial-boundary problem for (3.21) whose initial value condition and

Existence

and Uniqueness of Solutions:

Higher Dimensional

Case

35

boundary value condition are u(x, 0) = u0(x),

for x G Q

(3.22)

and u(x, i) = 0

for (x, t)edQx

(0,T),

(3.23)

where tt C RN is a bounded domain with smooth boundary. The existence and uniqueness of generalized solutions of the problem (3.21)-(3.23) can be proved in a similar way [CE]. One can also treat the problem with nonzero boundary value [SA1]. 3. Requiring u0 to belong to both L°°(RN) and L 1 ^ ) is too harsh. Fortunately this condition can be relaxed. It is not very difficult to prove the following result. Theorem 1.3.5 Assume that UQ G L°°(RN) and uo(x) > 0. Then the Cauchy problem (3.1) and (3.2) admits a generalized solution u G L°°(QT)Proof.

Choose 0 < u0n G Cg°{RN) such that llu0n||L~(E«) < 11Mo||L~(RJV)

and for any fixed R > 0 II"on -"oillii(B R ) -» 0

(n-too).

Letting uHtu be a classical solution of the problem (

dii

— =Aum

iov(x,t)eBRx(0,T),

< u= -

for (a;, t) G dBR x (0, T),

u = u0n + for x G .BR, n where R > Rn. The maximum principle implies ^

- < u„tR(x, t) <

||UO||L~(K^)

un,R(x, t) < un>K(x, t)

+ !>

for R > R, (x, t)&BRx

Thus the limit Un(x,t)

=

lim

R—>oo

UUiR(x,t)

(0, T).

(3.24)

36

Newtonian

Filtration

Equations

exists and un is a classical solution of the problem

(3.25) u = Uon H— for x e n

>N

Obviously we have -

< Un(x,t)

< ||lto|U~(RW) + 1-

Let w{x) = e x p ( - v / l + \x\2). Clearly for some constant C, < Cw(x).

\VOJ(X)\

Now we substitute u = un into the equation in (3.25) and multiply both sides by (m + l)u™u>. Then we arrive at Qum+1

"—§f~

=

(m + l)div(wu™Vu") - (m + l)w|Vu™| 2 -(m + l ) < V w . V C

Integrating over QT and using Young's inequality we obtain

<

/

u(x)u£+1(x,T)dx

f

u,(x)(uZ+1(x) \

JR"

<

cf

+ (m + l) f + l)dx nj

oj(x)dx + e f N

Jm

JO

+C [ Jo

f

w\Vu™\2dxdt

f JuN

W

| V,.m|2 < dxdt

u\Vu™\2dxdt

f N

JR

with small e > 0. From this it follows that, in particular / Jo

/ JR

N

u\Vu™\2dxdt

< C.

(3.26)

Existence

and Uniqueness of Solutions:

Higher Dimensional

Case

37

3li

Multiplying both sides of the equation in (3.25) by grating over R

N

- / Jt

/

'dvLm+1)/2*

<

C

d<

£

Jw

U)\Vu™\dxdT+\

8t

uj(x)\Vu™(x,T)\2dx

fl

dt

Uj{x)\Vu™(x,t)\2dx

f

dut+1)/2

Jt Jm

<

l

Lj(x)\Vu™(x,t)\2dx

II

C

2

JRW

uj\S7u™\dxdT

Jt Jm +i /

u>(x)\VvZ(x,t)\2dx

rT r

('dv(m+l)/2V

+C £

rT r

I, J C H ^ V - J ^ < >/ /,

w\Vu™\2dxdT

w(x)\Vu™(x,t)\2dx

+i [

with small e > 0. Hence (m+l)/2>

Jt JWL" \ <

cf

f

Jt

N

dxdr

dt

u\Vu™\2dxdT+l

JR

*

f

u;(x)\Vu™(x:t)\2dx.

N

JR

Integrating over (0, T) we further obtain

/dulm+1)/2Y

II < <

tuj

/o JR> ^i" Jo C f Jo

C f Jo

inte-

dxdr

dt

-£-V">-V
JRN

+\ f <

-—,

x (t, T) and using Young's inequality again yield

4m f1 f 2 (mi + I) Jt JR" ' =

ITKJB™

-1

f

JRN

f

JRN

—^—

V

dt

dxdt

J

tw\Vu™\2dxdt+\

f 2 Jo

cj\Vu™(x,t)\2dxdt.

[

JRN

uj\Vu™(x,t)\2dxdt

38

Newtonian

Filtration

Equations

This together with (3.26) yields du™ x

Jo

JMN

tu I ^ dt

2

I dxdt < C.

(3.27)

The estimates (3.26), (3.27) and the uniform boundedness of {un} imply the compactness of {u™} in Lfoc(QT)- From this it follows that there exists a subsequence of {un}, which converges to a certain bounded function almost everywhere on QT and the limit function is a generalized solution of (3.1), (3.2). The proof of Theorem 1.3.5 is thus completed. • In [BEN], [BFU], the existence of a generalized solution u of the Cauchy problem (3.21), (3.2) is proved for u0 G L 1 (K JV ) by using the semigroup method, u satisfies the following estimate: for t € (0, T)

K-.*)IU-(R") ^

tN/^t-m^i^r1^

(3-2§)

with a constant C independent of UQ. This estimate can also be obtained by using the regularization method [FR1]. A similar estimate holds for the boundary value problem [EV]. The estimate like (3.28) expresses a kind of regularization effect. As pointed out in §1.2.4, in some cases, the initial data might be a measure. [PIE] has proved the existence of generalized solutions for the case that the initial value is a nonnegative finite Radon measure. In [BCP] it is noticed that the finiteness of the measure can be relaxed; the authors have proved in [BCP] that there exists a nonnegative continuous generalized solutions of (3.1) if and only if the nonnegative Radon measure initial data fi satisfies the condition SU P uN+2/(m-i) / dnT Rn+V(rn 1) J^
for any R > 0.

(see §1.8.3). 4. It should also be pointed out that the uniqueness of generalized solutions of the Cauchy problem(3.1), (3.2) is valid not only in L 1 (Qr) H L°°(QT) but also in a more general class of functions, especially in the class of all bounded and measurable functions, (see §1.8.3). 5. All what we have discussed above is the case m > 1 which corresponds to the slow diffusion. In this case, the equation (3.1) or (3.21) degenerates at u = 0. Many authors have studied another important case 0 < m < 1 which corresponds to the fast diffusion. In this case, another kind of singularity

Regularity of Solutions:

One Dimensional

Case

39

occurs in (3.1) or (3.21) at u = 0 [VA3], [DD1], [DD2]. Some authors have studied (3.1) or (3.21) in the case m < 0 [ERV], [DD3], [DD4] and the equation -

= Ahm,

[HU1], [HU2], [DD5] which correspond to the super-fast diffusion. Both the slow diffusion equations and the fast diffusion equations are examples of equations of the form

where A(u) is increasing and continuous. Many people have studied equations under such general conditions. For details, see §3.5. 6. Many authors have been devoted to degenerate equations of the form — = AA(u) + *(u) and ^

= AA(u) + d i v ^ ( u )

which involve some lower order terms. In Chapter 3 we will discuss more general quasilinear degenerate parabolic equations of second order, which may have many points of degeneracy, even have a set of points of degeneracy including interior points.

1.4 1.4.1

Regularity of Solutions: One Dimensional Case Lemma

Generalized solutions of the filtration equation in one dimension obtained in §1.1.2, are all continuous, which satisfy the equation in classical sense in a neighborhood of any "point of non-degeneracy". Early in [KAl], Kalashnikov noticed that solutions of the equation du

d2um

40

Newtonian

Filtration

Equations

du with m > 1, may have discontinuous derivative ——, even if its initial data ox u(x,0) = UQ(X)

(4.2)

(x G

are sufficiently smooth. In this section a thorough investigation on the regularity of generalized solutions of the equation (4.1) will be made. To this purpose we first prove Lemma 1.4.1 Let m > 1 and u be a positive and smooth solution of (4.1) onG = (a, b) x (0, T}. Here by "smooth" we mean u G C2(G) flC(G), d3u such that —— G C(G). Then for any r G (0,T), a\, b\ G (a, b) (ai < &i), we have on G\ = (ai,bi) x

[T,T],

dum-1(x,t) dx

(4.3)


where C = C{T, m, M, a-^ — a, b\ — b), M = sup u. If G

du^'1 dx

Mi — sup (a,6)

(X)

<

00,

then (4-3) holds on (ai,6i) x [0, T] where C depends on M\ instead of T. Proof.

Let v = um~1.

Then v satisfies

dv

d2v

dt

2

dx

m

dv

m — 1 V dx

dv d 2v Here we have used the assumption u > 0, otherwise — and ——^ may not exist everywhere. dv Since our goal is to estimate ——, a natural approach is to derive and ox dv use the equation for ——. However it seems difficult to achieve our goal in ox this way. So we consider a transformation of v. v =
J:[0,1]->[0,M*],

¥/(r)>0,

(4.4)

Regularity of Solutions: One Dimensional Case

41

where M* = M m _ 1 . Then w satisfies d2w

dw dt

w" ('dw\ ip' \dx J

m . { dw^ m —1 \dx

Let p = ——. Then p satisfies ox d2p

dp

m2

/
/;

m(m + 1) .


771—1

dp dx

or after multiplying both sides by p, 1 dp2 2 dt

d2p I m2 rrvpp— + dx \ m — 1

+

Now we choose £ = ((x,t)

m

ip"

w"\

0dp

(4.5)

\^-r1*+2m*v)p/x

€ C2{G) satisfying

C=l

for ( z , t ) e G i ,

C= 0

for x = a,b and t = 0,

0 < C < 1 for (x, i) € G, dt T dx d2( < C 9a;2

< Cmax

a\ — a' b\ — b J

™*{T^2>Q^W}

and use (,(x,t) to cut off the function p2, namely to consider the function

42

Newtonian

Filtration

Equations

z = £ 2 p 2 instead of p2. Multiplying (4.5) by £ 2 , we can check that z satisfies d2z\

1 fdz 2\dt-m^dx2Jm.io

—

2 \


m —1 d

^.^d2C\

C „.J9CY

2 (4.6)

m(m + 1) , m• n2

+

-

/ JH m m

2

^

"x™,J?LI

/ dp\ \&)

1 /-2„4

^dC, dp *mvCd-xPdx-

A

~

Let (xo,to) G G be a point where z achieves its maximum. Then at this point

Since ^ = 2 C ^ aa; ax

+

2

C

^ ax

2

,

at (a;o,to) we have

^¥x

=

-Pdx-

(48)

Using (4.7) and (4.8), from (4.6) we obtain

dt

^\dxj

m—1

^dx2J'

if' J

'

ox

r

\dx

Regularity of Solutions:

One Dimensional

Case

43

namely 2„4

i^t^lliV

<49)

* {<%+*•*{£)'-"«&y /mjm + a

£ \ *,_

\ m—1
or ^"V - (f") < 0. Clearly ip(r) = ar2 + (M* - a)r with —M* < a < 0 satisfies both (4.4) and (4.10). In particular, we choose M*r Substituting this function into (4.9) and using Young's inequality, we obtain

CV < Ci + C2(\p\ < d + eCV + J

(e > 0)

(l-£)CV
— a, 6i — 6) (i = 1,2). Taking e = - yields

C V < C. Hence dv dx

dw
This complete the proof of the first conclusion.

onGi.

•

Similarly we can prove the second conclusion. To do this we choose <(z) € C$(a,b) such that 0 < C,(x) < 1, £ = 1 for x € (ax,6X).

44

Newtonian

1.4.2

Regularity

of

Filtration

Equations

solutions

Theorem 1.4.1 Suppose that UQ(X) is nonnegative and bounded, UQ1(X) is Lipschitz continuous and u(x,t) is a generalized solution of (4.1), (4.2). Then 1. for any r G (0, T), u G Ca'a/2(R

x [T, T]) with a = min i 1,

I

I

rn-1)

a,a 2 and u G C / ( R x [0,T]) provided u™ 1(x) is Lipschitz continuous; dum dum 2. —— exists and is continuous in x, in particular, —— = 0 whenever ox ox u(x,t) = 0; o ou r « , du 3. —— exists and is continuous in x if1 < m < 2, in particular, —— = 0 ox ox whenever u(x, t) = 0. Remark 1.4.1 The Barenblatt solution (1.37) shows that u G C a - Q / 2 (Rx [T, T]) is the best possible global result and the Holder exponent a =

min < 1,

I

> cannot be increased.

m-lj

Proof of Theorem 1.4-1- The case UQ = 0 is trivial. We suppose that sup uo > 0. 1. Prom the proof of Theorem 1.2.2 and also using the uniqueness theorem we assert that u(x,t) can be obtained as the limit of the classical solution un(x,t) of the problem f du

d2um

. .

M

n

m = ^~

Wo,

u(x,0) = u0n(x)

\x\ < n,

u(±n, t) = M = sup u0

0 < t < T,

and un(x,t) uniformly converges to u(x,t) on any compact subset of QTHere uon{x) > 0 and {uon{x)} uniformly converges to uo(x) on any finite interval. Given any r G (0,T) and a < b. By Lemma 1.4.1, for any large n and (x,t) GGT = [a,b] x [T,T], we have ° " dx

< C

with some constant C = C(T, m, M) independent of a, b.

(4.11)

45

Regularity of Solutions: One Dimensional Case

It follows by virtue of the uniform boundedness of un onGT


dx

(4.12)

Checking the proof of Theorem 1.2.2 we see that in fact this estimate has been established there. Let (x,t),(y,s) £ GT. Based on (4.12) we see from the proof of Theorem 1.2.2 that for some x* e[x,x+\ts\1/2}, \un{x*,t) - Unix*, s)\ < C\t - si 1 / 2 .

(4.13)

Therefore if m > 2, then

K ' - V , * ) - <-\x*,s)\ < C\t - si1/2. This and (4.11) imply \u™-\x,t)-u™-\y,s)\ <

+ \u™-l(x*,t)

K T ' O M ) -u™-\x*,t)\

-

u^ix^s)]

+\u™-l(x*,s)-uZ-1(y,s)\ <

+ \t- s| 1 / 2 + \x* - y\) < C(\x -y\ + \t-

C(\x -x*\

s| 1 / 2 ).

Using the inequality \a - b\a < \aa - ba\ (a > 0,6 > 0,a > 1) we further obtain

K( a ; ,i)- u „( 2 /, S )r- i
- un{y, s)\ < C{\x - y\a + \t- s| Q / 2 )

1 m —1 If 1 < m < 2, then using (4.11) again yields

w i t h a ••

dun dx

<

1

m-l

n

_u2-md<-X

dx

<

1 ,.2-m du,m—1
and hence \un(x,t)

-un(y,t)\

<

C\x-y\.

46

Newtonian

Filtration

Equations

This and (4.13) give \un(x,t) <

-un(y,s)\

\un(x,t)

-u„(x*,t)\

+ \un(x*,s) <

+ \un(x*,t)

-un(x*,s)\

-un(x,s)\

C(\x - a:*| + |t - si 1 / 2 + \x* - y\) < C(\x -y\ + \t-

s| x / 2 ).

Summing up, in either case we have \un(x, t) - un(y, s)\ < C(\x-

y\a + \t-

s\a/2)

I, which implies that u G Ca'a/2(R

with a = m i n i 1,

x

[T,T])

by

letting n —¥ oo. If u^ 1 - 1 is Lipschitz continuous, then by Lemma 1.4.1, (4.11) holds for (x, t) G [a, b] x [0, T]. Based on this fact we can prove u G 2. If u(xo,to) > 0, then from Theorem 1.2.2, we see that u is a classical dum solution in some neighborhood of (xo,to), in particular, —— exists and is ox continuous near (xo,to)If u(xo,to) = 0 (to > 0), then for any given 8 > 0, from the first conclusion of our theorem 0 < u(x, to) — u(x, to) — u(xo, to) < CSa for x G Is = [XQ - S, xo + S]. Hence for large n, we have un(x, to) < Ci6a

for x G Is,

and vZ(x,t0)-vZ(y,to)\ m TO

<

—1

m 771—1

=

fin"1-1

rv

J

L

dttf&to) d(L

UniUo)^-^,^)^ a

C15 \x-y\

for x, y G Is- Here we have used (4.11) again. Letting n -> oo gives \um(x,t0)

-um(y,t0)\

< ^ZTisa\*-v\

for

x,y £ Is

Regularity of Solutions:

One Dimensional

with some constant C\, which implies that x = XQ.

Case

47

dum(x,to) — - — is continuous at ox

3. Suppose 1 < m < 2. Since u is a classical solution of (4.1) whenever u(x,t) > 0, we need to treat only the case that u(x0,t0) — 0 for some (:co>£o)- We have du^-\x,t0) dx

< ClU2n-m(x, t0) < dS^2-™)

_, ,. ., , du(x,to) From this it is easily seen that ox 1.4.3

Some

for x e Is.

. is continuous at x = XQ.

[J

extensions

Kalashnikov [KA3] extended the above result of Aronson in two directions. On the one hand, he proved that if u is a generalized solution of the problem du

d2um

=

^t ~dx^~CUn

(c>°>m>1>n>0)

with initial data Uo which are nonnegative, continuously differentiable and bounded together with — u0n(x), then for any r G (0, T], the derivative —— is bounded on K x [r, T], where a = max < m — 1, — - — >. Examples dua~s show that — is unbounded no matter how small e > 0 is. (cf. [MP3]). On the other hand, he proved that if u is a generalized solution of the equation du dt

d2A(u) dx2

(4.14)

with initial data uo, which are nonnegative, continuously differentiable and bounded together with — A(uo)(x), where A(u) is appropriately smooth, A'{Q) = A(0) = 0 and _L. U) = / tf/((u) —^^-do< oo, Jo a then for any r e (0, T), — ox

is bounded o n R x k l ) provided

)—— is ax

48

Newtonian

Filtration

Equations

bounded. As a corollary, one has \u(x,t)-u{y,t)\<*-1(K\x-x'\) provided ^"(u) > 0 for u > 0, where \I>_1 is the inverse function of * and if is a constant. Examples show that this result is sharp. Gilding and Peletier [GP2] studied the Cauchy problem for the equation du

d2um

dun

,

and proved that the problem admits a generalized solution u G Ca'a/2(M. x [T,T])

n C 1 / m ' 1 / 2 m ( R x [0,71) for any r G (0,T) with a =

[l,—~j,

provided that the initial data UQ are nonnegative, continuous and bounded and u™ is Lipschitz continuous. The exponent a is the best possibility. In addition, he obtained a result similar to Theorem 1.4.1.

1.5

Regularity of Solutions: Higher Dimensional Case

In this section we discuss the regularity of generalized solutions of the Cauchy problem nil

^

= Au™

(m>l),

u{x,0)=uo(x).

(5.1) (5.2)

Compared with one dimensional case, the investigation of regularity of generalized solutions in higher dimension is a very difficult problem. Even the proof of continuity of generalized solutions of (5.1), (5.2) had been remained as an open problem in a long period. In 1979 Caffarelli and Priedmann [CFlJproved the interior Holder continuity and global continuity; their argument is based on the comparison principle. As shown in [LSU], it is effective to apply the properties of the so-called class Bi of functions to investigate the regularity of solutions of uniformly parabolic equations. Chen Yazhe [CH2] introduced a class of functions, called generalized class B
Regularity of Solutions:

Higher Dimensional

Case

49

In this section we will use Chen's method to prove the Holder continuity of generalized solutions. Assume that u is a generalized solution of the Cauchy problem (5.1), (5.2), which is bounded on QT = RN x (0, T). As we have shown in Theorem 1.3.5, we can obtain a bounded generalized solution as the limit of a sequence of positive and smooth solutions with positive initial data. We have also indicated in §1.3.3 that the uniqueness is valid in a general class of functions,especially in the class of bounded and measurable functions. By virtue of uniqueness, we may regard the given generalized solution u as the one obtained by regularization in Theorem 1.3.5. To prove the Holder continuity of u, it suffices to establish the uniformly Holder estimate for its approximate positive smooth solution. We simply assume that u is just the later one. 1.5.1

Generalized

class B-i

Let fi C R " be a bounded domain, w G Whp(n) number k, denote

(p > 1). For any real

Ak = {x G il;w(x) > k}. Then the set Ak is measurable. The function defined by

w^{x) =

(w{x)-k)+

is called a cut-off function of w(x). It is easily seen that w^k\x) and fti)W(a) dx ( k dw - \x) dx

dw(x) dx

G

W1^^)

for x G Ak,

0

elsewhere.

Let w = um, where u is the given positive smooth solution of (5.1) on QT- Then w satisfies

-5T = Aw where $(a) = s 1 / ;

(5 3)

-

Newtonian

50

Filtration

Equations

Let {xo,to) G QT, Bp = Bp(x0) = {x e l " ; \x - x0\ < p} and ((x) be a function on Bp which is continuous and piecewise smooth such that 0 < £(x) < 1 and ((x) = 0 for x £ dBp (also called a cut-off function). Denote Ak,p{t) = {x 6 Bp;w(x,t)

> k},Bk,P

= {x € Bp;w(x,t)

< k}.

Multiplying (5.3) by (2(w — k)+ and integrating with respect to x over Bp yield (2(w-k)+?^-dx

/

C2(w-k)+Awdx.

= f

vt

JB„

JBP

It is clear that

°t

JB„

ft

[ C2!" / JBP a

Aw-k)+

&{k + T)TdTdx

ut Jo p{w-k)+

/• 2

— / C / "t JB0 JO =

(2Xk{w-k)dx,

-£. / 0 1

&{k + T)Tdrdx

JAklP(t)

where Xk(s)=

[' &{k+T)TdT. Jo

Integrating by parts gives (2(w -

/

k)+Awdx

JBP

=

C2V(w - k)+Wwdx - 2 J C(w - fc)+VC • Vwdx

-f JBP

=

- [ JAkJt)

JB0 2

2

< |VH -2/" JAkJt)

C(w - fc)VC • Vwdx.

(5.4)

Regularity of Solutions:

Higher Dimensional

51

Case

Therefore we have d

<;2Xk(w-k)dx+

I

(2\Vw\2dx

f

JAk,p(t)

JAKp{t)

-2 /

C,{w - fc)VC • Vwdx

JAk,p{t)

(2\Vw\2dx + 2 [

-f 2

JAk.Jt)

Hence

1/

\V{\2(w-k)2dx.

JAk,M)

M*,„(t) Ot JA k,p(t)

exk{w~k)dx+\ f e\vw\2dx Z

2

< 2/

JAkk,.p0{t) (t)

(5.5)

2

\V(\ (w-k) dx.

JAk,Jt) P(*)

If we multiply (5.3) by (2(w-k)~ we can obtain |; /

instead of (2(w-k)+,

C2Xk(u-k)dx+±

Ot JBk,p(t) „>(*)

<2

Z2Ww\2dx

/

JBk,Pp(t) ^ • Bi.,

2

[

[ l

then, similarly

(5.6)

2

\V(\ {w-k) dx,

JB'Bk fc ,„(t) „(t)

where **(*) = / *'(fc-T)rdT. Jo if 0 < w{x,t) Definition 1.5.1 w G B2(QT,M,m,j) 2 L OC(QT) and w satisfies ~T /

(?Xk{w-k)dx+\

4 /

t2Xk(w-k)dx

(5.7) < M, \Vw\ G

J

Q2\Vw\2dx

(5.8)

+\ [

(2\Vw\2dx

(5.9)

for any cut-off function £(a;) where Xfc(s) and Xfc(s) are defined by (5.4) and (5.7). The above derivation shows the following proposition. P r o p o s i t i o n 1.5.1 Assume that u is a positive, smooth solution of (5.1) on Qx, which is bounded by a constant M. Then w = um G BI{QT, M, m, 2).

52

1.5.2

Newtonian

Some

Filtration

Equations

lemmas

We need the following lemmas. [LU1] Let Q, C RN be a bounded domain, w G HQ(SI).

Lemma 1.5.1 Then

f \w\2dx < C{mesS)2/N

[

Js

Js

\Vw\2dx,

where S = {x G fl; w(x) > 0} and C is a constant depending only on N. Proof.

By Holder's inequality we have / \w\2dx < (mesS)1-^-2^

\w\2NKN-Vdx\

( /

Using the embedding theorem yields /

f

(N-2)/2N

\Vw\2NKN-Qdx\ ,(N~2)/2N

f

/ <

\

f

( /

\w+\2N/(N-Vdx)

c(f\Vw+\A


which combining with the above inequality gives the desired conclusion. • Lemma 1.5.2

[LU2] For any w G W1'P{BP) (p > 1), there holds

(A - WmesAx,,,)1-1"'

<

f/! mes(Bp\AktP)

where X > k, Ak,p = {x G Bp;w(x) only on N. Proof.

I

\Vw\dx, JAk,p\AKp

> k} and P is a constant depending

Consider the following nonnegative function w(x) = 0

for x G

Bp\AktP,

w(x) = w — k

for x G

Ak!P\A\tP,

w(x) = X — k

for x G A\tP.

Regularity

of Solutions:

Higher Dimensional

53

Case

For almost all s, y £ Bp, we have _ _ w(y)—w(x)=

r\v-*\ QW / — (x + ru)dr, Jo Of where (r,u>) is the spherical coordinates centered at x. Integrating this formula with respect toj/G Bp\AktP and noting that w(y) = 0, we obtain (Bp\Ak,p)w(x

mes

-_f

f

JBP\AK,P

\v-*\ gw

^L (x +

JO

roj)drdy.

dr

Now we estimate the integral on the right hand side:

j

JBp\AkyP

< <

*L

~x\ dw (x + ru)dr dr

dyf

JO

y-x\

I dyf JB0

JO

I

x-y\N

dw (x + ruS) dr dr 1

\v-x\

dy\x-y\dw

i€B„ 2p

x-y\N (2p) N

N

L

1

dy\x-y

\Vw(£)\

L \x-t

dw (x + rui) dr dr

I|W(OI

Td£

\N •Zi<%-

Therefore mes (Bp\Ak,p)w(x)

< (2p)

N

N

LJ

|V«J(0I

N jdt. Bp\x-t\ -

Integrating both sides over A\tP and estimating the right hand side yield mes (Bp\Ak,p){\

<

C(N)pN(mesA^)1/"

-

k)mesAx,p

f

|V«J(0|d£.

JB0

Here we have used the estimate 1" JAX.0

\X

dx ctx .c\N-i

< r , , , „ _ n V m P = 4 , ^/Jv - ^N + l)(mes J 4 AiP ) 1

Newtonian

54

Filtration

Equations

which can be proved by noting f

dx

f

dx

f

dx

-t\<6

and choosing S = (mesA\,p)l/N.

O

Lemma 1.5.3 Xfc(s) and Xfc(s) possess the following properties: (i)ifn>k>~>0,0
£ ! < Xfc(fe) < A2 2 ~*'(/i) ' (ii)ifk>h>0,0
2

_

XfcW < _L X.W-/32'

then

$'(fc) ~

Xfc(ft) ^

^w-

,

1+max

( M l ^ )

1 / m

2(1-/32)

\—w—>-wr-

The proof is easy. We leave it to the readers. We need also the following iteration lemma which can be proved by induction. Lemma 1.5.4 bers satisfying

Let {yn} (n = 0,1,2, • • •) be a sequence of positive num-

yn+1

n

,,i+ < cb,ny,

a

where c> 0, 6 > 1, a > 0 are constants. If y0 then lim yn = 0. n—>oo


Regularity of Solutions:

1.5.3

Properties

of functions

Higher Dimensional

Case

in the generalized

55

class B%

First let us introduce some notations. For any fixed p G (0,1], e G (0,1], denote Qp = {{x,t);\x

- XQ\ < p,t0-aBp2


where B —
w = p — p,

a = 2 L''y, pi = — — p , p2 = — g — p , p 3 = crp, A=$'(p),

Bfc = $'(*).

The fact $"(s) < 0 implies that A < Bk < B,

if ps < k < p.

In this section, we will always assume that w G B2{QT,M,m, p > p c.

2) and (5.10)

In addition, we require that to — aBp2 > 0. The assumption (5.10) implies that t 0 ~ aBp2 < t 0 - aAp2. Lemma 1.5.5 There exist constants (3, a — a(j, N) G (0,1), b = 6(7, N) G (0,1) such that (i) if p> k > p/2, h = p, — k > 0 and mesAkiPl(t0

- aAp2) < -K/vpf,

(5-H)

then for any t G [to — aAp2, to] mes (BPl \Ak+0h,Pl(t))

^

bK

NP\]

(ii) if h = k — p> pe and mesBktPl(t0

- aBkp2) < -nNp^,

then for any t G [to — aBkp2, to] mes (BP1 \ Bk-ph,Pl(t))

> bKNp?.

(5.12)

Newtonian Filtration

56

Proof.

Equations

For T\ > r 2 > 0, let 1,

if |a; - x 0 | < r 2 ,

ri -

C{X;TI,T2)

\x-x0\ -,

n

if r 2 < |a: - x0\ < T\,

-T2

if |a; — sco I > r i -

0,

Choose £(a;) = C(a;;pi,Pi — o"iPi) in (5.8) with a\ € (0,1) to be determined, integrate over (£0 — aAp2,t) and use the condition (5.11). Then we obtain

J

C2Xk(w-k)dx

/

Ak,P1{t)

<

iaAp2„2u2 h2 N C2Xk{w - k)dx + ~, y2~^NPl

/ 1

iv

K

<

^ NPI

,,\

2"faAh2

N

j—KNPI •

Xk{h) H

On the other hand, C2Xk(w - k)dx >

/

C2Xk{w-k)dx

/

•> Ak,P1(t)

JAk+/3htP1_<,lf,1(t)

XkH3h)mesAk+0h,p1-
> Therefore for any t G [to — aAp2,to] A mesAk+0h,Pl^lPl(t)

M ^ Xk^ <^ y

l

• ^NPl

N , laiAh2 +^ ^ j ^

N P

Using Lemma 1.5.4 we obtain .. mesAk+0htPl-aiPl{t)

/ 1 Aaj \ N < I — j + - j ^ I KivPi •

Choose /3 = A / 2 / 3 and small b\, o\ and a such that

£ + ^ s <'-«<•-->"• Then for t £ [to — aAp2,t0] we further have mesAk+phtPl-aiPl(t)

< (1 - 6i)(l -

CTI)NKNPI-

l

•

Regularity of Solutions: Higher Dimensional

Case

57

Hence (BPl\Ak+phtPl(t))

mes >

mes(B / , 1 _ 0 . l P l \yl f c + / 3 h i / , 1 _ o . l P l (t))

>

(l-a1)wKjvpf-(l-61)(l-f71)^wpf

=

h(l-CT^KNP?,

and (i) is proved by choosing b = &i(l — a\)N. (ii) can be proved similarly.

•

Lemma 1.5.6 For any 6\ > 0, there exists s(#i) > 0 such that (i) if fi > k > /J./2, h = fi — k and 1 - aAp2) < -Kjvpf,

mesAk,Pl(to then (° 2

mesA^h/2,+liPl(t)dt<61Ap»+2;

(5.13)

Jtn—aAp

(a) if S1^

* r* t€[to

—a

A 2,

mes

>BP+"/2,pAt)^oKlfpir>

^ P >'o—"^4p ]

-s

^5'14)

then s+2 _£

w < 2' + V or /•to

mesBka+uPl

< 0i£ f c 3 + 2 pf + 2 ,

(5.15)

7t0-aBfca+2

w/iere A;s = p, + u>/2s, Bka = <&'(ks)/m. Proof. First we prove (i). From Lemma 1.5.5 (i), for any t € [to — aAp2, to] we have mes (Bpi \ Ak+0h,Pl(t)) Choose ro such that — 2r"

> bnNp^.

(5.16)

58

Newtonian

Filtration

Equations

where /3 is the constant in Lemma 1.5.5. Let ki = \i

j . Noting that

k + 0h < k + (l - ^A h = /i - A < p - ^ = hi,

for I > r0,

from (5.16) we have, for t £ [to — aAp2, to], mes (BPl \ AkuPl

(t))

> mes (BPl \ Ak+ph,Pl

(t)) (5-17)

> bKNpi , for I > ToUsing Lemma 1.5.2 we obtain < mt^m0fA W) f Ww\dx, JDl{t) mes {BPl \Akl,Pl{t)j

(kl+1 - k^mesA^^it))1-1^ where (3 > 0 is a constant and Di(t) =

AkltPl(t)\Akl+ltPl(t).

Thus from (5.17) we see that for t € [to — aAp2, to], h 2i+i

mes Akl+uPl(t)

< £- f 0KN

\Vw\dx.

JDt(t)

Noting that ( m e s A f e , ^ ^ ^ ) ) 1 ^ < K^ p\ and applying the Schwartz inequality to the right hand side of the above inequality, we further see that for t G [t0 - aAp2, t0], /

JLmeSAkl+uPl(t)

<-

^ U

\ 1/2 2

(mesA(*)) 1 / 2 -

^ \SJw\ dx\

N

Integrating the square of both sides of this inequality over [to — aAp2, to] and using the Schwartz inequality once again we arrive at h2 22(!+1)

( ft0 / ' 2

i

um

-jzrm) bnNN %

mes

Akl+uPl{t)dt)

J to—aAp2 i J

/

to

r

f° Jto-aAp2

f JAkl,pi(t)

\Vw\2dxdt

f°

mes Di(t)dt.

Jto-aAp2

(5.18)

Regularity of Solutions:

Higher Dimensional

59

Case

Now we use the properties of functions in the generalized class B? to estimate the integral of the right hand side of (5.18). Take k = ki, ((x) = C,{x;p,p\) in (5.8). Then we have ta

i r°

r r

2

2 Jt0-aAp

[I

< ^

J

. .,

Vw\2dxdt

JAH, •i

i

M J

C Xfc, {w - h)dx + -

l-a) ¥ '

zAp2) Ahl,pi{t0—a.

9aj

9ajA h2 N 2-^ 1 ^KNPi

2

\ Ah

N

where Lemma 1.5.3 has been used at the last step of the above derivation. Substituting this into (5.18) yields (f° \Jtn—aAp

2


mesAkl+uPl(t)dt\

f"

/

mesD^dt

Jta—aAp\

with constant C\ depending only on N, 7. Summing up the above inequalities for I from ro to s, gives (a - r0 + 1) ( f °

\Jtn—aAp2

mes Akm+uPl{t)dt)

<

C1KNaA2p21N+4.

/

Choosing s such that

s - r0 + 1 ~ we finally obtain (5.13) and complete the proof of (i). Next we prove (ii). Suppose w > 2s+2p£. We are ready to prove that (5.15) must hold. To this purpose, we take k = p,+ —,hh = k — p=—. It is evidently that (5.14) implies mesBji+CJ/2,Pl{to - aBkp2) <

-KNp?

and the assumption u > 2s+2p£ implies h = —gep£. Hence from (ii) in Lemma 1.5.5, we have, for t £ [to — aBp2, to], mes (BPl \Bil+{uJ(i-0))/2,p1(t))

> bKNpf.

(5.19)

60

Newtonian

Filtration

Equations

Now we choose ro such that for t G [to — aBp2, to], mes \BP1 \ B^pi(t)j

> bnNp^',

if I > r0,

which is possible from (5.19), where k) = fl + —r. Then similar to the 2' derivation from (5.16) to (5.18), we can verify that for s > I > ro,

221+2

<

N

rto

0J2

/ l

I ff* \bKN

zn.es B^

(t)dt

2 Pp

Jtn—aBr Jto-oBi

) f° f J Jt0-aB%ii+^JB~ki,P^t)

\Vw\2dxdtf°

mesD^dt, Jt0-aB%s+ip* (5.20)

where Di(t) =

B~kupi(t)\B~kt+upi(t).

Based on (5.20) we can obtain (5.15) and complete the proof of (ii).

•

Lemma 1.5.7 For any 62 > 0, there exists 6\ = 61(02) > 0 such that (i) if pb> k > p,/2, h = p, — k > 0 and

J J t0—aAp2

mesAktPl{t)dt<6iAp^+2,

(5.21)

then for any t G [to — aAp2/4, to], mesAk+h/2,P2(t)<62P%;

(5.22)

in addition, if mesAk,Pl{to — aAp2) = 0, then for any t G [to — aAp2,to], (5.22) holds; (ii) if h = k — p> p£, f ° Jt0-ap2Bk_h/2

mesBktPl (t)dt < 61 Bk_h/2p?+2,

then for any t € [to - aBk_h/2p2/4,

(5.23)

to],

mesBk_h/2jP2(t)<62p^;

(5.24)

in addition, if mesBk,Pl (to — aBkp2) = 0, then for t G [to — aBk_h/2p2, (5.24) holds.

to],

Regularity of Solutions:

Higher Dimensional

Case

61

Proof. We only prove (ii); the proof of (i) is similar. To this purpose, we use the property (5.9) of functions in the generalized class B2. Take ((x) = £(x;pi,p2), assume that t0 > t > T > t0 - ap2Bk_h/2p2 and then integrate (5.9) over [r, t]. We easily see that Xk(-^)raesBk_h/2iP2(t)

( # 7 - {Tp^^

-

w1

+

xk(h)mesBk,Pl(T)

(5 25)

-

l mesBk pAt)dL

)l

'

From (5.23), we have, in particular, ft0—ap

Bk_h/i

mes Bk,Pl(t)dt

Jtrto-ap2Bk_h/2/4,

0iBk_h/2pN+2 l

<

which implies that there exists T £ [to — ap2Bk-h/2i to — 0'P2Bk_h/2/A\ that

such

4 mesB k , Pl (T) < -w-OiPi • Substituting this into (5.25) and using the condition (5.23), we have, for t<= [t0-ap2Bk_h/2/A,t0}, R

us ^ f Xk(h)

4 , jBk_h/2

(

9h2

™sBk_h/2,P2(t) < { - ^ y ^ + ~ ^ y [j^^

,

2\\n

N

+P

))0lPl .

Since Lemma 1.5.3 implies Xk{h) Ak\ j ^ 1 Xk(h/2) ~

+ 1 6

'

Xk{h/2) AM i j Bk_h/2

h2 >

we can conclude the existence of 9\ satisfying (5.24). If mesB k t P l (to — ap2Bk_h/2) = 0, then we take T — to — ap2Bk^h/2 (5.25) and assert that (5.24) holds for all i e [to - ap2Bk_h/2,to\-

in •

Lemma 1.5.8 There exists 62 > 0 such that (i) if p> k > p/2, h = p — k > 0 and max te[t0-aAp2,t0]

mesAklP2(t)

<02p2 ,

(5.26)

then for any t G [to — aAp2/4, t0], mesAk+h/2iP2(t)

= 0;

(5.27)

62

Newtonian

Filtration

Equations

in addition, if mesAk,P2(to — aAp2) = 0, then for any t € [to — aAp2,to], (5.27) holds; (ii) if h = k — p> pe, max

mesBkiP2(t)

te[to-aBkp2,

then for any t £ [to —

< 62P2 ,

(5.28)

t0]

aBkp2/4,to],

(5.29) mesBk„h/2tP2(t)=0; 2 in addition, if mesBktP2{to — aBkp ) = 0, then for any t £ [to — aBkp2, to], (5.29) holds. Proof. We only prove (ii); the proof of (i) is similar. Let h h j = k ~ 2 + 2I+T'

k

tj =t

Pi = Ps + —TTi—.

°~

N =

Q(x) = C(x; Pj,Pj+i),

1 4aBkP

m a x

2

3 ,_, o ~ ^p^aBkP >

, m e s Bkj ,Pj (t),

Ij{t) = /

w)(2dx.

Xkj(kj -

J

Bkj,Pj(t)

Since kj > fc/2, by Lemma 1.5.3, we have Xkj(kj -w) < m$'(kj)(kj

- w)2 < 2mBk(kj

- w)2.

Hence Ij(t)<2mBk

I

(kj-w)2C2dx.

(5.30)

•>Bkj,Pj(t)

Using Lemma 1.5.1, we further derive /,-(*) < CBkp)/N

f

\Vw[2(2dx + CBkp)/N

•

JBkj,Pj(t)

^H.

(5.31)

(Pj-Pj+1)

On the other hand, (5.4)implies

m +U 2

JBkj,Pj(t)

Given t G [tj+i,t0].

Ww\2C]dx < 7 (

h

\{Pj-Pj+l)

]

+ l) ^. J

(5-32)

63

Regularity of Solutions: Higher Dimensional Case (a) If I'jit) > 0, then from (5.32), we have / \Vw\2C]dx < 2 7 ( 'Bki,Pj(t)

k2

+1)

H.

Substituting it into (5.31) yields /,(«) < CBtf™

((27 + »

i

^

~ + 27) •

(5-33)

(b) If IAt) < 0 and for some r e [tj,t], I'J(T) — 0, then we can take r such that Jj(s) < 0 for s € (r,t] and J,-(f) < ^-(r). Since (5.31), (5.32) imply (5.33) for t = r, (5.33) holds for £ too. (c) If for any r G [£j, i], 7j(r) < 0, then from (5.32) we have \ 1

f f \Vw\\]dxdt hi JBkj,Pj(t)

< Ij{tj) + 7 (* - *,-) ( 7 ^—T2 + l ) H\{Pj ~ Pj+i) J

Integrating (5.31) over [tj,t] and using the above inequality yield / ' 7,(r)dr < CBktfN 7tj

hl^)

L

+ ( 2 7 + l ) ^ ( * " * j ) ^ + 2 7 (* - i * ) ^ ' \Pj - Pj+i)

Substituting (5.30) into this inequality and noting that IJ(T) is increasing on [tj,t], we derive that for t € [t J+ i,io], « « ) < C B ^ » « ( ± r ^ v*j+i-*i

+ 7 ei±lSl + 2

(Pj-Pj+ir

7). )

Thus in all cases (5.34) holds for t G [t,+i,io]On the other hand, using (5.3) we have Ii(t) > Xkj(kj - kj+1 )mesB k . + 1 iPj+1 (t) >

^Bkikj - kj+1)2mesBkj+uPj+1

(t),

which combining with (5.34) yields ^ ori 2/JV+l920+l) ( ±™Bk 27 + l 27\ /ij+i < 2G/i/ 2w^' I —+ —+ — . 3 ytj+i-tj (Pj-Pj+1)2 h2j

(5.34,

64

Newtonian

Filtration

Equations

Hence from the definition of tj and pj, we see that the above inequality can be simplified as C12^p2/N+1 M+ ni 1

?

where C\ is a constant depending only on N, m, 7, T. If we set yj = p,jj pN, then the above inequality can be written as

yj+1oo

62 is chosen small enough and complete the proof of (5.29). If mesBk tP2 (to — aBkp2) = 0, then we take tj = to — aB^p2 instead 1 3 of tj = to — -aBkp2 — . „abkp2. The desired conclusion can be easily obtained. • Lemma 1.5.9 There exists s > 0 such that (i) if fi> k> n/2, h = p, — k > 0 and mesAk,Pl(to

- aAp2) <

-KNp?,

then for any t G [to — aAp2/16,to], mesA M _ /l/2 .+3 iP3 (t) = 0;

(5.35)

(ii) if max 2

2

mesBu+tll/2,p1{t)<7,KNPi,

te{to-aBp ,t0-aAp }

'

(5.36)

2.

then u < 2S+V

(5.37)

osc {w, Qp/4} < 11 - ^ 3 J osc {w, Qp},

(5.38)

or

Regularity of Solutions:

Higher Dimensional

65

Case

where QP/4 = {{x,t);x

- aA(p/A)2

G Bp/4,t0


Proof. Let 62, B\ and s be constants determined by Lemma 1.5.8, Lemma 1.5.7 and Lemma 5.6 successively. Then (5.35) holds. Similarly from the assumption (5.36) we can derive (5.37) or mesB^ +w / 23 +3 i/93 (t) = 0 for any t G [t0 - aS f e s + 2 (p/4) 2 ,t 0 ], where Bkii = &(ks), the later case, obviously we have

ks = p + w/2 s . In

1 + ^ 3 J < ( 1 - x-^r 2T+3 ) w,

osc {w, Qp/4}
which is just what we want to prove. 1.5.4

Holder

continuity

of

•

solutions

Proposition 1.5.2 Let p0 G (0,1], e G (0,1], QPo C QT and u(x,t) classical solution of (5.1) on QPo such that 0 < u < M. Denote p0 =swpw(x,t),

Jio = inf w(x,t),

be a

OJO=PO-PO,

where w = um. If for some constant Co > 1, Po>2CoPs0,

uo
(5.39)

then there exist positive constants a, C depending only on N, m, T, such that for (x,t) G QPo, \w{x,t) - w(x0,t0)\ Proof.

< C{\x - x0\a + \t-

t0\a/2).

Let , X

x - x0 po '

, "

t' =

t — to apl$'(po)

Then QPo is transformed into

Q; = {(a/,O;|z1
M,

66

Newtonian

and v(x',t')

= u(x,t)

Filtration

Equations

satisfies dv — =

dwx,(A(x',t')Vx
where A(x',t')

= am$'(/xo)w m_1 (a;,i).

Prom (5.39), jio> — and hence /lt)\^m_1^m

/]\(n»-l)/m

Thus using the estimate which holds for parabolic equations without degeneracy, we have | « ( a : , ) f ) - « ( 0 , 0 ) | < C 7 i ( | i r i + l*,ri/2) with constants C\ > 0 and a.\ G (0,1) depending only on N, m, M, T, which holds for any (x',t') G Q'1/2 = {(x',t'); \x'\ < 1/2, - 1 < t' < 0}. Returning to the original variables (x,t), we get

MM)-<*.,*.)! < C , ( f e ^

+

j^^)

(5.40)

for any (x,t) G Q Po /2, where Q ^ o / 2 - { ( a : , t ) ; | i - a ; o | < y , * o - |*'(«>)Po < * < *o} • Let E = j ( z , t ) ; |a; - x0\ < y , t o - ( f $ V o ) P o ) 2 < * < *o} • Then from (5.40), we see that, for (x,t) G S, |u(a;, i) - «(a;o, *o)| < Ci (|z - z o | a i / 2 + \t - t0\ai/i)

.

(5.41)

Regularity of Solutions:

Higher Dimensional

67

Case

If (x,t) G QPo \ S, then using the assumption (5.39) again, we have (|x-x0|2 + |*-io|)1/4 >

min{p 0 /2, ^ ' ( M o ) ^ / ^ 1 / 2 }

>

1 /a$'(Mm)\1/2 min<-,^ ] }p0

>

min<j-,(

y

—I)

} (C0

l

^Y'2.

Hence Vtnjrnw

1/2

c,o
|j

(\x-x0\* +

\t-t0\y/\

which shows that (5.41) holds on QPo \ E for some other a\ G (0,1). The proof is complete. • Theorem 1.5.1 Let u be a generalized solution of (5.1) on QT such that 0 < u(x,t) < M. Then there exist constants C > 0, a G (0,1) depending only on N, m, M, T such that \w(x,t) - w{xo,t0)\ < Cpo1 (\x - x0\a + \t- t0\a/2) , V(xo,t0)€QT,

(x,t)£Q*po,

provided to — ap\jvr? > 0, where w = um,

p 0 =mm|l,^-j

j,

Q*P0 = {(x,t);\x-x0\

< p0,t0

^Po < t
,

and a = a(2,N) is the constant determined in Lemma 1.5.5. Obviously, Theorem 1.5.1 implies that for any n G (0, T), the generalized solution u is Holder continuous in RN x (77, T). Proof. As we have indicated at the beginning of this section we may simply assume ihatu(x,t) is a positive classical solution of (5.1).

68

Newtonian

Given (x0,t0)

Filtration

Equations

G QT- Denote C = max[(2s+8m3)"\—j, 7] = 6 4 m 3 C 1 - 1 / m ,

where s is the constant determined in Lemma 1.5.9. Choose e € (0,1) small that

( - Y > 1 - 2"1-(S+3)AS

Nc = 4m3(71-1/"\

and let P°

Pi = —r, if

Qi = QPl = {(a;,i);|a;-a;o| < pi,t0 -a&(pf)pf fii = sup w(x,t),

Jli = inf w(x,t)

Qi

Ljt= in-fit,


(I = 0,1,2, • • •),

<2<

I* = 'mi{l;pi > 2Cpf}.

From the definition of C, it is clear that wo < Cpe0.

If I* = 0, then the conclusion for our theorem follows from Proposition 1.5.2. Now we suppose I* > 0 and prove ui < Cpi

when / < /*

(5.42)

by induction. Suppose (5.42) holds for I < I*. We want to prove that (5.42) holds for l + l, provided Z + l < I*. If p,i < 2pi, then it is easy to see that ui+i < m < Cpel+1 which means that (5.42) holds for I + 1. Now we suppose p,i > 2p\. If max *6|to— aBpf,t0—

m e s % i + a J l A p i l ( t ) < -KNp%, aApf]

£>

(5.43)

Regularity of Solutions:

Higher Dimensional

69

Case

where B = *'(pf).

^ = *'(W),

^0 = 2 - ^ ,

PU = ^Y^PI,

then from the definition of C and Lemma 1.5.9, we see that wi < 2 s+2 pf < 2 s + V p f + i <

^ C3 1 / " l ( 6 4 m 3 ( 7 1 - 1 / " i ) V f + i < Cpf+i m

or OBC{W;QPI/4}

< ( 1 - ^ 3 j osc{w;Q P l } < ( - j Cp\ < Cpf+1.

From the definition of 77 and noting that pi < 2Cpf, we can verify that ct, pi+x Q1+1 C Q Pl /4- In fact, pi+i<<— y and and aA{Pl/lf _ $'( W )p 2 a*'(pf + 1 )p 2 + 1 16$'(pf +1 )p 2 +1

>

^ ( } y-i/m> 16m

3

^1™

\2CT)*J

~

>x

32m3C1-1/m

Thus Qi+i C Qp,/4 and hence u>i+i < Cpf+1. If (5.43) does not hold, then there exists r € [to—aBpf,to — aApf], such 1 that mesB^ i+Wj / 2 ,p li ('r) > -KNp^ and hence 2'

m e s ^ + W j / 2 , P l i < 2 K ivPu or mesA w _ W i / 2 i P l l (r) < -KNp%.

(5.44)

Now we divide the interval [r, to] into K equal parts, such that ^aApf
t

^ - < aApl

Clearly K

<

2a^(pf)p?

< 2

/2^\

X 1/m

-

<

^ _

1 / m =

70

Newtonian

Filtration

Equations

Let tp = T + (p — l ) A i (p = 1,2, • • • , K), tK+i = to- Using Lemma 1.5.9 on [£1,^2], from (5.44) we obtain mesA^^l/2s+4iP3l(t2) where pzi = croPh Co = 2~1/N.

= 0,

Therefore

mes^ w _ Wi/ 2»+4 iP3i (i 2 ) < <

mes A w _ u , l/2 .+4 iP31 (t2) +

KN{PU

- P31)

2KNPU'

By induction, we finally arrive at 1 N mes^4w_a,1/2(»+3)(^-1)+1,p11(fe) < 7iKNP\i 2 and for t e (t0 -

—Apf,t0), m e s A w _ U l / 2 ( . + 3 ) K + i ] P 3 1 ( t ) = 0.

Therefore wi+i = o s c { w ; Q ; + 1 } < <

(l-

2(s+3)K+1)

osc {w;Q[}

v-eCp!
Summing up, we have proved (5.42) for all I < I*. In particular, we have UJI* < Cpf,. On the other hand, by the definition of I*, /f;» > 2Cpf,. Thus, by Proposition 1.5.2, there exist positive constants a, C depending only on N, m, T, M, such that \w{x, t) - w(x0, to)! < C (\x - x0\a + \t- i o | a / 2 )

(5.45)

holds on QPl,. For any p G [pi-,po), there must be an I < I* such that Pi+\ < P < Pi- Thus by the definition of I*, we have osc{w;Qp} = sup w(x,t) — inf w(x,t) < w;

< dpf = c(m+1y

< c(VPy = Cv'/f,

from which it is easy to see that (5.45) holds on Qpo for some positive constants a, C depending only on N, m, T, M. The proof is complete. •

Properties of the Free Boundary:

One Dimensional

Case

71

Remark 1.5.1 If the initial data UQ > 0 are nonnegative, bounded and Holder continuous, then by a similar argument we can prove that the corresponding generalized solution is Holder continuous down to t = 0.

1.6

1.6.1

Properties of the Free Boundary: Case Finite propagation

of

One Dimensional

disturbances

Consider the equation 8u dt ~

d2A(u) dx2

(6.1)

with initial data u(x,0) = uo(x)

(6.2)

i £ l

We always assume that A(u) e C^O, oo) n C 2 (0, oo), A(u) > 0, A'(u) > 0, A"(u) > 0

for u > 0,

,4(0) = A'(0) = 0, and UQ is nonnegative, bounded and continuous on R with A(uo) satisfying the Lipschitz condition. By virtue of Theorem 1.2.1 and Theorem 1.2.2, the Cauchy problem (6.1), (6.2) admits exactly one nonnegative, bounded and continuous generalized solution u on Q = R x (0, oo) with bounded weak derivative ^—-. ox In §1.4, we have discussed the regularity of generalized solutions for the equation du _ ~dt~

d2um dx2

(6.3)

with m > 1 which corresponds to the slow diffusion. In physics, slow diffusion should imply that the speed of propagation of disturbances is finite. The mathematical description of this fact is that if suppwo is bounded, then for any t > 0, swpp u(x,t) is also bounded. We have the following general result.

72

Newtonian

Theorem 1.6.1

Filtration

Equations

Assume that for any u > 0, tf («) = / -±lds s Jo

< +oo.

(6.4)

Let u be a generalized solution of the Cauchy problem (6.1), (6.2) on Q. If suppuo(x) is bounded, then for any t > 0, supp u(-,t) is also bounded. Proof.

Consider the function of the form u(x,t) = * _ 1 ( c ( c i + ii) - x ) + ) .

It is easy to verify that for any £i > 0, c > 0, u(x, t) is a generalized solution of (6.1) with initial data y~1(c(ct1-x)+).

u(x,0) = Let XQ = sup{suppuo(:r)}. Then

u(x, 0) = uo(x) = 0 < u(x, 0)

for x >

XQ.

Besides, since ty and ^ _ 1 are increasing, we have u{x,t)

=y-1(c{c{t

+

t1)-x0)+)

> ^ _ 1 ( c ( c t i - x0)+) >M = supw, 0 provided ct\ — x0 > 0 and c(cti — xo) > * ( M ) . Hence u(x0, t) <M < u(x0,t)

for t > 0.

Now we apply the comparison theorem on GT = (^0;°°) Theorem 1.2.4 and Remark 1.2.2) and then obtain u(x,t) < u(x,t),

x

(0, T) (see

on GT,

from which it follows that u{x,t) = 0 when x > Xt — c(t + ti), since u(x, t) = 0 for x > Xt = c(t +1\). Notice that in applying the comparison theorem on GT, we need to check u e L1(GT) (see Remark 1.2.2). From Definition 1.1.3 and Remark 1.1.2, for any T G (0,T) and

Properties of the Free Boundary:

One Dimensional

Case

73

enough, we have / u(x,T)ip(x,T)dx J®.

— I

uo(x)ip(x,0)dx

Jn

- l(»t + ««>£)**• In particular, we take

= 0, for \x\ > X + 1, K ( z ) | < C,

where the constant C is independent of X. Then we obtain / u(x,r)dx J\x\<x

+ // JJx-^x+1

A(u)ip'x(x)dxdt

< I uo(x)d: JWL

Since the right hand side is bounded uniformly in X, letting X —>• oo, we see that / u(x, r)dx is bounded and hence u £ LX{QT)Jw. Similarly we can prove that there exists X[ such that u(x,t) = 0 when x < X[. D Remark 1.6.1 Theorem 1.6.1 shows that (6.4) is a sufficient condition for (6.1) to possess the property of finite speed of disturbances. One can prove that (6.4) is also necessary for (6.1) to possess such property. Oleinik guessed the necessity of this condition at a symposium held in Moscow university. Since then people verified this supposition for some special cases or under some additional condition. Finally Peletier [PE3] gave a satisfactory result. When A(u) = um, the condition (6.4) is equivalent to m > 1, the slow diffusion case. Assume that suppwo = [£i,£2] (—oo < x\ < x% < +00) and u is a generalized solution of (6.3), (6.2) on Q = R x (0,00). Denote Q = {(x,t);u(x,t) n(t) = {x;u{x,t)

>0,t>0} >0}

Ci(t) = inf fi(t), C2W = supfi(t).

74

Newtonian

Filtration

Equations

By the continuity of u, 0 is an open set. Theorem 1.6.1 implies that for any t > 0, d(t) (i = 1,2) is finite. We call x = 0(t) (i = 1,2) the free boundary or interface of u. Theorem 1.6.2 creasing and

Assume that m > 1. Then (—l)lQ(t) (i = 1,2) is m-

lim (-l)'Ci(t) = + o o ,

(z = 1,2).

(6.5)

t—>oo

Theorem 1.6.2 means that in case of slow diffusion, disturbances will be propagated to infinite scope, although the speed of propagation is finite. Before proving Theorem 1.6.2, we first prove the following proposition which is also very useful in the sequel. Proposition 1.6.1 Assume that m > 1 and u is the generalized solution of the Cauchy problem (6.3), (6.2)on Q = R x (0, oo). Then ^ > - - , dt ~ t ' dv ^ (m - l)kv dt~ t

(6.6) ' . . [ '

y

in the sense of distributions. Remark 1.6.2 As will be seen from the proof stated below, this proposition is valid for any initial data UQ G L°°(M.N). Proof of Proposition 1.6.1. From the proof of Theorem 1.3.5 and point 4 in §1.3.3, the given generalized solution u can be obtained as the limit of a sequence of classical solutions which are positive, and uniformly bounded on Q and whose derivatives up to second order are bounded. To prove the proposition, we may simply suppose that u is just its approximate smooth solution. We first verify (6.8). Notice that v satisfies dv

,

. d 2v

1)V

dt-^-

o^

+

(dv^

[9-X

(6 9)

-

Properties of the Free Boundary:

Lw = -

- (m - l)v^

- 2m-

One Dimensional

75

Case

• - - (m + l)W2 = 0.

(6.10)

~ k Clearly the function w = — also satisfies the equation (6.10) on Q. Since w = -r—^ is bounded, we have ox* k wix.e) > — £

provided e > 0 is small enough. Thus we may use the comparison theorem on Rx (e, oo) to assert ~ k w(x,t) > w(x,t) = —

for (x,t) e R x (e, oo),

from which (6.8) follows by virtue of the arbitrariness of e. (6.7) follows from (6.8) and (6.9). To verify (6.6), it suffices to note that du

/m-l\

1 / ( m - 1 )

(82V

1

at

\

J

\ dx2

m — 1

m d2v

v(2-m)/(m-l)

\dxj

ku

^Udx~2^---

Proof of Theorem

( OV \ ^

• 1.6.2.

From (6.6) we have

!(**«>> o. From this and the continuity of u, it is easy to see that for any x &R, tku is increasing in t. Therefore fi(ii) C ft(£2), f o r £ i < £ 2 , in other words, (—l)Xi(t) {i = 1,2) is increasing. Next we prove (6.5). For simplicity we suppose that x\ < 0 < Xi, uo(0) > 0. Then there exist 6 > 0, £o > 0 such that UQ(X)

>

£Q

for |a;| < 6.

76

Newtonian

Filtration

Equations

Consider the function BTtL(x, t) = l}lm-xBm{x,

L{t + T))

{L,T > 0)

where Bm(x,t) is the Barenblatt solution of (6.3) (see (1.37)). It is easy to check by direct calculation that for any L, r > 0, BTIL{X, t) is a generalized solution of (6.3) with initial data Ll/(m-^B(x,LT)

=

BT,L(X,0)

m - 1 x> * X l / ( m _ 1 ) 2m(m+l)(Lr)2/(m+i)y+y

LV(«-D / / (Lr)V("»+i) ^ Obviously

L 2/(m

0
2

-l)

rl/(ro+1),

suppB T ) i (z,0) = L : |z| 2 < 2 " ^ m _ + 1 ) ( L r ) 2 / ( " > + i ) \ . Choose L, r > 0 such that L 2/(m

2

-l)

=

l/(m+l)

?m(m+_l)

2 / ( m + 1 )

=

^

m—1 It is easy to see that this is possible. For such L, r > 0, we have BTiL(x,0)L(x,t)

< u(x,t).

In particular

Q(t)

D{X:\X\><

2

^^{L{t

+

r)fl^

This completes the proof of (6.5). 1.6.2

Localization

and extinction

. •

of

disturbances

We will indicate in the sequel that for filtration equations with appropriate absorption term, the support of generalized solutions might be included in a

Properties of the Free Boundary:

One Dimensional

Case

77

bounded domain forever. In this case we say that the disturbances possess the property of localization. Consider equations of the form du 5 i

=

d2A(u) ^ ^ ~

, c ( u )

^

G

Q

'

/*,,>, -

(6 11}

where J4(U), C(W) are appropriately smooth. In addition to the assumptions on A(u) and «o given at the beginning of this section, we assume that c'(u) > 0 for u > 0 and c(0) = 0. Similar to the argument stated in §1.1.2, we can prove the existence and uniqueness of generalized solutions (which can be defined in an obvious way) of the Cauchy problem for (6.11). Also the comparison theorem is valid (cf. [KA2]). In the sequel, we will use these results without proof. Theorem 1.6.3 Let u be a generalized solution of the Cauchy problem (6.11), (6.2) on Q. Ifu0{x) = 0 for \x\ > X > 0 and f ( f

*(£Kj

where &(v) = c($(v)), $(v) = A~1(v), u(x,t) = 0 for \x\ >X,t>0.

dv<+oo,

(6.12)

then there exists X\ > X such that

Proof. To prove that for some X\ > X, u(x,t) = 0 for x > X, t > 0, it suffices to construct a generalized solution w(x,t) on the domain Gx = {(x, t); x > X, t > 0} such that u(x, t) < w(x, t)

for (x, t) e Gx

(6.13)

and w(x, t) = 0 for x > Xi, t > 0. We try to seek such generalized solution w(x,t) among functions which depend only on x. First we require w(x) to satisfy (6.11) on G x \ G x 1 5 namely, for X < x < X\, d2 •^A(w(x))

fM

= cmj(x)))

= c(w(x))

= ^(j(x)),

(6.14)

78

Newtonian

Filtration

Equations

where j(x) = A(w(x)) and Xi > X is a constant to be determined later. Denote the inverse function of j(s) by J(v). Then dj(x) dx 2 d j(x) dx2

1 J'(v)' _ J"(v) (J'(v))2

dj(x) dx

J"{v) {J'(v))3

( 1 \2(J'{v)y

and (6.14) turns out to be

1 2{J'{v))2

V = *(v).

Integrating this equality yields J(u) =

i, W o *(0 7

d?7

'

(6 15)

'

Prom the above analysis, it is natural to define J(v) by (6.15) and then to consider its inverse function j(x). The condition (6.12) ensures the definition of J(v) for all v > 0. Since J(v) is increasing and J(+oo) = +00, j(x) is well-defined for all x > 0. j ( s ) is a solution of (6.14) for x > 0; so is j(X\ — x) for any X\ and a; < Xi. Hence for any Xi > X, <J>(j(Xi — x)) is a (classical) solution for x < X\. We choose X\ such that u{x,t)<w{X).

(6.16)

This is possible; for example, we may take X\ = X + J(A(M)), M = sup u. Now we define ( $(j(Xi - x)) w(x,t) — w(x) = < [ 0

where

for X < x < Xlt fora;>Xi.

n, dA(w) , ,r . . , , , bince w and — equal zero at x = X1, it is easy to check that a n s a generalized solution of (6.11) on Gx- Using the comparison theorem for u and w on Gx and noticing (6.16) and that uo(x) = 0 for x > X, we arrive at (6.15) and that u(x,t) = 0 for x > X, t > 0. Similarly we can prove that u(x, t) = 0 for x < —X, t > 0. •

Properties of the Free Boundary:

Remark 1.6.3

One Dimensional

79

Case

For equations of the form

d2um

du

the condition (6.12) becomes 1/2

(m + n\ \

mc J

-i

f1

vHm+n)/{2m)dv

J0

< +oo

which is equivalent to n < m. Theorem 1.6.4 Letu(x,t) be a generalized solution of the Cauchy problem (6.11), (6.2) on Q = R x (0, oo). If

t^
(6.17)

c{y)

then there exists T £ (0, oo) such that u(x, t) = 0, for x S R, t > T. In this case we will say that extinction occurs for the solution u(x, t) at the time t = T. Proof.

To prove our theorem, we first choose a function w(t) such that ^ p

= -c(tu(t))

forO
(6.18)

w(T) = 0

(6.19)

with T > 0 to be determined later. Integrating (6.18) and using (6.19) we obtain

I

w(t)

dv -rr=T-t,

for 0 < t < T.

(6.20)

Here we notice that the condition (6.17) ensures that the integral / -u Jo

c(y)

is defined for any v > 0 and is increasing in v. Extend the function w(t) defined by (6.20) to (0, oo) with w(t) = 0 for t € [T, oo). Then it is easy to check that w{t) is a generalized solution of (6.11) on (0,oo). If we have u{x,0) =u0{x)

<w(0),

(6.21)

Newtonian

80

Filtration

Equations

then the comparison theorem gives u(x,t) < w(t)

for (x,t) G Q

and hence u(x, t) = 0 for x G R, t > T. Since

Jo

c(y) /•M

for (6.21) to be held, it suffices to take T = The proof is thus completed.

Jo

j

—r^where M > c

iy)

svpuoix). •

Remark 1.6.4 Kalashnikov proved in [KAl] that for the generalized solution of (6.11)to have the property of extinction, the condition (6.17) is also necessary. 1.6.3

Differential

equation

on the free

boundary

In what follows we will further investigate the properties of free boundaries x = d(t) (i = 1,2) of the generalized solution u of the equation (6.3). Naturally we expect the free boundaries to move with the local velocity. Set m

,m— 1

m —1 Then, in view of the equation of state, which we have used to derive the equation (6.3), v is essentially the pressure, and, by Darcy's law, we expect

(x,t)gn x-Ki(t)

where Q = {(x,t) almost true.

G Q;u(x,t)

Theorem 1.6.5

The limits

vx((i(t),t)

=

lim

OX

> 0}. The next result shows that this is

(*,t)en

^

^

OX

(t = l,2),

Properties of the Free Boundary:

exist for allt>0

One Dimensional

Case

81

and

C(t + 0) = -vx(d(t),t)

(i = l,2).

Proof. FVom Theorem 1.2.2, u e C°°(0). By Proposition 1.6.1, for any T > 0, there exists a constant /? depending only on r such that 0^2 >~P

for {x,t) e£l,

t>r

in the sense of distributions. Hence the function f(x,t) rj2 £

= v(x,t) + f3x2

t) £

satisfies 7-—^ > 0. This means that —— is an increasing function of x for oxz ox each fixed t > r. Since for any t > r, u has compact support, it follows from Lemma 1.4.1 that for any finite T > r , — and —— are bounded ox ox df(x t); on n n {{x,t);t e [r,T]}. Thus the limits lim ^ ' (i = 1,2) and (x,t)en dv(x i) lim ——-— (i — 1,2) exist for any t G [r,T\. Since T, r are arbitrary, (a;,t)6fi

these limits exist for any t > 0. We are ready to prove the rest part of the theorem for any t. For simplicity we take t = 0 and treat £ 2 (i) only. Set a = £2(0), vx(a, 0) = a. Then either a = 0 or a < 0. Case 1. a < 0. We will show that for any sufficiently small e > 0, there exists a 6 > 0 depending only on e such that

C2(Ai) -a At

+a < s

(6.22)

dv(x 0) whenever 0 < At < S. Since lim — ' = a, it follows that for any 0 < e < —a, there exists a do > 0 such that a-

dv(x,0) e < — ^ — - < a + e,

. . (a-60 < x < a).

Using the mean value theorem we get (a + e)(x - a) < v(x, 0) < (a - e)(x - a)

82

Newtonian

Filtration

Equations

and hence u(x,0) > u(x,0) <

l/(m-l)

m—1 (a + e)(x — a) m 771 — 1

m

l/(m-l)

(a — e)(x — a)

wherever a — <5o < £ < a. We will use the following pressure solutions of (6.3) m —1 (a + e)((a + £ ) t + ( a ; - a ) ) -

ui(x,t) u2{x,t)

m—1 (a — e)((a — e)t + (x — a))_

=

l/(m-l)

l/(m-l)

which are some modification of (4.37). From (6.23) it follows that ui(a;,0) < u(x,Q) < U2(x,0),

for x > a — <5o-

By continuity there exists r > 0 such that, u\{a — So,t) < u{a — 6o,t) < u^{a — 5o,t),

for 0 < t < r.

Therefore the comparison theorem gives u\{x,£) < u(x,t) < U2(x,t),

ioi x > a — So,0 < t < T

and from this it is easy to see that a - (a + e)t < (2(t)
e)t,

for 0 < t < r

or C2(*)-a + a <e, t

for 0 < t <

T,

which completes the proof in Case 1. 2°. a = 0. From the proof in Case 1, we see that we still have ui(x,t)


for x > a — 50,0 < t < r

and hence

C 2 (i)-a < e,

for 0 < t <

T.

(6.23)

Properties of the Free Boundary:

One Dimensional

83

Case

On the other hand, by the monotonicity of C2W, Ca

<*>-°>o

fori>o.

t

D

Combining these two inequalities completes the proof in Case 2. 1.6.4

Continuously

Proposition 1.6.2 [\ 1' continuous on

Proof.

Let t\ e

differentiability

of the free

For any 0 < S < 1, d(t)

'4

boundary

(i = 1,2) is Lipschitz

. Denote ao = ^2(^1) and C — max 5
xEB.

which is finite by Lemma 1.4.1. Consider the pressure solution of (6.3), m—1 c(c(t — ti) +a0 - x)+

U(x,t)

l/(m-l)

Since v(x, t) < C(cto — x) for x < ao, we have u{x,ti)


for x G K.

Therefore by the comparison theorem, u(x,t)
for x G R , i i
From this it follows that C2W
forti
and hence, since ^(*) is increasing, we obtain 0 < C2W " C2(*i) < C(t - ii),

for

h
The Lipschitz continuity of (1 can be proved similarly. In the sequel we will use the Barenblatt solutions (see (1.37)) l/(m-l)

w x t)

x t)

< > =^ > =

1

m{( -jQ+

dv(x,t) dx

84

Newtonian

Filtration

Equations

where

A(*)=cm(*+i)i/c«+i)>

Cm={^±iiy

{m+i)

An immediate calculation shows that for any c, L > 0, wc,L(x,t) =

cw(Lx,cm~lL2t)

is a generalized solution of (6.3). Denote ,,m—1 /

or

^•*

)

„m-i / I 1

^ 2

= ^I3S W( -AJIWJ

'

(6 24)

'

where Ac,L(t) = C m ( c m - 1 L 2 t + l ) 1 / ( m + 1 ) , *(<)(t) (i = 1,2) be the free boundaries of v £. Then Let a; = Q L Ci C W(i)

= % ( c — i L 2 t + l)1/{m+1),

C W(t)

= - % ( c ™ - 1 L 2 t + l)1/(m+1).

D

Proposition 1.6.3 For any 5 > 0 tfiere exisis a convex function &(*) (t > 0) and a C 1 ' 1 function rn(t) such that (-!)*&(*) =6(*)+»/i(*),

(*>*,* = 1 , 2 ) .

(6.25)

iJere C 1 ' 1 denotes the class of all functions whose derivatives are Lipschitz continuous. Proof. We only prove (6.25) for % = 2. We set £(£) = C2W a n d want to study the free boundary x = £(£) near a point (xo,to) with to > (5. For the sake of simplicity, we perform a translation of x, t variables so that, in new x, t variables, i 0 = 0,

x0 = ((t0) = - ^ .

Properties of the Free Boundary:

One Dimensional

Case

85

Then (xo,to) = (-p-,0) lies on the free boundary of both v(x,t) v

c,i(xit)

and

defined by (6.24). We wish to choose c, L so that C'(0 + 0) = C'(0),

(6.26)

VC,L{X^)
(6.27)

where £(£) = Q [(t). In view of Theorem 1.6.5, (6.26) is equivalent to

d2v{x,t) dx2

>-2P,

for z € K , i > < 5 .

(6.29)

If c, L are chosen so that d2vc>L{x,0) dx2

= -2P, whenever vCtL(x,Q) > 0,

(6.30)

then since also

we then deduce the inequality (6.27). By an immediate calculation, (6.26), (6.30) turn out to be C'(0 + 0) = -^—Cmcm-lL, m +1 m m-1

2L2cm~1 = 2p, C%+1

(6.31)

(6.32)

which clearly have a unique positive solution (c, L). For such (c,L), (6.27) holds and hence by the comparison theorem, Vc,L(x,t) <

v(x,t),

from which it follows that C(*) < C(*). Using this inequality and recalling (6.26) we obtain C(h) - C(0) - h('(0 + 0) > C(h) - C(0) - /iC'(0).

(6.33)

Newtonian

Filtration

Equations

By a direct calculation, using (6.26), (6.32) yields 2(m-l)L3

C"(0) = - C „

(m + 1) 2 2 (m-l)C-+ „_m_1 Pc^x^-^pc'to (m + 1) 2

where 7 =

+ o),

(m-l)C™+1 T . , . It is clear that m+1 C"(*) = C"(0) + O(t)

(*-^0+).

Therefore the right hand side of (6.33) is equal to i/i 2 C"(0) + 0(h3) = -^h2P('(0

+ 0) + 0(/i 3 ),

and (6.33) gives 1, ((h) - C(0) - h('(0 + 0) > -^/i 2 7 C'(0 + 0) +

0(h3).

Going back to the original (a;, t) coordinates, we obtain *h(t)>-YPC'{t

(6.34)

+ 0) + O(h)

at any point t = to > 0, where

Mt) = at Now, in any interval

+ h)

* ;

-^-hCit h?/2

+ 0)

(*>o).

and for any h £ ( 0, - ), $h(t) G £ c

since by Proposition 1.6.2, ((t) is Lipschitz continuous on S, - . Also, for any 0 < s < T < 00, rT

2

/ rT+h

j $h(t)dt= -^u n

rs+h

c(t)dt-j

/»T+/i

= V]T

\

at)dt-h(c(T)-as))\ o

ps-\-h

(C(*)-C(T))dt-^jf

(C(t)-CW)dt
Here and below C > 0 is a universal constant independent of h. Since, by (6.34), £(£) is Lipschitz continuous, $h(t)>-C,

for

te(s,T),

Properties of the Free Boundary:

One Dimensional

Case

87

we conclude that

I

\$h(t)\dt
Js

We can therefore choose a sequence h = hn —>• 0 such that $hn weakly converges to a measure /L*O- Using (6.34) we then have $/in (<) + 7-PC'(* + 0) -» M.

weakly asn -4 oo,

where fj, is a nonnegative measure. Since the weak convergence implies the convergence in the sense of distributions, we conclude from the definition of $fc, that C+lP(! = H

(6-35)

in the sense of distributions. Set £(*) = /

[Tdn(s)dT,

Js Js V(t) = CW + C'(* + 0)(* - 6) - 7 P |*(C(a) -

W))ds.

Then £(£) is convex, n € C 1 ' 1 , and C,(t) = £(i) + rj{t) is a solution of (6.35) with £(<5) = £(<$), C'(^) = C(^ + 0)- By uniqueness we must have C = C- The proof is complete. • Corollary 1.6.1

For any t > 0, £»'(* ± 0) eiriste and (-l)*Ci(* - 0) < (-1)*C*(* + 0).

(6.36)

Furthermore, for any 5 > 0, there exists P > 0 suc/i f/iat /or i 2 > *i > <5, (-iyCi(h

- 0 ) ^ > (-l)*C,'(*i + 0)eyPtl,

(6-37)

which implies that there exists a constant t* > 0 suc/i that £'(£) is strictly increasing for t > t* and Q(t) = x, for 0 0, then the free boundary x = £(i) remains stationary (vertical) for t* units of time after which it begins to move without further stops. We will call t* the waiting time. Proof. (6.36) follows from the convexity of £(i) and 77 G C 1 ' 1 . The existence of Ci(i — 0) can be proved similarly.

88

Newtonian

Filtration

Equations

From (6.35), we get

(t'e^y = e*ptn > 0, which implies (6.37). If £(£) =const. on the interval 0 < si < t < s2, then ('(t) = 0 on this interval. Using(6.37) we deduce ('(£) = 0 on 0 < t < si. This completes the proof. Now we are ready to discuss the continuous differentiability of Q(t) for t > t*. We will treat C,{t) = (2(t) only. Let (xo,to) = (C(^o)i*o) with io > *2 a n d Ng be the intersection of a ^-neighborhood of (x0,t0) and Q, = {{x,t) 6 Q;u(x,t) > 0}. Take 6 > 0 so small that N25 stays away from t = t\ and from the free boundary x = Ci(i). Denote by d(x,t) the distance from (x,t) to the free boundary. • Lemma 1.6.1 m, 5 such that

There exist positive constants C\, C2 depending only on

C : < ^ < C d(x,t)

2

inN6.

Proof. By Corollary 1.6.1 and our assumption on Ng, there exist constants c > 0,C > 0, depending only on 6, such that for all t with (C(£), t) € dN5, c < C(t ± 0) < C.

(6.38)

Hence there exist constants C\ > 0, C2 > 0 depending only on S , such that Ci < , ^ l * * , < C 2 ,

for^eiV*.

Since t>(a:, t) is Lipschitz continuous, v(a;,t) = v(x,i) - v(£(t),t)

< C'\x - C(t)| <

C2d{x,t).

d2v Next, since —-^ > —2P (see (6.29)), using the Taylor formula, Theorem ox*

Properties of the Free Boundary:

One Dimensional

Case

89

1.6.5 and (6.38), we see that for 6 > 0 small enough, > \vx(C{t),t)\\x - C(*)l - P\x - C(*)|2

v(x,t)

= |C'(t + 0)||x - C(*)| - P\x - CWI2 > c'\x - C(t)|

>

dd(x,t)

with a constant Ci > 0 depending only on S . The rest part of the lemma can be proved similarly. • L e m m a 1.6.2

There exists a constant C depending only on 5, such that

dv
Let (x,t)

7 = -d(x,t).

d2v dt2

<

C

d(x,t)

for(x,t)eNs.

,

G N$ and 5 be a square with center (x,t) and side

For simplicity, we suppose x = 0, t = 0. Consider the

function w(x, t) = ry~1v(jx, 7<), where (x,t) varies in a unit square SQ (so that (jx^t)

6 S). Clearly, in So

^ = (—D-^+(^J ,

(6-39)

and in view of Lemma 1.6.1, for some constants C\ > 0, C^ > 0, Ci < ( m - l ) w < C 2 . — I is a bounded function, we can apply the Nash-Moser estimate [LSU] and conclude that for some a £ (0,1), \Mc"(S') < C, 3 where 5 ' is a square concentric with So and with side - , and C > 0 is a constant depending only on C\, C2, dist (S', dSo). We can now construct a fundamental solution for the equation dW

.

,,

d2W

90

Newtonian

Filtration

Equations

and use it to express w (see [FR1]), from which we see that —— is Holder ox continuous. Applying Schauder's estimates [FR1] we conclude that -7—, d2w —^ (in fact, any derivative of w) are bounded by a constant C*, depending only on S, in a square 5"' concentric with So and with side - . Going back dvCx t) d vCx t) to the function v we deduce that ——-—, ——^— are bounded by J C*; dt dt2 this completes the proof. D Lemma 1.6.3

Let x0 = ((t0),t0

> 0,

v x

( °>to> = _£'( t(J + 0) = -b < 0.

Then there exists a neighborhood N$ of(xo,to), a,

such that for any constant

v(x,t) = Lb(x - x0,t - t0) + o(\x - x0\ + \t —-*oj) for (x,t) £ Ns with Proof.

X — Xr\

t — to

= a, where Lb{x,t) — b(bt — x)+.

For rj > 0, define vs(x, t) = ri^v^x

-xo,r]t-

t0).

Oi

Fix a £ (0, to)- Then v is defined in the —neighborhood of the origin and V v is also a generalized solution of the equation (6.39). (This means that \ V("»-l) m _ l vo J is a generalized solution of (6.3)). By Proposition 1.6.1, for t>

--, V d2v„ d2Vr,, l ^ =^ ^

+

xo,Vt

+

. r]k rik to)>-^>-^-v.

, . (6.40)

From the proof of Theorem 1.4.1, we see that vn G C 1 ' 1 / 2 with the Holder coefficient independent of 77. Therefore there exists a subsequence {vnn } with T]n —*• 0 such that vVn —> v in M2 uniformly on compact sets and v is a generalized solution of (6.39). We may simply suppose that vv —> v in R2 as 77 —>• 0 uniformly on compact sets. It is easily seen that all of the derivatives of v„ converge to the corresponding derivatives of v.

Properties of the Free Boundary:

One Dimensional

Case

91

In view of our hypothesis, for each fixed x < 0, — (z,0) = lim ——{x,0) = -z-{x0,t0) dx 17-+0 ox ox

= -b;

and for each fixed x > 0, v(x,Q) = lim Vr,(xo,0) = lim -v(rjx + x0,t0) n—>0

= 0.

T;-+0 T]

Thus —bx

for a; < 0

0

for x > 0.

v(x,0)

Moreover, (j

v„{x,t)

< — \r]x + xo -C(vt + to)\ < -{r]\x\ + \x0 - Civt + to)|) < C(\x\ + 2bt) V

provided i £ l and 0 < t < e with e > 0 small enough. Therefore we can use a uniqueness theorem in [KA3] mentioned in §1.2.4 to conclude that for ieR,0
Lb(x,t).

(6.41)

Repeating the argument we can further prove that (6.41) holds on the whole Q. By Lemma 1.6.2, it is easy to see that there exists a constant C depending only on S, such that d2Vj](x,t)

dt

2

d2v Tjp(r]x + x0,rit + to) <

Ci

s2

for x < 2bt < 0 with t2 + x2 < —^. Therefore d2vTI(x,t) dt2

< — —x

for x < 2bt < 0.

(6.42)

92

Newtonian

For x <2bt<

Filtration

Equations

0, -,

x

-/

„N

dv(x,0) dt dLb(x,0) Lb(x,0)+t dt

t2 2

d2v(x,t) dt2 t2d2v(x,t) 2 dt2

where t < t < 0. Thus v(x, t) - Lb{x, t) = — — g - t — and it follows by using (6.42) that lim (v(x, t) - Lb(x, t)) = 0

for t < 0.

(6.43)

X—> — OO

Next we show that v{x, t) > Lb(x, t)

for t < 0.

(6.44)

Since Lb(x, t) = 0 for x > bt, it suffices to prove (6.44) for x < bt, t < 0. d2v In view of (6.40), we have —^ > 0. Thus for x < bt, t < 0, dv(x, t) dv(bt, t) dx ~ dx v{x,t)

=v(bt,t)+

—{y,t)dy

bt dx Jbt

> v(x, t) - b(x - bt) = v{bt, t) + Lb(x, t) from which (6.44) follows. Now we prove that (6.41) holds for t < 0. Consider a fixed point (x,t) with x < bt. Let S be a rectangle with center (x, t) and boundaries parallel to the coordinate axis, such that (x,t) G S implies that x < bt and S contains points with t > 0. In S, v — Lb > 0 and achieves its minimum value, 0, on S D {t > 0}. By the strong minimum principle, we must have v = Lb in S. Therefore (6.41) holds for x < bt, t < 0. Suppose that v(x, t) > 0 for some (x, t) with x > bt, t < 0. Then from Theorem 1.6.2, we have v(x, t) > 0 for all t > t. However this contradicts the fact that the line x = x must intersect the line x = bt for some t > t and v = Lb = 0 at that intersection. Therefore (6.41) also holds for x > bt, t<0.

Properties of the Free Boundary:

One Dimensional

Case

93

Given a > 0. Let rj = t - to ^ 0. Since v\v\ -» v = Lb as |7y| ->• 0, given e > 0 there exists a % = %(£) > 0 s u c h that —-v(?7a + a:o,7? + to)-i6(asgnr?,sgnr/)

<e

H X — Xc\

provided \rj\ = \t — t0\ G (0,770), - — — = a. Namely t — to \v(x,t) - Lb(x -x0,t-

t0)\ < e|t — *o|-

The proof of Lemma 1.6.3 is complete. • T h e o r e m 1.6.6 £*(£) (i = 1,2) is continuously differentiable at any t ^ t* (i — 1,2), where t* (i = 1,2) is defined in Corollary 1.6.1. Proof. We will prove the assertion only for £(i) = (^2(*)• Since a convex function which is differentiable everywhere on an interval is necessarily continuously differentiable there, by Proposition 1.6.3, it suffices to prove that £(£) is differentiable everywhere at any t = to > t^, i.e C(to - 0) = C'(*o + 0) = b.

(6.45)

If b = 0, then by Corollary 1.6.1, (6.45) is trivial. We suppose b > 0. Denote x0 = C(£o)- From the definition of t\, there must be a r e (*2,*o) such that C'(T + 0) > 0. First we prove that for any a > b, there exists e > 0 such that C(t)> x0+a(t-t0),

t0-e
(6.46)

If (6.46) is false, then there exists a sequence en with en I 0 as n -4 00, such that C(*o - £n) <x0-

e na

and hence, by the definition of £(£) v{x0 -£na,t0

- £ „ ) = 0.

However, from Lemma 1.6.3 we have v{xo - ena, t0 - e„) = Lb{-ena, =

-b2en +baen + o(e„).

- e n ) -f o(e n )

94

Newtonian

Filtration

Equations

Hence -b2 + ba + o(l) =-b(b - a) + o(l) = 0

(n -> 00)

which contradicts the fact that a > b > 0. Therefore (6.46) holds and we conclude that hm — < a. t->t0-o to — t Since a > 6 is arbitrary, it follows that hm ^ ^ < 6 . t-tto-0 to — t Next we prove that for any c < b, there exists T] > 0 such that C(i) < z 0 + c ( t - i o ) ,

for t0-r]

(6.47)

(6.48)

If (6.48) is false, then there exists a sequence {rjn} with rjn 4. 0 as n —»• 00, such that C(*0 - »?n) > Z0 - Cqn > X0 - fofoBy the mean value theorem there exists x G (XQ — br\ni x§ — cr)n), such that ,~ dx[X't0

Vn)

-

s _ ^(zp - c??n, t0 - -qn) - v(x0 - brjn, t0 - rjn) {b-c)Vn

Since c < b, we have, by Lemma 1.6.3,

^'t°-77")=(V^)^

= 0(1)

(n

-*°0)-

According to Proposition 1.6.1,

v(x,t) + ±(C(t) - xf is a convex function of x. Thus ^(C(*0-»?n),<0-»7n) >

9^ .„ . k(C(tn — Tln) — X) ,„. . . _(a:,to-^l) + -^-2 ^ ^ = o(l) ( n - > o o ) , ox t0 - r\n

Properties of the Free Boundary:

One Dimensional

Case

95

so that dv l i m s u p — (C(t),*)>0. t - v t o - 0 OX

On the other hand, by Corollary 1.6.1,

|j(C(t),i)

=-C'(*o + 0) < -C(T + 0)e^p(r-*) < -C{T + 0)eT p ( T - to ) < 0 for t G (r, to).

This contradiction shows that (6.48) holds. Since c < b is arbitrary, we conclude that hminf^^^>6. t-*to—0 to — t This, combining with (6.47) gives (6.45). 1.6.5

Some further

•

results

1. The property about the finite speed of propagation of disturbances is valid for more general equations. Gilding [GI] studied equations with convection term du

d ( . ,du\

.du

and proved that if 6 2 ( u ) = 0{a(u))

{u -> 0+),

ua'(u),

ub'{u) G L^O, 1),

then the disturbances possess the property of finite speed of propagation if and only if

I

1

<*(«),, du < oo.

2. The property about the localization of disturbances was first discovered by Martinson and Pavlov [MP1], [MP2], [MP3], who studied equations of the form

du d2um = m -dx^-cu{m>1>c>0)'

96

Newtonian

Filtration

du d 2u CU m=0^-

Equations

^0).

Theorem 1.6.2 and Theorem 1.6.3 were obtained by Kalashnikov [KA1]. Afterwards, Kershner [KE2] studied the property of localization and also the property of extinction for the equation du d2um ,dun , -b2 — cu dt dx dx with m > 1, n > 1, I > 0. 3. The result given in Theorem 1.6.6 is the best possible. Aronson, Caffarelli and Vazquez [ACV] pointed out that, for some initial value,

£(*:+o)^£(*,*-o). This shows that in general, &(£) is impossible to be continuously differentiable on the whole (0, oo). Knerr [KN] discussed the problem that when t* will be zero or not. Quantitative investigation on the waiting time for some equations was done by Aronson, Caffarelli and Kamin [ACK], Lacey, Ockendon and Tayler [LOT] and others. They gave some sufficient conditions for the waiting time to be positive although the free boundary is continuously differentiable on (0,oo). 4. Using the iterative weighted norm, Hollig and Pilant [HPI], Hollig and Kreiss [HKR] proved the C°° smoothness of free boundaries after the waiting time. Such result was also obtained by Aronson and Vazquez [AVA]. Semi-group method was applied by Angenent [ANG] to prove the analyticity of free boundaries. For recent works concerning propagation of disturbances and free boundaries, see for example, [AB1], [DS], [GSV1], [GSV2], [GSV3], [GV], [KAM3]. 1.7

1.7.1

Properties of the Free Boundary: Higher Dimensional Case Monotonicity boundary

and Holder

continuity

of the free

Consider the Cauchy problem

£-*.-.

(7.D

Properties of the Free Boundary:

u{x,0)=uo(x)

Higher Dimensional

xeRN,

Case

97

(7.2)

where m > 1 and UQ > 0 is Holder continuous on R^ with compact support. Prom the discussion in §1.3 and §1.5, the Cauchy problem (7.1), (7.2) admits a unique generalized solution t i o n Q = B.N x (0, oo) which is Holder continuous down to t = 0. Denote fi = {(x,t);u(x,t)

>0,

t>0},

il(t) = {x;u(x,t)

>0},

r = dnn{t>o}, T(t)

=dSl(t).

By continuity, D, and 0(i) are open sets of Q and RN respectively. In §1.1.6 the condition for disturbances to possess the property of finite speed of disturbances is presented for one dimensional case (Theorem 1.6.1). It is not difficult to prove that a similar result is valid for higher dimensional case, from which it follows that for any t > 0, il(t) is a bounded domain and r ( t ) is a bounded set. We will call T the free boundary or interface of the generalized solution u. This section is devoted to a study of properties of the free boundary. To this purpose we first point out that we can extend Proposition 1.6.1 to higher dimensional case. Proposition 1.7.1 Assume that m > 1 and u is the generalized solution of the Cauchy problem (7.1), (7.2) on Q = RN x (0, oo). Then du

>

dt dv dt ~ Av>

ku

T' (TO —

i

l)kv

'

k -• t

in the sense of distributions, where v =

- u m _ 1 , k = (TO— 1 + 2/iV) - 1 .

Below we will always assume that TO > 1 and u is the generalized solution of the Cauchy problem (7.1), (7.2) o n Q = R N x (0, oo).

98

Newtonian

Theorem 1.7.1

Filtration

Equations

For any t2 > h > 0, fi(*i) C 0(i 2 ),

and for any t > 1, mi{\x\;xGT(t)}>Ctk/N, where C > 0 is a constant and k = (m — 1 +

2/N)~1.

We will omit the proof, since the first conclusion follows from Proposition 1.7.1 immediately and the proof of the second conclusion is similar to that of Theorem 1.6.2. Theorem 1.7.2 Assume that <90 £ C 1 . If there exist positive constants CQ, S, 7 £ (0,1] such that for x € D with dist(x, dD) < 5, u0(x) > C0(dist{x, dD)V^m~l\

(7.3)

then for any t > 0, O(0) = D C fi(i). In other words, the support of u expands from the very beginning, or x G D = Q(0) implies that for any t > 0, u(x, t) > 0. Proof. Since from Theorem 1.7.1, O(0) C O(t) (t > 0), it remains to prove that dD C O(i) {t > 0). Let xo € dD. Since dD £ C 1 , D satisfies the inner ball condition, there exist p > 0 and XQ £ D such that Bp(xo) c D, BP(XQ) n <9.D = {zo}. We wish to prove that XQ € O(i) or u(xo,t) > 0 (t > 0). To this purpose, we use the Barenblatt solution of (7.1): wc
c S m ( a ; - X o , c m - 1 ( i + T))

= c(c-Ht + T))-*(i-k(m-V fclm-DV^,,.,^^ ( i + T) 2mN J f 2mN \k{m-l)

2k[m_1)/N

!£zBJ!_V /(m_1) +

" +

2k/N

_

1/( 2\ '

J

Properties of the Free Boundary:

Case

99

2/N)'1.

where k = (m - 1 + Observe that

w

Higher Dimensional

*(m-i)V /(m -y1/fa,_„

^(*>°)= , 2mN

2mN

2k{m_1)/Nif3k/N

_

_

fc(m-l)

2\

y+

Choose r > 0 large enough such that

fc(m-l)V/(m-y) ,/(„_!) < Co, 2mJV

and then choose c > 0 such that 2miV c2k(m-l)/NT2k/N fc(m - 1)

_

2_

Then ^ C r ^ . O ) < C 0 (p 2 - | x - S o | 2 ) + / ( m _ 1 ) . On the other hand, from (7.3), u0(x)

>Co(p2-|x-xo|2)X/(m_1) ^Co^-lx-xol2)1/^"1)

provided that p is small enough. Therefore wCtT(x,0)


for x G 1 ^ .

By the comparison theorem, wCtT{x,t) < u(x,t)

for (x,t) € Q.

In particular u)C]T(xo,t) < u(xo,t)

for t > 0.

Since |so - ^ o | 2 = P 2 < - ^ - c 2 f e ( — ^ / ^ ( i + r ) 2 ^ , KiTTl

from the definition of

for * > 0.

U>C,T,

L)

we have wCiT(xo,t)

> 0 and hence u(x0,t)

>0

•

100

Newtonian

Filtration

Equations

Now we are ready to discuss the Holder continuity of the free boundary. To this purpose we need some auxiliary propositions. Proposition 1.7.2

Givenrjo > 0. Assume that the constants7/, C satisfy

3am^(^)(2»C +

^)<1,

(,4,

and x0 € RN, t0 > r]0, R > 0, 0 < a < r). If v(x,to)=0,

for

(7.5)

X&BR(XQ)

and v(x,t0 +o~)dx < BR{X0)

,

(7.6)

o-

then v{x,t0 + o-)=0,

forxeBR/6(x0).

(7.7)

Here v = ^—Um~\ m

/ f(x)dx

=-!-/"

f(x)d2

The physical interpretation of the proposition is that if the gas reached the ball BR/Q(XQ) at time to + & and there was no gas in BR(XO) at time to, then a considerable amount of gas has entered the ball BR{XO) at time to +
Proof.

1) Let 1

x'=—(x-x0), ti

v{x', t') = j»v{xo

1

t' =

-(t-t0),

0~

+ Rx', t0 + at').

It is easy to check that fm-l xVC"1-!) /aNi/(m-i) U= ( V{x',t') J = (-^ J U(X0 + Rx', t0 + (ft') is a generalized solution of (7.1) on Q. Dropping the mark "'", we obtain v(x, t) — Jy2V(X0 + Rxi *0 + °"*o),

Properties of the Free Boundary: \1/(m_1)

(m-\_

u(x,t)=l

Case

101

/ (J \ l/(m-l) =

v(x,t)\

Higher Dimensional

( R2 )

u(x0 + Rx,t0 + at).

Thus after changing variables, (7.5), (7.6) turn out to be v(x, 0) = 0 /

for x € Bi = Si(0),

(7.8)

v(x,l)da;
(7.9)

and (7.7) becomes fora;GBi/6(0).

tJ(a:,l) = 0

(7.10)

2) Set ,

x l/(m-l)

/ ,

m —1„\

^(r^j

'

V

Tr,

=^

,N

,A/ t

= U

-.

+r

'

1

-3

where r = |a;|, A = l/(6mN). It is easy to verify that {/ is a generalized super-solution of (7.1), i.e U satisfies ^ > AUm dt ~

on R * x y(0,1) '

in the sense of distributions. 3) The basic idea of the following argument is to compare u with U on B\/2 x (0,1). If we can assert that u{x, t) < U{r, t)

for (x, t) G B1/2 x (0,1),

v{x,t) < V(r,t)

for (x,t) G B1/2 x (0,1),

i.e (7.11)

then, since clearly we have V{r,t)=0

for(i,t)GB1/6x(0,l),

(7.10) follows immediately. To prove (7.11), by the comparison theorem which is valid also for generalized super-solutions (sub-solutions), it suffices to verify v(x, 0) = 0 < V(r, 0) v(x, t) < V{r, t)

for x G B1/2,

for {x, t) G dB1/2

(7-12) x (0,1)

(7.13)

102

Newtonian

Filtration

Equations

4) (7.12) follows from the assumption (7.8). However the proof of (7.13) is a little tortured. By Proposition 1.7.1 and an immediate calculation, we can check dv - > -*,,

(7.14)

Av>-e,

(7.15)

7T7 nTl

where e =

. (7.15) implies that

m A(*+^M a )>0, that is, v+ T—r|a;|2 is subharmonic in x. Hence by the mean value theorem for subharmonic functions, for any x € Bi/2,

"('•D + S f H ' s / f t / i W ( K « . i ) + ^ K l " ) « <2"/ J Bi

v((,>M+W

Using (7.9), we conclude that v(x,l)<2NC+^,

ioYxeB1/2.

(7.16)

From (7.14) it follows that v(x,t)<ee^-t)v(x,l)

f o r t e (0,1).

Combining this with (7.16) and using the assumption (7.4) we obtain v{x,t) < e* (lNC+^)

<^

for x G B1/2,t

e (0,1).

Therefore (7.13) holds. The proof of our proposition is completed. Corollary 1.7.1

Under the assumptions of Proposition 1.7.2, if v(x, t0) = 0

for x £

BR(x0),

and (xo,to + a) € T, then

I

v(x,to + a)dx > BR(XO)

CB? G

•

Properties of the Free Boundary:

Higher Dimensional

Case

103

Let x0 € RN, t0 > r)0, R > 0, v > 0, 0 < a < 77. If

Proposition 1.7.3 f

f E>2\

m

f

u {xM)dx>v

•/ BR(X 0 )

m W

/(m-i)

(— ) V

i/ien i/iere exists \o = Xo(m,N,i/)

m

CT

,

(7.17)

/

> 0 such that

/ T->2\ W ( m - ! ) (z 0 ,io + A a ) > c ( — J ,

(7.18)

provided that A > Ao, rj < ——-, where c = —-—. A mk\ The proposition expresses the physical fact that if a large mass of gas was in BR(XQ) at time to, then the gas covers a neighborhood of xo at time t0 + ACT. Proof. 1) For simplicity we replace x — XQ by x and t — to by t. Let / a \1/(m—i)

_

u{x,t) = f —j J

u(Rx,at).

Then (7.17), (7.18) turn out to be / um(x,0)dx>u, J Bi

(7.19)

um(0,X)>c.

(7.20)

2) Let $ ( t ) = / um(a;,i)c/a;. is, We wish to prove / $(s)ds c .

104

Newtonian

Filtration

Equations

ri-N

_ p2-N _ ^ _ ? ( p 2 _ r 2 ) ; for AT ^ 2,0 < r < p, 2pGP(r) 1 for N = 2,0 < r < p, logr 2p2 Gp(r) = 0 for r > p. Obviously GP(p) = G'p(p) = 0,

Gp(r) > 0, for 0 < r < p

AGP = aNxsp

-

(7.22)

7NS(X),

where ajv = N(N — 2), rjv > 0 is a constant depending only on N, \A denotes the characteristic function of the set A, and S(x) denotes the Dirac measure. Note that (7.22) is valid in the sense of distributions. If 0 < t < 1 and u is smooth, then /

Gi(r)u(a;,t)da;> /

JB-L

=

J0

[ [ Jo i s i

Gi{r)dU^,S^dxds

/

Vs

JBi

G1(r)Aum(x,s)dxds

~7AT / um(Q,s)ds + aN / / Jo Jo JBi Hence for t G (0,1), =

um(x,s)dxds.

[ $(s)ds<^ [ um(0,s)ds + — [ G1{r)u(x,t)dx. (7.23) Js aN J 0 aN JBl In general case, we substitute the approximate smooth solution of u into (7.23) and then pass to the limit to conclude (7.24) for u. We will consider the following three cases separately. Case 1 N N < 2 or — —- > N-2 We have, for q

m

m-1

(7.24)

m —1 Gi(r)u(x,t)dx JBI

<

/

(7.25) G\{r)dx\

I

m

u (x,t)dx

Properties of the Free Boundary:

Higher Dimensional

Case

105

Using (7.24) it is easy to verify /

G\{r)dx < o o .

Combining this with (7.23), (7.25) yields (7.21) with C 2 = 0. Case 2 (7.24) is not satisfied, but N<4

r - ^ - > - ^ - . (7.26) v N -4 m-1 ' Without loss of generality we suppose that u is smooth as above. For x G B1/2, using (7.22) we can express u"1 by Green's function Gi/ 4 : n lNvJ {x,t)

=

0

aNAN /\

um{y,t)dy l/4(z)

G1/4(\y-x\)Aum(y,t)dy. >B1/i(x)

By Proposition 1.7.1 du _ A_m Aum = — > -eu. at Substituting this into the previous formula and using Young's inequality yield u{x,t) = {vr(x,t))1/m <

2(31Jmmt))1/m

+ 2filmeVm

<

2(31Jmm))1/m

+ C^/m+

( f G 1/4 (|l/ x\)u(y,t)dy] \JB1/t(x) J f G1/4(\y x\)u(y,t)dy, JB1/i(x)

(7.27) (ANaN I \ m where pjy = max < , — >, q = . V IN IN J m- 1 Now we calculate the second integral on the right hand side of (7.23). Using (7.27) we have /

Gi(\x\)u(x,t)dx

J Bl/2


106

Newtonian

+ [

Filtration

[

JB1/2

Equations

G1/4(\y -

xDdUxDuiy^dydx

JB1/4(\X\)


/

JB1/2

=C^{t)f'm

Gyidy

-

x\)G1(\x\)u(y,t)dydx

JB3/4(\X\)

+ CGe"'m + f

G(y)u(y, t)dy,

(7.28)

3 J B*\ 3 /IA 4

where G(y)=

f

Gf 1 / 4 (|y-ar|)Gi(|i|)dx.

JB1/2

It is easy to see that G(y) is continuous on B3/4 when N < 4, |G(y)| < (X

a log \y\ + (3 with some constants a, (3, when N = 4 and |G(?/)| <

with some constant a when N > 4. Thus under the condition (7.26), we have \G{y)\qdy

/

<

00.

J B* * 3 /14A

Similar to the proof of (7.25) we can use Holder's inequality to estimate G(y)u(y,t)dy

and then substitute into (7.28). Therefore we obtain

3/4

G{\x\)u(x, t)dx < C 7 ( $ ( i ) ) 1 / m + C6eq/m.

f

(7.29)

•> B1/2

Since Gi(r) is bounded away from r = 0, /

G!{\x\)u(x,t)dx

u(x,t)dx
< C» I

•JBl\Bi^2

>'-Bl\-Bl/2

which combining with (7.29) yields G1(\x\)u{x,t)dx

< C10($(i))1/m +

Ceeg/m.

Substituting this into (7.23) we obtain (7.21). Case 3 (7.26) is not satisfied, but AT

N<6

m

or-^->—?—. N-7 m-1

(7.30)

Properties of the Free Boundary:

Higher Dimensional

Case

107

In this case, we can proceed as above, but in evaluating the integral

/

b3 ""'3/4

:G(y)u(y, t)dy, we first express u(y, t) in terms of Green's function

Gi/8- Finally we also obtain (7.21). If (7.30) is not satisfied, then we can repeat the above procedure step by step until (7.21) is proved for given positive integer N. 3) Now we come back to the proof of (7.20). By the assumption (7.19), $ ( 0 ) > i/0 = i/|.Bi|.

Since from Proposition 1.7.1, dum -gf

> ~eum,

(7.31)

we have <J?'(i) > — £$(£). Hence $(*) > voe~eX > e'1^

for t £ (0, A).

(7.32)

Suppose that (7.20) is not satisfied, that is u m (0,A) < c . Then using (7.31) gives « m (0, t) < um{0, A)ee < ce

for t e (0, A), eA < 1.

Combining this with (7.21), (7.32) we obtain / ${s)ds Jo
+ C2es +

= dev1/™

C3mt))1/m

+ C2es + C3($(t))l/m

(7 33

'

)

„l/m

= (dev1/™ < c($(t))1/m

+ C2es) • -TJ^

• ( e - ^ 0 ) 1 / m + C3($(i))1/m

+
where p

l/m

m 1 emiM 77 — (r., i C={C +-L Cn„J\. leV ' 2e*)--T7-,

l/m>

1

is_ = C + C3. B

(7.34)

108

Newtonian

Filtration

Equations

Set ip{t) = f $(s)ds. Jo Then (7.33) can be written as ip'{t) >

Bm(${t))m.

(7.35)

A = e~1v0-

(7.36)

Notice that from (7.32), ^(t)>At,

We will compare ^f(t) with the solution of the equation x'(t)

= BmX(tr,

(7.37)

satisfying (7.38) From (7.35), (7.36) we can conclude that (*) > x(*).

^

(7.39)

t €

On the other hand, the solution x(i) satisfying the condition (7.38) is determined by (m-l)Xm-l(t)

1 C-B t' m

with C satisfying C -

1 = BmX/2

m /1\ / o \ m —1 m—1 (m-l)A - (\/2)

(7.40)

C Since x(t) —>• oo as t —>• — - , we also have, by (7.39), ip(t) —• oo, C provided A > -—.

as t

C 5™'

However this is impossible. The contradiction shows

that if A < 7 ^ - , then (7.20) must be satisfied.

Properties of the Free Boundary:

Higher Dimensional

Case

109

From (7.40), noting A = e 1UQ and (7.34) it is easy to check l/m

(2e \m—1 C Bm

A

= -2 +

(m - l j i / j 1 - ^ ™ - 1

Thus we can find a Ao = A0(ra, JV, ^) > 0, such that (7.2) and hence (7.18) hold provided A > Ao, e= Proposition 1.7.4

< -r- The proof is complete. % A Under the assumptions of Proposition 1.7.3, if

f -f-

R2

^

,

r-i ^

v{x,to)dx > vo • —

J BR{X0)

a

with vQ = z/™-1)/™, then

v(x0,t0 c^m-^'m.

with c0 = Proof.

+ ACT) > C0 • —

Since v(x,t0)dx=

/

um

-•+

BR(XO)

r n - l j

<

1

(x,t0)da

BR(X0)

um{x,t0)dx)

C* ( / BR(X0)

where C* is a constant depending only on m, N, Proposition 1.7.3 can immediately be applied to deduce our conclusion. • Denote

a(x, t) = {(x, r); 0 < r < t}, Tm(x,t)= Ts(x,t)=

{ ( i , t ) G r ; t r ( a ; , t ) n r = 0}, {{x,t)eT;a{x,t)cT}.

Clearly T m n T s = 0. Theorem 1.7.3 T = Tm UTS, that is, for any (xo,t0) e T, either (i) o-(x0,t0) C T or (ii) a(x0,t0) flT = 0. Thus if the free boundary contains a vertical segment, then the extension of this segment down to t = 0 also belongs to the free boundary.

110

Newtonian

Filtration

Equations

Combining Theorem 1.7.3 with Theorem 1.7.2 shows that under the condition (7.3), either T = T m or Ts = 0. Proof. If the assertion is not true, then there exist t\, t? with 0 < t\ < ti < to such that

(x0,h) er,(io,*i)^r. Without loss of generality we may assume that a = t2 —1\ is sufficiently small and A =

is sufficiently large, (otherwise we may replace h — *i (xo,t2) by another point at T). Theorem 1.7.1 implies that XQ £ Cl(t\).

However (xo,ti) G- T. Thus u(xo,ti) u{x,t\)

= 0,

= 0 and there exists R > 0 such that for x €

BR{XQ).

Applying Corollary 1.7.1 with a = ti — t\ we obtain cR2 v{x,t2)dx > / B (X ) h - h BR(xn0)

Next, we apply Proposition 1.7.4 with VQ = c, a = t^ — ti and a = to—t2, and then obtain v(x0, t0) = v(x0, t2 + ACT) > 0, which contradicts the fact (xo,to) 6 F .

•

T h e o r e m 1.7.4 Given any r/o > 0, there exist positive constants C, 7, h, such that for any {xo,to) € Tm, to > 770, if\x-xo\
(7.41) to — h < t < to, ij\x-x0\


u{x,t)>0,

(7.42) t0
+ h,

Proof. First we prove (7.41). Fix r G [0,to)- Let h = t0 — r. Since (xo, to) € r m , there must be an R > 0 such that u(x, T) — 0 for x G BR(XO)Let tx = r + (1 - X)h = t0 - \h,

Properties of the Free Boundary: Higher Dimensional

Case

111

where A S (0,1) is to be determined later. Suppose dist(x 0 i r(fi))
(7.43)

where a e (0,1) is also to be determined later. Applying Corollary 1.7.1, we deduce a)2R2

^ e(l / , '(I-JHC*!)

(1-A)/I

where c > 0, x\ € T(ii) with |xi — zo| < ai?. Hence

I

c{l-a)2R2, v(z, ti)da; > - ^

xAr

TTT—(1 - a) •

Given any large C > 0 we can choose a, A > 0 so that

But if C is sufficiently large (depending on m, JV, 770), then Proposition 1.7.4 implies u(xo,to) > 0, which is of course impossible. Thus, if (7.44) holds, then (7.43) cannot be valid, that is, d i s t ( z 0 ) r ( t i ) ) >aR.

(7.45)

Notice that (7.44) is implied by (1 - a)N+2

> C(l - A)

with another large constant C, provided A > - . Taking a = A7, we reduce (1 - A 7 ) w + 2 > C(l - A). This inequality is valid for some A near 1 and 7 sufficiently large (for instance, A = l— - , 7 = fc, k sufficiently large). k From (7.45) it follows that u(x,t\)

= 0,

for x e Ban(xo),

a = A7.

We can now repeat the previous argument with R replaced by aR, r replaced by t\ and h replaced by A/i. Thus we deduce that dist (x0, T(i 2 )) > a2R,

t2 = t0-

\2h.

Newtonian

112

Filtration

Equations

Proceeding step by step, we may obtain dist {XQ, T(t)) > X^nR,

t = t 0 - Xnh.

Now we take h = —— and let h vary on .

W .

h

h

h

Then to — h varies

'x

Xrj

0 on *o-y.*o--2- . It is easily seen that the corresponding values of R are bounded from below by a positive number RQ > 0. Thus for any

h€

and n = 1, 2, • • •, we have dist (x0, T(i)) > \~tnRo, t = t0-

Xnh.

(7.46)

Taking C such that c(j)


(7.47)

we must have u(x,t) = 0,

if | x - a ; o | < C(t0 -t)1,t0

-h
In fact, for any t £ (to — h,to), we may choose a positive integer n(t) such that

to-te Set h(t) = X-nW(t0

(xn^h,Xn^-^\

- t), that is, t0-t

=

Xn^h(t).

Then h(t) G (~h, - ). Therefore from (7.46), (7.47) we conclude that if \x — XQ\ < C(t0 — i ) 7 , then \x - xo\

<

<

C(Xn^h(t))i

tf0A7n(t)


Properties of the Free Boundary:

Higher Dimensional

Case

113

Next we prove (7.42). Consider the set - x0\ < C ( t - t 0 ) 7 , t o
E = {(x,t);\x

+ h}.

For any (x,i) G E, if (x,t) € T, then the segment a(x,t) interior points of the set defined by \x-x0\<

C(t - t0)'y,t0

must contain

-h
that is, 0, which contradicts (xo,to) G T. Therefore (x, t) $: T,that is, E does not contain any point of I\ This shows that on E either u = 0 or u > 0. Since (xo, to) € V, we must have, for any (x, t) G E, u>0. Thus (7.42) is proved. D As a corollary of Theorem 1.7.3, we have Theorem 1.7.5 For any (xo,to) G Tm, there exists a neighborhood in which r can be expressed by a Holder continuous function t = S(x)

for x G Ux,

where Ux is the projection on t — 0 ofT in the neighborhood. Proof. Applying Theorem 1.7.3, it is easy to see that there exists a neighborhood, such that any point of T in this neighborhood must be a point of T m . Hence the portion U{XQ, to) of T in this neighborhood can be expressed by a function t = S(x). For any x G Ux, since (x,S(x)) G V, we see from Theorem 1.7.4 that (C\S(x)-S(x0)\'y or \S(x)-S(x0)\<(^j

''ix-xo^.

Theorem 1.7.6 Under the assumptions of Theorem 1.7.2, T can be expressed by a function t = S(x)

for x G

RN\D.

114

Newtonian

Filtration

Equations

Moreover, S(x) is uniformly continuous on any compact subset K which does not intersect with D. Proof. By Theorem 1.7.2 and Theorem 1.7.3, V does not contain any vertical segment. Hence the whole T can be expressed by a function t = S(x), which is defined on RN\D in virtue of Theorem 1.7.1. From the continuity of u it is easy to check that S(x) is uniformly continuous on K, that is, for any h > 0, there exists r)(h) > 0 such that \S(x) - S(x0)\ < h,

for x,x0 e K,\x-x0\


(7.48)

Since K n D = 0, there exists T]Q > 0 such that S(x) > 770 for x e K. From the proof of Theorem 1.7.4 we see that C, h, 7 appeared there can be chosen to be independent of XQ E K,to = S(XQ) (> 770)- Applying Theorem 1.7.4, we conclude that for x,xo G K, \x — XQ\ < T](h), \x-x0\>C{t-tQ)'<, or

\S(x) - S(x0)\ < f i )

1.7.2

V-zo|1/7-

Lipschitz continuity of the free boundary

Without any additional condition besides those stated at the beginning of this section, one can prove the Lipschitz continuity of the free boundary for large time. Precisely, one can prove that the free boundary T is Lipschitz continuous for t > To where To > 0 is such a time that for t > To, BRO(0) C Cl(t) where -Bfio(0) is the smallest ball containing D = ft(0). For the proof of this result, see [CVW], in which the authors also proved that under certain conditions on the initial value uo(x), the free boundary T is globally Lipschitz continuous. Theorem 1.7.7 Assume that uo £ C. Then the free boundary T can be expressed by a function t = S{x)

forxeRN\D

and S(x) is locally Lipschitz continuous on RN\D. We say that UQ € C, if D = {x; UQ(X) > 0} is a bounded C1 domain and

Properties of the Free Boundary:

(iJvo^^-u^eC'iD); m —1 (ii) v0 + \VvQ\ > Kx > 0,

Higher Dimensional

Case

115

for x G -D;

(wj in £/ie sense of distributions, Av0 >

-K2,

where K\, K% are positive constants. It is easy to check that the condition (i), (ii) imply that VQ satisfies the Lipschitz condition near dD and v0(x) > C*dist (a;, dD)

for x G D

with a constant C* > 0. This shows that functions in C satisfy the condition (7.3). To prove Theorem 1.7.7, we first prove some auxiliary propositions. Proposition 1.7.5 Assume that UQ G C. Then there exist constants A, B depending only on K\, K2, R, such that {A -2)v(x,t)

+ x- Vv(x, t) + (At + B )

d v

^ >

0

(7.49)

in the sense of distributions. Here R> 0 is a constant such that D c S/j(0) j

m

m-l

andv =

u

.

771—1

Proof. Let us first sketch the basic idea of the proof. Suppose that u is smooth and u > 0. Then — = (m-l)vAv

+ \Vv\2.

(7.50)

For £ > 0, (7.50) admits a family of the following solutions vs{x, t) =

1 + As

t;((l + e)x, (1 + Ae)t + Be),

where A > 0, B > 0 are constants. In fact one can easily check that for any constants a^=0,b, a, j3, —v(ax + b, at + (5) is a solution of (7.50). If we can prove that ve(x,t) > v(x,t), then

±v£(x,t)

116

Newtonian

Filtration

Equations

However dv(x,t) = (A - 2)v{x, t) + x0Vv(x, t) + (At + B)-

±v.(x,t)

dt

e=0

Therefore (7.49) holds (in classical sense). Unfortunately, u is not smooth and positive everywhere in general. It is natural to consider its approximate smooth and positive solution and then pass to the limit. For any 5 > 0, let vos(x) = (v0 * ps)(x) + 5a, where a € (0,1), ps is the kernel of a mollifier with compact support contained in -8,5(0). Since vos G C°°(RN) and vos(x) > 6a, there must be a smooth solution vs(x,t) with initial data vog(x), which satisfies vg > Sa, and vs —^ v as S —> 0. To prove that vg satisfies (7.49), as stated above, it suffices to prove veg(x,t) > vg(x,t), where ve8(x,t)

1 + Ae

= ^

|

s2vs((l

+ s)x, (1 + Ae)t + Be).

By the comparison theorem, to prove vsg > vg(x,t), it suffices to check veg(x,0) > vg(x,Q). To do this, we need a careful discussion. For simplicity below we will drop the mark "6". The readers need to keep in mind that v, VQ and v€ are just vg, VQS and veg respectively. Denote le =

^{ve{x,0)-v(x,0)).

Then we have J

e

1 f 1 + Ae

= - ( „ , ,2v((l

-l((1

+

l

e)x,Be)-v(x,0)

¥A~ 2)£ ) v^1 + E)x'Be) ~ ^Z'0))

-{A-2)v({l

+Bjtv{{\

+

+ e)x,Be) + e)x, 6e) + x- Vv{£, 0)

Properties of the Free Boundary:

Higher Dimensional

Case

117

with some 9 G (0, B) and some £ on the segment joining the points x and (1 + e)x, provided that A > 2 and e > 0 is small enough. Since e is sufficiently small, there exists a constant C depending only on m, N and vos such that h

> ^ ( 4 - 2 M i 1 0 ) + B^t;(i,0) +x- Vv(x,0)

-Ce.

Notice that here we have used the fact that for given 5 > 0, the derivatives of v up to second order are bounded. Since v satisfies (7.50), we have h

> Q ( A - 2) + B(m - l)A V (z, 0)) v(x, 0) +B|Vu(a;, 0)| 2 + x • Vv{x, 0) - Ce,

provided that A > 2 and e > 0 is small enough. If |x| < i? + <5, then Ie>

(±(A-2)-B(m-l)K0S\v0(x)

(7.52)

+|V«d(a;)|(B|Vwo(a;)| - R - S) - Cs, where we have used the fact Ai>o > —K2, which follows from the condition (hi). Denote S = {x G D; dist (a;, dD) < 5}. The condition (ii) implies that for small 5, we have |Vi*| >

^

on S. In fact, for small S, this inequality holds also on Uc>s = {x€RN;

dist (x, 3D) < c8},

where c G I 0, — 1 is independent of S. Now we are ready to prove Ie > 0 on RN. We will divide HN into several parts and prove Ie > 0 on each part of RN. Denote £>i = {x G RN; dist (1, U) > 6}, £>3 = D\S,

D2 = Is G K^; | Vv 0 | > —

D4 = {x(= RN; c6 < dist (a;, D) < 6}.

118

Newtonian

Filtration

Equations

Since S U UCts C D2 (but D2 may contain points outside S U Ucj), we have MJV = fl1ui>2UD3U D 4 . Case lOnD1 = {xG RN; dist (a;, D) > 5}. Since v(x, t) > 5a, we have

Then noting that on D\, v(x, 0) = Sa, we have vE(x, 0) > v(x, 0) or Ie > 0 on Z?i. Case 2 On D 2 = j z e R^; |V«o| > ^

j.

On D22 wwe must have |a;| < R + 8. Hence, since on D2,

\WVQ(X)\

< —,

from (7.52),

h

>\Vv0{x)\(B\Vv0(x)\-R-6)-Ce

>*(*£- fi -,)- C .>0. 4.R provided that B > — , .4 > 2 + 2B(m Case 3 On £>3 =

1)K2.

D\S.

It suffices to consider those points in D3 such that |Vvo(^)| < — • Since vo{x) is bounded below by a constant a > 0, from (7.52) we have

h

>(^(A-2)-B(m-l)K2\a -^-{R

+

provided that A > 2 + 2B(m — l)K2 H

S)-Ce>0, 2RK

and e, S are small enough.

w

Case 4 On D4 = {x G R ; c<5 < dist (a;, D) < 5}. This case is a little difficult to treat. In order to prove Ie > 0 on D4, we need to choose the kernel p$ of the mollifier such that 0 < ps(x) < Ps(0),ps(x) ps(x) = 0,

for \x\ > d,

= ps{0),

for \x\<8-


Properties of the Free Boundary:

Higher Dimensional

Case

119

where 0 < 7 < 1 is to be determined. Of course, we also assume that /

ps{x)dx = 1. We will further divide D4 into two parts: D^

= {XGRN;S-

JT+1

D^

= {xe RN; cS < dist (x, D) < 6 - <57+1}.

< dist (x,

D)<6},

Since dD e C 1 , from the condition (i) we can verify (but not clearly) that for S small enough,
IVVOSW

where K3 is the upper bound of |Vi>o| near dD and C > 0 is a constant depending only on N. Using this we obtain from (7.52) that {^{A-2)-B{m-l)K^j5a

h>

-(R + 5)CKMl+{N-1)/2

- Ce.

Hence, if we choose

-— r-r- < 7 < 1, A > 2 + 2B(m — l)Ko and S v l + (N-l)/2 ' small enough, then for small e we have I£ > 0 on D±'. (2) Finally, we prove that Ie > 0 on D\'. Since we have proved the conclusion on D2, it remains to consider those points in D\ such that |Vi>o(z:)| < —r- For such points, from (7.51) we have IE > B(m - l)v(x, 0)Au(a:, 0) (7.53) -(R +

S)\Vv(x,0)\-Ce

provided A > 2.

2 It can be proved (but not easily) that for 0 < 7 < —

and small 6,

Av(x,0)>CK1S-^1^N-1^2\ where C > 0 is a certain constant. Using this in (7.53) and noting that v(x, 0) > 6a, we derive Ie>

B(m-

VCW-V-W-ViM

-{R + 6)^-

Ce,

120

Newtonian

Filtration

Equations

2 (N - 1)7 provided 7 < — — - , 0 < a < 1 and we first choose 6 and then choose £ small enough, then Ie > 0 follows. The proof of our proposition is thus completed. • Proposition 1.7.6 denote x(s) = x0es,

Under the assumptions of Proposition 1.7.5, if we t{s) = j ({At0 + B)sAs - B) ,

then the function v(x(s),t(s))e^A~2^s Proof. have

s> 0

is decreasing.

Since x'(s) = x(s), t'(s) = At(s) + B, from Proposition 1.7.5 we

=

e(A-21s((A-2)v

>

0

+ x-Vv + (At + B)vt)

in the sense of distributions.

•

Theorem 1.7.8 Assume thatvo £ C. Then for any (x,t) £ Q, there exist positive constants A, B, C depending only on VQ, t, R\ = sup{dist(x,y); y G D} such that v{x,t) > vfaty-W-1),

(7.54)

where (x,t) £ Q, t
2

t-t

;

^ < ^ -

~ At + B

(7-55)

Proof. Let A, B be the constants in Proposition 1.7.5 corresponding to R = 3.Ri and C = A — 2. To verify (7.54), it suffices to prove that any point (£, T) in the cone (7.55) can be joined with (x, t) by a curve in Proposition 1.7.6. Since Proposition 1.7.6 is based on Proposition 1.7.5, we have to check D C BR(0). Suppose (£, r ) is a point in the cone (7.55), t < r < t + e. Choose XQ such that

(Ar + BV'A

Properties of the Free Boundary:

Higher Dimensional

Case

121

For simplicity, we change coordinates so that XQ becomes the origin but the notations (x, t) and (£, r ) do not change. Then the above formula turns out to be e

_(AT

+ B \

1 / A

i = X

\-AUB)

•

Consider the curve At + B\1/A ,At + B/ which can be expressed in the form of parameter x{s) = xes,

t(s) = \{{At + B)eAs - B)

{s > 0).

This curve joins (x,t) and ( £ , T ) . By Proposition 1.7.6, if we can verify that D C -Bfi(O), then we obtain the conclusion of our theorem. To prove D C BR(0), notice that B\1/A

Ar +

= *

At + B \£,\<\t,-x\ + \x\ R

^ATVB

+

^

that is,

A B ,A

^ ^

At + B J

i IW 'At- +*B ~

Clearly, there exists e = e(A, B) > 0, such that for t < r < t + e, AT + B\1/A

At + B J

1 T-t ~ 2 At + B

Substituting this into the previous formula yields 1, . r — t „ r —t - | x | - = — - < Ri-r=T 'At + B ~ 'At + B' that is, \x\ < 2R\. Thus for y S D, we have \y\<\y-x\

+ \x\<3R1

= R.

122

Newtonian

Filtration

Equations

Hence D C .Bij(O). The proof is complete. Proof of Theorem

•

1.7.7. By Theorem 1.7.8, if (x,t) G I \ then

u(x,t) = 0,

when \x — x\ < —=——(i — t),t — e
u(x,t) > 0,

when \x-x\< ,_ l (t-t),t< i ' ~ At + B


t
provided that e = e(A, B) > 0 is small enough. From this, it follows that r can be expressed by a function t = S(x). Theorem 1.7.1 implies that Furthermore, since for (x,t),(x,t) € T with S(x) is defined on RN\D. t - e
\x-x\> 1 f

Rl

~At

A

Jt-t\,

+ Bl

"

we have \S{x)-S(x)\


for some constant C > 0. Thus we obtain the conclusion of Theorem 1.7.13 Remark 1.7.1 Under certain conditions, Caffarelli and Wolanski [CW] further proved that the free boundary is a Cl,a surface. C°°-regularity of the free boundary was proved for two dimensional case by P. Daskalopoulos and R. Hamilton [DH1]. See also [DH2]. 1.7.3

Differential

equation

on the free

boundary

We have proved that on the free boundary x = Q(t) there holds CKt + 0) =

-^MM,

(7.56)

u m _ 1 and u is a generalized solution of the equation (6.3) —1 with compact support for any t > 0. This result can be extended to the higher dimensional case. Let r > 0, y & fl(r) be fixed. Denote

where v =

TO

h(t) = max{i?; v(x, t) = 0, x £

Bn(y)}.

Initial Trace of Solutions

123

From the monotonicity and continuity of the free boundary, it follows that h(t) is decreasing and there exists So > 0 such that h(t)>0,

for 0 < i < r +
The following theorem is an extension of the formula (7.56) to the higher dimensional case. Theorem 1.7.9

For 0 < t < r + 6Q, limits

7t(0 + 0 ) = lim 7t(e) = lim

max

v(x

ti\

-—

and ,,, ^ h'it + 0) =

,lim

h(t v +

/

At)-h(t) —

exist and /»'(* + 0 ) = - 7 t ( 0 + 0).

(7.57)

For the proof, see [FR2]. If N = 1, then h(t) = y — ((t), where x = ((t) is the free boundary, for example, the right free boundary, then

h'(t + 0) = C'(0 + 0) and (7.57) is just (7.56). Remark 1.7.2 Topics related to propagation of disturbances and the free boundary for (7.1) and equations with convection term and absorption term are studied by many authors. For example, the readers may refer to [AB2], [AB3] for the property of instantaneous shrinking and [AA1], [AA2], [BBA] for the focusing problem. 1.8

Initial Trace of Solutions

The Widder-type theorem shows that , if u(x, t) is a nonnegative solution of the heat equation

£ = *.

(8..,

124

Newtonian

Filtration

Equations

on QT = RN x (0, T] (T > 0), then there exists a unique nonnegative Radon measure //, such that lim /

u(x,t)ip(x)dx

= /

(p(x)d(j,

(8.2)

for any