The initial-boundary value problem for a nonlinear degenerate parabolic equation

Nonlinear Anolysrs, Theory, Printed in Great Britain.

Methods

& Applicatrom,

Vol. 16, No.

11, pp. 997-1009,

1991. 0

0362-546X/91 $3.00+ .OO 1991 Pergamon Press plc

THE INITIAL-BOUNDARY VALUE PROBLEM FOR A NONLINEAR DEGENERATE PARABOLIC EQUATION ISAMU FUKUDA Department of Mathematics, Kokushikan University, Tokyo 154, Japan

and MASAYOSHI TSUTSUMI Department of Applied Physics, Waseda University, Tokyo 160, Japan 13 March 1989; received for publication

(Received

Key words and phrases:

1990)

Degenerate parabolic equations, uniqueness, semiconcavity.

1. INTRODUCTION LET a < b AND Iz > 0.

6 November

AND MAIN

RESULTS

We consider nonnegative solutions of the initial-boundary

value problem

U, = UU,, - A1#,12

(a < x < b, t > 0)

(1.1)

u(a, t> = u(b, t) = 0

(t > 0)

(1.2)

4x9 0) = %(X)

(a < x < 6)

(1.3)

where initial data u0 satisfy and

U&X)2 0

(a I x 5 b).

(Hl)

In order to construct a solution to the problem (1 .l)-(1.3), it might be natural to employ the well-known viscosity method. Let E > 0 and let uE(x, t) be a unique classical solution of the initial-boundary value problem for the uniformly parabolic equation: u Et = (% + c)&r - G,12

(a < x < 6, t > 0)

(l-l),

U,(U, t) = UE(b, t) = 0

(t > 0)

(1.2),

(a < x < b).

(1.3),

1 &(X, 0) = u&)

We call u the viscosity solution of the problem (1 . l)-( 1.3) if U(X,t) = liiy u,(x, t). Let us consider solutions with compact support and define the interface c,(t) for t > 0.

[* (t) = + sup1 f x: U(X,t) > 0)

Differentiating u(c+ (t), t) = 0 with respect to t and using (l.l), face [+ (t) satisfies formally

dC+(t)

A

dt

=

~~X,W, 991

by

f),

we easily see that the inter(1.4)

998

I.FUKUDA and M.TSJTSUMI

provided u,(<+_ (t), t) # 0. Thus we might expect that the support of solutions shrinks if u,([, (t), t) # 0. Indeed, for A > i we have a special weak solution of the form

t)“@-I)

24(x, t) = (To -

L

c; -

1

2(2/l - 1)

X2(7-0_ t)2M2X-‘) 1+

(1.5)

where T, and C, are positive constants such that (-2(22

- l)c()Td.“2”-i),

2(212 - l)c&~‘(2~-l))

c [a, 61

and [*I+ = max(*,O). Apparently its support shrinks to one point. But this conjecture is not true for viscosity solutions. In [l], Bertsch et al. show that every viscosity solution of the Cauchy problem for (1.1) has a property that supp u(t) = supp ug for I > 0. (1.6) It is a striking result. If ,l < 0, equation (1 .l) is called the pressure equation, porous medium equation and the support of solutions spreads out as time goes, by the interface equation (1.4). Another curious property of the problem (1 .l)-(1.3) is the nonuniqueness which was discovered by Dal Passo and Luckhaus [2] (A = 0), Ughi [5] (A = et al. [l] (A 1 0). The existence of our special weak solution u also suggests the phenomenon. We now define weak solutions of the problem (l.l)-(1.3) as follows.

related to the as is suggested phenomenon 0) and Bertsch nonuniqueness

Definition 1. A nonnegative function u E L”([O, co) : W’,“([a, 61)) is called a weak solution of (1 .l)-( 1.3) if for any T > 0 u, E L2([a, bl x K4 Tl) and b

b a,

ia

thw,

0

d-x

=

~,(X)u/(X,

0)

dx

\’ L1 -

u(x,

Gu,(x,

Wx(x,

f

b

SI’0

a

+

(W

4

-

@

W,(x,

+

l)lu,(x,

s)

s>i2w(x,

s)J

d.xdt

(1.7)

for all t 2 0 and any function I,VE C2,‘([a, b] x [0, 00)) with compact support in (a, b). Note that u E L”([O, Q)): W1p”([a, 61)) with u, E L2([a, b] x [0, T]) for any T > 0 implies that u is continuous in x and t. In this paper we establish the global existence of (weak) solutions of (l.l)-(1.3) and investigate the uniqueness of solutions. We propose a new uniqueness class of solutions which is different from [l], [2] and [5]. As to the existence theorem, we have the following. THEOREM

1. Let u. satisfy (Hl). Then the problem (1 . l)-(1.3)

has at least one weak solution.

Before introducing a new class of solutions, we define semiconcavity

of functions.

999

Degenerate parabolic equation

Definition 2. A function w(x, t) (or w(x)) is semiconcave if the inequality w,, I c

in D’((a, b) x (0, a))

(or D’((a, b)))

holds for some constant C. A function w(x, t) is semiconcave almost everywhere if the inequality a.e. in [a, b] x (0, 00)

%(X9 t) 5 C holds for some constant C.

THEOREM2. Let I > i. Assume that u,, satisfies (Hl) and %(X) lim -
and

lim 2 &l(x) < 00. xTb(b - x)

U-W

Then u satisfies

(1.8) and, in particular, u E L”([6, 43) : W’*“([a, b])) as well as U, E L”([a, b] x [a, m)) for any 6 > 0. Moreover, if we assume that u,, is semiconcave, then u is also semiconcave almost everywhere. Remark 1. In theorem 2, the hypotheses (HI) can be weakened as follows: u. E L”([a, bl),

uo(x) 2

0

a.e.

(Hl),

Remark 2. Let U(X,t) be a weak solution which is obtained by the viscosity method. Then U(X0,to) > 0 * 24(x,, t) > 0 for all t 1 to 2 0 and x0 E (a, b). This has been proved by Bertsch et al. [l] for the Cauchy problem in R”. Moreover if U(X,t) is semiconcave a.e., we can find that 24(x,, to) > 0 H u(x,, t) > 0 for all t 2 to 1 0 and x0 E (a, b). Indeed the equation (1.1) and semiconcavity of u(x, t) yield that u, I -cu which follows that 24(x,t) 5 ec@-%4(x, to). We now define a new class of weak solutions. Definition 3. Let 6 be a arbitrary positive constant. A nonnegative function u E L”([a, b] x [0,00))flL"([d,00): W’*“([a, 61)) n C([O, 00) : L*(a, b)) is a semiconcave weak solution of (1 . l)-( 1.3) if u satisfies (1) U, E L’([a, b] x [a, T]) for any T > 6 > 0 (2) u is semiconcave a.e. and (3) the weak form (1.7).

1000

I.FUKUDA and M. TSUTSUMI

3. Under the assumptions (Hl), and (H2) the problem (1 . l)-(1.3) semiconcave weak solution which has properties in theorem 2.

THEOREM

Concerning the uniqueness and continuous-dependence-on-data following.

has at least one

of solutions, we have the

4. Let 2(and u be two semiconcave weak solutions of (1 .l)-(1.3) corresponding initial data u,, and u,,, respectively. Then the inequality

THEOREM

to the

b (u(x, t) - v(x, t)( dx 5 ect b luO(x) - v,(x)1 dx 5(1 s (I holds valid for any t > 0 and some positive constant c. Let u. satisfy (H l), , (H2) and be semiconcave. Then the problem (1 . l)-( 1.3) has an unique semiconcave weak solution u which depends on initial data continuously in L’(a, b).

COROLLARY.

Remark 3. Our special solution (1 S) is not semiconcave.

valid for the problem (l.l)-(1.3)

Uniqueness theorem does not hold

with initial data

1 uo(x) = (To)1’(zA-1) co2- 2(2k - ,)X2(~) 2X/(2X-l) +

L

1

2.EXISTENCEOFWEAKSOLUTIONS

Before proving theorem 1, we shall obtain a priori estimates of u,. LEMMA

1. Let u. satisfy (Hl). Then tt%~tL”([O,-a):

W’.“([a,b]))

5

(2.1)

c

and m b (z&(x, t) + E)IU,(X, t)lJ?&(x, sj0 (I

t)Qx

d c

(2.2)

for any p L 1, where the sequel C denotes various positive constants independent

of E.

Proof. The maximum principle gives 0 5 4(x, t) 5 mXyb u,(x). Multiplying (l.l), by (l/p)(lu,(x,

t)lP-‘u,,),

Iu,,Ip+r dx +

+ p+

and integrating by parts on [a, b], we have

b (u, + s (2

lu&&,OIPU&, t)

(2.3)

- p+

dIhxIP%x dx lu,@, t)lPu,(b,

t) = 0.

(2.4)

Degenerate

parabolic

1001

equation

Here and from now on, we abbreviate x and t variables in the integrand. negative, we easily see that u&&z, t) 2 0

Since u, is non-

U&b, t) I 0.

and

Hence integrating (2.4) from 0 to t, we obtain that, for any p 2 1 b(~, + ~))u,~P-luf&l.xdt

dpcp:

1) .i’ Iu,,~~~+’dx a

form which it follows that

IIhxwlIL~+’ (o,b)

s

for any t > 0

hOxllLP+‘(a,b)

(2.5)

and ss0 Since

b(u, + c)~u,~P-lu~xdxdt

I C~(U~~~~;$~~,~~.

(2.6)

a

is bounded, we can easily get (2.1) from (2.5). II~oxllL-~~a,b~~

LEMMA2. Let 0 < E < so where e. is a fixed number. The inequality

(2.7) holds for any T > 0. Proof. Using (1. l), and integrating by parts, we get

ss T

u,:dxdt

0

s T

b

(u, +

=

E)~z&dx

dt -

+A

&Ah O3- ~(a, 031dt

0

a

ss T

+

(+a + a2)

u&dxdt

0

5 (b&-([a,bl

b

0

x [O,T]) +

CO)

ss T

(u, +

0

+

+~~TIb,llt-(~a,b~ x [o,m))

b

+

E)&

dx dt

a

($a+ a’)@- ~)Tbdl$‘(~a,b~ x 10,c.z)).

From (2.1) and (2.2) with p = 1, we can easily obtain (2.7). Proof of theorem 1. From (2.1), (2.3) and (2.7), we see that there exists a nonnegative function u E L”([O, 00): W’.“[a, b]) with ut E L2([a, b] x [0, T]) (for any T > 0) and we can extract a subsequence of (u,), which is denoted by (uJ, such that as ai + 0, strongly in C([a, b] x [0, T])

uci,-+ ux

weakly star in ,?‘([a, b] x [0, 00))

ucit - ut

weakly in L2([a, b] x [0, T]).

and

1002

I. FUKUDA and M. TSUTSUMI

In order to show that u is a weak solution of (l.l)-(1.3),

it suffices to show that,

strongly in L’([a, b] x [0, T]),

l%;X12-+ l&l2 for any T > 0 and this implies

strongly in ~?([a, b] x [0,00)).

U,$ + UX From (2.1) and (2.2), we have

ll~&llL~~[o,b] x [O,T]) 5 4lw4,xx+ &IIL~([a,b] x [O,T]) 5 c* We also have

Il(u,Z),llL2~0,T:H-l(o,b)) 5 CIl(&llL2([a,b] x [O,T]) 5 c. By virtue of Aubin’s compactness theorem (see Lions [4]), we may assume that (U& = 2U&,

+ 22.424, = (uZx

strongly in L2([a, 61

x [O, T]).

Hence we may also assume that 4iu,,,

+ UK

a.e. in [a, b] x [0, co)

from which it follows that a.e. in [a, b] x [0, co)

uq* + %

(2.8)

since 8~1~3~= 0 a.e. in E = (x E [a, b]; u = 0) (see Kinderlehrer and Stampacchia and a.e. in ‘E = (x E [a, b] : u > 0). nq, + %

[3, p. 531)

In view of Lebesgue’s bounded convergence theorem we can easily obtain that Iu,iXj2dxdt = On the other hand, from L2([a, b] x [0, T]). Hence

’ b h.# dwdt. i’0 a

(2.1) we may assume that

%$ + u*

u,;, converges

strongly in L2([a, 61 x [0, T]).

(2.9) to U, weakly in (2.10)

This completes the proof of theorem 1. 3. SEMICONCAVE

WEAK

SOLUTIONS

First of all we shall give estimates of U, on the boundary. LEMMA

3. Let u. satisfy (Hl),

and (H2). Then (3.1)

and (3.2) for any t > 0.

Degenerate

Proof.

parabolic

1003

equation

We only show that (3.1) holds valid. From (H2) we see that for some 6 > 0 and

c, > 0 0 5 u&V) I C,((x - a)2 + 6(x

for any x E (a, a + 6).

- a))

(3.3)

Let T > 0 be fixed. For any (x, t) E [a, a + 61 x [0, T], set ii(x, t) = A((x - a)2 + dF(x - a))

(3.4)

where A is chosen so large that A 1 C1

(3.5)

and (3.6) Note that n E C2,‘((a, b) x [0, T]). Direct calculation gives ii, - (ii + e)ii,, + I)Q(~ = 2(211- 1)A2(x - a)’ + 2(21 - 1)A2&(x 20

- a) + 2wl(lA

in (a, a + 6) x (0, T)

- 2) (3.7)

provided that 2 2 t and that A is so large that IA - 2 > 0.

By virtue of (3.3)-(3.7), we apply the maximum principle to obtain 0 5 UE(X, t) I

in [a, a + 61 x [0, T].

27(x, t)

(3.8)

Hence 0 5 uor(a, t) = lii

u,(a + h, 0 - da, h

0 < lim ii(a + h) = Afi h h10

Thus we have (3.1). we have the following. As for uEXXr

LEMMA 4.Under the same assumption in lemma 3 Iu,I

for any t > 0.

5 :

Moreover, if uoXX_-= C2 in the sense of distribution, u .?XX 5

then (3.10)

c3

for all (x, t) E [a, b] x (0, a) where C3 is a constant independent of E. Proof, Putting p = u,,/(u,

(3.9)

+ E), we have

Pt = (u, + E)Pm + 2(1 - ~)%XPX + P2

(x, t) E (a, 6) x (0, a)

p(a, t) = P@, 0 = 0

t E K4 00)

x e (a, b).

1004

I. FUKUDA and M. TSUTSUMI

The standard comparison theorem yields that

Using (1. l), , we easily see that uEXX 1 --.

1

(3.11)

t

We put q = u,, to obtain that 4t = 04 +

As for the boundary conditions,

da, t) =

m, + w we utilize (l.l),

1Iu&,

o12,

a)%&

+ (1 -

w72.

(3.12)

to get

4GJ9t) = p I&@, t)12

(3.13)

0 5 q(b, t) 5 x2.

(3.14)

for any t > 0. In view of lemma 3, we see that 0 5 q(a, t) I AC2,

Hence the comparison theorem yields that, if L > $

4(x9t) = %x*k 0 5 for some constant C > 0. If I.4 0XXI C, in the sense of distribution,

c (2a

_

l)t

A L t and (3.12)-(3.14) yield that

%X(x, t) 5 G where C3 = max(AC2, C,) which is independent

(3.15)

of E.

Proof of theorem 2. Because of lemma 4, we see that {u,) is bounded in L”([a, b] x [6,00)) for every 6 > 0. Hence we can assume that - UXX UQXX

weakly star in ~!,“([a, b] x [a, 00))

and

Ik(X9 01 5

;

If u ,,XXI C in the sense of distribution, &X(x, t) 5 C

for any (x, t) c [a,blX [d,00). we have from (3.15) for a.e. (x, t) E [a, b] X (0, 00).

This completes the proof of theorem 2. Proof of theorem 3. Using the comparison theorem with (Hl), and the facts in theorem 2, we see that {u,) is bounded in L”([a, b] x [0, a)) and L”([6, 0~) : W2s”([a, b])) for any 6 > 0 and (u,,) is bounded in L2([a, b] x [0, 00)). Indeed, integrating (1 .l), over [a, b] and noting that U_(Q) t) 1 0 and uEX(b,t) I 0, we have

1005

Degenerate parabolic equation

from which it follows that

In a similar way to lemma 2 we have is bounded in L’([S, T] : L”([a, b]).

l%]

for any 6 and T such that 0 < 6 < T. Hence we can extract a subsequence (u,J of (u,) such that weakly star in ,?‘((a, b) x [0, 00))

UEi- u

in L”([6, co) : UW -

W’*“([a,

b]))

in L2([a, b] x [0, 00))

weakly

Ux

and

and weakly star in L’(S, T: ,?‘([a, 61))

u,,t - ut

for some u E L”((a, b) x [0, 00)) n ,!_“([a, a) : W2s”([a, b])) with u, E L2([a, b] x [0, co)) and u, E L2(S, T:L”([a, b])) (0 < 6 < T). In order to prove the strong convergence of u,~, in the topology of L2((a, b) x [0, T]), we shall first show that, as i + 00 h&h((a,b)

x [O,T])

+

h&2((a,b)

x [O,T])*

Writing ~4,~as ui, we have from (l.l), U mt -

unt =

(urn

+

G?z)u*,

-

(4

+

042,

-

an*12 - l%A2).

Integrating this equation over [a, b] x [a, T] (6 > 0), we get (lUm,(x,

t)i2

-

IU,x(x,

U,(%

l))dw

t)i2)

dxdf T

b

I

=-

(U,(X

t)

-

+

&I

a

-

E”

turnAh s

s)

-

%x(6

s)] ch

6

T (Unx@,

4

-

u,,(a,

@I

d&Y.

s6

From lemma 3 we have b[um/“dxdl+Cj.b]u,LI (I

u,] d.x -I- CT(ez2

+ ~2’~).

(3.16)

In much the same way as in the proof of theorem 1, we can show that for any 6 and T such that 0<6
strongly in L2([a, b] x [a, T]).

Then as n tends to infinity in (3.16), we have (u,,12

dxdt I

T bIuX]2dxdt+C s.i0 II

bl~,-~l~+CT~~2 s L1

(3.17)

I. FUKUDA and M. TSUTSUMI

1006

from which it follows that lim = b JU*,lQlxdt I T blU,IQxd~. m--)- 3‘.r 0 D 0 (1 1.l’ On the other hand, noting the lower semicontinuity of L2-norm, we have T

(3.18)

b

lux)2 dxdt I lim inf T b )u,12 dxdt. m-- s!;0 a .i!0 a Equations (3.18) and (3.19) yield that

(3.19)

lim ’ b ]n,j2dxdt = ’ b ]UX)2dxdt. m-m si0 n is0 a Since U, converges to U, weakly in L2([a, b] x [0, T]), we conclude that strongly in L2([a, b] x [0, T]).

r&X + n,

(3.20)

is nonincreasing with respect to t, it follows that I1u(t)llLZco,b)is continuous at at t = 0 in L2(a, b). It is clear that u is continuous on (0, 00) since U, is bounded in L2([a, 61 x [a, T]). This completes the proof of theorem 3. Since

iU(f)k(a,b)

t = 0, and then u(t) is continuous

4. UNIQUENESS

OF SEMICONCAVE

Let u and v be two weak solutions of (l.l)-(1.3)

WEAK

SOLUTIONS

with initial data no and no, respectively. Let

T > 0 be fixed and put w(x, t) = u(x, t) - u(x, t) and we(x) = u,(x) - uo(x). Then we have b

b

c

w(x,

TMx,

T)

dx

=

i

JO

woWw(x,

0)

d-x

+51’ T b

(wwt - (uu, -

0

uo~x -

(A +

l)(lu,12- bx12h4~dt

(I

(4.1)

for any w E C2.‘([a, b] x [0, 00)) with compact support in (a, b). For each n E N, define

g,(x)

=

i

-1

1

1

ifs<-

ns

if Is] I f ifs<

n

--

1 n

and Y = (g,((u2

-

u2)8/&9,)* py* CT@*

p, *

a@,@,

where p, and or are the standard molifiers with respect to x and t, respectively; ek(x/k) where 0 E C,“((a, b)) with 0 5 0 5 1 and B(x) = 1 in a neighborhood of 0 (we may assume 0 E (a, 6)) and o,(t) E C,“((O, 00)) such that 0 5 0, 5 1, o,(t) = 0 near 0 and o,,,(t) tends to the indicator function of [si , s2] (0 < s1 < s2) as m -+ 00. Then Y E Cr((a, b) x (0, m)) and Y(x, t) 1 0 for any (x, t) E (a, b) x (0, 43).

Degenerate

parabolic

equation

1007

Substituting Y for a test function w in (4.1), we have T

b

o=

(-WY’,

+

(uu,

-

vv,)‘l”,

+

(A

l)(~~,~~ - ~~,~~)‘I’)d.xdt.

+

jj0 (1

(4.2)

One can calculate the first term as follows: -

wY’,dxdt = -

(we, (0,)~ * P, * a&(g,,((u2 - u2)& 0,) * P, * 0,) d.x dt

+

(we,

@,A

* pv

* a,, k,

((u’

-

v2)e,

@,)

* P,

* apI

dx

a.

Letting v and p tend to infinity in (4.2), we can easily see that wek(0,),g,((u2

-

(we, o,),g, ((~2 - 2Y)eko,) dwdt

dxdt +

v2)ek0,)

w)kn((~2

-

u2)ek

wl,

d-x dt I

(1~~12- W)e,o,g,W

- u2)e@o,) dxdt

1

= Z,(k, m, n) - Z,(k, m, n) - Z,(k, m, n) = 0.

(4.3)

As n tends to infinity, we find that I,&, m, n) tends to T I;@,

m) =

T

b

since sgn((u2 - u2)Ok0,) we have f,(k,

= sgn(w&@,)

= sgn(w8,).

0 a

(4.4)

ss

Moreover, 0, (t) = 0 near 0 and T, then

dW~k%l)f~df -

m) =

b

we, sgn(wf3,)(0,), du dt

(we,@,), sgn(w&@,) dxdt is0 (1

T

b

0 a

Iwe,Iw,

dxdt

ss T

b

.Is

=-

0 (I

I

wbl(w, dxdt.

(4.5)

As for Z,(k, m, n), using the chain rule, we get T

b

(UU, - nux)2g,((n2 - u2)ek~m)ekOm dxdt

Z,(k, m, n) = -2 1s0 (I -

-

(2424, -

u~x)g,((t42

-

u2)ekc9m)(242

(UU,

uu,)g,

-

3)e,

-

((~4~

0,)02,

-

u2)02,ek(ek)x

ek (e,),

du

dt.

ti

dt

I. FUKUDA and M. TSUTSUMI

1008

Since the first term on the right-hand side is nonpositive and

we have

Mk m,n) 5 ~Cll&= + II42 + I141L-+ I141L-)(ll~xllL~ + Il4l2> where Lp = LP([a, b] x [0, T]) (p = 2, 00). Since IIuIILm,IIz&-, llu,llL2and IlvxllL2are bounded, we get I,#,

m,

n)

5

(4.6)

z

where C depends on IlullL-, II&-, \Iu,llLz and llu& and independent of m and n. Since sgn((u’ - v~)~J~@,) = sgn(w&@,), letting n + 00 we see that I&, m, n) tends to I;(k,m)

= -(A

+ 1)

(lux12 - )u~~~)&o~ sgn(w&@,) dxdt.

Recalling that u and u are semiconcave, we have T f3(k,

m)

=

-(A

-

1)

+

(A

T

b

.ii0

a

T 5

(A

+

+

(1

1)

a T

SC

1)

T

b

ii 0

a

(U -

b

1s0

+

hm,Im a

is0

1)

+

b

+ 0,)d.xdf

u)(u,

-

~,)(e,),o,

sgn(w&@,) dx dt

Iwe,0, I(u, + u,,)b dt b

(1~1+ I4)d4 + I~xl)l(~,),ld-x& ss0 (1

Iweko,l

dxdt + ;.

(4.7)

Hence equation (4.3) with (4.9, (4.6) and (4.7) implies that -

Iwe/J(@,),dxdt

T

b

0

a

Iweko,l

5 c ss

In (4.8) letting k,

m

-+

00,

we

dxdt + ;.

find that

b

a

Iw(x,s2)I

dx

-

sa

for any sr and s2 (0 < .sr < s2). As s2 = t and s1 tends to 0, we have

s

blW(X,t)ldx-

a

SC sbIWo(X)mxC a

c o f 1 Iw~x,d dx*

(4.8)

Degenerate

parabolic

equation

1009

from which it follows that, for any t 1 0

3b

Iw(x, t)l dx

5

I

ecr :

a

Iw,(x)l dx.

(4.9)

This completes the proof of theorem 4. The corollary is easily obtained from (4.9). REFERENCES 1. BERTSCH M., DAL PASSO R. & UGHI M., Nonuniqueness and irregularity results for a nonlinear degenerate parabolic equation, in Nonlinear Diffusion Equations and Their Equilibrium States I (Edited by W. M. NI, L. A. PELETIER and J. SERRIN). Springer, Berlin (1986). 2. DAL PASSO R. & LUCKHAUS S., A degenerate diffusion problem not in divergence form, J. diff. Eqns 69, 1 (1987). 3. KINDERLEHRER D. & STAMPACCHIA G., An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980). 4. LIONS J. L., Quelques M&hodes de RPsolution des Probltimes aux Limites Non Linkaires. Dunod, Gauthier-Villars (1969). 5. UGHI M., A degenerate parabolic equation modeling the spread of an epidemic, Annali Mat. pura appl. 143, 385

(1986).