An elliptic-parabolic free boundary problem

AN ELLIPTIC-PARABOLIC

FREE BOUNDARY

PROBLEM

J. HULSWX and L. A. PELETER Mathematical

Institute, University of Leiden, Leiden. The Netherlands

(Received 27 IWay 1985: received for ~u~f~cur~on21 October 1985) Key words nncf phrases: Elliptic-parabolic of weak solutions, asymptotic behaviour.

free boundary problem. existence, uniqueness and regularity

1. INTRODUCTION IN THIS

paper we study the problem ~~~~~~~ = @xx

Cl>

xE(O,1).

ap(i, tf -I- f-l)“*‘b,u,(i,

8) =fJr)

i C(~fX, 0)) = 4.4

tEt(O,T]

Cl. 11

f E (% q

WI

x E fO*If)

(W

in which c is a continuous nondecreas~Rg function defined on R such that C(U) is strictly increasing when u < 0 and constant when M5 0 (fig. 1). The ~oef~~~~~ts a, and bj are nonnegative constants with the property ai -i- bi > 0, Ui= 1 if bi = 0 and bi = 1 if ai = 0 for i = 0 and i = 1 and fO, fi and t’. are prescribed functions. Problem I arises in the theory of fluid Aow through parGaIly saturated media. Then t6denotes the potential due to capiitary suction and c the moisture content. The dependence of c on IE is found empirically to be as in Fig. I, c being bounded above by the saturation value c, = 1. Thus, in regions where the medium is saturated, the flow is of potential type, and described by an ehiptic equation, and in regions where the medium is unsaturated, the flow is of diffusive type and described by a parabolic equation. At the boundary between these regions-the interface-one expects U’= 0

and u, continuous.

Fip. I, The function c. 1327

1328

J.HULSHOF and L. A. PELETIER

More detailed accounts of the derivation of problem I in the context of filtration problems can be found in [3, 7, 161 and the literature cited there. Problem I was first studied by Van Duyn and Peletier [7, 61, but only for constant Dirichlet boundary conditions, i.e. for ai = 1, bi = 0 andf, E (- 1)“’ (i = 0, 1). A class of weak solutions was introduced and global existence, uniqueness and regularity of such solutions was established, a comparison principle for the concentration was shown to hold, the interface was studied and shown to be continuous, and for large times the concentration profile c(t) was shown to converge to a unique equilibrium profile e exponentially fast. At about the same time, a local existence result was obtained using classical methods by Fasano and Primicerio [8] for the case a0 = 0, bi = 0 and f. > 0, fi = 1. The existence, uniqueness and regularity of solutions of equation (1.1) and generalizations thereof, in bounded cylindrical domains S2 x (0, T], where R C RN and N 2 1, was established by Hornung [lo] and Alt and Luckhaus [l]. They imposed the boundary conditions

u =fD

on

rD

x (0, T]

and

au = f,\. au

-

on

rN x (0, Tl

where Po iS a nOnempty subset Of dR and P,v = an - I& and fD and fNare independent of t in [lo] and fn = 0 in [l]. Subsequently the study of problem I was taken up by An [2], Xiao, Huang and Zhou [19], Su and Li [15,18], Kriiner [12] and Bertsch and Peletier [4]. They proved existence, uniqueness and regularity for solutions of equation (1.1) as well as of

in which K is a nondecreasing function, under a variety of boundary conditions. In particular, An studied problem I when a0 = a, = 0, i.e. with Neumann boundary conditions, establishing the existence of a solution u(t) for t E [0, T], and giving estimates for the final time T. That problem I may not have a solution u(t) for all time if a0 = a, = 0 can be seen if we integrate equation (1.1) formally over (0, 1) X (0, t). This yields

{fo(s)+f1(s)ld = I1<44x70) - uo(x>)~ Iof 0

< =

l{l- u&)}dx.

(1.4)

i0

Thus, if we define T* =sup ( t>O :I’(fo+fl)
(1.5)

0

(assuming u. s 1 ($ l)), we may only expect problem I to have a solution for t E [0, T], where TS T*. In the hydrological context, the term on the left-hand side of (1.4) gives the amount of fluid having entered the medium up till time t. The term on the right-hand side expresses the amount of fluid the medium can take up at t = 0. Clearly, no more fluid can enter the medium than can be taken up. Finally, in a recent study, Kroner and Rodrigues 1131, studied the large time behaviour of the solutions discussed in [l].

An elliptic-parabolic

free boundary problem

1329

In this paper we prove under mild assumptions on the functions c, fi and u. that problem I has a unique solution u(r) for 0 ~5t 5 T, where -if 00 + al > 0, T may be any positive number, and -if a0 = a,=O, TE (0, T*), where T’ is defined by (1.5). Thus, here we prove existence for the optimal length of time. Following the methods of [7] we establish the uniqueness and regularity of solutions, and prove a comparison principle for the concentration. But in addition we prove here the continuous dependence of solutions on the boundary valuesfi and the initial concentration uo. All this is done in Section 2. In Section 3 we briefly discuss the structure of the solution, the unsaturated and the saturated region and the interface between these regions. In Section 4, we turn to the large time behaviour of the concentration profile. Assuming that the solution exists for all time, and that the boundary values stabilize, i.e. fi(t)*fi

(i = 0, 1).

ast+m

we prove that the concentration profile stabilizes too, provided either a0 + al > 0 or, if a0 = ai = 0, the fluxes at the lateral boundaries converge to the same value: c(u(t)) --, c(c) as t -+ =

in C(G),

where 52 = (0,l). Here 6(x) is a linear function which is uniquely determined by Ui, bi and j, and also by u. if a0 = al = 0. If a0 + ai > 0, the convergence is proved, as in [7], by multiplying equation (1.1) by a suitable weight function and thus establishing the convergence in a weighted L’-norm. Together with a uniform bound on U, this implies convergence in C(G). This method also yields a rate of convergence. If u. = a, = 0, the solution can only stabilize if -jJ =!I(=

ru).

The method we use here to prove stabilization is based on a Lyapunov function argument, the Lyapunov function being V(z) =

I0

* {Q+(x))

+ =+)I

h,

where Q(y) = /“”

SC’(S)d.r

0

in which c(c-l(y))

= y if y < 1 and c-‘(l)

= 0. This method yields no rate of convergence.

2. BASIC

RESULTS

In this section we shall prove the existence and uniqueness of solutions of problem I, establish some results about regularity and continuous dependence on data, and prove a comparison principle. About the functions c, fo, fl and V. we shall make the following assumptions. (Al) c : 53+ R is a nondecreasing Lipschitz continuous function with Lipschitz constant K,

1330

J. HULSHOF and L. A. PELETEER

such that c(s) = 1 for s Z 0 and ii? inf t[c(s + h) - E(S)] is bounded away from zero on every bounded interva1 [-A, -61. where A > S > 0. (A2) For every T > 0, f; : [0, IT]- 12 is Lipschitz continuous. (A3) 3~~1: [0, l]W, which is Lipschitz continuous, such that c(uO) = v. on n. (A4) If bi = 0, then (with ai = l), uO(i) =fi(O). (A5) If a0 = a, = 0, then (with b, = b, = l), (i) lim c(s) = -x; 2-+--I (ii) no, fo,fI and T satisfy the compatibl’iity condition 1

I0 ~~~~~~~~~.

uo(x)

d.x

+

( tfo(f) i

-t-f*@))

df

<

1

vt

E

[O,

T].

0

In the future we shall usually write !J = (0, 1) and Qr = Q x (0, T].

Definirion. A solution u of problem properties: 0) c(u) E C(&%

I on [O, T] is a function u E L’(O, T; H!(Q))

with the

fora11@E C'(8 T) which vanish at x = i if bj = 0 and at I = T. Here .tii = l/k+ if b; > 0 and pi = 0 if bj = 0 (i = 0, I). TO prove the existence of a solution we proceed as in f7], cons~~cting a solution as the limit of a sequence of functions u,, which are smooth solutions of approximate uniformly parabolic problems fG&>),

= @XX

(Iti) aju(i, t) i- f-l)“‘b,u,(t, i n(x, 0) = %I (x)

f) =fin(t)

in QT

(2.2)

O
(2.3)

x E 5%

(2,4)

Here the functions c,, fin and uon are chosen so that: (i) c,, fitf and uDnare defined and in~niteiy differentiable on, respectiveiy, R, 10, Tf and 6; (ii) l/n S c;(s) I if for all n 1 1 and s E R; (iii) CA is bounded away from zero in every compact subset of (--x1 0). uniformly with respect to n 2 1; (iv) c,- c as n- = uniformly on bounded subsets of R, and c,(O) = 1 for all n 2 1; fv) fin-fi and ~a~-, ug as n - 3~uniformly on, respectively, [O, T] and a; (vi} there exist positive constants tt and L2 such that for all n Z 1, ]f>Jr)i 5 Z.,t and j&(x)/ S Lz for, respectively t G [O, T} and x E a;

An elliptic-parabolic

free boundary

problem

1331

(vii) the following conditions are satisfied at the points (x, r) = (i, 0). i = 0, 1: =fin(0)

i=O,l,

u&(i) = cA(uOn(i))f:n(0) if bi = 0

i = 0,l.

uruon(i) + (-l)‘+‘b,u&(i)

The construction of these functions proceeds essentially along the lines given in detail in [7], and we shall not reproduce it here. In view of the properties of c,, fm and uOnrwe may conclude that problem I, possesses a unique solution u,, E C’.l(Qr) n C”(Qr) [9. 141. be satisfied. Then there exist positive numbers M and L. which only depend on the data such that

LEMMA 1. Let (Al)-(A5)

Iu,(x, t)l I M

and

)u~~(x, f)I 5 L

for all n Z 1 and all (x, t) E (2 r. Proof. We shall treat the cases a0 + a, > 0 and a0 + a, = 0 separately. (i) The case a0 + a, > 0. Consider the function w(x) = Ax + B. It is a solution of equation (2.2) for every A, B E R. In view of the boundedness of uO,,and fm, tt . is p ossi bl e t o c h oose A and B so that w becomes a supersolution and --w a subsolution of problem I for all n 2 1, whence for that choice of A and B, ~u,(x,t)~SA+B

for

(x,t)E&

for all n 2 1. To obtain a bound for u,, we differentiate equation (2.2) with respect to X. This yields, using the equation again to eliminate u,, and writing U, = p: c”(U)

c’(rl)P,= Pxx - -c’(u) ppx. Thus, by the maximum principle. IpI assumes its maximum value on the parabolic boundary I-r of QT. By property (vi) of the functions uOnrIp/ s L2 at t = 0. At x = 0, we can estimate IpI by means of the boundary condition and the bound for K, if b. # 0. If b. = 0 (and hence a0 = 1) we use a barrier function argument, the barrier function being w(x, t) = fo,,(f) + (KL, + L&c - $KL,x’. At x = 1, we can estimate IpI in a similar manner. Thus jpl is bounded on I-r, and hence in er. (ii) The case a0 = a, = 0. Be-cause now we have b. = b, = 1 we obtain, exactly as in part (i), a uniform bound on / u, / in Q r. To obtain an upper bound on cl,, we proceed as follows. We integrate (2.2) over Q,, where 0 < t 5 T. This yields I

1 c,(%l(x,

t>) dx =

I0

I Cn(~on(X>) dx +

I0

{fon (s) + fh 6) 1 dJ.

(2.5)

1332

J.HULSHOF and L. A. PELETIER

In view of the compatibility condition (A5) (ii) on uo, fi and T, and the uniform convergence of the sequences {c,}, {no,,}and {fi,} we may conclude that for n large enough 1

I0

c,(u,(x,

t)) dx C 1

for all

r E [0, T].

Since c,(s) > 1 for s > 0, this implies that for n sufficiently large, there exists for each t E [0, T] a point &(rj E R such that &(E”(99 t> < 0

t E [O, T].

Because 1u, 1S L in Qr for all n B 1, we may conclude that for n large enough for

u,(x, t) 5 L

(x, t) E Qr.

Finally, to obtain a lower bound on u,, we use the property have

I

(A5) (i) of c,. For by (2.5) we

1

c,(u,(x,f))dXzl-M

t E [O, 7’1

(2.6)

0

for all n Z 1 and some constant A4 > 0. By (A.5) (i) there exists a number s,,, < 0 such that c,(s)Cl-M

forall

s
Thus, by (2.6), there exists for each t E [0, T] and each n 2 1 a point rln(l) E n, such that 4(77&),

t) >sM

t E [O, 7’1,

whence, by the bound on 1uru 1, u,,(x, r) Z sM - L for (x, r) E Q T for all n B 1. This completes the proof. As in [7] we may prove now that the sequence {c,(u,)} is uniformly bounded in C’+l(Qr) and find a function u E L*(O, T; H’(Q)) and a subsequence {u,} such that as ,uL-, r up + u cr(uP)-,

c(u)

in L*(O, T; H’(Q))

weakly

in Co+@@ r)

strongly

with /3 E (0,l). In fact we have u E L”(0, T, W’~s(Q)). It is now easily verified that u is a solution of problem I. Uniqueness follows also as in [7], by considering the test function

f’{u(x, s) -

6(x, s)} ds

xEfi,rE[O,r,) x E S=i, rE

[rl, T]

where rl E (0, T) and u and fi are solutions of problem I. Theorem 1 summarizes our results so far. 1. Let (Al)-(A5) any admissible T > 0.

THEOREM

be satisfied. Then problem I has a unique solution u on [0, T] for

An elliptic-parabolic

free boundary problem

Remark 1. Because the solution u of problem

smooth solutions {u,} of problem I,, converges, u, + u c,(u,)--,

c(u)

1333

I is found to be unique, the entire sequence of i.e.

in L’(O, T; H’(R))

weakly

in C”+B(Q r)

strongly.

Remark 2. Let a0 = al = 0 and b. = bl = 1. Then by taking as a test function ~(x. t) = t - t, with 0 c t < T, we deduce, by differentiating with respect to t, the conservation law for weak

solutions of the Neumann problem:

Remark 3. If a0 + al > 0, any T > 0 is admissible,

for which the compatibility

but if a0 = a, = 0, only those values of T

condition (AS) (ii)

1

I0

uo(x)

b

+

+fl(s)Ib

Vt E

< 1

[0, T]

holds. THEOREM

2. Let u be a solution of problem I on [0, T], in which the data satisfy (Al)-(A5).

Then u E L”(0, T; W1*2(!i2)) II L’(0,

T; H*(R)).

Proof. In view of lemma 1, we need only prove that u E L’(0, T; HZ(n)). This will be so if we can find a uniform bound for the sequence {u,} in Lz(QT). For E E (0,4) define Q$ = (E, 1 - E) X (E, T]. Then u, E Cx(e ;) and hence, if we multiply (2.2) by u,~ and integrate over Q$ we obtain T

II

l--E

l--E c;(u,)&

dx

dt =

E E

? unrun$-’ I E

IE --

l--E

1 c

ux(x, E) dr

I

&(x,

T) dr.

If we now let E* 0 we obtain, since u, E C2~1(~T), c; (u,)u$

dx dt 5

By (A3), and the uniform convergence uniformly bounded. As to the first one we have at x = 0: (i) when b,, = 0 (and hence a0 = 1):

I0

uIu~]b

T ~,(O,0~,t(O,

(2.7)

of {uon} to uo, the second integral on the right is

T

I0

1 l L&(x, 0) dx. dr + 2 I0

4 dt=

u,(O, I

0

t)f b,(t) dt;

J. HULSHOF and L. A. PELETIER

1334

(ii) when bO> 0: T

i0

T

~~(0, t)unr(O, t) dr = ;

{W,(O~ r) - fOn(f)~4l~(O,r) dr

0 I0

= $ 0

T 1 ui(O,r) o -&fo.(f)uo~(O.

4 IT 0

+ k ~oT/h.(r)~n(O~ r)dr.

Clearly in both cases the right-hand side is bounded in view of lemma 1 and the assumptions on fo. At x = 1 the situation is similar, so we may conclude that the first integral on the right of (2.7) is uniformly bounded with respect to n E N. However, because 0 < CA~5 K in W for all n E 1,

Hence the sequence {u,} is uniformly bounded in L2(0, T; H2(Q)). Thus, there exists a subsequence {u,} which converges weakly to an element U* E L’(O, T, H2(Q)). It is clear that u* = u. In the next theorem boundary data fi.

we show how the solution depends on the initial value u. and the

THEOREM3. Let u and ri be solutions of problem I on [O, T] with data uo, fi and co, i (i =O, 1) respectively, which satisfy (Al)-(A5). Then: (i) there exists a constant %, which only depends on the data such that

ii

(u - ti)[c(u) - c(C)] dx dr 5 %T [~‘l~o-~ol~+~~Tlf~-~jl~~}~ 0

QT

(ii) (comparison

principle) let u. 5 O. and fi Si

(2.8)

0

(i = 0, l), then c(u) % c(a) in or.

Proof. It will be enough to prove (2.8) for the smooth solutions u, and Li, of problem approximating u and li. It will cause no confusion if we omit the subscript n. Consider the function n(x, r)=

T {u(x,s)- ti(x,.s)}ds. II

Then 1 - q,c(u)} d_~dr = {??XUX

I0 +

17(x.O)uo(x) h

,io

(-l)icl

ITq(i,r)u,(i,r)dr. 0

I,,

An elliptic-parabolic

Subtracting

the corresponding rlI(x,

1335

equation for ri we obtain

1 ’ 20I

free boundary problem

0) dJc+

II

- c(C)] dr dt

(u - a)[c(u)

QT

=

I rl(~,W~O(~) I0

+ LiO (-I)'+'

JT

- boll

(2.9)

q(i,t){u,(i, t) - &(i,t)}

dt

0 = Ii + Z*(O) + Zz(l). If bi # 0, we can write

~(i, t){fi(t> -fi(t>l dt -

Z,(i) =fy ’ 0

1

=-

bi

1 zG

q(i, t){u(i, t) $1’ 1 0 T

T I0

fi(i, t)} dt

~(i, t>{fi(t>-_fi(t)ldt + 2

q(L t)v,(L 4 df '0I

T rl(C r){fi(r) - ~~(0~ dr o I

S%lT

’ Ifi

-fi(t)I dt,

where %, depends on L”-bounds for u and li in or. On the other hand, if bi = 0 (and Ui= 1) we have

where se2 depends on Lx-bounds for u, and ii, in 0 T. Using these bounds for Zz(i) in (2.9) we obtain the desired estimate. (ii) Let {u&~ &5,) and {fin), be app roximating sequences of respectively ldo, irg and fi, pf where a; = r&J+ E,

p; =fi + &

for some E > 0. Since uon + u0 and ii&, + c?g as n + x uniformly on a, it follows that uO,, < ii&, for n large enough. Similarly, fin
9 ~,(a;)

in

Qr.

136

J.

HKLSHOF and L. X. PELETIER

Taking the limit through a convergent

subsequence

c(u) 5 c(c?“)

we obtain in Q r.

Now let E+ 0. Then IIQT

{c(ri’) - c(Li)}’ dx dt Z K =

iiQT

O(E)

(a’ - ti){c(ri’) - c(6)) dx dt as

(2.10)

~-0

by (2.8). Because the sequence (~(6’)) is equicontinuous

[7], (2.10) implies that

c(fi’) ---*c(a) in C(Q r) as E+ 0. Thus c(u) 5 c(ii) in 0 r, which completes the proof.

3. SATURATION As we indicated in the Introduction, the region in which c < 1 will be called unsaturated and denoted by 9. Thus, if u is a solution of problem I on [0, T], Gr?= {(x, t) E Qr : c(u(x, t)) c 1). On the other hand, the region in which c = 1 will be called saturated and denoted by 9: 9 = {(x, t) E 0 r : c(u(x, t)) = 1). Since, by the definition of a solution c(u) E C(&), THEOREM4. Suppose (Al)-(A5)

the set Gi?is open relative to gr

are satisfied, and u is a solution of problem I on [0, T]. Then:

(i) u, = 0 a.e. in int 9; (ii) suppose that for some ro E (0, T] and 0 5 x1 < x2 5 1, (xi, to) E 5%i = 1, 2, then (x,to)EEbforallxE[x,,x2]. th en u is a classical solution of problem I in 9.

(iii) if c E C2((-=,O]),

Proof. (i) Write 9’ = int 9, and choose I; E Cz(9’)

0=

IIB’

By theorem 2, u E L*(O, T; p(Q)).

(crux - f,)dxdt=

was arbitrary,

IIB’

&K, dr dt.

Hence (3.1) can be written as IIa’

Since 5 E Ct(9’)

as a test function. Then

[u,dxdt=O.

u, = 0 a.e. in 9’.

(3.1)

An elliptic-parabolic

free boundary problem

1337

(ij) Choose 6 E (0, to) such that u(xi, f) < 0 for i = 1, 2 and c E [to - 6, to]. Since c(u) E C(Qr) it is possible to find such a number 6. Now define the rectangle R = (x,, x2) x (to - 6, to] and let ti : I?+ II? satisfy c(a), = Li,

in

ti = min{u, 0)

on

R

and dpR,

where dpR denotes the parabolic boundary of R. Thus, by assumption, ri = u in the lateral part of apR and at t = t,, - 6, c(G) = c(u). Therefore the solution ti coincides with u in R. By [17] ri < 0 in R, whence u C 0 in R and in particular u(x, to) < 0 for x E [xi, x1]. For the proof of part (iii) we refer to [7]. Suppose the sets 9 II {t = lo} # 0 for t E I C (0, T]. Then on I we can define the functions c-(t) = inf{x E Sz : (x, t) E 9} 5“(f) = sup{x E 52: (x, t) E 9). It follows from theorem 4 (ii) that for t E I: c(u(x, t)) = 1

0 c x < l-(t),

c(u(x, t)) c 1

I;-(t) < x < g+(t)

c(u(x, t)) = 1

P’(t) < x d 1.

It was shown in [6] that the functions c’(t) are continuous in the case b. = bl = 0. It will be shown in a subsequent paper that t- and 5’ are continuous at every to E I where fo(to) # 0 and fi(t,J # 0 respectively [ll]. 4. LARGE

TIME

BEHAVIOUR

OF THE

CONCENTRATION

PROFILE

Suppose the functions h(t) stabilize as f-+ 35, i.e. there exist numbers i;: E W such that fi

Cf) +

_fi

as

t+=.

Then we ask the question as to whether the solution u(x, r) of problem I stabilizes as well. Let ii be an equilibrium solution of problem I, then by equation (1.1) 6” = 0, whence ri is of the form p,qER,xESZ.

G(x) = P + (4 - P)X By the boundary equations.

conditions

(1.2) the coefficients

p and q must satisfy the pair of linear

HOP- (4 - p)bo = !o [ a,q + (4 - p)bl =fi. Thus, they are uniquely determined

by f. and f; if and only if

(a0 + bo)(al + b,) - bobI # 0. Since ai Z 0 and bi 2 0 (i = 0, 1) this condition is equivalent to CIo+a,

>o.

(4.1)

J. HLZSHOF and L. A. PELETIER

1338

Therefore, except when a0 = a, = 0, and th e b oundary conditions are of Neumann type, there exists a unique equilibrium solution a. If a, = a, = 0, then either there exists no equilibrium solution (f. + fr f 0) or there exists a continuum of equilibrium solutions (f. + fi = 0), given bY CE R.

G(X) =pix + c

We shall discuss the cases a, + a, > 0 and a0 + a, = 0 separately. Case 1. a0 + a, > 0. In this case we have a unique equilibrium solution ti. We shall show that if u is the weak solution of problem I, then C(U(X,f)) ---, c(fi(x))

as t+ =

first in a suitably weighted L’-space, and then in C(a). As a weight function, we shall use the principal eigenfunction

q of the eigenvalue problem

in R

11”+ Arj = 0

(II){

0

Uiq + (-l)'+'bi~'=

atx =i (i= 0, l),

chosen so that ?j > 0 in !2 and sup ?j = 1. It is easily verified that the principal eigenvalue i is positive, and that q(i) > 0 if and only if bi > 0. In what follows we shall drop the tilde again. Suppose y and U are solutions of problem I corresponding to the initial values lo and Go, and the boundary data_f, and fi. Write F(r) =

1q(x){c(U(x,

i0

If _oos Go and& 5’fi then by the comparison give an upper bound for F. LEMMA

t) - c(g(x, f)) dx.

principle F(t) 2 0. In the next lemma we shall

2.

where

44 = r’(Oko(9 - rl’(lkl(O

ifuo = u, = 1,

(4.2)

w(t) = q(O)go (9 + rl(l)g, (0

ifbo = 6, = 1,

(4.3)

40 = rl’mo(o + rl(lk,(O

ifao = b, = 1,

(4.4)

and g&) = E(r) -J(t). Proof. Let a0 = ai = 1, and let {_u,}and {E,,}be sequences of solutions of problem I,, which approximate, respectively, _uand E. Then, proceeding as in (71 and writing

An elliptic-parabolic

free boundary problem

1339

we obtain

where gin(t) = fin(t) -_fi,Jt) and K is the Lipschitz constant of the function c. Thus F:,(t) + &Jr)

S o,(t)

where (I),,is defined as in (4.2), and hence F,(t) d (F,(O) + 1’ ecYns w,(s) ds} e-@/m*. 0

Passing to the limit completes the proof for the case a0 = a, = 1. The other cases are proved in a similar manner, and we omit their proofs. THEOREM

5. Let (Al)-(A4)

be satisfied, a0 + a, > 0 and ~fi(t)--~;~Sp(t)

t>O(i=O,l),

where p(t) + 0 as t ---, m. Let u be the solution of problem I and fi the equilibrium solution determined by the numbers A. Then there exist positive constants A and B which only depend on ~0, fi, ai and bi such that 1’ tj$~)lc(u(x, t)) - c(Li(x))j dr S (A + B

i’ p(s) e(“a

0

d.s) e-(l/K)t,

(4.5)

0

and A the principal eigenvalue in problem II and K the

where 77is the principal eigenfunction Lipschitz constant of the function c.

Proof. Define functions fi,_fi and gi (i = 0,l)

Y;(t) =

max{fi(O~~i~

by

and

giCf>

=fi(l>

_fii(t)

=

midfi(O,fi>

-_fiif>.

Then, by the assumption on the functionsfi(f): Igi(

5 p(f)

for

I > 0.

In addition, choose functions co and O. such that u&)

2 max{u&),

c(W))

and

which satisfy (A3), (A4). By theorem 1 there exist solutions U and u corresponding

to the data

1330

J. HULSHOF and L. A. PELETIER

60, ~0 and fi, _fiTand by the comparison

principle

c(U) B max{c(u), c(i)) and c&J 5 min{c(u), c(C)}. Thus It(u) - c(iql S c(U) - c(g) and hence, by lemma 2, the desired estimate (4.5) follows with A=

1 17(x){Oo(x) -vo(x))~

and

r1’(0)+ 17’(l)

if a0 = a, = 1,

V(O) + V(I)

ifbo = b, = 1,

i V’(0) + 770)

ifao = 6, = 1.

B=

Remark.

The rate of convergence

is given by

o(e -(V0)

I + 0 e -(Y0 p(s) e@lflS d.s . ) ( i0

Loosely speaking, the convergence rate of the second term is more or less that of p(r), but it cannot tend to zero faster than e-ldK. In theorem 5, the expression

may be replaced by

where p = 2 if 7(0)7(l) # 0, that is, if b. = bi = 1 and p = 3 if q(O)q(l) = 0, that is, if bob1 = 0. This can be seen by using proposition or a variant thereof.

2 in [7]

Case 2. uo = al = 0. In this case we shall discuss the problem

(III)

x (0,3~)

c(u), = UXX

in Q = (0,l)

u,(i, t) = U + (-l)i+‘gi(t)

i = 0, 1, r > 0,

in G, 0)) = uo(x) 1 ,447 where (YE R and gi E L’(0, ~j), i = 0, 1. W e sh a 11assume problem III has a solution for all time. This means, by theorem 1, that the functions gi and u. satisfy (A2), (A3) and (A5) for

An elliptic-parabolic free boundary problem

1341

all f > 0, i.e. 1 uO(x)dx+

I0

forallt30.

r(gO(s)+gl(s)}&
I0

Observe that if go = g, = 0, then problem III has a one-parameter solutions

where p E IR in (uniquely) determined

III in which uo, go and g, satisfy (4.6). Then in C(sZ),

ast+x

c@(t)) + c(@)

family of equilibrium

/3E w.

rip(x) = (Yx+ p THEOREM 6. Let u(t) be a solution of problem

(4.6)

by

(4.7) As a first step we consider the autonomous

case.

PROPOSITION. Suppose go = g, = 0, and I

i0

uo(x) dx < 1.

Then the solution u(t) of problem III satisfies in C( 0)

ast+=

c(u(r)) ---, c&3)

where Lib(x) = (YX+ /3 and p is chosen so that (4.7) is satisfied. Proof. We define the space X = {u E W’s”(Q) : u 5 1) endowed with the supremum norm. Then X is a (not complete) metric space. By theorem 1 and 2, and (4.6), problem III defines a continuous semigroup s(t) on X: for uo E X, s(t)uo = c(u(. 90) If x is any initial function in X, the corresponding

ost<=.

orbit

Y(X) = u S(t)x PO is precompact by the ArzCla-Ascoli theorem, whence the omega-limit set wc;C) is nonempty. In what follows we shall show that o(uo) consists of the single element ti,, where p is determined by (4.7). We do this by constructing a Lyapunov function. Write for y d 1, Q(y) = /c-1’y’ SC’(S)ds, 0

1342

J. HUSHOFand L. A. PELETIER

where c(c-‘(y))

= y if y < 1 and c-‘(l)

= 0. For u E X we now define the function

V(u) = j’ {Q(u(x)) 0 LEMMA 3. The function

V: X+

- ax/.+)} dr.

(4.8)

W defined in (4.8) has the properties

(i) V E C(X, 62); (ii)letXEXandO5s
- I’ j’ (u, - a)’ drdt. 5 0

Thus, lemma 3 asserts that V is a Lyapunov function. Accepting lemma 3 for the moment we may conclude from [5] that, given uu E X, asi+=

W)u, ---*~(Ull)

in C(h),

and that, if D E w(uO), then V(.S(t)o) = constant and hence, by lemma 3, the solution Li of problem III with initial value d can be written as ti(X, t) = ax + h(t) a.e. By theorem

1 we have j1 C(crX + h(r)) dX = j1 q)(X) dX, 0 0

whence, because c is nondecreasing,

/z(t) = p, p being defined by (4.7). Thus o(u0) = M&N

and ast-,x

S(9uo + c(fi,)

in C(G).

Proof of lemma 3. (i) In view of the definition of c-l, Q E C((-=, (ii) Let U, be a solution of the regularized problem (C&))I

= u1.z

u,(i, t) = cy

l]), whence V E C(X, W).

in Q = (0, 1) x (0,x) i=O,l,t>O

i u(x, 0) = &n(X)

XES=2

where the sequences {c,} and {u,} are defined in Section 2. Let for y E W Q,(y) = [““’ where c,(c;‘(y))

SC;(S) ds,

= y for all y E W, and for w E C(G), V,(w)

=

{Q&W>

- ~xw(x>) dr.

An elliptic-parabolic

free boundary

problem

1343

Then, writing u, = c,(u,), we obtain !R

{Q,(u,(~, =

0) - ~,~u,(Jc>41 dx -

iR

{Q&(X,

s,) - ~yxu,(,~. 3)) cb

dx dt ’ I {Qn(on> - LYXU,,}~ ii5 0 I

I

=

li5 II =

(u, - (YX)LI,, dxdt

’ (u, - cyx)(u, - cux), dxdt

’

li5 0 =-

’ (unx - c+ dx dt,

[

li5 II

v,(o,(~,t))

We now

let

n+

3~

-

V,&,(~,S>)

(4.9)

in (4.9). To do this, we need a technical lemma.

LEMMA4. Let {yn} C R be a convergent (i) y := !lmy”

= - /-‘I1 (u,,, - c+dxdr. 5 0

sequence such that

5 1;

(ii) M := sup{c;l(y,) Then

: n 2 1) < x.

!I-“I Qn(rn) = Q(Y). Proof. If y < 1, the proof is straightforward, Q,(Y,)

=

so let y = 1. Define x, = c-~(Y,J. Then

i*’ SC;(S) CIJ 0 X”

=xnyn

-

I

C,(J)d.s

0

=

II”

{Yn

0

+O

as

- c,(s)) d.s n-+x

x, d M for all n 2 1, lim inf x, P 0 and yn - c,(s) - 0 uniformly on ?I-= [0, M]. Since by definition Q(1) = 0, this proves the lemma. We now return to (4.9). By corollary 1 and lemma 1. u,,- u as n--* x and {c;‘(u,)} is bounded in C(Qr) for every T > 0. Hence, by lemma 1 and the dominated convergence because, by assumption,

J.HKLSHOF and L. A. PELETIER

1314

theorem hninV,(W,S)

= +(*A)

for every f 2 0. Since the sequence {u,} converges weakly in L*(Qr) to u, for every T > 0, we have (u, - CU)~clr dr 5 lim inf n-x

- cu)’ drdr

or V(u(.,t))S’(u(.,s))-/(II

(u,-cu)*&dt. 5 R

This completes

the proof of lemma 3.

Proof of theorem 6. Let us first suppose that 1

I UO(X)hf Ixko(~)+g*(41~=l. 0

(4.10)

0

Then it is easy to see that c(u(t)) ---, 1 as t --, 2 C(a). If (4.10) does not hold then for all E > 0 small enough there exists T = T(E) such that 1

I uob) dx + If (90(s)+ g, (41d.v< 0

1- E

0

for all t Z 0, and such that * Ir Now let u;, given by

&!o(s)l + lg, ($I} d.s < fc.

u”, and u: be weak solutions of c(u), = u,, for t > T with initial data for t = T c(u;(x,

and Neumann boundary

T)) = c(u;(x, T)) = c(u:(x, conditions given by u,(i,

t) =

N -

(-l)'+'Igi(t)l

uo,(i, t) = a uL(i,

t) = a+ (-l)‘+‘(g,(t)(

T)) = c(u(x, T))

An elliptic-parabolic

respectively.

free boundary problem

1345

Then we have c(u;)

s c(u) s

+& -

c(ui)

and thus (4.11)

IC(U) - c(uo,)l s c(uZ) - c(ui). Since I {C(r6 (x, 0) - +;

(x, 0)) ~

I0

and because of the uniform bounds on /u:l, it follows that 0 zz c(ul) where b(&)--* 0 as E-+ 0. By the previous proposition

- c(u;)

(4.12)

z5 b(E)

we have

C(~~(f)) -+ c(k)

in C(Q)

as f+ =

(4.13)

where OEk) =&7-l-B, and B, is determined

by

uo+ i

7x0 +g1)*

0

Observing that I&> - cg9>1 5 k(u) - CW

+ I&4 - c(fi,)l + I&) - ct~@l

we conclude from (4,11), (4.12), (4.13) and from c(C,) * c@Y)

as E--, 0

in C(G)

that +c0> which completes

+ c(&>

as f 3 32 in C(ii)

the proof. REFERENCES

*

1. ALT H. W. & LUCKNAUSS., Quasilinear elliptic-parabolic differential equations, Math. Z. 183, 311-341 (1983). 2. AN LIANZUN,Some properties of the solution of filtration problem in partially saturated porous media, Report, Tsinghux University, Beijing (1984) ACM Norh. A&. Sinica 1, 44-56 (1984). 3. BEAR J., Dynamics of Flukis in Porous Media, American Elsevier, New York (1972). 4. BERTSCH M. & PELETIERL. A., Nonstationary filtration in partially saturated porous media, in A~u~yzjcul and Numerical Approaches to Asymprofic Problem in Analysis (Edited by 0. AXELSSON,L. S. FRA.~ and A. VAN DER SLUIS), North-Holland Math. Studies Vol. 47, pp. 205-212 (1980).

J. HL.LSHOFand L. A. PELETIER

1346

5. D.~FER.LIOS C. &VI..Asymptotic behaviour of solutions of evalution equations. in ,Vonlinear Evolution Equations (Edited by 41. G. CRASD;\LL). pp. 103-123. Academic Pres. New York (1978). 6. VAS DL’YNC. J., Nonstationarv filtration in partially saturated media: continuity of the free boundary. Archs ration. ,Ctech. Analysis 79. 261-265 (1982). 7. VAN DCYN C. J. & PELETIERL. A.. Nonstationary filtration in partially saturated porous media. Archs ration. .Uech. Analysis 78. 173-19s (1982). S. FASANOA. & PRI~~ICERIO%I.. Liquid flow in partially saturated porous media. /. /rut. math. Applic. 23. 503-517 (1979). 9. FO[
porous

media.

Report,

Tsinghua

University,

Beijing

(198-1); Actn .Ctath. Appl.

Sinicn 1. 108-l-126 (1984).