Homogeneous diffusion in ? with power-like nonlinear diffusivity

Homogeneous Diffusion in tt with Power-Like Nonlinear Diffusivity

JUAN R . ESTEBAN & JUAN L . VAZQUEZ

Communicated by HAIM BREZlS

Abstract

We study the nonnegative solutions of the initial-value problem

ut ---- (u r ]Ux I p-I Ux)x, u(x, 0) E ZX(R), (1)

where p > 0, r q- p > 0. The local velocity of propagation of the solutions is identified as V = --vx lv~l p-1 where v = cu ~ (with ~ ---- (r + p - - 1)/p and c = (r + p)/(r q- p - - 1)) is the nonlinear potential. Our main result is the a priori estimate

1 (Vx _-> (2p q- r) t

which we use to establish: i) existence and uniqueness of a solution of (1), ii) regularity of the free boundaries that appear when r Jr p > l, and iii) asymptotic behavior of solutions and free boundaries for initial data with compact support.

Introduction

In this paper we are concerned with a nonlinear parabolic equation of the form

u t - = ( O ( u , Ux)Ux) x, ( x E R , t > O ) , (0.1)

where the diffusion coefficient D(u, ux) is given by

D(u, ux) -= Cu r luxl p - ' (0.2)

for some constants C > 0 and r, p E R. Equations like (0.1) with coefficients as (0.2) arise in different physical situa-

tions, some of which we will present below. To begin with, let us remark that for u ~ 0, (0.1), (0.2) is an equation of

parabolic type which, at the points (x, t) where u(x, t ) = 0 or Ux(X, t ) - ~ O,

40 J.R. ESTEBAN (~ J. L. VAZQUEZ

can be either degenerate or singular, according to the values of r and p. Thanks to the particular structure (0.2) of the diffusion coefficient we can write (0.1) in what is called doubly nonlinear form

8u 8 (8u m 8u" p-l~ 8--t" = 8--x ~ ~ ] (0.3)

r where m = 1 + - - and we have scaled out constant factors. Also we will assume

P that both exponents m and p are positive.

An important quantity in our study of equation (0.3) is the local veloeity o f propagation V(x, t), whose expression in terms of u can be obtained by writing the equation as a conservation law in the form

u, + (uV)x = O.

In this way we get

V = --vx Ivx ]P-', (0.4)

where the nonlinear potential v(x, t) is the following function of u

1 _ m p m - - - |

I v m p - - 1 u P if rap4= 1,

1 v = - - l o g u i f m p - ~ 1.

P

(0.5)

Equation (0.4) is usually referred to as a nonlinear Darcy law and appears in different physical settings. For instance, in the study of water infiltration through porous media, Darcy's linear relation

V = --K(O) grad 4> (0.6)

satisfactorily describes flow conditions provided velocities are small. Here V represents the seepage velocity of the water, 0 is the volumetric moisture content, K(O) is the hydraulic conductivity and ~ is the total potential, which can be ex- pressed as the sum of a hydrostatic potential ~o(0) and a gravitational potential z,

= ~v(0) -r z. (0.7)

However, (0.6) fails to describe the flow for large velocities. To get a more accurate description of the flow in this case, several nonlinear versions of (0.6) have been proposed. One of these versions is

V ~ = --K(O) grad (h (0.8)

where o~ ranges from 1 for laminar flow to 2 for completely turbulent flow, cf. [AhS], [Kr], [Vo] and their references. If it is assumed that infiltration takes place in a horizontal column of the medium, the continuity equation

80 8V 8-7 + T ; = 0 ,

Homogeneous Diffusion with Nonlinear Diffusivity 41

(0.7) and (0.8) give

e--? = ~x (n (oy 10~l ~- ' 0~)

with p : 1/~ and D ( 0 ) = K(O)w'(O). Choosing D ( 0 ) : DoO m ~ (cf. [M], [GP]) we obtain (0.3), u being the volumetric moisture content.

Another example where equation (0.3) appears is the one-dimensional turbulent flow of a gas in a porous medium, according to LF.IBENSON [L]. Here u stands for the density and the pressure is proportional to um-~; see also [EV]. Typical values o f p are again 1 for laminar (nonturbulent) flow and �89 for completely turbulent flow.

Notice that in the laminar case p = 1, we get

u, = (um) x, (0.9)

which is the porous medium equation if m > 1 ; cf. [M], [A1]. In this case, the pressure is given (up to a constant factor) by the potential v and the velocity follows the linear Darcy law

V : --Vx;

cf. (0.4). Equation (0.9) occurs in a variety of other settings. It describes, for instance, radiative heat transfer in ionized plasmas [RZ, Ch. 10] and crowd- avoiding population spreading, [GM]. If 0 < m < 1, (0.9) is usually known as the fast-diffusion equation, of. [AB], [BH], [HP]. When m = p : 1, (0.3) is the classical heat equation, c f [W].

Finally, the case m = 1, p :# 1 is of interest in Mechanics of non-newtonian fluids; ef. [MP]. See also [HV], where the propagation properties have been studied and its references.

This paper studies the nonnegative solutions of the Cauchy problem for (0.3). In the case of the porous medium equation a fairly complete theory has been developed in recent years; c f [A4], [AV]. Among the mathematical techniques used in overcoming the difficulties originated by the degenerate character of the equation, a key role is played by the a priori estimate

1 V~x >= (m + 1) t ' (0.10)

due to ARONSON & B~NILAN, lAB], and valid for all nonnegative solutions of (0.9) in Q = R • (0, oo).

In this work we obtain an estimate which plays the same role for the nonnegative solutions of the Cauchy problem for (0.3). With the above notations (0.4), (0.5) our crucial estimate has the form

1 V~ <~ p(m § I) t '

which can also be written as

1 (vx [vxlP-l)~ > p(m -~- 1) t" (0.ll)

42 J .R. ESTEBAN & J. L. VAZQUEZ

Observe that for p ---- 1 we obtain again (0.10). On the other hand the estimate is best possible since equality holds for a family of explicit solutions. In fact, these solutions are the asymptotic profiles to which all other solutions with compact support will be shown to converge as t -+ ~ , and the exact constant 1/p(m + 1) is determinant in establishing the rates of convergence.

Using estimate (0.11) as the main tool, we construct a theory for the Cauchy problem for (0.3) with nonnegative, integrable initial data. In particular, we answer the following questions:

i) Existence, uniqueness and regularity of strong solutions; cf. Theorem 1 in Section 1.

ii) Existence and regularity of free boundaries; cf. Theorem 2 in Section 4.

iii) Asymptotic behaviour of solutions and free boundaries; cf. Theorem 3 in Section 5.

As for the mathematical difficulties involved, it should be remarked that in case p =[= 1 the equation is degenerate or singular even at points where u is positive, whenever ux vanishes. One of the nice points of the theory of the porous medium equation, p ---- 1, is the fact that every positive solution is smooth. This is no longer true in the general setting so that, whenever we need to approximate with smooth solutions we are forced to modify the equation somehow. In particular, a very delicate construction is needed to obtain the crucial estimate (0.1 l) with the best constant.

We end this introductory section with a comment on preceding works. One of the first mathematical contributions to the study of the initial-value problem for (0.3) is the paper [B], where BARENBLATT constructed a class of self-similar source-type solutions which have the property of finite propagation if mp ~ 1. These solutions will be extensively used in the sequel as a paradigm for the behavior of all nonnegative solutions with integrable initial data uo, especially when Uo has compact support.

A class of weak solutions with bounded velocity was constructed by KA- LASHNIKOV [K1] for mp ~ 1 and initial data whose velocity is bounded. The uniqueness question for this class of solutions remains open. The study of propagation was continued in [K3].

In [EV] we constructed a class of unique strong solutions for the Cauchy problem with nonnegative and integrable initial data and showed that whenever mp ~ 1, the velocity is bounded for t ~ ~ ~ 0 with a bound depending on T and [[ Uo [[~. This result was then used to obtain H61der continuity of the solutions with an optimal exponent and also Lipschitz continuity of the free boundaries.

A number of authors have studied equations similar to (0.3) in one or several spatial variables; cf. e.g. [An], [AtB], [B61], [B~2], [K2], [PR], [R]. They are not directly related to our Cauchy problem. An interesting nonlinear diffusion equation o f t h e f o r m ut = (O(u) cb(ux))~ with O(u)= u(l - - u ) and ~ ( s ) = s/(1 + s2), arising in a problem in hydrology, has been recently studied by VAN DUIJN & HIL- HORST [DH].

1. The basic result

We are concerned with the one-dimensional initial-value problem

S t - - e x ~ O x l - ~ - x | ] in Q : R • co), (P)

u(x, O) : Uo(X) for x E R,

for m, p > 0 and nonnegative Uo E L~(R). Our first contribution to problem (P) consists in showing that a unique strong

solution of (P) exists and it obeys the crucial a priori estimate.

Theorem 1. Let m, p > O. For every Uo E Lx(R), Uo >= 0 a.e. there exists a unique u E C(Q), u >: 0 such that

a) uE C([O,~x~); La(R)) A L~(R• [r, oo)) for any r > O,

b) a.e. t > O, u m E WL~176

~u 0 [ ~ u ~ l ~ - ~ ~ c) -~, ~ \--~-x i-~x / /ELL((0, oo);L'(n)) a n d

eu ~ ( 0 u m e~m~,-,~ tg-'-}" : c2"-~ ~ -~--x I ] a.e. in Q,

and

d) u(x, O ) : Uo(X) a.e.

Moreover, i f v(x, t) is as in (0.5) then

l (v~ ' ' "jvxt'-')x>: p ( , n + l ) t" (I.1)

This theorem will be proved in Section 3 after estimate (1.1) has been derived in Section 2 for a family of smooth approximate problems.

The solutions satisfy moreover a series of other properties that we list next.

Corollary 1. a) We have

in D'(Q), and

U u, ~ (1.2)

p(m + 1) t

2 Iluol[~ []ut(" t)[la <~ p(m + 1) t" (1.3)

b) The total mass is conserved, i.e. for every t > 0

+ o o + o o

f u(x, t) dx = f uo(x) dx. - - o o - - o o

c) As I x l --* oo , we have (um)~ (x, t )-+ O uniformly on tE(r,T), 0 - < z < T < o o .

44 J.R. ESTEBAN • J. L. VAZQUEZ

d) I f Uo E L I (R) /5 L~(R) for some 1 <_ r <-- o0, then u(', t) C L~(R) I1 u(., t)lit < l[ no I!, for every t > O.

e) Moreover, for r ---- 1 we have the following contraction property:

f [u(x, t) -- u(x, t)]+ dx <= f [U(X, O) -- (t(x, 0)1+ dx, - - 0 o - - o 0

where u, ~ are an), two solutions of (P).

and

Existence and uniqueness of a strong solution was shown in [EV] for mp ~ 1, by using Nonlinear Semigroup Theory. In this paper we give a different and more direct proof which applies to all positive m and p; besides, (1.1)-(1.3) are new.

Estimates (1.1) and (1.2) are motivated by the knowledge of a class of explicit solutions to (P), called Barenblatt solutions; cf. [B]. This i~ a family ~(x, t; M) of selfsimilar solutions with a Dirac mass as initial data, ~(x, 0; M) = M O(x), (M > 0). They conserve the total mass

M = f ~ ( x , t ; M ) d x for t > 0 , (1.4) - - o o

and read as follows:

if m p > 1,

1 f P+.--!1 p . ( 1 . 5 ) -if(x, t; M) = t p(,.+1~ [ C _ k(rtl, p) [~'[ p ]+ rap--l;

if mp = 1,

l [ v + l I ( 1 . 6 )

~(x, t; M) = Ct vgl exp [--ko(p) ]~ rT] ;

if rap< I, (1.7)

1 [ ,+l]m " f f ( x , t ; M ) = t p(m+i) c+k,(m,p) l#l'~ ~-,,

where ~ = xt -lIp(m+1), k ( m , p ) = - - k ' ( m , p ) - - m p - - 1 ( 1

m(p %- 1) ,p (m + 1)1 > 0, ko(p) = pZ(l/(p %- 1)) (p+O/v and C = C(m,p, M) > 0 is computed from (1.4).

The Barenblatt solutions serve as a model (particularly in studying the asymptotic behavior) for the nonnegative solutions of (P) with initial data of finite mass. They evidence two different kinds of behaviors:

i) if mp ~ 1, -if(x, t; M) is positive everywhere in Q,

ii) if m p > 1, -u(x, t ; M ) is compactly supported in x for every t > 0. Two curves x = • where

r(t) = c(m, p) (M "v-1 t) l/p(m+l) , (1.8)

usually called interfaces or)bee boundaries, separate the regions where ~ is positive, [ ~ > 0 ] - - - - - ( ( x , t ) : J x I < r ( t ) } and where ~ vanishes: [ ~ = 0 ] = { ( x , t ) : [ x I r(t)}.

If mp ~ 1, these solutions have a maximal regularity, given by the Lipschitz continuity of the function

1

v(x, t; M) -- mp -if(x, t; M) m - 7 . m p - - I

The velocity V(x, t; M) of the Barenblatt solutions is given by

Therefore,

V(x, t; M) -- X

p(m q- 1) t for (x, t) E [-ff > 0].

is discontinuous at x = i r ( t ) . Moreover

p(m -k 1) t '

which shows that the constant I/p(m q- 1) obtained in (1.1) is sharp. This is the fact on which the study of the large-time behavior (Section 5) will be based.

We end this section by pointing out some consequences of Theorem 1. A simple computation on u(., t) for t > 0 shows that as a consequence of (IA) we have

Corollary 2. For x E R and t ~ 0,

u(x, t) <= C(m,p) Jlu0[['~ t -a,

with o~ : (p q- 1)/p(m q- 1) and fl = l i p ( m + 1).

In the same way, it follows that whenever m p > 1 the solutions have bounded velocity V(x, t) for any t ~ z- > 0 even if the initial data do not have this property.

Corollary 3. For every x E R and t ~ 0,

p4-1

I W(x,t)[---;---- FvAx, t)! p+l ~: - - p + l Ilv(., t)l!~

p p(m Jr 1) t" (1.9)

For another proof of these results see [EV]. In fact, the derivation of (1.9) by means of the Bernstein technique is the main object of that paper. H61der continuity of u is then obtained from (1.3) and (1.9).

The boundedness of the velocity implies the property of Finite Propagation of disturbances from rest, as exhibited by the Barenblatt solutions. Free boundaries appear separating the regions [u > 0] and [u = 0]. They are Lipschitz continuous curves, as it can be easily deduced from the boundedness of the velocity; c f [EV]. They are also monotone curves since u(xo, to) > 0 for some to > 0 implies U(Xo, t) > 0 for t _> to, thanks to (1.2). The properties of the free boundaries will be studied in detail in Section 4.

We remark that if mp ~ 1, the velocity is not necessarily bounded, as the Barenblatt solutions (1.6), (1.7) show. The Finite Propagation property does not hold in this case; on the contrary, u is positive everywhere in Q (see Proposition 3.4).

46 J.R. ESTEBAN & J.L. VAZQUEZ

2 . T h e c r u c i a l e s t i m a t e

To prove estimate (1.1) we will essentially differentiate equation (0.3) twice. Since the equation can be either degenerate or singular (according to the values of m and p) at the points where u = 0 or ux = 0, direct differentiation is not justi- fied, and we have to use an approximation process and estimate instead of the original velocity gradient, the one corresponding to the new problem.

We first avoid the level u = 0 in the usual way by considering smooth, initial data Uo such that 0 < ~ =< Uo(X) <= Mo holds for some constants 8, Mo. More- over we choose u~' to be Lipschitz continuous.

The problem at ux = 0 is more difficult. We circumvent it by replacing (0.3) by

~u ~ ( ~ ) --~- = ~x qS. u,-~-x , (2.1)

where #,(u, z) is a smooth approximation to the original nonlinearity # ( z ) = z [ziP-l, such that

<ee (.,z) __< c O < c = Oz

all d _ < u _ < M o ,

< + ~ (2.2)

and some constants c, C 8u"

then 6 = --

OV To this aim, we show that 0 = -- 0--~ satisfies a uniformly parabolic partial

differential equation of the form

and

0 t = A(x, t) 0,, x + B(x, t) 00 x + C(x, t) Ox

+ F(x, t) 02 + G(x, t) O, (2.5)

where the coefficients A, B, C, F, G are bounded. If moreover we can show that

F ~ p(m + 1) (2.6)

p(m + 1) t

G ~ 0, (2.7)

will be a subsolution for (2.5) and the estimate (2.4)

satisfies the estimate ~V 1

- - - - > ( 2 . 4 ) ~x = p(m + 1) t"

holds for fixed e > 0, ]z I ~ Mt

depending only on e, 8, Mo, M1. (The dependence of ~, not only on z = Ox

but also on u will be essential in some of our constructions.) One then obtains a smooth solution u --= u(x, t) of (2.1) with initial data

uo(x) as above. For this solution, we will prove that the corresponding velocity, defined as

' ( ~ ) V(x, t) ---- ----u ~ u , -~x ' (2.3)

will follow from the maximum principle. Conditions (2.6) and (2.7) are in a way natural since formal differentiation of (0.3) gives for

the equation

0 t = mp lA m - I /(/gm)xlP-I Oxx

+ m(p -- 1) u 'n (u")x ](um)x1-2 O0 x

1 (2.8) + (p(m + 2) -- 1) - - (u")x I(u")x I ~-, 0x

U

+ p(m + 1) 0 2 ,

so that F = ~ p ( m + 1) and G ~ 0 . Finding a function q~,(u, z) such as not only to make equation (2.1) be uniform-

ly parabolic but also satisfy (2.6) and (2.7), turns out to be rather delicate. In fact, we need to have recourse to separate constructions based on different ideas

1 1 for the ranges of p : 0 < p = < 1, 1 < p < 1 + ~ and 1 +m+-------~ =< p"

First Case. 0 < p ~ 1. We try ~ ( u , z) of the simple form tP,(u, z) = ~,,(z), where ~p,(z) is smooth, increasing and ~(0) = 0. In this way (omitting the subscript e for simplicity) we get

V = - - - - ~ o U

8V We now formally compute the equation satisfied by 0 = -- ~--~ and show that

the coefficients A, . . . , G of equation (2.5) are given by (we set z = (u")x) :

A = m u m-I ~o'(z), (2.9)

B = m u m v/'(z) v/(z) ' (2.1o)

z [ ,/,(z) "'(z) C = - [2m V,'(z) -~ ~ ] (2.11)

zr (2.12) F = ( m + l ) 1 + V,'(z)_]'

( [ (z, = 1 + ~ 1 + r'(z~ r(z) J" (2.13)

We leave the computations involved in (2.9)-(2.13) to the reader. Later, in the next case, we will indicate how to proceed in a similar computation with a more complicated ~,(u, z).

48 J .R. ESTEBAN ~; J. L. VAZQUEZ

In view of (2.12) and (2.13), inequalities (2.6) and (2.7) are respectively equivalent to

z~p"(z) ~ P (2.14) 1 + ~p'(z'------~

and

z~."(z) ~ z~p'(z) (2.15) 1 + W'(z) - - ~o(z----5-

for all z E R . Notice also that (2.2) implies 1 ~ p when we set z = 0 in (2.14). In the following lemma we construct a function ~p~(z) satisfying (2.2), (2.14)

and (2.15). For further convenience, we state this lemma in a general form valid for any p > 0, though only the case 0 0 there exists an odd, C~176 function %(r) such that %(0) = 0 and

i) ~l)te(r) ~ - ~p--(l+2k) r2k for all ]r I ~ e

ii) ~o'~(r) : p [rlP-I for all [r[ ~ 2e.

r~p~'(r) ~ P for every r E R. iii) 1 + ~ :

r~p"(r) mp',(r) < 0 for every r E R. iv) 1 -~ ~p~(r) ~v,(r) =

and some fixed 2 : 2(p) > 0.

Moreover, %(r)-+r [riP-l, uniformly o n R as ~-+0.

Proof . Let us consider an even, C~(R) function r satisfies r : l + 2 k for I r i l l , r for 2 ~ l r l and decreases for l ~ r _ < 2 . We define ~p(r) through the identity

r~o"(r) 1 + - - - - ~(r). (2 .16)

~0'(r)

Direct integration of (2.16) gives

~P ' ( r )=2r2kexp[ - i o ( l + 2 k ) - - ~ ( S ) d s ] s

and 2 = 2(p) > 0 can be sochosen a s tomake ~p'(r) = p Irl p - ' hold for 2 ~ ]r I. Clearly, iii) is satisfied. As to iv), we have

r~p'(r) : f ~(s) W'(s) ds 0

H o m o g e n e o u s Dif fus ion with Nonl inear Diffusivity 49

for every r E R. Therefore

r~v"(r) r~p'(r) 1 1 q- ~v,(r~- S ~v(r) = ~v(r) f (~(r) -- ~(s)) v/(s) as

is never positive since ( ( r ) ~ ~(s) for [ s ] ~ l r ] . Finally, we set ~0~(r) = eP~p(r/e), which satisfies all the requirements of the

lemma. [ ]

Now that we have constructed suitable functions ~p~(z), we will carry out the program outlined in (2.1)-(2.7). We begin by solving (2.1) for tb,(u, z) ---- ~o,(z) and smooth initial data as above.

Lemma 2.2. Let uo(x) be a C~176 with bounded derivatives of all orders and such that

0 < ~ ~ Uo(X) <= Mo

holds for some ~, Mo. Let also be 0 0 there exists a unique u = u(x, t) E C~176 solution of

~u ~ [ ~u"~ ~--~ = ~-'~ ~P" l~ Ox ] in Q,

(2.17)

u(x, O) = Uo(X) for x E R ,

satisfying in O_ 0 < ~ ~ u(x, t) ~ Mo (2.18)

and ~um(x, t) <

= I[ (U~')' [1~- (2.19)

Proof. We follow a classical procedure; ef [ O K C ] , [A2]. First , we introduce v = u m as a new dependent variable; it transforms (2.17) into

v t : m v l-(I/m) ~3~(Vx) UXX in Q, (2.20)

v(x, O) = Vo(X) for x E R,

where Vo = u~'. Then we choose ~a(s), a nondecreasing and C~(R) function such that u ~ ( s ) = s for s ~ O m and ~n(s)=Om/2 for s ~ 0 . For any T > 0 , we consider the initial value problem

/)t : m ua(v) l --(I /m) ~/)~(Vx) Vx x in Qr : R • (0, T] (2.21)

v(x, O) = Vo(X) for x E R.

Notice that for a ' ~ s ~ M ~ and I z l = < M t =lJ(ua')'l]~ we have

w~(Mx) min (~m 1, M~,-1) < u~(s)l-(llm) ~o'(Z) =< 2e p-I max (hm-1, M~,-1).

This means that (2.21) is a uniformly parabolic problem. By the general theory of quasilinear partial differential equations (eft [LSU, Th. 8.1, p. 495]) one gets

50 J.R. ESTEBAN 8s J. L. VAZQUEZ

the existence of a unique solution v(x, t) of (2.21) which satisfies Iv(x, t) l :< M~'

o n QT, and

v E

for some t 3 E (0, 1). By the maximum principle,

0 ~ 0 m ~ v(x, t) ~ M~' on Qr,

and v satisfies (2.20) by the definition of un. The regularity v E C~ follows

by a standard bootstrap argument and the estimate Ivy(x, t) ] ~ M~ on Qr follows from the maximum principle applied to the equation satisfied by z = v~, i.e.

Zt : ~X (m v(x, t)'-(l/m) ~)te(Z) Zx),

Finally, we define u = v z/m, which clearly satisfies all the conclusions of the lemma. [ ]

We can now establish the main estimate for the velocity defined in (2.3), which in the present case reads

1 ~cou"'[ V(x, t) -- u % ~-'~x ] " (2.22)

Lemma 2.3. Let Uo(X) and u(x, t) be as in Lemma 2.2 and V(x, t) as in (2.22). Then (2.4) holds in Q.

As explained above, the proof consists in applying the maximum principle to the equation satisfied by 0 ~- --Vx, i.e. equation (2.5) with coefficients given by (2.9)-(2.13). We will give in Appendix A a rigorous justification of this use of the maximum principle.

To continue, let us remark that using (2.17) and (2.22) we may write equation (0.3) in the form

co ~V 1 ut = ~x (uV) = --u "~x + mu --'-~ (um)x ~Pe((Xm)x)"

From this, (2.4) and zT)~(z)>: 0 for all z E R, we obtain the following

Corollary 2.4. Let Uo and u as above. Then in Q

6~U U - - > (2.23) ~ t = p(m + 1) t"

Therefore, the function t ~ u(x, t) t 1/p('+a) is nondecreasing in (0, oo) for every x E R .

1 - - < p . Second Case. 1 + m + 1 :

We rewrite (0.3) in the form

Ou ~um~ -1 ~2um

x2' 1

we put V - - (um)z[(um)~] p-I and O : - - V ~ , U

Let us explain the key argument of this case.

(2.24)

and rewrite (2.8) as

0 t = m p u m - 1 I(u% t ~ Oxx

Imp(urn-1 ](Um)xf l)~ + +

+ p ( m + 1) 02 .

p ( m + 1) ] u (um)x I(Um)xl p- ' j O~ (2.25)

Now, i fp is large enough, the functions ~(z) = z Izl p-1 and q~'(z) = p Iz] p 1 entering in the coefficients of (2.24) and (2.25) are not singular at z = 0. Hence these coefficients are bounded for bounded values of z = (um)~. Nevertheless, it remains the problem of the degeneracy of (2.24) and (2.25) at z = 0. We circumvent it by adding to (0.3) a porous -med ium- l i ke term. Actually we consider

On ~ (SU m ~Ump--I~ t3t - - 8x ~--~x -~x ] + e(td)xx (2.26)

and choose n so as to obtain 1/p(m + 1) as the constant in ARONSON & BI~NILAN'S estimate (0.10), namely

n = p ( m + 1 ) - 1. (2.27)

Equation (2.26) is of the form (2.1) with

n E ~e(U, z) : z I z l p - 1 . A f _ _ _ U n - m Z . (2.28)

m

Let us check condition (2.2). If ~ _ m , i.e. p > 1. As for the corresponding velocity, c f (2.3), we have

where

V(x , t ) : --r t) , (Um)x (X, t))

Z ]z[P--I r/E Un_(m+l) Z ~(u, z) - - -F - - �9 (2.29)

U m


We next compute the equation satisfied by 0 ---- - -V x -= ekx.

4, = r162 + r

z = ur + ur ~ r + r

Z t = m(u m-l(wb)x)x

= (m + 1) zO + m u m O x + dpz x ,

and then

We have

(2.30)

(2.31)

~b t = mu~r + [ueku + (m + 1) z~bz + ~b] 0. (2.32)

Now we use (2.29) to work out the term within brackets, and obtain

u~b~ + (m + 1) zr + dp = p(m + 1) 4.

Hence

dpt = mumdpzOx -}- p(m + 1) ~bO,

and differentiation with respect to x gives finally

O, = mu~bzOxx

+ [m(umqb~)x + p(m + 1) ~b] 0 x

+ p(m + 1)02.

Surprisingly, conditions (2.6) and (2.7) are satisfied and equality holds. Now, if p > 4, all the steps performed in the first case can be repeated with the above choice of approximation, i.e. (2.26) and (2.28), and Lemmas 2.2 and 2.3 and Corollary 2.4 hold true.

This method would not work for lower p's because of the lack of sufficient regularity of the solutions. Nevertheless, the idea of the porous-medium perturba-

1 tion can be made to work in a slightly modified form for the range p >-- 1 +

Our approximate equation will now be m + 1"

~t - - ~x ~p~ ~ + e(U")xx. (2.33)

As in (2.27), n + 1 ---- p(m + 1). Therefore, q~,(u, z) is

n6 #,(u, z) : ~/,,(z) + - - d ' - " z (2.34)

m

and the velocity V = --4 is given by

~(u, z) ---- ~0~(z___~) + n___ee u~_(,,+l ) z. (2.35) u m

Observe that n - - ( m + 1)--> 0. To find the equation satisfied by 0 - - --Vx = 4~x we can use again (2.30)-(2.32). Differentiation of (2.32) with respect

to x gives an equation of the form (2.5) with (we omit the subscript e for simplicity)

A(x, t) = m u ' % , (2.36)

B(x, t) = mu" ~ , (2.37)

C(x, t) = u4,, 1 r ] + (m + 2) Z4,z -r uZdOuz + 4', (2.38)

1 r(x, t) = (m + 2) + ~z [u4~,z + (m + 1) Z~zz], (2.39)

G(x, t) = ~Z [--murb,-{- u24,,, + (m 4- l) uz4,,z -- ~u~O" [u4,,z + ( m + l) zrbzz] ] .

(2.40)

We look for F >~ p(m + 1) and G =< 0. By use of (2.35) in (2.39) and (2.40), these inequalities turn out to be respectively equivalent to

zv?"(z) > P (2.41) 1 + v:(z----T =

and

[ re(rune + 1) u "-m zT~(z) ~o'(z) 1 + ~'(z) ~o(z) ]

~ (m-t- 1 ) + ( m + 1 ) ( n - - ( m + 1)) l - k - - ~ - ] (2.42)

- (n + 1) (n - (m + 1))] z2~/(z) - (n + 1) z~o(z).

Let us remark that (2.41) amounts to saying that the function z -+ z 1-p ~o'(z) is nondecreasing. Therefore, since p > I, ~o'(z) must necessarily vanish at z = 0. This is why we need an extra porous-medium term in (2.33).

We shall take as vA(z ) the function constructed in Lemma 2.1. We see at once that (2.41) is satisfied. Moreover, it can be easily verified that (2.2) is also satisfied for the choice (2.34) of ~,(u, z).

Let us now verify inequality (2.42). Its left-hand side is nonpositive thanks to Lemma 2.1.iv). The right-hand side in (2.42) can be estimated from below by

(m + 1) [z2~/(z) -- pzq,(z)] (2.43)

( 1 ) here the condition p=> 1 - S i n + 1' i.e., n > = m § 1, is used . Finally, since

+ r~"(r)~ z2~v'(z) ~ z J 1 ~o'(r) dr > o ~--~Tfl = I, z~o(z),

(2.43) is nonnegative and (2.42) follows.

54 J .R . ESTEBAN 8s J. L. VAZQUEZ

We can now repeat the procedure followed in the first case and obtain Lem- mas 2.2 and 2.3 and Corollary 2.4 in this case.

1 Third Case. 1 < p < 1 -F m +'-----~" This last case combines the different

arguments presented above. We try yet another very special approximation,

St -- ~x ~o~ + euH, ,

where ~p,(z) will be constructed as in Lemma 2.1, and He(z) will be a regularized version of z [z[ ~-l, with

1 O < ~ = p - - < 1 .

m + l Therefore, with the usual notations,

Or(u, z) = ~o,(z) + euH,(z) (2.44) and

~o(z) r + el l (z) . (2.45)

U

The equation for 0 = -- Vx = q~x is of the form (2.5) with coefficients given by (2.36)-(2.40). It is not difficult to see that, when we introduce the choice of 4~ (2.45) into these coefficients, the desired inequalities F >= p(m + 1) and G < 0 are respectively equivalent to

~o'(z) I-~ p § 1 + I-I'(z) ~ > 0 (2.46) u v,,'(z) =

and

z~"(~) z~'(z)] z r ( z ) V/ (z ) 1 + - - u 2 ~o'(z) W(z) J

(2.47) 1 zH"(z ) zy)'(z)] + zH'(z) y)(z) I

u + ~ + H'(~) ~ J <= O, for every z E R.

Now we define ~p~(z) as in Lemma 2.1, which renders nonnegative the first term in (2.46) and nonpositive the first term in (2.47).

We construct in Lemma 2.5 below a function He(z) such that the second term in (2.46) is nonnegative and the second term in (2.47) is nonpositive. In this way, (2.46) and (2.47) will be satisfied.

1 1 L e m m a 2 . 5 . Let r e > O , l 0 there exists an odd, C~176 function He(r) satisfying

i) There exists # ~- t~(m, p) > 0 such that for all [r I =< e/2, n ' ( r ) = / ~ e '~- ~.

ii) Hi(r) = or [r I ~-1 for all I r 1>~ e.

rH" ( r ) iii) 1 + ..-777-7,, -> ~x for every r E R .

Hi(r) :

iv) I f ~p~(r) is as in Lemma 2.1, then for every r E R ,

1 rHi'(r) < rw'~(r) l + ~ + - -

m + 1 H' (r ) = ~o~(r)"

Moreover, eH~(r) -+ 0 uniformly on R as e -+ O.

Proof. Let us consider an even, Coo(R) function a(r) which satisfies aft) = 1 for i t [<_1/2 , a f t ) = o ~ for t r i a l and decreases if � 8 9 1.

As in Lemma 2.1, we define H(r) by means of

rH"(r ) 1 + H'(r----S - a(r),

and then set H,(r) = e~H(r/e). Let us prove iv) for ~p(r):

- - - - - - + a ( r ) - - ~ ( s ) ~ v ' ( s ) ds, m + 1 + a(r) ~v(r) ~v-~) o

where r is as in Lemma 2.1. This expression is never positive since ~(r) and aft)

have been chosen because 1 < p < 1 + so that ~ + a(r) ~ ~(s)

holds for all O--<s~<r. [ ]

Notice that the nondegeneracy condition (2.2) is clearly satisfied by ~,(u, z) as in (2.24) with the present choice of ~,,(z) and H,(z). With this construction, Lemmas 2.2 and 2.3 and Corollary 2.4 are also true and the proof of estimate (2.4) is complete for the whole range of m and p.

3. Existence and Uniqueness

We devote this Section to proving Theorem 1 and Corollary 1. The uniqueness statement in the theorem is an obvious consequence of the Ll-contraction property, part d) of Corollary 1, which we establish first.

Proposition 3.1. Let u(x, t) and t~(x, t) satisfy a), b) and c) o f Theorem 1. Then for every t >_ s >_ O

-boo -boo

f [u(x, t) - - fi(x, t)]+ dx <: f [u(x, s) - - ~(x, s)]+ dx. (3.1) - - o o - - O O

Proof. It is a rather standard argument. One subtracts the equations for u and t~, multiplies by ~(u m -- ~m), where r/(r) is a smooth approximation of the function sign+ (r) ( = 1 if r > O, = 0 if r ~ 0), integrates by parts with respect to

56 J: R. ESTEBAN 8l; J. L. VAZQUEZ

x and finally lets ~/ tend to sign+. Since necessarily bo th u and ~ satisfy

lim (Um)x (X, t) = 0 [ x l ~ O o

uniformly in t E (w, T), 0 < w < T < ~ , thanks to ut E L]oc((0, oo); LI (R)) and (0.3), and since sign+ (um - - {tin) = sign+ (u - - ~), one gets

d + o o

_ f ~ [u(x, t) - {t(x, t)l+ dx <= O. --d?

N o w (3.1) follows by integrat ing with respect to t. [ ]

As for the existence s ta tement in Theo rem 1, it will be obta ined by passing to the limit e = 8--~ 0 in the approx imat ions constructed in Section 2. In doing this, the tools we use are i) the crucial est imate (2.4) and its corollary, (2.23); ii) an energy est imate (see L e m m a 3.3 below) which will be essential to conclude the s t rong regulari ty o f the solut ions; and iii) and extended version o f the L 1- contract ion p roper ty (3.1), suitable for appl icat ion to the nonintegrable funct ions obta ined in L e m m a 2.2. We begin by proving this proper ty .

Lemma 3.2. Assume that Uo(X) and {to(X) satisfy all the hypotheses in Lemma 2.2 and are such that

Uo - - 8, {to - - 8 E L I ( R ) .

I f u(x, t), {t(x, t) are respectively the solutions of (2.1) with initial data Uo(X) and {to(X), then

u(., t) -- 8, {t(., t) -- 8 E LI(R)

and -t-oo q-eo

f [u(x, t) - - t~(x, t)]+ dx ~ f [Uo(X) -- {to(x)l+ dx (3.2) - o o - o o

hold for t >= O. In particular,

-~-oo q -oo

f I u(x, t) - 81 dx ~ f ]Uo(X) -- 81 dx. (3.3) - o o - o o

Proof . The p r o o f relies on the fact tha t the solution u(x, t) of (2.1) can be obta ined as the limit o f the solutions of a sequence of second boundary-va lue problems. As in L e m m a 2.2, we only write the details o f the p r o o f for 0 < p ~ 1 ; the other cases are entirely similar.

Fo r any n = 3, 4 . . . . we consider a nonnegat ive and smoo th cutoff funct ion ~n(x) such tha t (n(x) = 1 for Ix I ~ n - - 2, ~ ( x ) = 0 for I xl ~ n - 1. We define

uo,n(x) = 8 + Cn(x) (Uo(X) - 8)

and Vo..(x) --- Uo~(X) m. For each n we consider the solution v.(x, t) of the second boundary-value problem

v,,,t = m vln -(l[m) ~Ote(Un,x) On,xx

Vn,x(• , t) = 0

As

in (--n, n)• T],

for 0 < t ~ T ,

v.(x, O) = Vo,.(x) for Ix I < n.

n---~e~, v .(x , t)---~ v(x, t) uniformly on compact subsets of Q, v(x, t) being the solution of (2.17). This assertion is shown in [LSU, Ch. V, w 8] for a sequence of first boundary-value problems. The same arguments apply for Neu- mann boundary conditions. See also [LSU, Ch. V, w 7].

We now set u,(x, t) = v~(x, t) l/m, which satisfies

u~,t = ~ W, ~-~-x] in (--n, n)• T],

~u7 ~x (• t) = o for 0 < t ~ T ,

and

u,(x, O) = Uo,,(x) for I x [ < , .

Therefore, for any fixed t we have

d f u ,(x , t) dx 0

dt -n

f [u.(x,t) - ~[ d x : f [Uo,.(x) -- O [ d x ~ Iluo - - OlE~(R). ~ n - - n

As n -+ cx~, Fatou's Lemma implies (3.3). The proof of (3.2) is obtained along the lines of Proposition 3.1. [ ]

We divide the actual existence proof into several steps. First step. Let Uo E LI(R) A L~(R) and be nonnegative. We also assume that

u~, 7 = min( l ,m} is Lipschitz continuous and Uo(X)> 0 for all x E R . For any e > 0 we consider an strictly positive averaging kernel 9,(x) ~ C ~

and define

Uo,,(x) = e + [ f 9 , (x - - y) uo(y) v dy] xle �9

As is easily verified, Uo,,(x) satisfies

0 < ~ <= Uo,~(x) <- ~ + I[ no II~

[(u~,,)' (x)l =< I[(u~)'ll~

Iluo,. - ~lh <= Iluo[ll,

and u0,, -- e ~ Uo in LI(R) and uniformly on R as e ~ 0.

58 J.R. ESTEBAN & J. L. VAZQUEZ

Therefore, Lemmas 2.2 and 3.2 can be applied with e = 6 to obtain a unique

u, = u,(x, t)C C~176 solution of

in Q , (3.4)

for x E R ,

~,, = ~ ~ u~,-~-x/

u,(x, o) = uo:(x)

satisfying

and

on L), and also

for all

satisfies

0 ~. e ~ u,(x, t) ~ e -4- II Uo [Ioo

(x, t) ~ II (U'go)'lloo

l[u~(., t) - - t i l l =< Iluoh

t ~ o. By Lemma 2.3,

1 (.. V,(x, t) - - ~ u~ ~x/

and by Corollary 2.4,

(3.5)

(3.6)

(3.7)

t9 V~ > 1 (3.8) ~x ~ p(m -r l) t '

~u_._2 > u, (3.9) ~t = p(m -[- 1) t"

We next let e ~ 0. By Aubin's Lemma, [Au], it turns out from (3.4)-(3.6) that (for some subsequence, also denoted by u, in the sequel for simplicity)

u~ ~ u uniformly on compact subsets of Q- (3.10)

Therefore u C C(Q). From (3.6), (3.10) and Lemmas 2.1 and 2.5, it follows that

~ ~u ~ Ox ---~ --~-x weak* -- Loo(Q) (3.11)

~um] and also Z~ = ~ , u~, --~-x] converges in the same sense to some Z(x , t). We next

set ~(s) ---- s Isl ~ - ' and show that in fact

Z = ~ -~x a.e. in Q. (3.12)

To this aim we fix t > 0. The family {Z~(., t)}0<~_l is uniformly bounded and

oZ.(x, t) ~x - - u~.t >= --Co

holds for some Co ~- Co(m, p, t, Uo) > 0, thanks to (3.5) and (3.9). Then, for any bounded interval I Q R, the total variation on I of Z,(x, t) is uniformly bounded. Therefore,

Z,(x, t) -+ Z(x, t) a.e. x E R . Since we also have

I i

uniformly on ~) (cf. Lemmas 2.1 and 2.5), we obtain

~ \--~x! ~ Z(x, t) a.e. x e R ;

(3.12) follows from this and (3.11) by Br~zis' Lemma; cf. [Br, p. 126]. On the other hand, (3.9) and (3.10) give

U

u, >~ p(m + 1) t (3.13)

in D'(Q). Actually, ut is a function as a consequence of the following energy estimate.

Lemma 3.3. Let be Uo, Uo., and u, as above. Then for every T > O, T +oo

lim~o o f -foo t l(u(*m+')12)t[2 dx dt ----< Cm I "0 m+lm+l" (3.14)

This type of estimate is due to BI~NILAN; cf. [B61], [B62]. The difficulty in our case consists in having to derive the estimate for the complicated approximate problems. We give a full proof of Lemma 3.3 in Appendix B. Since u is positive by (3.13), it follows that u t E L~oc(Q) and (0.3) is satisfied a.e. in Q.

To continue with the proof of Theorem 1, we show that for t > 0,

lim (u")x (x, t) = 0. (3.15) Ixl-+oo

We first remark tht u(., t)E LI(R) and

!lug t)lh < Iluoll~ (3.16)

by virtue of (3.7), (3.10) and Fatou's Lemma. Then (3.13) and (0.3) imply that

, / ~ ( x , t ) + p ( m + 1) t u(y, t ) dy

is nondecreasing in x. Therefore, the limits

lim ~ ( ~u~ ) x-~• ~o -~x (x, t)

must exist. Since u is bounded, both limits must be 0 and (3.15) is proved.

60 J . m . ESTEBAN & J. L. VAZQUEZ

We can now integrate (0.3) and get

+oo

f u,(x,t)dx<=O, - - o o

which together with (3.13), (3.16) yields

2 Ilu0111 Ilu,(., t)ll~ =< p(m q- 1) t"

In particular, we deduce:

i) u is a.e. differentiable from (0, cx~) into L~(R), so u E C((O, o0); L~(R)).

ii) urn(., t) E W~'~176 a.e. t > O, with

2 I!uo![l ](um)x (., t)]P ~ p(m + 1) t"

We next show that

+oo

lim f ]u(x, t ) - - Uo(X)l dx = 0. (3.17) t-+0 --oo

Notice first that {u(., t)}t~0 is precompact in L~or since it is a bounded family in L~(R) on which translations are equicontinuous because

q - o o - } - c o

f l u (x+h, t ) - -u(x , t ) ]dx<= f l u o ( x q - h ) - u o ( x ) - - o o - - ~ o

dx.

(Use (3.2) to prove this, taking into account that u0,~ ~ u0 in LI(R).) Therefore, given any sequence t, ~ O, there is a subsequence tj such that u(., tj) ~ Uo in

Lloc(R) (recall that u E C(Q)). In fact this convergence takes place in LI(R) because of (3.16). Since the limit is independent of the sequence, thus (3.17) is proved.

As a consequence of (3.17) and the flux condition (3.15) we obtain the property of conservation of the total mass:

q - c o -}-oo

f u(x, t) dx = f Uo(X) dx for all t > 0. (3.18) --~30 - -oO

We complete this step by displaying the main estimate for V. We recall that

I 1 V, -- Z,, and V = -- - - Z , (3.19)

U e U

where Z,, Z are as in (3.12). We know that Z~---~ Z a.e. and weakly in L~oc(Q). Moreover, since Uo is continuous and positive everywhere and u E C(Q), it follows from (3.13) that u is positive in Q and l/u, ~ 1/u uniformly on compact subsets of~9. Applying this to (3.19) we conclude that F~ ~ F a.e. and weakly in L~oc(Q). We may now pass to the limit in (3.8) and obtain (1.1). [ ]

Second step. Let us assume that Uo E LI(R) A L~176 Uo ~ 0 a.e. and let be 7, P, as in the first step. We define

UoAX) = I f q,(x - y) uo(y) ~ dy] 11~.

As is easy to verify, 0 < Uo,,(x) <= Iluolloo, liuo,~lh =< Iluol[, and Uo,~-+ Uo in L~(P0 as e ~ 0. Moreover, for any e > 0, ug.~ is Lipschitz continuous.

Therefore, the results in the first step of the proof give the solution u,(x, t) of (0.3) corresponding to initial data u0,,. As e -+ 0, Theorem 1 is obtained with only small changes with respect to the proof of the first step. Thus the estimate

2 Iluolll [(u~n~ )x (x, t ) f ~ [lug,t(-, t)111 ~ p(m + 1) t (3.20)

must be used instead of (3.6). (3.20) implies H61der continuity also in t. To be precise: for every ~ > 0 , all t , t ' ~ z and x , x ' E R , we have

]u~(x, t) -- u~(x', t ') I <- C ( I x - x ' ? + It - t ' f ) ,

where 0~ = min {1, 1/m}, fl = o~/(o~ + 1) and C = C(m, p, 3, Uo) > 0 is independent of e. See [EV, Th. 2, b] for a proof. In particular, the convergence of the u~ takes place uniformly on compact subsets of Q.

On the other hand, the energy estimate (3.14) still holds in this step. If u is positive everywhere we easily conclude that ut E L~oc(Q). I f u is only nonnegative, the same conclusion can be reached from a result of B~NILAN, [B62, Lemma 4].

To prove (1.I) when mp ~ 1, we proceed as in the first step using the fact that u(x, t) is positive everywhere, (this is proved independently in Proposition 3.4 below). If mp > 1, we use the boundedness of the velocity (1.9) to pass to the limit. Now v~, v~,~ and V~ are bounded for t ~ T > 0 and V~,x ~: I/(p(m + 1) t). Arguing as in the proof of (3.12) we conclude that V~ ---> V a.e. and weakly in L~o~(Q), V being related to v by (0.4), (0.5), and V x <: 1/(p(m + 1) t). [ ]

Remark. There is no loss of generality in assuming that Uo is bounded. Indeed, from Corollary 2, any solution with initial data in LI(R) is bounded for any t > 0 with bound depending only on Iluolll. Therefore, if we suitably approximate Uo by UO,n E L~176 LI(B) all the above estimates will be independent of n for t ~ ~ > 0 and in the limit n -+ cx~ we will get a solution of (P).

We end this section by showing the following

Proposition 3.4. Let mp <= 1 and let Uo(X) and u(x, t) satisfy a)-d) in Theorem 1. Then

u(x, t) > 0 for all x E B and t > 0. (3.21)

Proof. If mp < 1, the operator

the homogeneity A(2u) = 2 mp A(u)

t u- A ( u ) - - x ,--ffx IT;/ z

(for all 2 > 0 ) of

62 J.R. F_,STEBAN & J. L. VAZQUEZ

implies ut <= u/((1 -- mp) t) a.e. in Q, by the results of B~NILAN • CRANDALL, [BC]. This leads to

u

O <= (1- - mP) t lax/ /

and the Strong Maximum Principle--as stated in [V3]--implies that either u > 0 or u ~ 0. Finally, mass conservation (3.18) gives no opportunity for u ~ 0 to occur.

If mp = 1, we cannot apply [BC], so we argue as follows. The mass conservation and the continuity of u(x, t) imply that given ~" ~ 0 there exist ~ E 1%, 0 t o < r < T and M > 0 such that u ( ~ , t ) : > M holds for all t, to<~t<--71". By translation we may put ~ ---- 0 and to : 0, so that (3.21) reduces to proving the following

Lemma3.5. Let m p = 1 and Uo, u as above. I f u ( O , t ) ~ M > O for all O~t<- -T , then u(x ,T)>O for all xER .

Proof. We use the maximum principle and compare with a suitable positive subsolution. Let be 0 0 fixed. We set c = (1 -F Xo)/T and for x < ct define

w(x, t) = M exp [--2(ct -- x)-~],

where 0 ~ = p / ( 1 - - p ) , 2 = f l ( 1 - - p ) - ~ c -l/O-p) and ),---- lift satisfies V + V P = (1 -- p) -~ (1 + Xo) -(p+I)/O -p) T j/O-r).

It is easy to verify that

--E I / in $ 2 = ( O < x ~ c t , O < t < : T } .

Since also 0 < w ( 0 , t ) ~ w ( 0 , T ) < M ~ u ( 0 , t) and w(et, t ) = O ~ u ( c t , t) hold for 0 --< t --< T, we obtain w(x, t) _< u(x, t) on ~ . In particular 0 < W(Xo, T) <: U(Xo, T) holds.

I f p 2> 1, the same argument applies with

w(x, t ) : A ( e t - - x ) ~

for (x, t)E .Q, e = (1 + Xo)/T, 2 = M(1 q-Xo) -~ and o~> p such that

~x (1 p) l /p pT[(1 -]- Xo)/T] fv+I)/p. []

Remark. Notice that (3.21) can be deduced aposteriori from (1.1) by integrating it twice in x starting from a point Xo of maximum of u(., t).

As for the proof of Corollary 1, we have begun the section with a proof of part e). Parts a), b) and c) have been established in the course of proving existence,


though we remark that they can be easily derived a posteriori. We are left 0nly with part d), i.e. that the map Uo -~- u(., t) is bounded in L'(P~) for all r E [1, cx~]. This can be proved in a standard way by multiplying (0.3) byj(um), where j E C 1 [0, e~) is a suitable increasing function, and then integrating by parts. We omit the easy details.

4. Regularity of the free boundary

In this Section we consider solutions u of (0.3) with mp > 1 and initial data that vanish in some interval, so that there is an interface or free boundary which separates the nonvoid subsets of Q, [u > 0] and [u --- 0]. Without loss of generality we assume that the initial data Uo satisfy

0 ----- ess sup (x E R: Uo(X) > O)

and consider only the right free boundary ((t), given by

r : sup {x E R: u(x, t) > 0} if t > 0,

and ~'(0) ---- 0. We study here the regularity of the free boundary r and of the solution near

~(t). In the porous-medium case, p ~ 1, this study has been performed by a number of authors, e.g. [A3], [Kn], [CF], [ACK], [ACV], IV2]. In this program a key role is played by the estimate vxx >~ --C/ t .

As we will show, the same program can be applied to the doubly nonlinear equation (0.3) using as a foundation the crucial estimate (1.1). Many of the arguments, in particular of [CF] and [ACK], will have counterparts in this general case. Though our equation can be either degenerate or singular at points where Ux = O, we will be able to exclude such points in a neighborhood of the moving part of the free boundary, which will be enough for our purposes.

We have already mentioned that ((t) is continuous in [0, e~), locally Lipschitz continuous in (0, oo) and nondecreasing. Since, for fixed t > 0, the function

X x - + p(m § 1) t V(x, t) is nondecreasing, the following limit exists:

Moreover, we have

lim V(x, t) : V(~(t), t ) . x-+ r

u (x , t ) > 0

Lemma 4.1. For every t > O, the right derivative D+~(t) ( = ~'(t) a.e.) exists and

D~((t) ~- V(((t) , t ) . (4.1)

This is proved in [Kn] for the porous medium equation (p = 1) and the same arguments apply for the doubly nonlinear case, provided (1.1) is known.

An important consequence of the main estimate (1.1) is

64 J.R. ESTEBAN ,~ J. L. VAZQUEZ

Lemma 4.2. The inequality

1 ~'(t) > ~ " ( t ) + ( 1 p ( m + l ) ' ) t =

is satisfied in the sense o f measures on (0, oo).

0 (4.2)

We remark that the free boundaries r(t), cf. (1.8) of the Barenblatt solutions, satisfy (4.2) with equality sign. (4.2) is proved in the case p ---- 1 in [CF] and [V1]. The same proof applies without major changes for p =~ 1.

1 Let fl : 1 > 0. The differential inequality (4.2) means that the

function p(m + 1) t

F(t) : f sa('(s) ds (4.3) 1

is convex in (0, oo). Therefore, it has side derivatives D+F(t) at every time t > 0 and D-F( t ) <= D+F(t). Writing (4.3) in the form

t

F(t) : ta((t) - - ((I) - - fl f sa- l ( ( s ) ds, 1

we conclude that D • ((t) exists for t > 0 and

D• : taD • ~(t)

for every t > 0. From this and the convexity of F(t) we get

O+((t) ~ O - ( ( t )

and also D • ~ (z/t) a O• (4.4)

for every t > z > 0. In particular, we have the following result for the waiting time:

t* = sup {t ~ O: ~(t) = 0}.

L e m m a 4.3. t* is finite (possibly zero) and ~(t) is strictly increasing on (t*, oo).

Examples of solutions with a positive waiting time are constructed in the case p ---- 1, c f [Kn], [ACK], and a precise characterization is given in [V2]. All of these results can be adapted without difficulty to the case p ~ 1.

The main result of this section is the following.

Theorem 2. The free boundary (( t) is a C 1 function in the interval (t*, ~ ) , v is a C ~ smooth function in a neighborhood o f x : (( t) , t > t*, restricted to x < ~(t), and

v(x, t) = v~(x -- ~(to), t -- to) + o(I x -- ~(to) [ -[- It -- to I) (4.5)

where 7 : ~'(to) and v~ is the constant-velocity front

Vr(X , t) : yllp[yt - - X]+.

Moreover, v x and vt admit limits as (x, t) -+ (Xo, to) with x < ((t), Xo - ~'(to) and t o > t * . We have

and

~'(t) = --vx I Vx i v - ' (4.6)

v, = Ivxl

on the moving part of the free boundary (x = ((t), t > t*}.

(4.7)

The C ~ regularity of the free boundary after the waiting time has been established for the porous medium equation, i.e. the case p = 1, by CAFFARELLI & FRIED- MAN, [CF]. Subsequently ARONSON, CAFFARELLI & KAMIN [ACK], provided a new proof centered around the fact that near a point Po = (((t0), to) on the moving part of the free boundary, v behaves like the constant velocity front v~.

Below we give a complete proof of Theorem 2 based in this method. As mentioned above, a problem that we encounter when analyzing the behavior of the function v around Po is the possible degeneracy of the equation satisfied by v, i.e.

( 1 ) - - - m Ivxl (4 .8 ) v t : m - - (v ]vx[p-l vx)x + l + p

near Po, even at points where v > 0. This is avoided by proving that vx is bounded away and above from 0 at points (x, t) near Po such that v(x, t) > 0; cf. Lemma 4.4. We proceed then to obtain the linearization (4.5), from which the rest of the proof follows. On the other hand, even for p : 1, our proof is short and simplifies the argument in [CF], [ACK].

Before going into the proof of Theorem 2, let us remark that the C ~ regularity of ((t) in (t*, oo) can be reduced to proving that

D-~(t) : D§ for t > t*,

since in that case, F(t) would be convex and differentiable in (t*, oo), hence continuously differentiable and so would be ((t).

We begin by establishing bounds for V(x, t) near a point of the free boundary, Po = (Xo, to) with Xo = ((to) and to > t*. Let 7 = D+((to) > 0 and let N~ ---- N~(Po) = B~(Po) A [u > 0], where B~(Po) is the ball in R 2 of radius e > 0 and center Po. We also fix w and zt such that t * ~ w ~ w ~ .

Lemma 4.4. As (x, t) --> (Xo, to) with x ~ ((t) we have

lim sup V(x, t) ~ 7,

lira inf V(x, t) >= D-((to).

Moreover, for w ~ to <: wl we have

(4.9)

(4.10)

V(x, t) ~ Co > 0 a.e. in N~o(Po),

where eo and Co depend only on u, w and wl.

(4.11)

66 J.R. ESTEBAN ~; J. L. VAZQUEZ

Proof. (4.9) can be obtained as in [CF, Lemma 4.3] arguing by contradiction and using estimate (l.1).

As to (4.10), we take tC (to -- e, to q- e) with to -- e >_-- ~ > t* and integrate (v x [vxlP-l)x >= - -K(z ) , this time between x and ((t), to obtain a.e.

V(x, t) ~> D+((t) - - K( ( ( t ) - - x) ~= (by (4.4))

((to - - e)/t) ~ D+((to - - e) - - K ( ( ( t ) - - x) (4.12)

>= ((to - - e)/(to q- e)) ~ O+((to - - e) - - K( ( ( t ) - - x ) .

Since D+((to - - e ) -+ D-( ( to ) when e - + 0, (4.10) follows. The same calculation above implies (4.11). [ ]

The limits (4.9) and (4.10) have to be understood for the moment as essential limits. Later in Lemma 4.6, we will show that Vis C ~ near x ----- ((t) so that these limits are pointwise limits. From the proof of (4.10) we may derive another result which will be of use later on.

Corollary. V(x, t) -+ ~ as (x, t) -+ (Xo, to) with x < ~(t) and t >: to.

The proof consists in observing that in this case we may replace to -- e by to in (4.12).

Lemma 4.5. There exist positive Constants A, ~o, C1 and C2, depending on and ~ , such that

C~(~(to) - - x) ~ v(x, t) <: C2(~(to) -- x) (4.13)

holds f o r every (x, t) in the triangle

.Q : ((x, t): 0 <= to - - t < A(~(to) - - x) < t~o}.

Proof. The right-hand inequality in (4.13) follows from the boundedness of the velocity (1.9) and the monotonicity of ((t). The left-hand inequality is deduced from (4.11) and the Lipschitz-continuity of r [ ]

A careful study of the constants that appear in Lemmas 4.4 and 4.5 shows that (4.13) is uniform when to moves in any closed subinterval of ( t*,oo).

Lemma 4.6. v is Coo smooth on sets o f the f o rm

N : (~(t) - - 0 , < x < ~'(t), ~ < to < ~1}- (4.14)

Moreover, v is a Lipsehitz continuous function o f x and t in N and

C3 Ivtt(x, t) l =< r - x /n U. (4.15)

The constants ~1 and C3 depend only on the solution, ~ and 71.

Homogeneous Diffusion with Nonlinear Diffusivity

Proof. Let to E (z, zl) and A, ~o and ~ as in Lemma 4.5, and put

O : ((x, t): A x < t ~ O, - - 5 / 4 < x < --1/4}.

For any ~ E (0, 4~o/5A) and (x, t) E D we define the function

i v~(x, t) = -T V(Xo + ~x, to + ~t).

It turns out from (1.9), (4.11) and (4.13) that

0 < C~/4 <: v~(x, t) <- 5C2/4

and

- -C~ <= V~,x(X , t) <= --C~/" < 0

hold a.e. in D. Since v~ satisfies a.e. in D

vo,t = (a(x, t) v~,~) x + b(x, t) Vo,x,

with

and since w6 : v,~,x

with

a(x t ) : (m-- v.,.xl '

) - - - - m V~,x [V~.xl " -~ b(x, t) = 1-ff P

is a weak solution of

wt : ~ A(x, t, w, Wx)

67

Lemma 4.7.

We need the following estimate for vt, analogous to the estimates of Lemma 4.4 for Vx.

lim sup v,(x, to) <: ?r X~Xo

Proof. The proof is the same as in [CF, Lemma 4.4]. [ ]

We are now ready to prove the linearization part in Theorem 2.

Lemma 4.8. There exists a neighborhood No o f (xo, to) such that (4.5) holds with 7 : D+~.(to) > O.

A(x, t, w, z) = (mp -- 1) v~ I w I " ' z + V~,x I v~,x I~w -1 ,

the results in [AS, Thin. 4] apply to obtain the H61dcr continuity of v~ and V~,x in D, uniformly in ~. The rest of the proof follows standard arguments; cf. [ACK, Lemma 2.2]. []

68 J .R. ESTEBAN • J. L. VAZQUEZ

Proof . i) Fix ~ /> 0 sufficiently small and define

I v~(x, t) : -~ V(Xo + 6X, to + 6t)

f ~ t $ > 0 and ( x ' t ) E ( ~ ~ - ) - 6 ' • ( ' 1 6 , 0 . ' q ) . The func t i~176

satisfies a.e. the equation (4.8) and is Lipschitz continuous in x and t. Therefore, given any sequence t$, ~ 0, there exists a subsequence t~j--> 0 such that

v~j(x, t)--> v*(x, t)

uniformly on compact subsets of a 2. By virtue of the lemmas above

0 < Co ~ V*(x, t) ~ 7, (4.16)

Ca 3A I v*(x, t) [ ~ if x ~ t ~ 0, (4.17)

=--X ~

and

On the other hand,

v*(x, O) <= 70'+l)/p if x =< 0.

implies

and

(V ,x ] V ,x lp-l)x 6

p(m + 1) (to + tSt) : p(m + 1) (t o -- ~])

v,j,x(X, t) -+ v*(x, t) a.e. x,

for t fixed. As a consequence of(4.1), v*(x, O) : --71/p for x ~ 0. Since we also have v*(x, 0 ) : 0 for x > 0, we conclude

v*(x, O) = v~(x, O) for x E R .

ii) We next show that

v*(x, t) >= vr(x, t) for x E 1% and t ~ 0. (4.18)

From (4.16), v*(., t) -- v~(., t) is nondecreasing. Thus the limit.

lim (v*(y, t) -- v~(y, t)) = L >= -- oo y ~ - - oo

exists. Now, if t ~ 0 and y ~ yt:

t a v*(y, t) = v*(y, O) + tv*t (y, O) + --f v*(y, 7)

t 2

__> v~(y, t) + T v*(y, 7),

where t < T < 0 . As y - ~ o ~ , ( 4 . 1 7 ) y i e l d s L ~ 0 , Actually

v*(x , t) = vv(x, t)

which implies (4.18).

holds for every x E R and t ~ 0. This is proved with the Strong Maximum Principle when (x, t) is such that x =< yt. On the other hand, if (x, t) satisfies x > yt and v*(x, t) > 0, then necessarily v*(x , s) > 0 for all s ~ t. This leads

to a contradiction sincewe knowthat v * ( x , O ) ~ - O for x ~ 0 and v * I x , X } = 0 for x < 0 . \ y l

iii) Let us finally consider the case t > 0. Since D + ( ( t ) - + 9, as t ~v to, it follows (use also Lemma4.4) that v*(x , t ) > 0 if and only if x < 9,t, i.e. ~*(t) = 9,t is the free boundary for v*. Moreover, by the Corollary to Lemma 4.4, V~(x, t) ~ 9' if (x, t) ~ (0, 0) and x < ((t), hence V*(x , t) ---- y for x < yt. This means that v*(x, t ) = v f lx , t) for x E R and t > 0.

iv) Since the limit v* is unique, the whole family vn(x, t) converges to v* as 6 -+ 0. Formula (4.5) is a way of writing this limit with 9, = D+((to). [ ]

L e m m a 4 . 9 . D - ~ ( t o ) == y.

P r o o f . This is a simple consequence of the result of linearizing. Assume that D - ( ( t o ) < ~'. Then there exists a sequence e~ -~ 0 such that

~'(to - E . ) => ~-( to) - 7'1e, , > X o - 9,e,,

for some Yl "< V- Then for some ~n E (Xo - - 9,e., Xo - - ylen) we have, from Lemma 4.8,

v ~ ( ~ ' . , to - e . ) = (Y -- 7'I) e.

[V(Xo - - 9,1e., to - - ~.) - - V(Xo - - 7'e., to - - e,,)]

1 - - (9, - - Yl) e,, o(e,,) ~ 0 as n -~- o o .

On the other hand, the estimate (v x [I)xlP--I)x (X, to -- en)=> --K(z) implies

v ~ . , to - ~ . ) > v ( ~ ( t o - ~ . ) , to - ~ . ) - I C ( ~ ( t o - ~ . ) - ~ . )

--- o + ~ ( t o - ~ . ) - K ( ~ ( t o - ~ . ) - 2 . )

~ C o - O(~ ) ,

(where Co is the constant in (4.11)). We have arrived at a contradiction. [ ]

End o f P r o o f o f T h e o r e m 2 . W e have yet to establish the continuity of vx and vt up to the moving free boundary and formulae (4.6) and (4.7).

Firstly, the continuity of vx is a consequence of Lemmas 4.4 and 4.9. This also proves (4.6). As for v t, a bound from above is given by Lemma 4.7. The other

70 J .R. ESTEBAN • J. L. VAZQUEZ

bound is a consequence of the inequality

Vt = ( m - - - ~ ) v(,Vx]P-I Vx)x-~ [Vx] p+!

> I Vx I " + ' - K ( v ) v

which clearly implies

lim inf vt ~ y(v+o/,. [] (x , t ) -~ (xo , to )

x<~(t)

5. Asymptotic behaviour

The most important consequence of estimate (1.1) with the exact constant concerns the asymptotic behavior of the solutions and their free boundaries that correspond to compactly supported initial data Uo(X). Let

at = inf {x E R : Uo(X) > 0} > -- ~x~,

a2 = sup {x E B : Uo(X) > 0} < + cx~,

and define the outer (left and right) free boundaries by

~t(t) = inf {x E K: u(x, t) > 0}, ~t(0) = a l ,

r = sup {x 62 R: u(x, t) > 0}, ~2(0) = a2.

We also define the center of mass at time t > 0 as

1 +oo x(t) = "-~ f xu(x, t) dx,

- -oo

where + o o + ~ o

M = f u(x,t) d x = f Uo(X) dx - -oo - -oo

is an invariant of the equation, called total mass, cf. (3.18). The sharp form of estimate (1.1)and inequality (4.2)for the free boundaries,

i.e. 1 ~ ]

allow a precise description of the asymptotic behavior of u(x, t), ~i(t) and ~(t), ( i = 1,2) as t ~ c ~ .

Theorem 3. Let Uo(X), u(x, t) be as above, and set k = 1/p(m + 1).

a) I f t > O, we have

k k k ( t ( t ) ('l(t) < -'7" x(t) x'(t) < -T (2(t) (~(t). t - =

b) There exists an asymptotic center o f mass, i.e. a point x~ E (a~, a2) such that

x( t ) ~ xoo and t Ix'(t) l ~ o,

as t -+ oo.

c) Let fi(x, t) = ~(x -- xoo, t; M ) be the Barenblatt solution centered at xoo with mass M. As t ~ oo, we have

i) (-- 1)" ((,(t) -- ~(t)) ~ 0,

ii) r ~ 1, t ](~(t) -- ~(t) l--~ 0,

iii) t ~ Iv(x, t) -- b(x, t) l ~ 0

1 ( I ) m uniformly on R , where oc = - - 1 i f 0 < p = ~x p (m + 1) i f l < p. p p(m + 1) < 1, and --

When p = 1--the porous medium equation--Theorem 3 was proved by VAZQUEZ in [Vl ], and much of the method of proof applies without major changes. Nevertheless, there is an interesting difference between the cases p = 1 and p =]= 1. Namely, in the latter case the center o f mass is not invariant in time for a large class of initial data. Assume in particular that Uo E L~(R) is compactly supported and that (u~)' I(u~)'l p-1 is a function of bounded variation on R. Then

+oo ~um p--I +co limt_+o _-~" -~x gum dx = .[" [P-' dx.

-fix --oo

This is easily deduced from

~u" t) p -fix(X, =< r[u,C, o/It = VAR [(Um)x ](Um)x p-llj

VAR [(u~o)' I(u~o)' [P-'].

In particular, if this last integral is not O, then x ' ( t ) =t= 0 at least for small t > O, since

1 f 1 fSu" ~-~-~-'dx. x ' ( t ) = " - ' ~ 3 xu t (x , t) dx = C X [

On the other hand, the center of mass is invariant if for instance Uo is sym- metric, i.e. uo(a + x ) - - - - -uo (a - x) for some a ER. Indeed, by symmetry x( t ) - - - -a for all t > 0 .

Proof of Theorem 3. Repeating the proof of Theorem A in [V1] we obtain the existence of x~ with al ~ x~ =< a 2 and c) of Theorem 3 holds.

In fact, x~ E (a~, a2). For if, say xoo = a2, then necessarily

~2(t) = ~2(t) = x ~ + r(t) for all t > 0.

72 J. R. ESTEBAN & J. L. VAZQOEZ

Integra t ion of (1.1) gives u(x, t) ~= fi(x, t) in Q, and since u and fi have the same

mass, u ~ ft. In part icular, (~(t) = ~l(t) = xoo -- r(t), which leads to the con- tradict ion al = a2.

We obtain a) as follows (we may assume M = 1):

x'(t) = f xut(x, t) dx = f u(x, t) V(x, t) dx

= v ( G ( t ) , t) - f V~(x, t) ( u(y, t) dy dx r r

f u(y, t) dy dx => r -- p(m + 1) t cx r

1 = ~2(t) p(m -}- 1) t [r - - x(t)] .

Finally, notice tha t a) is equivalent to

r + r(t) - - p(m + 1) t[~'~(t) + r ' ( t ) ]

<: x( t ) -- p(m + 1) tx '( t)

<: (2(t) - - r(t) - - p(m + 1) t [~ ( t ) - - r ' ( t ) ] .

As t - + ~ , c) i) a n d i i ) imply t l x ' ( t ) [ - + 0 . [ ]

Appendix A. Proof of/ . ,emma 2.3

We present a p r o o f covering all the three choices of (b~ made in Section 2. We use the me thod of ILIN, KALASHNIKOV & OLEfNIK, [ IKO]. Inequali t ies (2.6) and (2.7) will be essential in correctly est imating the different coefficients.

Let u(x, t) satisfy (2.1), V(x, t) by given by (2.3) and 0 = - - Vx satisfy L[O] = O, where

L[w] = A(x, t) Wxx + B(x, t) ww x + C(x, t) Wx + F(x, t) w 2 q- 6 (x , t) w -- wt,

cf. (2.5). The coefficients A . . . . . G are given by (2.36)-(2.40). All o f them are bounded, with bounds depending on e, 8, Mo : lluo[Ioo and M1 : II(u'g0)'[Ioo. By L e m m a 2 . 2 , for any T > 0 and some t o > 0 we have

O(x, t) >: - - to in 0T-

To prove (2.4) we fix 0 < r < T o such tha t 7 o < m i n T , p ( m + t)~, and define

1 ~'o O(x, t) = -- p(m + 1) (t - - 7) R 2 (x2 + Kt) e at

for x C R and t ~ 7 0 > 0 , We first r emark tha t

0(x, 70) -<_ -

and some positive constants 0~, R, K to be chosen.

1 < --~o < O(x, ~o)

p(m + 1) (To - - 7) = =

and

0(• t) < --%e~' < --% <= O(~k R, t).

Assume for a while that we have proved the following assertion:

There exist o~, K > O, which depend on m, p, To and the bounds o f the coefficients

A(x, t), B(x, t) and C(x, t), such that L[0] > 0 Jor every R > 0. (A.I)

Given (Xo, to) E Qr, we take 30, R such that (Xo, to) E (--R, R) • (30, T]. Then the maximum principle gives

-O(x, t) <= O(x, t) on [--R, R] x [30, T].

In particular,

1 ~o -- p(r~ + 1) (to -- z) R 2 (x2 + Kt~ eat~ ~ 0(Xo, to).

Since R does not depend on o~ and K, we obtain

1 - < O(xo, to )

p(m + 1) (to -- ~') ~-

by letting R-->oo. As r - + 0, we get

1 0(Xo, to).

p(m + 1) to

Since (Xo, to) is arbitrary, (2.4) holds. To complete the proof of the lemma, we have yet to show (A.1).

We compute L[0] and use (2.6), (2.7) to obtain

L[O'] ~ Sl q- S2 -k S3 + S,,,

where

1 [ F(x,t) ] $1 =p(rn + 1) (t -- ~.)2 Lp(m + 1 ) - 1 ,

~2 ( x2 § Kt) e z~t [2xB(x, t) + (x 2 § Kt) F(x, t)], S2 = --~

1 2Vo e~ t [xB(x, t) + (x 2 + Kt) F(x, t)], S3 = p(m + 1) (t - - ~:) R -'-~

~o e~ t [--2A(x, t) -- 2xC(x, t) + o~(x 2 + Kt) + K].

$1 is nonnegative because of (2.6). $2 has the same sign as

S~ = 2xB(x, t) + (x 2 + Kt) F(x, t)

>--_ --p(m + 1) x 2 a(x, 0 2

p(m + 1) Jr (x 2 + Kt) F(x, t)

>~ IF(x, t) -- p(m + 1)1 x 2 B(x, t) z

p(m + I)

>= p(m + 1) K% Bo

p(m + I)"

+ p(m + 1) K'co

Here we have used (2.6) again and set [B(x, t)] <__ Bo. Therefore $2 ~ 0 is implied by

= ~ o p ( m ~ - l ) " (A.2)

The same argument shows that $3 ~ 0, with the same choice of K. To study the sign of S+, we consider

S* = --2A(x, t) -- 2xC(x, t) + o~(x 2 + Kt) + K

and use the boundedness of the coefficients A(x, t), C(x, t): 0 < A(x, t) <= Ao and [C(x, t) I < Co. We find that, if x 2 __> 1,

which is positive when

S~ > --(2Ao + Co 2) + (o~ -- 1) x 2

> ~ x - - 1 - - ( 2 A o + Co2),

c~> 1 + 2 A o + C 2. (A.3)

On the other hand, if x 2 ~ 1,

which is positive when

S* >~ --(2Ao + C 2) -- 1 + K,

K > I + 2 A o + C o 2. (A.4)

The choices (A.2)-(A.4) show (A.1) and this ends the proof of the lemma. [ ]

Appendix B. Proof of Lemma 3.3

To begin with, we introduce some notation that will be useful in the sequel. We write ~+(u, z) in the form

~+(u, z) : ~/,,(z) + f~(u) H~(z),

where %(z) is as in Lemma 2.1 and the functions f,(u), H~(z) are defined as follows.

a) if 0 < p - - < _ l , f~(u)=O and H~(z)=O.

1 b) if 1 < p < 1 + m +-----1' then f~(u) = eu and H~(z) is as in Lemma 2.5.

~ m u~-m and c) if 1 + m + 1 ~ p ' we set n-----p(m+ 1 ) - - 1, f~(u) m

H~(z) ~- z.

Let us consider a cutoff function r E C~(R) with 0 ~ r ~ 1, and fix T > 0. Our aim is to estimate the integral

4m r +oo I (m + 1) 2 f -co f t~(x)2 ](u(m+O/2)t 12 dx dt.

Here and in the sequel we omit the subscript e for simplicity. Thanks to the identity

4m /,,(m+1)]25 2 ( l lm) t 14 t

(m + 1) ~ ! '" ,,L =

we have I = f f tr (u% u, dx dt.

We now use (2.1), integrate by parts with respect to x, and get

I = f f t~(x) 2 --~ -~x qbe u, -~x dx dt

-- 2 f f tr r (Um)t ~e (U, ~-~X ) dx dt

~ I1 +12 .

We next estimate 11 and 12 separately.

Estimate o f It. We set ~(z) = ~p(r) dr and [-I(z) = f H(r)dr. Then we have o o

11 = - f f tr 2 -[i ~ + f(u) --~ I-I ~ dx dt,

and integrating by parts, with respect to t,

I, ~ I3 + I , , where

h = f f r 2 ~ ~ ax at,

#) I, -- f f r 2 (tf(u)), h ~ dx at.

76 J . R . ESTEBAN & J. L. VAZQUEZ

Let us est imate these integrals:

I3 <= f f r 2 ~ v, t-~x ] dx at

( um) = - 2 f f r r um~j ~x dx dt

<_

- f f r ~ u" [ u , - ~

1 m q- 1 rrjj r m ,tu_+l), dx dt

- 2 f f r r umq~, (U, ~x ) dx dt

- f f r H ~ dx tit.

N o w observe tha t the last integral is nonnegat ive and use u m+l =< 2m[(u - - e) m§ + e " + l ] in the first one to obtain

/~_<-- 2 m 2 m

^llm+l @. /~m§ - - ellm+l m + 1 Iluo,, f ~(x) 2 dx

In order to est imate the last integral, we use (3.5), (3.6). Indeed,

( qb u, <= C~ -= ~p(Ml) + f(Mo) H(M~)

with Mo = 1 + Iluol[~, n ~ = II(~)'llo~. Therefore

2 m 2 m ^Hm§ ~ 8m+l

m + 1 Iluo,, - ~IFm§ ~ f r ~ dx

+ 2Ct f f g(x) I r l u m dx dt

2m 2 m < II Uo,~ o,m+l e,.+I = m + l - - ~iJm+l + ~ - - ~ ' f f r 2 dx

§ ~ f f r ~ e n+* dx dt + ~ f f r ~ u m-' dx tit,

where 2 > 0 is arbitrary. Finally

m m+!

= m + l

§ (m -[- 1)~.Mg/[uolh

( 2m T) + ~ + 2 , ' + ' f r 2dx

+ f f r u ~-~ dx dr,

where we have used (3.7). We next estimate L . To this aim, we remark that (tf(u))t => 0 holds in all the

cases, because of (3.9). Then we put C2 = /~ (M1) and get

I , <= C2 f f r 2 (tf(u))t dx dt

= C 2 T f ((x)2f(u(x, T)) dx

= C~r [f $(x) ~ [f(u(x, T)) -- f(e)] dx + f ( e ) f r ~ dx].

In case b), we have .f(u) = eu and then

L <= eC2T [f r T) -- el dx + e f r 2 dx]

< eC~T [f (u(x, T) -- e) dx + s f r dx]

< ec2r[l[uo[h + e f r dx],

He from (3.7). In case c), we have f ( u ) = - - d '-m with n - - m ~ 1, and

m

I , <= nC2Te[(n -- m) M~ - ~ - l IIUo[ll + e"-m f ~-(X) 2 dx], m

where we have used (3.5) and (3.7). This completes the estimate o f /1 .

Estimate o f 12.

]/21 ~ 2Ct f f tr I r (~ ) , I dx dt

< �89 + 2mC~,ff tr 2 ~ - 1 dx dt.

We now put together the estimates of I~ and I2, and obtain

� 89 where

Is = 2mC21 f f r 2 d " -~ dx dt.

Let us now pass to the limit as e -+ O. We take ( , ( x ) = (o(eX), where (oEC~(R) , 0 ~ ( o ( X ) = < 1, (o(X)= 1 for [ x l ~ 1 and ( o ( X ) = 0 for ] x [ ~ 2 .

78 J . R . ESTEBAN • J. L. VAZQUEZ

We have

where

Then, if m ~ 1,

rtl

/3 < I1Uo,~ _ ilmq- 1 = - - e l l m + t m 4 - 1

4- (m 4- 1) 2(M~ n 4- e m) [luolh

4- 4 + 2 e m

4- 13,~

c? , z3, = T f f r dx dt.

CTC~M~ '-~ I3'~ ~ 2 e--~ O

as e - -+0. I f 0 < m < I, we use (3.5) and get

C T C 2 e m --~ 0 1 3 . ~

2

as e--~ O. Therefore, as e---~ O, we have

m ,, IIm+ 1 lim 13 < II"ollm+l -}- (m -21- 1) 2 HUoll~ HuOHl.

= r n 4 - 1

As for the integral /4, we have in all cases /4 ---- O(e). Is is similar to I3,6. Since 2 ~ 0 was arbitrary, we finally get

l i m I ~ Cm FJUolI,~_~. [] ~ -'~-0

[AhS]

[An]

[A1]

[A2]

[A3]

[A4]

R e f e r e n c e s

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Departamento de Matemfiticas Universidad Aut6noma de Madrid

28049 Madrid, Spain

(Received December 23, 1987)

Homogeneous diffusion in ? with power-like nonlinear diffusivity

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