Degenerate parabolic equations with general nonlinearities

Nonlinear Analysis, Theory. Methods & Applications, Vol. 4, No. 6, pp. 1043-1062 Q Pergamon Press Lid. 1980. Printed in Great Britain

DEGENERATE PARABOLIC EQUATIONS WITH GENERAL NONLINEARITIES

ROBERT KERSNER

Computer and Automation Institute of the Hungarian Academy of Sciences, Budapest*

(Received 15 June 1979; revised 10 October 1979)

Key words: Nonlinear degenerate parabolic equation, filtration with absorption.

1. INTRODUCTION

IN THE present paper we shall investigate the Cauchy problem for the equation

a, = c441,, - 44 (1.1) defined in the halfspace R: = {(t,x): t > 0, - 00 -=c x < co> with initial value

u(0, x) = z.+)(x). (1.2)

It is meant by the adjective ‘general’ in the title that (1.1) is not supposed to be close to the model equation

u, = (us),, - c&, (1.3)

where ,u > 1, v > 0 and c0 > 0 are constants. To be precise, we shall impose only the following restrictions on functions a(u) and c(u):

Assumption k a(u) E C’([O, M]) n C3+‘((0, M]) for any M > 0; CC E (0, 11.

a(u) > 0, u(0) = u’(0) = 0,

44 > 0, u’(u) > 0, u”(u) > 0 when u > 0.

Assumption C. c(u) E C([O, M]) n C'((0, MI),

c(u) 3 0, c(0) = 0,

44 > 9 c’(u) > 0 when u > 0.

The function L+,(X) is nonnegative, continuous and it has compact support. We shall assume that supp u,,(x) c [ - 1, I], I > 0.

* H-1014 Budapest I, Uri utca 49.

1043

1044 R. KERSNER

It is to be noted right now that the results of this paper are valid in the case when a’(O) > 0 and c(u) s 0 or for the more general equation

n, = [441,, - v441x - C(U)> (1.4) where b(0) = 0, b(u) E C’([O, M]) n c?+~((O, M]).

When c(u) z 0, (1.1) is called the filtration equation (or the equation with nonlinear heat conductivity). When c(u) + 0, it is called the filtration equation with absorption (or the equation with nonlinear heat conductivity and absorption). The term [b(u)lX in (1.4) corresponds to the transport of matter or heat. One considers such a terminology for equations of the form (1.1) reasonable, although problems from other fields of natural sciences (e.g. plasma theory, magnetic hydrodynamics, boundary layer theory, etc.) may lead to these equations, too.

Equation (1.1) is not bound at all to degenerate under assumptions A and C, i.e. the solution of problems (1.1) and (1.2) may not turn into zero anywhere in R:. Distinguishing the class of really degenerate equations is not a trivial task at all.

It is known (see [18, 4, lo]), that problems (1.1) and (1.2) have no classical solution in general even if c(u) 3 0, i.e. the derivatives in (1.1) do not exist for sure.

Definition for the generalized solution will be given in the next paragraph We mention that continuity is needed.

The basic idea for proving theorems on existence in the theory of degenerate equations has appeared in the paper by Oleinik, Kalashnikov and Chzon Yui-Lin [17] where the filtration equation was considered This idea consists of the following:

Function u&x) is approximated by positive smooth functions uOn(x). Then in the domain Q, = (0, n) x (-n, n), the first boundary value problem is solved for u,(t, x):

U nt = P4TLx - C(U,)> (1.5)

u,(O, x) = u,,(x), xE(--,n),

u,@, +n) = q)“(k$, t E (0, n). (1.6)

Equation (1.5) is proven to be a non-degenerate quasilinear parabolic equation. The theory of such equations states classical solvability of problems (1.5) and (1.6). Then the existence of lim u,(t, X) = u(t, x) is verified Due to its construction, function u(t, x) satisfies the appropriate integral identity. Finally the continuity of u(t, x) is shown

Accomplishing this program one encounters two serious difficulties. The first of them is proving that (1.5) is non-degenerate. For this purpose we have to show that

u,(t, x) 3 E, > 0 (1.7) in Q,.

A simple analysis of the corresponding proofs of [2] shows that the witty method of Aronson invented for the filtration equation u, = (I&‘),.~ p > 1 is applicable to the problems (1.5) and (1.6) provided that function c(u) satisfies the Lipschitz condition at zero. For (1.3) it is equivalent to the inequality v > 1. The fulfillment of inequality (1.7) was considered to be obvious in case of v < 1, too.

Unfortunately this is not the case. It turns out that the following unexpected fact arises: in case of ‘v < 1’ when n 3 no the solution of problems (1.5) and (1.6) turns into zero in Q. on a set with positive measure.

Degenerate parabolic equations with general nonlinearities 1045

Here we shall not prove this. But we give a counterexample which shows that such a phenomenon may arise. The details are found in Section 5.

Thus when proving the existence of solution for the problems (1.1) and (1.2), a new idea is needed In the next section we adhere to one of these ideas that serve the purpose.

Roughly speaking the regularization found here is as follows:

u, = [441,, - 44 + C(E)>

u(0, x) = uo(x) + E. (1.8)

The solution of problems (1.1) and (1.2) is yielded as the limit of the sequence of solutions u,(t, x) to problem (1.8). The sequence u,(t, x) plays an important role when proving uniqueness and comparison theorems (see Sections 3 and 4). In proofs of these theorems, we apply Holm- grem’s method in the form given by Kalashnikov [9, 121.

The second difficulty is to prove continuity of the generalized solution Sections 6 and 7 are

devoted to this problem. In order to tell the preliminaries of the results obtained here we introduce the functions

44 44 G(u) = ~

u

Here and further on h-’ denotes the inverse function of h: h-‘(h(x)) z x. Suppose that A(u) < GO when u > 0 (‘finiteness of pressure’). The case of A(u) = co will be

discussed shortly at the end of the introduction. It turns out that the modulus of continuity of the generalized solution depends on the be-

haviour of function G(u) in the neighbourhood of u = 0. When u > 0, G(u) supposed to be monotone (decreasing, increasing or equal to a constant). In detail: provided that G(u) d const. when u 3 0, the generalized derivative (a/&x) [A( u )] is b ounded when t > T P- 0, while the generalized derivative (a/&x) [f-‘(u)], p rovided that lim G(u) = co when u -+ 0, is bounded when t > z > 0. Here f(z) = a-‘(F-‘(z)) (The precise formulations are given in Theorems 6 and 7).

To make things clear we ponder on what these results mean in the case of model equation (1.3). We have

A(u) = Ku”-‘, F(u) = (p - v)- lKu@-V)Q~, G(u) = ICuP+“-‘,

where K = K(p, v, cO) denote different constants. If p + v 3 2 then G(u) d const. In this case the derivative (up-‘), is bounded which is the

same as for filtration equation It is known that this result is sharp (due to Aronson [l] for c0 = 0 and due to Kalashnikov [li] for c0 > 0).

If ,U + v < 2 then lim G(u) = co provided u -+ 0. Now v < 1 hence v < p and the function F(u) is defined We have in this case that the generalized derivative (u(~-“)‘~)~ is bounded This result is also due to Kalashnikov [ll].

Naturally, any power of function u(t, x) is not differentiable in general, i.e. u(t, x) is not bound to satisfy Holder’s condition.

1046 R. KERSNER

If A(u) = 00 and certain condition on monotony of a-‘(u) is fulfilled then u(t, x) > 0 when t > 0 provided G(u) < const. [12], hence u E C”.

If A(u) = co and lim G(u) = 00 when u -+ 0 then, in general, this does not hold For example, if u(t, x) is the solution of problems (1.3) and (1.2) and p = 1, v < 1 then it is known that

supp u(t, x) c [0, T] x C-L, L] for sufficiently large T > 0 and L > 0. But if moreover F(u) < co (when ,U = 1 and v < 1 it holds) then the result is analogous to the case of A(u) < co. Logically, the case of A(u) = co, F(u) = cc and lim G(u) = cc remained unconsidered Due to our results, one easily obtains boundedness of the generalized derivative [a(u)], that is valid in any case. Undoubtedly, this result can be improved.

As regards the multidimensional case here the strongest results belong to A Friedman and L A Caffarelli [7].

They have proved the continuity of the solution for equation u, = Au” (indicating moduli of continuity). This paper refers to Aronson and Benilan [3], where similar problems are considered for equation u, = Acp(u). Recently Brezis and Crandall [S] published a preprint that presents a new proof of the uniqueness theorem for this equation

2. EXISTENCE THEOREM

Definition We shall say that the nonnegative continuous function u(t, x) is a (generalized) solution of the problems (1.1) and (1.2) in R: if ~(0, x) = r+,(x) and for any t,, t,, x0, x1 such that

t, < t,, x0 < x1 and P = [t,, tl] x [x0, x1] c R:, the following integral identity holds:

I(& f, P) = ss

*; x1 [a(u)f,, + uf, - c(u)j] dx dt - x0

~~~u~dx(::-~~a(u)/;dr/::=O. (2.1)

Here f(t, x) E C:;“(P) is an arbitrary function that equals zero when x = x0 and x = xi.

THEOREM 1. There exists a generalized solution u(t, x) of problems (1.1) and (1.2). In the points (t, x) E R: where u(t, x) > 0, it satisfies (1.1) in the usual sense.

Proof Let M = max u,,(x) + 1; E E (0, I). For fixed E, construct the sequence of smooth functions u,(x; E, n) (belonging to Cm) with the following properties:

(i) E < u,(x;s,n) < M, whenxE(--,n); (ii) uJ+n;s,n) = M

(iii) u,(x; E, n) is strictly monotonically decreasing with respect to n and tends to u,,(x) + E

when n---f co.

We recall that supp u,,(x) c [ - 1, I]. Assume that n > 1 + 1. The following statement is basic for constructing the generalized solution of problems (1.1) and (1.2).

LEMMA 1. Let Q, = (0, n) x (-n, n). The first boundary value problem of the form

L(t), E) E -I), + [a(~)~~ - c(u) + C(E) = 0 in Q,,

l’(0, x) = &&x; E, n), x E ( - n, n),

o(t, + n) = M, t E (0, n),

(2.2)

(2.3)


has unique classical solution in Q, such that for any n

0 < E < u(t, x) < M in Q,.

(2.4)

Proof of Lemma 1. Following [2], for r E R’ denote by q(r) an infinitely differentiable function such that q(r) = r when r 2 E, q(r) = &/2 when r < 0 and cp increases in the interval [O,E].

Consider the first boundary value problem for the equation

U1 = u’(&:))ll,, + u”(&))t’; - c@(r)) -t C(E) (2.5)

with boundary values (2.3). Due to Theorem 4.4 in [15], this problem has a unique classical solution such that 1111 d M. We shall show that v 2 E in Q,.

Let z(t, x) = e-‘(0 - E). We have ~(0, x) B 0 (see (iii)), z(t, &n) = e-‘(M - E) < 0, hence z(t, x) 2 0 on the parabolic boundary of Q, Suppose that z(t, x) takes negative values in (0, n] x (- n, n). Then it must have a negative minimum in a point (to, x0) for which the inequality

z, - a’(&))zxx d 0 (2.6)

holds. As u = e’.z + s, from (2.5) the identity

z, - 4’(&))zX,, = -2 + e-‘[c(s) - c(qo(e’z + E))]

follows (as a”(q(o)) elzz = 0 at the point (t,, x,)). Due to properties of function q(r), the right-hand side of this identity in the point (t,, x0) is

strictly positive which is in contrast to (2.6). Lemma 1 is proved.

Let E = l/n. We show that sequence u, = u(t, x; E, n) where u(t, x; E, n) is the solution of the problems (2.2) and (2.3X is monotone in Q,.

LEMMA 2. If s1 > s1 then u,, > uEZ in Q,.

Actually, due to (iii), ~~~(0, x) > u,(O, x). Further ~,~(t, kn) = M > uE,(O, k n) for, if it held uE2(t0, _+n) = M in a point t, then we would have t?, 2 0, u,, < 0 where r = u,? in the same point.

But, as 0 = L(v) E - 11~ + u’(u)uxx + u”(u)i:~ - c(u) + C(E) in Q,,, and so one obtains

0 = - 11, + a’(o)i!,, - c(M) + C(E) d -c(M) + C(E) -=c 0

for the points (to, +n). Further, in Q, we have the relations

L(“,i, &i) = ‘7 i = 1,2,

that follow from the conditions. Thus Lemma 2 is a consequence of Friedman’s comparison

theorem [6]. According to the Theorem on the convergence of monotone bounded sequences, for any point

(t, x) E Rt there exists lim U, if n + 00. Denote u(t, x) this limit.

1048 R. KERSNER

The functions u,(t, x) satisfy (2.2), hence the integral identity

f(t, x) dt dx = 0

holds. Following Lebesque’s theorem we can change over to the limit in this identity when n + co. Identity Z(u, f,P) = 0 shows that the constructed function satisfies (2.1). Continuity of u(t, x) will be proved in Sections 6 and 7.

For the points where u(t, x) > 0, one obtains smoothness of u(t, x) by standard methods. Theorem 1 is proved.

3. UNIQUENESS

THEOREM 2. The generalized solution of problems (1.1) and (1.2) is unique.

Proof Denote ui(t, x) the solution constructed in the previous paragraph It is known that u,(t, x) = lim u,(t, x), where un are positive smooth functions (n-i d u, d M) which satisfy the integral identity

I(un, f, P) + c x) dt dx = 0. (3.1)

Let u2(t, x) be another solution of the same problem and let u1 # u2 in the point (t2, x2) E R: (ul and u2 are not to be confused with the first two members of the sequence u”). Due to continuity of the functions u,(t, x), there exist E > 0 and an interval t = t,, x2 - E d x d x2 + E such that ui # u2 in this interval. Denote by o(x) an infinitely differentiable function with closed support such that supp o(x) = [x2 - E, x2 + E] and w(x) (u, - UJ > 0 on the above defined interval. If we show that

s cc [U&, 4 - UJt,, x)-j44 dx d 0 -cc

(3.2)

then the statement of the theorem follows from the contradiction obtained. Let T E (0, n), r ~(0, n) such that T > t,, r > (x21 + E. Set (see (2.1)) t, = 0, t, = t,, x0 = -r,

x1 =randsoP=(O,t,) x (-r,r). Introduce the notations

s 1

A, = A#, x) = a’(Ou, + (1 - t?)u,)d@ 0

s 1

c, = cI,(t, x) = c’(Ou,, + (1 - &.Q) do. 0

The function u,(t, x) satisfies the integral identity (2.1) hence

12 I I(u,, f, P) - I@,, f, P) + c(n-- ‘)

ss f(t, x) dx dt = 0

0 --r


or what the same is:

s ’ ’ [%(L x) - %(t,, x)] f(rz, x) dx = [u,(O, x) - ~~(0, x)]f(O, x) dx

--I s --r

s

f2 C&J - 4~JIL

I s dt + t2 [4qJ - 4u,)lf, dt

0 X=l 0 x=--I

12 I

+

ss

12 r

[A,& x)fx, + f, - C,(c 4fl (u, - ~2) dx dt + c(n- ‘)

ss

f(t, x) dx dt. 0 --* 0 --I

Further we shall suppose that c(s)/s and c’(s) are monotone and

lim+ = ~0. s+O

(3.4)

If s- ‘c(s) < const when s E [0, M], then the constructions given below can be simplified In the rectangle (0, T) x (-r, r) c Q, we construct two sequences of smooth positive func-

tions &Jr, x) and Cnkr(t, x) with the following properties : {Ankr} is monotonically decreasing and uniformly tends to A,, when k -+ a3; {C,,,} is monotonically increasing and uniformly tends to C,, when k -+ co. Remark (as a” > 0)

that a’(&, + (1 - 0) UJ 2 d(B(l/n)) h ence A,, 2 nu(l/n) = 6, and so we have Ankr 3 6,. Assume

that Ankr d M,. From (3.4) one obtains that c’(s) decreases, thus c’(Bu. -t (1 - ~9) UJ d c’(@(l/n))

and

Cllkr < C, < nc(n-‘). (3.5)

Rewrite the identity (3.3) in the form

s r Cu,(rz> x) - uz(r,, $1 f@,, 4 dx = --I s

; r [u,(Q 4 - @> 41 f(o, 4 dx

s

12 ” ’ - [ a(~,) - 4Q-j f,j;I’, dt + (24, - Ankr) (un - u2) f,, dx dt o

ss 0 --I

f! I

+

ss

” * (Cnkr - C,) (u, - u2) f dx dt + b4dxx - f, - C,,,fl (un - 4 dx dt. 0 --I ss 0 -r

It is known from the theory of linear parabolic equations [6] that the following first boundary value problem has unique classical solution f = fnk*:

Lf - Ankrfxx + f; - Cnkrf = 0 inp, (3.7)

f(t*, xl = 4% f(t, +I) = 0. (3.8)

We will give some properties of the function f(t, x). To prove them, the results in [16] are used

LEMMA 3. F(t, x)( d max [w(x)1 = M, in P.

The statement is a consequence of the maximum principle.

1050 R. KERSNER

LEMMA 4. There exist numbers y > 1 and M, = M,(y) > 0 such that

If(t,x)l < M&l + Ix~)-~ inP.

ProoJ: Denote by x3 the number which belongs to (0,r - 1) and let [x2 - E, x2 + E] c [ - xg, xJ. Moreover, let P, = (0, r2) x (0, r).

Set z(t,x) = M&l + xJ(1 t ~)-~e~(‘~-*~. W e shall compare the functions f(t, x) and z(t, x) in P,. Let w = z ) f: We have

w&,x) = M&l + xJ(1 + x)-? f o(x) b 0

as for x > xs, o(x) = 0. Further, w(t, r) > 0 and w(t, 0) = M&l + x~)~ eO@-‘) + f(t, 0) 3 M, i- f(t,O) > 0 if M, > M, (see Lemma 3). Thus w 3 0 on the parabolic boundary of P,. Hence from inequality Lw < 0 in P, it follows that w 3 0 everywhere in P,. We have

Lw = Lz = A,,,M,(l + xJy(l + x)-?(l + ~)-~e~(~~-~)y(y + 1)

- /3M,(l + x&Y (1 + x))’ eB(f2-t) - Cnkrz < 0 ?

if, for example, j3 = My(y + 1). From w 2 0 it follows that

IfI < z d M&)(1 + x)-' inP+.

In P_ = (0, tJ x (-r, 0) we can proceed similarly.

LEMMA 5. laf/axI( ,X, = I d M,rpY, where M, = Me(y).

Proof Consider the auxiliary function

w(t, x) = f(t, x) + M,r-‘(x - r)

in the cylinder P(r, r - 1) = (0, t2) x (r - 1, r). Obviously, w(t, r) = 0. Moreover, we have (see Lemma 4)

w(t, r - 1) = f(t, r - 1) - M,rvY < M,rPY - M,rpY < 0 if M, = M, + 1.

w(t,, x) = M,FY(x - r) -c 0 as xg < r - 1.

If Lw 2 0 then w assumes its maximum either at t = t, or at x = r - 1 or at x = r, i.e. w < 0 in P(r, r - 1) hence (8w/ax)l,,, > 0, i.e. (8f/ax)l,,, 3 - M,rWY.

We find

Lw = Lf + L(M,rdY(x - r)) = Cnk,M4rpY(r - x) 2 0.

In order to obtain the upper estimate, one has to use the function w = -f + M,r-7(x - r).

In case of x = --I we proceed similarly.

LEMMA 6. f, r

ss (f,,)” dx dt d M,,

0 -I

where M, = M&n, r) (and M, does not depend on k).


ProoJ Multiply (3.7) by fXX and integrate the obtained identity over P:

12 I

ss AnkrfX; dx dt = - *’

ss ’ f,f,,dxdt + ”

0 -r 0 --I ss ’ Cnkrffxx dx dt = I, + I,.

0 --I

tl

6, ’ ss f,‘X dx dt d M, + I, where M, = ’ dx 0 -r

It follows from (3.5) and Cauchy’s inequality hat

and therefore

which was to be proved. Now we substitute function f = fnkr(t, x) into (3.3) and apply Lemma 34 (with y = 2):

+ M4t2 z; la(u,(t, 4) - a(u2(t,r))lre2 + M4t2 max (a(u,(t, - ~9) - a(u,(t, - d)(~-~ rCr2

+ “,“X IA, - A,,krl max [u, - u21Mlo(n)$ P

s I

+ max jCnkr - C( max (u, - u21t2 dx

P P

_-r (1 + lx\)2 + c(n-‘)Mw

When proving (3.2) (i.e. Theorem 2) it is sufficient to change over to the limit in the latter inequality, at first with respect to k + co, then n -+ co and finally I -+ GO.

4. COMPARISON THEOREMS

In this paragraph we shall formulate and prove statements of the type “theorems on monotone dependence of the solution on problem data”. This field is interesting in itself, besides, it is important when investigating qualitative properties of the solutions to problems (1.1) and (1.2).

Let G be a closed subdomain in ?i”, (G = @ is allowed).

Definition. A nonnegative, bounded, in G continuous function u(t, x) is called the (generalized) supersolution of (1.1) in G, if for u(t, x) the intgral inequality (see (2.1))

I(l4f ; P) d 0

holds for any in P c G nonnegative function f E C:,;“(P) such that f(t, x0) = f(t, x1) = 0.

In the following a simple sufficiency condition will be given that ensures a function to be a supersolution.

1052 R. KERSNER

LEMMA 7. Let ~(t, x) be a continuous nonnegative function in G and let it be smooth (at least let it belong to C:;‘) outside of a finite number of continuous curves of the form x = i(t) and let it satisfy the inequality

- 1:, + [@)lXX -C(U) < 0

in G. Besides, we assume &(u)/& is continuous when x = i(t). Then ~(t, x) is a generalized supersolution of (1.1) in G.

Integrating by parts yields the proof.

THEOREM 3. Let u(t, x) be the generalized solution of problems (1.1) and (1.2) and ~(t, x) a generalized supersolution of (1.1) in R:. If u& x ) < ~(0, x) when x E R’, then u(t, x) d u(t, x) holds everywhere in R:.

Proot Suppose that u(t,, x2) > ~(t,, x1) in the point (t2, x2) E R:. The continuity of u(t, x) and ~(t, x) ensures that u(t, x) > o(t, x) in some interval E = {t = t,, x2 - E d x < x2 + E).

Due to the properties of the sequence u,(t, x) constructed in the second section, we have u,(t, x) 3 u(t, x) > v(t, x) in E.

Let the numbers 7; x3 have the same meaning as in Theorem 2. When I < n - 1, the functions u,(t, x) satisfy (2.2) in P in the usual sense (E = n-l), therefore

I@“, fi P) + c(n- ‘)

Since I(u, f ; P) < 0, we have

f(t, x) dt dx = 0.

I(v,f; P) - I(u,,f; P) d c(n_1) ss f(t, x) dt dx. 0 --*

As it was done in Section 3, starting from (4.1) we come to the inequality

(4.1)

s ’ [q,(t,, xl - @,, 41 f(t,, x) dx d -, s 1, [u,(O, x) - #2x,] f(O, 4 dx

s *’ [u(u) - u(u,)] fxl’Lr dt + ” ss * + (A,, - &J (u, - 1,) f,, dx dt (4.2) 0 0 --*

fl. r t2 ?

+

ss

(Cnkr - C,)(un - u)f dxdt +

0 --I ss

(&,Ji + f, - C,,,f)(u, - r)dxdt 0 --I

+ c(n_') f2

ss

' f(t,x) dx dt. 0 --*

Here the sequences A,, and C,, are the same as in Section 3. We also suppose the fulfillment of inequality (3.4).

Let o(x) be an arbitrary smooth function which satisfies the conditions: o(x) > 0 when XE(X2 - &lx2 + E) and o(x) = 0 outside of this interval.


Consider the following first boundary value problem for function f = fnkr(t, x):

-&f = Ankrfxx + f, - c&f = 0 in p,

j-(&,X) = 4x), f(t, +r) = 0.

This problem has a unique solution f(t, x) with continuous patial derivatives f,, f,,, f,, in P (SW [6]). It follows from the maximum principle that f(t, x) 3 0 in P. Therefore, the inequality (4.2) have to be fulfilled when we substitute the solution of this problem for thef(t, x) of (4.2). Since UJX) d ~(0, x), we have u,(O, x) - ~(0, x) d ~~(0, x) - U&X), hence

s ’ ’ [u,&, x) - I’(& x)] W(X) dt d [u,(O, x) - I,] f(0, x) dx --I s --I

t2

” * + s

[u(v) - u(u,)] fzlLr dt + ss

(A, - A,,J (u, - ~:)f,, dx dt

0 0 -r

” ’ + (cnkr - C,) (un - zt)f dx dt + c(n-‘) ss

f(x, t) dx dt. 0 --I

(4.3)

Proceeding similarly as we did at the end of the proof of Theorem 2, we come to the inequality

XZ+f

s [~(t,, x) - I@,, x)] o(x) dx d 0

X,-E

which is in contrast to the positivity of the integrand.

THEOREM 4. Let u(t, x) be the generalized solution of problems (1.1) and (1.2) and z)(t, x) be a generalized supersolution of (1.1) in

G={(t,x):O<t<co, s<x<co}

(or H = {(t, x): 0 < t < co, - co < x < -s>)

where s is an arbitrary number. Let uo(x) < ~(0, x) for s < x < co( - cc < x < -s, respectively) and let u(t, s) < z(t, s)/u(t, - s) < ~(t, -s), respectively for t 3 0. Then u(t, x) ,< z!(t, x) everywhere in G (or in H, respectively).

Proof Assume s = 0 and maintain the notations of the previous theorem. Similarly as getting to the inequality (4.3), we can obtain the inequality

s ’ L-u&p xl - 4~ 41 f(tz, xl dx G

0 s

: [qj(O, xl - ~O(41 f(Q ~1 dx

+

s ” [a(Hk 9) - a(u,(t, r))] f,(t, 4 dt +

0 s

J’ [a(~,,(6 0)) - a(4t, O))] _f,(t, 0) dt

12

+

ss

’ (Cnkr - C,) (u, - 11) f dx dt + c(n- ‘) 0 0

1054 R. KERSNER

For f(t, x) we substitute the solution of the following first boundary value problem:

Lf = 0 in (0, tJ x (0, r),

f(t2, x) = o(x), supp o(x) = (0, r),

f(t,O) = f(t,r) = 0.

0 3 0,

This problem has a solution and it is unique and has the required smoothness. According to the maximum principle, f(t, x) 3 0 in (0, tJ x (0, r).

After substitution we would have completed the proof as it was done in the previous theorem if the third integral were not in the right-hand side of (4.4). Generally, it is not small. But it can be shown that it is nonpositive when n 3 IZ~ hence we may omit it.

Actually, as u(t, 0) < t(t, 0) by assumption and u,(t, x) tends to u(t, x) from above uniformly (due to Dini’s theorem). We have u,,(t, 0) < t~(t, 0) when 0 d t d t,, n > n,. Since f(t, x) 3 0 in (0, t2) x (0, r) and f(t, 0) = 0, we have f,(t, 0) 3 0. But a’ 3 0 thus a(u,(t, 0)) - a(o(t, 0)) d 0.

This completes the proof.

THEOREM 5. Denote 9 the curvilinear trapezoid which is specified by the inequalities 0 < t, < t d t, < cc and t,,(t) < x < cl(t), where ii(t) are continuous functions on [t,, tl]. Denote d9 the union of the lower base (t = to) and the lateral sides (x = ci(t)) of the trapezoids.

Let u(t, x) be the generalized solution of problems (1.1) and (1.2) and let u(t, x) E Ci,;2(9\d9) n C(9) and u(t, X) 2 u(t, x) on d9.

Assume the inequalities

r(t, x) > 0, L2’ 5s - 21, + [@)],, - c(r) > 0

hold in g\a9. Then u(t, x) 3 z~(t, x) everywhere in 9.

Proof For the functions u,(t, x) constructed in Section 2, the relations Lun = -c(n-‘) < 0 in $3\&3 and un(t, x) > t(t, x) on 89 are valid when n 3 IZ@ According to Friedman’s comparison

theorem [6], it follows that u,, > v in 9\d% Nothing else is left but the change over to the limit when n --f co.

5. A COUNTEREXAMPLE

Consider the first boundary value problem

(p > 1, v > 0, c0 > 0):

Lu 5% -u, + (l.QX - couy=O in Q,=(O,n) x (-n,n),

u(O, x) = u&) when -nIIxcn,

u(t, 5 n) = u& + n) when 0 < t -+ n.

(5.1)

(5.2)

It offers no special difficulty to prove that if t+,(x) 3 E > 0 for x E C-n, n] and v >, 1 then u(t, x) b const. > 0 in Q,. Therefore, (5.1) is non-degenerate in this case and the problems (5.1) and (5.2) has classical solution

Now let v < 1. We wish to show that the solution u(t, x) can turn into zero in Q, for this case. We try to explain why this phenomenon is hard to discover.


Naturally, a real counterexample would be convincing But finding a real one seems to be hopeless. It is enough to remark that even in the case of the Cauchy problem for equation (5.1) with v < 1 and U&X) that has a compact support, there exists only one explicit solution and what is more, it is essentially non self-similar (see [ 131).

Another, more practical possibility is to estimate u(t, x) from above by a function which itself turns into zero in Q,. For this purpose one needs a comparison theorem (besides the existence and uniqueness theorems). But if the fact to be proved is valid then with the help of the theory that has existed until now, one may not prove this existence theorem.

In order to prove that u(t, x) can turn into zero in Q, we should develop a complete theory of boundary value problems for (5.1). After having the results of the previous paragraphs, one might do so but in comparison with the Cauchy problem this theory would have a number of peculiari- ties and we shall not build it here.

Assume that such a theory has been developed i.e. it has been proved that the problems (5.1) and (5.2) has the unique generalized solution u(t, x) and the following (“physically obvious”) lemma holds:

LEMMk Let v(t, x) be a generalised supersolution of (5.1), let ~(0, x) 3 uo(x) and v(t, + n) 2 u(t, f n). Then r(t, x) 2 u(t, x) in Q,.

Consider the problem LU E -u, + (u5/4)xx - 4u3L-4 = 0 in Qls, (5.3)

UJX) = u(t, * 18) = 5.

THEOREM. u(t,x) z 0 in the domain {(t,x): 17 + &x2 < t < 18) c Qls.

Pro05 In Qls consider the function

i 0 k(17 - t) + &x2 1 4

if t<17+324x2,

z,(t, x) =

if 17 +$x2 4 < t < 18.

We have

As &~“/dx is continuous, o(t, x) will be a generalized supersolution of (5.3) provided that we show Lu < 0 when u > 0 (see Lemma 7). We find

[ f(l7 - t) + &x2

3

Lu = I{ 2 + 4 x ; x 2 x & [

$17 - t) + hx2 1 +4X;(4x~-1)x4x(&~x2-4}.

1056 R. KERSNER

The quantity in braces is less than

5X4X4X4

324 = -2+g+;<0.

Hence the theorem follows from previous lemma

6. REGULARITY C(u) < const.

Here and in the next paragraph we shall prove that the function u(t, x) in section 2 is continuous. The moduli of continuity which we obtain here, cannot be improved in general.

The idea of proof of the theorems stated below belongs to Bernstein. It was perfected by Aronson (for equation u, = (&)_I in [l] and by Kalashnikov (for equation u, = [cz(u)],J in [ll].

A(u), F(u) and G(u) are the functions defined in the introduction. We shall carry on considera- tions in the region S = [0, T] x R’ where T > 0 is an arbitrary number. Set S(z) = S n (0 < z d t Q T}.

THEOREM 6. Suppose that G(U) is a non-decreasing, bounded function when 0 d u < M and A(u) < co. The the generalized derivative [A(U)], is bounded in S(z). Moreover, if sup l[A(z+,)],\ < co then [A(u)], is bounded in S.

Proof: The generalized solution u(t, x) of problems (1.1) and (1.2) was constructed in Section 2 as the limit of the sequence of positive smooth functions u(t, x; E) which satisfy the equation

u, = c441,, - c(u) + 44 (6.1) in the rectangle Q, = (0, EC’) x (-&I, E-l).

To get the first statement of Theorem 6 it is sufficient to show that

I[A(u(t, x; 4)1,j d M, in Q, n S(z) (6.2)

where the constant M, > 0 does not depend on E. Denote by a(s) the inverse function to ,4(u): A(a(s)) = s for s 3 0. Set

We have the obvious

where primes denote

u = a(u).

identities:

A’(u)u = u’(u), A”(U)U2 = a”(U) u - u’(u),

A’(a(~:))a’(u) = 1, A”(a(u))a’2(u) + A’(a(u)) a”(2)) s 0,

the derivative with respect to argument.

(6.3)

(6.4)

Let (t, x0) E Q, n S(z) be a point such that lx01 < E- ’ - 2. From (6.3) and (6.4) it follows that the function v = o(t, x; E) satisfies the equation

21, = u’(a(v)) vxx + 0: - ___ &(L9) I 44

a’(u) a’(V) (6.5)

in P, = (z, T) x [x, - 2, x,, + 21.


Denote f the Aronson’s function:

f(y) = 2 Y (4 - Y), where N, = A(M,), M, = max no(x).

When 0 9 y < 1 we have

Define the function w(t, x; E) by r = f(w). It satisfies the equation

w, = 44fhN w,, + + fYwq Wf - u,;;;;;;~;w) + u,(f($;f,(w) (6.7)

in P, Denote [(t, x) a smooth function having the following properties (Mi > 0 are constants non-

depending on E): 0 d i Q 1, [ = 1 in P, = (T, T) x (x0 - 1, x0 -t l),

[ = 0 in the neighbourhood of the lines c = 0, x = x,, + 2, ([,I + I[,[ + li,,\ d M,. Consider the function z(t, x) = [‘$ in P,. In the maximum point of z(t, x) the following re-

lations are fulfilled (the arguments of functions are as in (6.7)):

ZX = 0, z, - a’zxx b 0,

i.e.

IW,WXX = -i& (6.8)

Pw,(w,, - a’w,,,) Z (-ii, + a’iixx + 3a’4$) wf. (6.9)

Differentiate (6.7) with respect to x and multiply the result by t2wX and apply relations (6.8) and (6.9). We obtain (in the maximum point of z(t, x)):

- [CC(T) - ii, + u~iixr - 2u+:.

(6.10)

Due to the properties off, the quantity in braces in the left-hand side of (6.10) is strictly positive (see (6.6)). It is obvious that (2 + .“a’) (1 + ~“a’)-’ d 2.

Let

fqw) = 44fW) - 44 4fha f’(w) .

We shall show that aHlaw 2 0 when w > 0, i.e.

c'cd2f '2 - (c - C(E)) (d'f ‘2 + df”) 2 0. (6.11)

1058 R. KERSNER

Recall that u = a@(w)) and u 2 E in Q, therefore c(u) 3 C(E) in the point in question If CC” d 0 then (6.11) holds. If cl” > 0 then for the fulfillment of (6.11) it is sufficient that

c’(u) CP 2 c(u) a”. (6.12)

We have (see (6.4)):

a” A”(a( 21)) A”(u) -Z=-m=-p= a A’(4

--&lnA’(u) = -&In%. I.4

Hence the inequality (6.12) can be rewritten in the form

and it holds on the assumptions. H,,, 3 0 always holds and the corresponding term in the right- hand side of (6.10) can be omitted From (6.10) the inequality

i2w; < MJW,I + M, (6.13)

follows. It is valid in the maximum point of z = c”wz. Thus Iw,( 6 M, in P,. Since A(u) = f(w), we have /[A(u)],/ = f’(w)(w,( ,< M, which was to prove.

One can show the second statement of Theorem 6 similarly: instead of [(t, x), a function c(x) must be taken.

7. REGULARITY: lim G(u) = m

Begin with a simple remark

LEMMA 8. If lim G(u) = co when u + 0 and A(u) < co then F(u) < co.

Indeed, in this case u d const. a’(u) c(u), i.e.

s z

const. c(x) a’(x) dx 3 z2, i.e. 0

&z)[~~c(x)01(x)dx~1’2 < const.?.

Therefore

F(u) = ~~-““‘af(p)[Z~~c(x)d(x)dx]-1~2d~ < const.~~~l(Y’~di < 00.

As F(0) = 0 and F’(u) > 0 when u > 0, F-’ is defined and F- ‘(0) = 0. Set f(z) = a-‘(F-‘(z)). It is not to be confused with the Aronson’s function in the previous

section, We suppose G(u) tends to infinity strictly monotonically when u + 0, (d/du) c(f(u))[f’(u)] - 1 c 0

and the function a’(u) c(u) is strictly monotone (Assumption M).

THEOREM 7. Let lim G(u) = + co when u + 0 and A(u) < co. Then the generalized derivative [f- l(u)], is bounded in S(z). Moreover, if sup 1 [f- ‘(u,)].J < co then [f- l(u)], is bounded in S.


Proof: It is sufficient to prove that

I[_/-‘(~0, x; s’)1,1 d M in Q, n S(z), (7.1)

where u(t, x; E) satisfies (6.1) and the positive constant M does not depend on E. Define the function w = (t, x; E) by

U = f(w).

It satisfies the equation

f’(W) w, = bm4L,~ 4 + 4fbw-‘(w) w,, - m4) + 44. (7.2)

Since

F’(y) = [21c(a-l(s))ds]1’z, F”(y) = -c(~-‘(y))[Z~~c(a-‘(s))dsl~3’2,

we have

[a(f(w))l,,,,” = [F- Wl,,~,” = - F”(f- ‘64) CF’F ‘(w>)l- 3 = +- ‘(a(f(w)))) = cv‘(w,). Hence (7.2) can be rewritten in the form

co- (4) w, = a’(f(w)) w,, + __ w* - c(fW) - 44 f’(w) x f’(w) .

(7.3)

Proceeding similarly as we got to the basic inequality (6.10) from (6.7), (6.8) and (6.9), here we deduce the following inequality from (7.3), (6.8) and (6.9):

As before, the inequality (7.4) is valid in the maximum point of the function z = [“ws. Set

H _ c(fW) -- f’(w) .

In order to prove inequalities of the type (6.13), from which inequality (7.1) and hence, the statement of Theorem 7 follows as well, it is enough to prove the following inequalities:

1. -%M >0 aw’ 1 f

2. a”’ (w,) f'(w) Q M, M, > 0,

1060 R. KERSNER

4. a’(_@)) Q M,g, I I

M, > 0,

a 1 5. G f’Cw) G 0

L-1

Here the constants Mi do not depend on E.

LEMMA 9. lim H(w) = cc when w -+ 0.

In fact,

H(w) = c(u-l(F_l(w)))a'(a-'(F-'(w)))

Since F.-‘(w) = a(u) by definition, we have

1 -l/2

c(u-i(s)) ds .

-l/2 l/2 c(a- l(s)) d.s

1 [

= lim (Cd)2

u-to 2 I$“’ ~(a- l(s)) ds 1 ’ Due to Assumption M, lim a’+ = cc results in lim (cu’) = co.. Thus,

u-0

;\y 2 p

(Cay2

~(a- l(s)) ds = lim (~a’)’ = cc

u-0

and the proof of Lemma 9 is completed If 0 < 6 < w d M, all functions arising in l-5 are continuous and - dH/aw 2 yl > 0 by means

of choosing constants M,, M,, M, one can obtain 2-4. When 0 < w < 6 the inequalities l-5 follow from Lemma 9 and Assumption M.

Theorem 7 is proved

We obtained a uniform estimate with respect to E for the modulus of continuity of u(t, x; E) with respect to x. A theorem by Kruzhkov [14] ( see also [8]) establishes that the estimate for the modulus of continuity of u(t, x; E) with respect to t follows from this fact. Now we formulate this theorem in the most general form and produce its proof changed slightly. We proceed in this way since Kruzhkov’s Theorem is directly applicable in our case only if for 0 d u < M the function c(u) satisfies Lipschitz condition Now we do not require that.

Let R c R” be a bounded domain, Q = (0, T] x CI and u(t, x) be a classical solution of the equation

Lu = u(t, x, u, u,, u,,) - 2.4, = 0 where

u, = grad U, u,, = (uxixj), 1 d i, j d n, [u( d M.

Suppose that the function a(t, x, u, p, r) with p = (pl, . . . , p,), r = (rij) satisfies the following assumptions.

1. The matrix a, E (~,<~(t, x, u, p, r)) is nonnegative. 2. [a(~ X, u, p, r)l < A((pl, . . . , Irijl, . . .I where IPI = (pipi)1i2

and the function A does not decreases with respect to (pi.


3. a(t, x, u, p, r) is a continuous function of y it satisfies Lipschitz condition with respect to p and it has continuous partial derivatives with resect to rij

Let the function w(s) be continuous, non-decreasing function for s 2 0 and o(O) = 0.

THEOREM 8. Let (to, x0) E Q and (to + At, x0 + Ax) E Q. At > 0, d = d&(x,, an). Assume that for the solution u(t, x) of (7.5) the estimate

I@,, x) - a@,, x&l d c+ - x01) holds. Then,

Ju(t, + At, x0 + Ax) - u(t,, x0)1 d ,~x,cp~d [&4 + 144 At + 2MiAxi2 F21 min

provided that d > 0 and /AxI G d. Here

p(p) = A(4Mp_‘,. . ,dij4Mp-2,. . .). In particular,

(u(t, + At, x0) - u(t,, x,,)( < cod(b) = mm [o(p) + ,u((p) At]. O<pSd

Proof Take a number p E [[AxI, 4 and consider the functions

i?‘(G x) = a@,, x0) 5 [w(p) + P(P)@ - to) + 2qx - Xo12P -‘I

in the cylinder Q’ = (t,x): to < t d to + At, lx - x0/ d p}.

Due to Assumption 2 we have

Lv+ < -Au(p) -I- A(4MIx - x0/ p-2,. . .) djj4Mp-2,. . .)

< -p(p) + A(4Mp9,. ..,hij4Mp-2 )...) = 0.

Denote I the lateral sides and the lower base of Q’. It is easy to see that u(t,x) < u+(t,x)

on I. According to Friedman’s comparison theorem (Theorem 16, Chapter 2 in [6]) it follows that u(t, x) Q zl+(t, x) in Q’. Proceeding similarly we get u-(t, x) < u(t, x) in Q’.

From Theorem 8 the estimate uniform with respect to E of the modulus of continuity of the function u(t, x; E) with respect to t follows. Hence u(t, x) is continuous with respect to t as well.

One can make sure for himself that the moduli of continuity with respect to x, stated in Theorems 6 and 7 may not be improved, in general. Example [l 1, p. 671, is suitable for verifying the accuracy of Theorem 6. Recalling that [a(f(w))],, = c(f( w )) , one may construct the exact stationary solution of (1.1) and verify the accuracy of Theorem 7.

Acknowledgement. The author expresses his deep appreciation to A. S. Kalashnikov for his help and criticism.

REFERENCES

1. ARONSON D. G., Regularity properties of flows through porous media, SIAM J. Appl. Math. 17 (1969). 2. ARONSON D. G., Regularity properties of flows through porous media: a counterexample, SIAM J. Appl. Math.

19 (1970). 3. ARONSON D. G. & BENILAN PH., Rtgularite des solutions de I’bquation des miheux poreux dans R”, C.R. Acnd. Sci.

Paris (to appear). 4. BARENBLATT G. I., On the self-similar solutions of the Cauchy problem for nonlinear parabolic equation of non-

stationary filtration, Prikl. Mat, i Mekh. 20 (1956).

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BREZIS H. & CRANDALL M. G., Uniqueness of solutions of the initial value problem for u, - dq(u) = 0, J. Math. Pures Appl. (to appear). FRIEDMAN A., Partial DifferentialEquations of the Parabolic Type. Prentice-Hall, Inc., Englewood Cliffs, N.J. (1969). FRIEDMAN A. & CAFFARELLI L. A., Continuity of the density of a gas flow in a porous medium (to appear). GILDING B. H., Hiilder continuity of solutions of parabolic equations, J. London Math. Sot. 13 (1976). KALASHNIKOV A. S., Cauchy’s problem in a class of increasing functions for the equations of the nonsteady-state filtration type, Vestnik Mosk. Univ. Math. 6 (1963). KALASHNIKOV A. S., On the occurrence of singularities in the solutions of the equation of nonstationary filtration. Zh. vychisl. math. i math. phys. I (1967). KALASHNIKOV A. S., On the differential properties of generalized solutions of equations of the nonsteady-state filtration type, Vestnik Mosk. Univ. Math. 1 (1974). KALA~HNIKOV A. S., The propagation of disturbances in problems of nonlinear heat conduction with absorption, Zh. vychisl. math. i math. phys. 14, 4 (1974). KER~HNER R., On behaviour of heat fronts in media with nonlinear heat conductivity with absorption, Vestnik Mosk. Univ. Math. S (1978). KRUZHKOV S. N., Results on the character of the regularity of solutions of parabolic equations and some of their applications, Math. Zam. 6 (1969). LADYZHENSKAYA 0. A., SOLONNIKOV V. A. & URAL’CEVA N. N., Linear and Quasilinear Equations of the Parabolic Type, A.M.S. Translations of Mathematical Monographs, Vol. 23 (1968). OLE~NIK 0. A. & VENTZEL T. D., The first boundary value problem and the Chauchy problem for the quasilinear equation of parabolic type, Matem. Sbornik 41 (1957). OLEINIK 0. A., KALASHNIKOV A. S. & YUI-LIN CHZOU, The Cauchy problem and boundary value problems for equations of the type of nonstationary filtration, Izv. Akad. Nauk. SSSR, Sot. Mat. 22 (1958).

18. ZELDOVICH YA. B. & KOMPANEETS A. S., On the theory of heat propagation where thermal conductivity depends on temperature, Collection Published on the Occasion of the Seventieth Birthday of Academician A. F. loffe, Izd. An. SSSR, Moscow (1950).

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