A free boundary problem for degenerate quasilinear parabolic equations

Nonlinear Analysis. Theory. Printed in Great Britain.

Mcrhods

& Applicomm,

Vol.

9. No. 9, pp. 937-951.

A FREE BOUNDARY QUASILINEAR

Department (Received

0X2-546&‘85 f3.00 + .CM (!$I 1985 Pergamon Press Ltd.

1985.

PROBLEM FOR DEGENERATE PARABOLIC EQUATIONS

ZHUOQUN WV* of Mathematics, Jilin University, Changchun, China 14 April 1984; received for publication 9 October 1984)

Key words and phrases: Degenerate quasilinear parabolic equations, free boundary problem, discontinuous solutions, jump conditions, existence and uniqueness, structure of solutions.

1. INTRODUCTION

paper, we are concerned the form

with a free boundary

IN THIS

au -=at

problem for quasilinear

a2A(u)

equations

of

U-1)

ax*

with A’(u) = a(u) 3 0. Our problem arises, in particular, from the investigation of the structure of discontinuous solutions of equation (1.1). It is shown in [ 11 that discontinuity occurs in a generalized solution only if there exists an interval (ui, ~2) such that a(u) = 0

for

u E (ui,u2).

Things are simple if the initial data &I

= uo(x)

(1.2)

happen to either fall in an interval where a(u) > 0 or fall in an interval where a(u) = 0. What we need to investigate is the case that u(u,&)) > 0 for some x and a(~&)) = 0 for some other x. The following situation seems to be typical: u(u) = 0 uo(x) < 0

for for

24CO

and

x CO

and

In this case, one could conjecture that the a line of discontinuity x = A(t) with A’(r) < the discontinuity emerge and develop? To the jump conditions which the generalized presented, which are u(u) = 0

u(u) > 0 us(x) > 0

for for

u>O x>O.

(1.3)

corresponding generalized solution u would have 0 starting from (0,O). The question is: how does get an answer, we need to make essential use of solutions satisfy. In [I], two jump conditions are for

24E Z(u-, u’)

* Sponsored by the United States Army under Contract No. DAAG29-80-C-0041. 937

(1.4)

938

ZHUOQUN

WV

and [sgn(n+ - k) - sgn(u- - k)](ti - k) V,G 0

for

kER

(1.3

for (l.l), where ti = (u- + u+)/2, U- and u+ denote the approximate limits at the points of jump, (I+, vX)the normal to the set of points of jump and I(u-, u+) the interval with endpoints u- and uf. In particular, (1.5) implies (u+ -u-)2+=0

or

r+=O,

(1.6)

which means that all of the normals to the set of points of jump are parallel to the x-axis. Hence the line of discontinuity starting from (0,0),if any, should be a straight line perpendicular to the x-axis. However, this assertion is not true in general; things are not really so simple. The wrong assertion comes from the incorrect jump condition (1.5). The correct form of this condition should be

bgn(u+-k)-sgn(u--k)][(ri-k)v,-$!$v,]sO

for

kER

(1.7)

and

or (uf

-u-$+

(&$))-I

[(!g+_

=o

(1.8)

where dx/dt denotes the slope of the tangent to the set of points of jump. Suppose (1.3) holds and x = A(t), where A’(t) < 0, is the line of discontinuity of the corresponding generalized solution u(t, x). Then, since U&K)< 0 (x < 0) and a(u) = 0 (u < 0), we first have u(t, x) = L@(x)

for

x C A(t).

Secondly, from (1.4) it follows that u+lX=qI)= 0, and hence, noticing that (t3A(u)/d~)-~,=~~,~ = 0, from (1.8) we get

-Uo(W))W + Thus (4, ~1,W)) is a solution of the free boundary problem for (1.1) with boundary conditions Ul,=A~,)= 0,

aA - ax

x=L(,) =

(1.9)

UoMm’(o~

(1.10)

and the initial condition &o

= uo(x)

for

x > 0.

(1.11)

939

A free boundary problem for degenerate quasilinear parabolic equations

We will prove the inverse in Section 2, namely, if (u(t,x), A(t)) is a solution of the fre boundary (l.l), (1.9), (l.lO), (l.ll), then we can immediately obtain a generalized solutio of the initial value problem (1.1). (1.2) with x = A(t) as the line of discontinuity. The rest of this paper (Sections 3-5) is devoted to a detailed study of the special case: A(u) = 0

for

Uo(X)= 1y< 0

for

A(u) = u”’

and

u
and

for

u > 0, for

UO(X)= p > 0

(1.12 x > 0,

where m > 1 and (Y.p are constants. The possibility. in this case, of reducing to a problem ii ordinary differential equations simplifies matters. The main results are included in the theoren in Section 5. By the way. since the proof of the uniqueness theorem in [l] is based on jump condition (for (1.1). they are (1.4), (1 .i)) and one of them is not true in general, a revision of the proo is required. We will derive the correct form of the jump condition and complete the proof o uniqueness in another paper [2]. 2. GENERALIZED

Suppose A(u) E C’(R).

uo(x) E L”(R)

SOLLTIONS

OF (1.1).

(1.2)

n C’(R\{O}).

1. If (u(t, x), i.(r)) is a solution of the free boundary problem (l.l), then the function

PROPOSITION

(l.ll),

w(t,x) =

u(t;x)

for

i uO(x) for

(1.9), (1.10)

x > A(t) x < A(t)

is a generalized solution of the initial value problem (1 .l), (1.2) with x = A(r) as the line 0’ discontinuity. By a solution of the free boundary problem (l.l), (1.9), (l.lO), (l.ll), we mean a pair 01 functions (u(t, x), A(t)). such that u E L”(G) n C(G\{(O,O)})

n C2(G),

G = {(t,x);O
for any bounded domain D with D qg

E

C d - {t = 0},

C(G\{(O,O)}),

>A@)},

(2.1:

(2.2)

(2.3)

/i(t) E C[O, T] n C’(0, T],

(2.4)

A’(t) < 0

(2.5)

and (1.1). (1.9). (l.lO), (1.11) are satisfied. Here C’.‘(G) denotes the class of functions which are continuous on G with their first order derivative with respect to t and their first and second order derivatives with respect to x. Proof. We need to check all of the conditions in the definition of generalized solutions given in [l]. Since u E L”(G). uo E L”(R). we have LVE L”(Q), where Q = (0, T) x R. The assumption

940

ZHUOQUN WV

u. E C’(R\{O)) and (2.2) imply that w E SV(Q). By SV(Q), we mean the class of integrable functions on Q of locally bounded variation. To prove aR(w)/ax EL:,(Q) where R(u) = .J: y(t) dt and y(t) = a(t)Y2, we multiply (1.1) by @u with $5 0 and I$ E C;(Q) and get

Integrating this relation over G, = ((t, x) E G; x > A(t) + E, E > 0} (notice that u need not be differentiable up to L:x = A(t)) yields 1

m

($j2drdx

zI

u~+~bu’lr=rdx-~i,~~+~$~2l~=i~,~+rdx-fjl’$Ui.i=odx+~~G~~~~~)

=-

1 2

Wu) +u-- ax

dt -

.

-U-

x= l(r)+E

(2.6)

Since @~,=o,J= 0, the first and third term of (2.6) are zero. By virtue of (1.9), (l.lO), the second and sixth term tend to zero as E--, 0. The fifth and seventh term are bounded uniformly in E. Thus, letting E+ 0, from (2.6) we get /I,@(u)

(g)2dtdx<

+m.

(2.7)

Denote

g=

Y(U)2

for

x>A(t)

0

for

x < A(t).

1

Then by (2.7), g E J%(Q). Clearly, for any @E Co”(Q),

Ww) @-ax

Integrating

$gdtdx+

dtdx=

#gdtdx

as

E--,0

by parts gives

Ww) ax dtdx= e-----

-

This means that aR(w)/ax = g E L&(Q). Now we prove that w satisfies the integral inequality ~(w, k, $1 =

11sgn(w

- k) [(w - k) 2

- 921

for

kER,

dt dx 2 0

Q

+EG’(Q>,

$20.

(2.8)

A free boundary problem for degenerate

941

quasilinear parabolic equations

Divide J(w, k, qb) into two parts:

J(w, k, @I=

sgn(uO - k)(uo j-i
+ =

Jl(uo,

k,

$)

+

J2(u,

w

k) xdf

dx

(u - k) $f - F$]

k

dt dx

@I.

(2.9)

Obviously

=

O Iuo -kl#W’(x),x) I W-J

dx = - brluo-

kJ@l,=~$(t)

dt,

J2(u, k, $) = lim J2,Jw k, $) q-o+

(2.11)

where P

~(,) sgn,(u

sgn&) =

w (u - k) x

- k)

for

t>q

[

for

ItlSq

I_-1

for

tC -71.

I

1

aA(u)w -axax

1

dtdx

We have sgn,(u - k) (u - k) $ [

- $!$$$I

= $ (sgn,(u - k)(u - k) $) - sgn,(u - k) 2 $ - sgn’,(u - k)

%(u

-

k) t$

+ sgn,(u - k)- a2tJ:) + sgn;(u “k) #(u) ($)2. Since u satisfies (1.1) whenever x > A(t), we have

-

sgnG(u - k)$(u

=IT%%(u - Wu - k) 0

- k)@dtdx T

9 x=A(,,W dt +

I0

(2.10)

w,(u

-

k)

i%UOQUN

942

il

-

=

sgn;(u - k)$(u x>d(t)

WU

- k)@dtdx T

rsgn,(W@ I0 -

x=A(IjW) dt -

i0

sgn,(W

X=i(,,W

dt

sgn;(u - k) $ (U - k) @dt dx.

For any fixed E > 0, since u E C’(G,), we have sgn;(u - k)$(u

- k)$drdx-,O

as

q-,0.

By (2.2), sgn;(u - k) $ (U - k) $Jdt dx

as

E+O.

as

r+O

Thus sgn;(u - k) $ (U - k) $dt dx + 0 and hence

Combining (2.9)-(2.12)

gives

J(w, k @) 2 ~r(-lllO

dt z- 0 - kj + ]kj - sgn(k) ~u~)~J~=~~~~A’(~)

since A’(f) < 0. This proves (2.8) and the proof of proposition 3. ESTIMATES

ON THE

SIMILARITY

is complete.

SOLUTION

From now on we consider the special case (1.12). In this case, the free boundary becomes

au -=at

problem

a2c a2 ’

(3.1)

&A(t) = 0,

(3.2)

at4m = d’(r), ax x = L(t) uI,=o = /3.

(3.3) (3.4)

A free boundary problem for degenerate

quasilineaf parabolic equations

943

We seek solutions of the form (35) Then we arrive at a free boundary

problem for an ordinary differential

equation,

namely:

(Urn)” = -j@‘, UI+$ = 0,

Let u = Urn.

(3.6)

The problem for ~(5) is

vv= --2mvE L&

U/m)-lv~

(3.7)

)

= 0,

(3.8)

u’l:=.& = q,,,

(3.9)

v If=z = pm.

(3.10)

We seek solutions v E C’[&, w) n C2( 50, co), which are positive for 5 > 50. We will first solve (3.7), (3.8), (3.9) for any fixed 50 < 0 and then choose 50 such that the solution satisfies (3.10) as well. Integrating (3.7) and using (3.8), (3.9) we get (3.11) (3.12) If we write (3.7) as v” = -$(vi/m)f = -+(Evi/m)f + tvilm, integrate

and use (3.8), (3.9), then we get u’(E) = +

v(g) =+-

-

!&‘/“(~) + ;j--vl~m(, &,) + ;j--(,-

dt,

2r)vVm(t) dt.

In order to solve (3.7), (3.8), (3.9), it suffices to solve the integral equation

(3.13) (3.14) (3.12) or

ZHUOQUN WV

944

(3.14)-to find a solution u E C[&, ~0) which is positive for 5 > &. We remark that the integrand in (3.14) is not singular at u = 0 in contrast to (3.12). Before we discuss the solvability of (3.7), (3.8), (3.9), we first derive some estimates on the solutions. From (3.11) we see that u’ is strictly increasing for 50 < $ G 0. In particular, V’ B c$0/2 for &Gg
b(E) Using (3.11), (3.15) we further obtain u’(E)

~~exp[-~f~~(~)‘l’m’-l(‘-

#ll.)-ldr]

&I +Texp

_ &)(r/m)-r dr]

L A [ 2m 0 2

(ym’-1 1,&/r/mli(r

for

go< ECO,

or (3.16) where R =

[(!$1(l’m’-lj~ol~‘“]~

exp i

(3.17)

Hence (3.18) Combining (3.18) with (3.15) gives for

~(5-5o)~U(5)++50)

&IS EG 0.

(3.19)

Next we prove that (3.20)

u(E) < oM50 whenever u(g) exists. Here M is a constant such that 2

RI601 +

ISexp(-q2)

Rm@/21 4(” - *)/2mljzol(m-*)/2m

dq

M(m-1)/2mG t.

(3.21)

0

Let [go, &) be the maximal existence interval of u(g). If !$Yr 6 0, then (3.18) implies (3.20). Suppose 5; > 0 and there exists f2 > 0, & < & such that u(5) < ~MEO

for

OsE<&

and

~(52) = CUM&J.

(3.22)

A free boundary problem for degenerate quasilinear parabolic

945

equations

Then, we divide the right-hand side of (3.12) into two parts:

= Zl + z2

and estimate each of them. Using (3.15) we get Z,=~~exp(-~~s~(l/m)-l(r)ds)dr~nb~. Using (3.15) and the first part of (3.22) we get for 0 G 5 < &, Z*=~iiexp(-~~~~(l/m)-l(S)dS)d~ (l/*)-r(,) ds) exp (- & [sv(‘~~‘-‘(~)

,

ds) dt

YE

I

exp(-s*)

ds

0

exp(-s2)

ds - IvI(~-~@”

Hence, by the choice of M (see (3.21)) 4‘52) s do

2RlEoI + Rm’/2[alW)/2

~~~o~(~-~)~~~xexp(_s2) 0

ds .M(m-l)bm

s!!&Eo,

which contradicts the second part of (3.22). Furthermore, using (3.11), (3.15) and (3.20) we get (3.23) 4. EXISTENCE

FOR (3.7),(3.8),(3.9)

Now we prove PROPOSITION 2. The problem (3.7), (3.8), (3.9) has a solution u E C’[co, ~0) n C2(co, 01) and c(f) > 0 for g> co. Solutions of (3.7), (3.8), (3.9) are unique and depend continuously on ,kcn 2nd fv.Yn

i%UOQUN

946

WV

Proof. First we prove the existence. By virtue of (3.20), (3.16) and (3.23), it suffices to prove that (3.7), (3.8), (3.9) has a solution on & d 5 G 0 which is positive except at 5 = &. Denote by & the class of functions v E C[&,, 0] such that 0 < u(f) c N

for

E>

f0

where N is a constant satisfying A((Ly(+ 3Nl’m) (j. I2s N. Clearly, At is a closed convex subset of C[ 50,01. Consider the operator w=Tv=-

aF (5 - Eo) + 1 1; (f - 2r)u”“(r)

dt.

For any v E JU, w = TV E c[&,OJ and w = TV > 0 except at 5 = go. Moreover w+~2+~]~o]2N1~m~

N.

So w E At. It is easy to see that TAt is compact and T is continuous on fixed point theorem gives the existence of solutions to (3.14) and hence To prove the uniqueness of solutions, let vl, v2 be two solutions of suffices to prove that v = v1 - v2 = 0 in a neighborhood of Eo, say 6 > 0 small. From (3.14) we have v(E) = 2’ /F(e50

A. Thus, Schauder’s to (3.7), (3.8), (3.9). (3.7), (3.8), (3.9). It &, G 5 < go + 6 with

2r)(~;‘~(r) - &“/m(r)) dt

(e-- 2t) ~~ti(l~m’-l(r) u(r) dt m where fi(r) is a certain point between v,(r) and u2(r) and hence from (3.15)

It is impossible for 6 small enough, unless u = 0 on &JG 5~ co + 6. Finally, we prove the continuous dependence of solutions upon 50 < 0, and (Y< 0. Denote by ~(5; 50, CY)the solution of (3.7), (3.8), (3.9). Let go < 0, cy< 0 be fixed. What we want to prove is that for any 5-i> & and E> 0, there exists a constant 6 > 0 such that

145;k-6,d) - 45; 60,4 I < E, whenever

Iu’(REb,d ) - o’(E;50,No)I< E

156 - eol < 6, Id - &yI< 6 and max(g6, 50) 6 5 G &.

A free boundary problem for degenerate

quasilinear parabolic equations

947

Suppose it is not the case. Then there exists ~0> 0 and Et’, aCn),/$“) such that E!’ --j &, c#+

tin+

m), max(.$),

and

&) C E@)G &, _

or 1uyp;

p,

c+)) - uyp;

50, ag) ( 2 E.

Suppose, for example, the case is the former: j u( 5’“‘; @,

a@)) - u(p);

&J, (Y) 13 E.

(4.1)

We have

From (3.20), (3.16), (3.23) we see that ~(5; &$, a’) and u’(E; 56, CY’)are bounded uniformly in (E’, cu’) in a small neighborhood of (&, a) and hence from (4.2), it is eas (~(5; ,$‘I, a(“))} and (~‘(5; Ef’, CY@‘)}are equicontinuous. Thus (~(5; EF {u’(gF;$“‘, ax”))} have subsequences converging uniformly. Suppose they are and (~‘(5, $“‘, CX(“))}themselves and lim u(& Et’, CX(~)) = w(t), lim u’(E; Et’, LX@))= G(g). “+*

(4.3)

n-+x

From (4.2), (4.3) we get

+Ew”“(g)+ ;

G(c) = +

-

IV(~) = 9

(E - &) + ; j’(j 5

/--wlim(r) dt, - 2r)~“~(r)

whence w(g) is a solution of (3.7)-(3.9) and C(c) = w’(E). u(5; 50, m). On the other hand, we may suppose that {EC”)}converges:

dt By uniqueness,

w(E) =

lim 5’“) = 5. n-a: Letting n + CQin (4. l), from the uniform convergence

we get

Iw
means that what we want to prove is true. The proof of proposition

2 is

948

ZHUOQUN

WV

Remark.

So far we have proved that for any & < 0 and (Y< 0, there exists a function u E C’[&, co) fl C2(&, m) with u(c) > 0 for c> 50 satisfying (3.7), (3.8), (3.9). Since u(g) is bounded and increasing, lim u(g) exists and is positive. Let E;-=

Then it is clear that u(x/~“~) and &l/2 satisfy (2.1)-(2.5) W= is a generalized

@It@ ) 1

(Y

and hence, by proposition

for

x > $t1’2

for

x < &ty2,

1,

solution of the initial value problem au -=at ul,+J =

a2A(u) ax2

’

Pa

for

x>O

( cy

for

x < 0,

where A(u) = max(Iulm-rU, 0). The fact that this generalized solution discontinuity x = &* shows, in particular, that .(1.6) is incorrect.

w has a line of

5. EXISTENCEANDUNIQUENESSFORTHEFREEBOUNDARYPROBLEM Now we return to the free problem (3.7)-(3.10) The free h
THEOREM.

Proof. Denote

and state our main results in the following:

boundary problem (3.7)-(3.10) has a unique solution (u(g)>, &) with C’[&-,, ~0) fl C2(&, a). Moreover, if we denote (u(E), 50) by to express explicitly the dependence on cz and /3, then ~(5; (Y,/3) and decreasing in (Yand strictly increasing in /3.

by ~(5; 50) the solution of (3.7)-(3.9)

for the moment. For any &, < 0, the

limit lim 45;

50)

=

v(b)

5--t-

exists. Let E = (50~ 0; ~(50)
Let

i. = inf E. We want to prove lim ~(5; $0) = u
(5.1)

Suppose ~$6) # 6”. Denote

6=

IYGOO)-

PI.

(5.2)

A free boundary problem for degenerate

quasilinear parabolic equations

949

From (3.21), (3.23), it is clear that we can choose positive constants ci and cr such that 0 < u’(R for go E [3&j/2, k/2].

(60)

=5 cl

520

(5.3)

Choose & such that exp(-c2t2)

Cl

By the continuous such that

for

exp(--c25*)

dependence

dg< !.

upon &J, there exists n E (0,1&l/2)

of solutions of (3.7)-(3.9)

I u(E1;

60)

whenever E E [& - q, & + r,r]. Now for go E [& - q, go + q], integrating

-

45’1;

50)

(5 -4)

s 4

I<

(5.5)

-

(5.3) over (4, w) and using (5.4) we get 6 4

0 < r(50) -

451;

50)

<

-;

0 < Y@o) -

u(h;

Eo)

<

$.

in particular,

Hence / Y(50) - Y(50) I <

Mf1;

Eoo)

-

4f1;

go1

I +

$

and by (5.5)

IYGOO)- AGo)I< p.

(5.6)

If ~(5~) = /3” - 6, then (5.2) and (5.6) imply

for E. E [& - n, $ + q]. This means that [& - q, & + q] C E, which contradicts the definition of 4. If y(&,) = pm + S, then (5.2) and (5.6) imply Y(E0) > P” + ; for 50 E [go - q, & + ~1. This means that no point on [&J- 7, & + ~1 is in E, which also contradicts the definition of 50. Therefore (5.1) holds and the existence is proved. Now we prove the uniqueness. Suppose (3.7)-(3.10) ha; two solutions (us, &,,l) and @$:jia5gi and 50.1 f b.2. (If EOJ = &JJ then, by proposltlon 2, ud5;) = ~451). Then from

TU(~‘~)-~(~) dr),

(5.7)

ZHUOQUN

950

WV

which holds for any solution u(g) of (3.7), it is easily seen that the curves u = ~~(5) and u = u*(E) cannot intersect at any point (~0, ug) with ~0 2 0. In fact, if Ul(rlO)

=

uz(rlo)

=

uo,

then we must have ui(rlo) + &(llo); otherwise the standard uniqueness theorem would imply that u,(E) = u:(E). Suppose ui(rlo) > &(ro). Then we have, in a right neighborhood

of qo, VI(E) ’ uz(0

(5.8)

vi(E) ’ ui(E)

(5.9)

Note that (5.7) implies

whenever (5.8) holds. Thus (5.9) would be true for all E> no and hence

which is impossible. In addition, a similar argument shows that if, for example, h(O)

>

(5.10)

u2m

then (5.11)

ui(0) < u;(o). Let g(u) be the inverse function of u(g), u(E) being a solution of (3.7)-(3.10) positive except at t= Eo. Then from (3.7)-(3.9), we obtain uU/m)- 1 Ef’ )

which is

(5.12)

El”=0= 50,

(5.13)

&.

(5.14)

5’l”=o=

If we denote the inverse functions of ~(5) and ~(6) by 61(u) and &(u), then from (5.10) we can assert that 5x1

<

(5.15)

Eo.2 ;

otherwise there exists ug, 0 < uo < uz(O) such that 51(o)

’

K(u)

for

0 s u C uo,

5!(Q) = Ez(uo),

and hence, noticing that 5i(O) s&b&=

G(O),

A free boundary problem for degenerate

we may conclude (0, uO) where

quasilinear parabolic equations

that g(o) = E,(u) - &z(u) ac h’Ieves its maximum k-> 0,

E’ = 0,

at a certain

951

point in

6” s 0.

On the other hand, from (5.12)

since E; > 0, E > 0, 5’ = 0. The contradiction proves (5.15). By the way, what we have just proved is that the curves u = ul(Q and u = Q(E) cannot intersect at a point in the halfplane E< 0. Now we consider the function E(u) = &(u) - 52( u) a gain on the interval [0, u*(O)]. We have 5X0) < MO) and from (5.10), (5.11)

Therefore the function g(u) = El(u) - E*(u) on [0,u*(O)]would achieve its minimum at some point in (0, Q(O)), which is impossible. This completes the proof of uniqueness. The rest of the theorem

can be proved by a similar argument.

Acknowledgement-The

author would like to thank Professor Michael Crandall for several interesting and helpful discussions during the preparation of this paper. REFERENCES

1. VOL’PERTA. I. & HUDJAEVS. I., Cauchy’s problem for second order quasilinear degenerate parabolic equations, Mar. Sb. 78, 374-396 (1969). 2. ZHUOQUNWV, A note on the uniqueness for degenerate second order quasilinear parabolic equations, to appear.