^ i IC/90/47

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

fON THE FIRST INITIAL BOUNDARY VALUE PROBLEM

FOR PARABOLIC EQUATIONS

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

Martin Lopez Morales

1990 MIRAMARE-TRIESTE

T " T" "T

IQW47

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

ON THE FIRST INITIAL BOUNDARY VALUE PROBLEMFOR PARAnOLIC EQUATIONS *

Martin Lopez Morales"

International Centre for Theorclical Physics, Trieste. Italy.

ABSTRACT

In (he present paper we establish the solvability (existence and uniqueness of the solution)

of the first initial-boundary value problem for linear and nonlinear parabolic equations of second

order in a cylindrical domain, under the assumption of Holder continuity of the coefficients only

with respect to the space variables. The corresponding solutions and its derivatives up to the second

order inclusively satisfy the Holder condition with respect to the space variable and with respect lo

the time.

MIRAMARE-TRIESTE

March 1990

To be submitted for publication.Permanent address: Institute* de Matemalicas, Academiadc Cicncias, Callc"0"N.X, Hntre 17 y 19.

Vctlado, Mabiina, Cuba.

t. INTRODUCTION

In (he present paper we establish new a priori estimates in anisotropic Holder norms for

the solutions of the linear parabolic equation

)tl = f(t,x) (1)

in a cylindrical domain QT = \0,T] x ft with the inliial condition

u||.o = (9(z)

and the following condition on the lateral surface Sr ofQr

(2)

(3)

I lere x = ( i t , . . . xn) is a point of ft , ft is a bounded domain in n- dimensional Euclidean space

En,t e ( 0,7*1. Sr » | 0 , r i x flft.where 3ft is the boundary of ft. ft = ft Udft .

The novelty in these estimates consists in the fact that, unlike the known results 11 J-|6|.

which are proved under the fulfilment of a Holder condition with respect to the totality of variables

( t , x ) on the coefficients of equation (1), our estimates are obtained only under the assumption

of Holder continuity of the coefficients in | 0 , 7"] x ft with respect lo ihe space variables, in this

connection, however, we also obtain an estimate of the module of continuity with respect to ( of

the higher derivatives u,.*, (but not u().

On the basis of new a priori estimates for the solutions of the problem {I \-Q) we establish

in this work the corresponding theorem on the solvability for this problem in Holder anisotropics

spaces.

With the aid of Ihe results of the linear theory we obtain a theorem on local solvability

with rcspcci lo ihe time ( of the first initial-boundary value problem for the nonlinear parabolic

equation

u, = ^ ( £ , i > t ( , u J , t i I I ) (4)

in Qr with the initial condition (2) and the boundary condition (3), where uz = ( « „ , . . . u , , ) ,

« n = ( • • • , U I , x / ( - • • ) , I < » , j < n .

In the present article E(j.(4) is linearized directly. No conditions are imposed here on ihe

nature of ihe growth of ihe nonlinearity of the function .4, which is defined for( ( , i ) € Qr and any

u, n,,!! , , , The main assumption concerning the function A(t, x,u,p,r), where p = (p i , . . . ,p , )

and r = ( . . . , r,; t . . . ) , I < i, j < n, is the parabolicity condition: for any nonzero vector f =

(fi i- •• ,tti> £ E* and any ( t , i ) € Qr, u,P,f

r r -

.-. n-'!>"-"-

where A,{ - • •) is the m.ilrix |Hr.>(«, x, u, p, r)J| and (£, n) denotes here and below the usual scalar

product in En.

We assume that the coefficients o^ ( t , x ) of Eq.(l) satisfy the uniform parabolicily con-

dition: for any nonzero veciorf = ( f i , . . .£„) G En and (t,x) € QT

IJ . X = const. (6).j-i

We suppose in all the woric that in Eq.(l), the function / = /i + / j ; av . h,c and /isatisfy the Holder condition in QT of exponent a 6(0,1[ with respeci to the space variables andft satisfies the Holder condition in Qr of exponent 0 G]0, l[ respect to the space variables, 0 <a < 0 < 1. All the coefficients and the independent terms are continuous in the cylinder Qr.

On the other hand, as complementary conditions, we require that the coefficients a,, ((, x)satisfy the Lipschiii condition with respeci lot on the lateral surface ST °fQr- Ihe other coefficientsarid the function / I ( 1 , I ) saiisfy the Holder condition in Sr of exponent ct/2 respeci to t and thefunction / j ( | r i ) of exponent 0/ 2.

We require less smoothness conditions from the functions A(t,x,u,p,r), ip(x) andi/>( (, z) than in the works |71-(9|.

2. BASIC NOTATIONS. AUXILIARY PROPOSITIONS

Tor the continuous functions ti(t ,i) in the cylinder Qr. we introduce the following

Hi ' t - ^p MSi0,|iiftlJ.= sup H ; O P (7)

wheresup |«(t,sr)|, « „ (8)

Por the functions thai have continuous derivatives with respect to x up to the order i, (I - 1,2)

inclusively in the cylinder QT we define Ihe norms

H J B I < « 0 , 1 , 2 (to)

i I' + V I I'

We will denote hy C ' ^ ( Q ( ) , € = 0 , 1 , 2 the Banach space of functions u( (, i ) that are continuous

i n Q , = [ 0 , 1 ] x f l together with all derivatives respect l o i up to the onlerf , £ = 0 , 1 , 2 inclusively,

and have a finite norm (10).

We define the parabolic distance between each two points P = (t,x), R = ( T , y) o f Q r .

x = (x,,,..xj.y = ( ) / i , . . . , V . ) . O < t < r by the magnitude

( O )

Por the functions thai have continuous derivatives wilh respect to x up to the order 1,(1 = 0,1,

inclusively in the cylinder QT, we consider the usual norms

- sun (14)

(15)

(16)

(17)

We denote by Ct,a(QT), t = 0 , 1 , 2 the Banach space of functions u( t, i ) that arc

continuous in QT together with all derivatives respect to x up to order t inclusively, £ = 0 , 1 , 2

and have finite norms <15>—(17) respectively.

It is possible to consider all the preceding definitions in the layer n r » f O.TI x En (see

110|) or in Sr.

Wilh respect to the coefficients of Bq.(l) we assume that 0^(1, x),b,(t,x),c(t,x) e

Clr*(QT).iJ= 1 nand

ij-t

Moreover, on the lateral surface ST of QT <•<« coefficients a,j(t,x) satisfy the Lipschitzcondition will) respeci to (. ihe coefficients *,-( t, x), c(t, x) satisfy on Sr the Holder condition wilhrespect tot of exponent «/2 and

We will make the following assumption concerning the domain (I: there exists a number

pn > 0 such dial the part of the boundary d t l lying in the neighbourhood { | i - xa | < pa} of an

arbitrary point xi» of dil is rcprcsentable by an equation of the form

in which the function h has second derivatives that are bounded and Holder continuous of exponent

rt G (0 , I) uniformly with respect toxo.

For Eq.(4) we consider in addition to the parabolicity condition (5) that there exists ;t

domain / / / = { ( t , i ) £ QT', |U | < J, \p\ < J\r\ < J} in which the function A(t, x,u,p,r),

together with its derivatives wilh respect to u, p ,̂ r,j up to the second order inclusively, is continuous

and satisfies (he Lipschitz condition wiih respect to u,p,r and a [folder condition of exponent

or £ (0,1) with respecl to x and the constant By Moreover A(t,x, 0 ,0,0) 6 Cl^tQr)

\A(t,x,0,0,0)\[T^ < Bo, 0 > a , (20)

on ihe lateral surface ST of QT the functions j ( r ^ { l , i ( u , p , r ) i , > = I , . . . n satisfy the Lipschitz

condition wiih respect to t and the functions A(t,x,u,p,r) Ap,{ttx.u,p,r) satisfy on ST Ihc

I lolder condition of exponent 0/2 with respect to ( and the constant Bj Consider

B; = [ i e£ , ; i , >0 } and GT = |0,r| x {x e En; x» = 0}

The a priori estimate for the solution of the heal equation

u( - Ati = (21)

considered in the half-layer wf = [ 0,T] x E* with the zero initial condition

u|i-o = 0 ,

and Ihe boundary condition on GT

will be used in (his work to solve Ihe first initial-boundary value problem (l)-(3)

(22)

(23)

Lemma 1 Suppose / = / , + / 2 , / , € CJ^(ir^), h e C^(7rf), 0 < a < 0 < I,l / i |« I , < «•• \h\?tfi < coandu( t , i ) € C ^ ' ( w f ) is a solution of Ihe problem (2IH23) in jr*.

Then (lie inetjnalily

(24)

is satisfied forO < ( < T. Here n^ = ) 0 , t | x /?;, G t = [0,(1 x { i G /'7n; i « = 0 } and i f is a

constant depending only on n, a, 0, T-

5

Proof For every solution ir( t, i ) G Ci,a( wf) ° ^ l n e problem

14.0=0, (2ft)

(27)

,,...,!^,),;^ 1,2

considered in a half-layer w£, where

holds the estimate (see 1111)

The function U I ( ( , I ) = u ( t , i ) — v(t,x) is a solution in irj iof the equation

wilh the zero initial condition (22) and the boundary condition (23). Note that (he functions F\it,r) t

CirJ(fff) and Fiit,x) e Cj^'drf). Moreover,

' , 0 ) = 0 , > = 1 ,2

Now we extend the function w{(, i ) into the domain i n < 0 by putting

-r» ^ -r \- fw(( , i ' , i . ) , i , >0

The function ui( t,x',x*) is a solution in irj-= [ 0, r | x En of the equation

(30)

with the zero initial condition (22), where

i. >0

Under (hese assumptions, the function t l i ( ( , i ) has continuous derivatives with respecl to

i up to the second order inclusively in ihe layer 1 0 , r | x En. Fn>m the conditions a), b) it follows

that /'", e Cn^J(wr) J |n ( l Ft € C^^d f r ) - F<>r 'he solution ur((, i) e C2,o(n7-) of the Cauchy

problem (30), (22) holds the following estimate (see Lemma 4 in [ 12]) for 0 < t < T

i < (31)

r -

Using the inequalities Mi ' i < M i ' i . Ifi C\h |Sj,, we obiain from (31) ihe estimate for 0 < t < T

< |/i C + l/> lo.!..1A C < l/i l!,'i •

+ 1/2 l?>) I (32)

fiy the eslima(cs of the moduli of continuity wilh respect Co the time for (he derivatives of ihc solu-

tion of ihc Cauchy problem for linear parabolic equations (see [ I3|), we obtain for (he ilcrivalivcs

ZJjw(t , i ) , £ = 0,1,2 f o r i G E* and* i,t2 6 | 0 , ( | (he inequality

m • \h&>i

Combining the inequalities (32) and (33) we can gel the eslimale

(34)

Since u ^ m t i i w obiain from (28) and (34) the eslimale (24).

Now consider the equation

= U , - ".,,, = f\{t,X) (35)

in the half-layer wf = | 0 , r | x E*. i 0 G fi-

We assume thai the coefficients a^(t,zo) satisfy ihe uniform parabolicity condition (6),

M t , i o ) | < A , te fO. r ] (36)

and ihe Lipschiiz condition in [{), 7*] with the constant A.

2 Lei u( t, x) 6 C [ ^ ( wf) be a solution of the problem (35), (22), (23) in (he half

layer irf. Under ihe same conditions of Lemma I the inequality (24) is satisfied for 0 < I < T,

where ihc constant K depends on n,a,0,X, A ,T*.

Proof Using a Isncitr transformation

,p J t= l , . . . ,TU( = T. (37)

» < C i (n ,A r >) < dci(/4t,(0) < Ci(n,ii,\), (Aki = 0 for Jt > »"), we can transform

the operator £™,, a,,U, x«) %iz,tix, into the operator £)",, U(,(,(T,0. £ = ( ( ! , • • • { • ) , U<T,£) =

U ( T , I ( T , O ) and ihc half space/7* inlo itself (fB = JT",, J 4»( ( ) i , = / I™(0 i« . >!,»,(0 > 0).

In Ihe new variables ( r , f i , . . .£„) Rq. (35) is transformed into the equation

O8)

where

Noting that

— |(..o = 0 t = l , . . . , n - I ,

we obtain from (3K) and (40) that Fi(T,f ' ,0) = / i (T, z'( T,( ' ,0),0).

(40)

Applying Lemma I lo the solution U ( T , ( ) of the problem (38), (22), (23) and using (3H),

0')), (40) we obiain ihe estimate (by going back lo the original coorinates)

for 0 < ( < 7'.

Applying the interpolation inequalities (see Lemma 2 in (12|)

with e = j L in (41) we find that

(42)

Ho,«l

Using the initial zero condition (22) we derive from Eq. (35) the following inequality

then arguing as in the work |13J we can eliminate the term |u|[,'}| in (43). Therefore we gel the

estimate (24). Under the assumption concerning the dii there exists (by the Heine-Borel Lemma)

anumherpo > 0, such that ihe part of Ilie boundary dCl lying in Ihe ball £ w = {x; \x~xn\ < pn}.

Hi e c)Cl is rcprcscnlable by one of the equations

Xi = M i , ) , x. = ), t = 1,... N,

ihe point i , belongs to a domain E>( c Sn-i and |ft*|2iO < const., a 6 (0,1) .

Tor the functions ip(t,x), defined in ST = I 0,7*| x dii we intrmlucc Ihc followingnorms

where the norm | ||^,,, m = 0 , 1 , 2 in the right hand side is considered in the cylinder \ 0 , t] x

Df, S, = 10, ( | x an. If Iv^lUli* < oo we will say thai f> £ Cj | J , |S i ) m .o ,u • Moreover, we will

consider in S, the norm

Now we shall consider Eq. (I) with the initial zero condition (22) and the zero boundary

condition

«|ST = 0 (44)

X A PRIORI ESTIMATES FOR SOLUTIONS OF LINEAR PARABOLIC EQUA-

TIONS

Theorem I Letu(t,z) £ C2,«(QT) be a solution of the problem (1), (22), (44) in the cylinder

QT. Assume that / , e Cj^(Or). h € C^(Qr) . 0 < a < 0 < t and | / , |*"n < oo,

l/z $0 < oo. Then ihere exists a constant K, depending only on n, a,f),\,T, dO., M and B such

thai(O <t<T)

l«l?i < ff I I/' l ( 1/2 li'J, + 1/2 |& (45)

Procif l.ci xo be an arbitrary point ofdii. There exists two numbers pa > 0 and AT such thai

ihe part of the boundary dQ lying in the ball C™ = { i ; | i — i o | < po } is rcpresentable by one of

Ihe equations

* i ) , \ht\2,a< const., l~\t...N

Denote by Q" the cylinder 10,T] x K£. The transformation

Vi = Xi, •. - , y - i = i , _ i , V, = i , , • • , Vn-i = x* ,

f, = i ,-Mi!),«=l, . .JV, (46)

maps (df i ) U IC^ onto a set which lies on yn = 0 and, as we may assume, it maps il U K.™ onto

a domain lying in (he half space yo > 0 and some half ball £JJ" n {j/n > 0 } , JJO = y(x0) is

contained in this domain.

tion

The mapping (46) i s o n e - l o - o n e , nondcgeneratedand il transforms EQ. (1) into the equa-

(47)

where u( l ,y) = u( t, i< « ) ) , the coefficients a,;( I, !/),b,(t,y) c ( t ,y ) , / i ( t , y ) and/jU,!/) satisfy

the same assumptions of continuity that a,j(.t,x), b,{t,x) ,c(t,x) / i ( ( , i ) and/2<t , l ) .

Let d be a number in ( 0 , l | , tj(j/) = ni\y — Jtol). where j*(r) is a infiniicly differentiablc

funclion on r > 0 , ;i(r) = 1 for 0 < r < rf; ̂ ( r) = 0 for r > 2d and satisfying the inequality

3

£V| l i ( > )(»-) | < const (4R)• -1

We will assume that 2d < a, then the function w(t,y) = Tj(y)u(t,y) satisfies the

equation

where

s w, - (4'J)

•J-I

in the half layer TT£ = [ 0, T\ x E* with the zero initial condition (22) and the boundary condition

wL-o = 0 (50)

Now arguing as in Theorem I of [10) we find that

where | t | ,2ct indicate that the norms in the right hand side of (51) are considered in the cylinder

Qiid = 10 ,1 | x {JCJ^n£*} and the norm | • | j ^ ' i s considered with respect to all variables irt

I" St,id ~ [0,11 x {y; |y — jrn| = 2d} the following inequality is satisfied

Applying Lemma 2 to the solution w((,j/) of Ihe problem (49), (22), (50) and using (51).

(52) we obtain the estimate (taking d = min( I Ay)", j))

(53)

10

T r "

For some <4(, d\ Uh < d-\) we obtain the estimate (by going back to the original coordinates)

With the aid of Theorems 1 and 3 of f I3| we derive the interior estimates

lu\i'& i f f l/'lo'a + ' l/itoji + l"li,o 1< (55)

where the norms on the right hand side are considered in the cylinder fO,( | x Q * ^ , f l * ' 4 =[x E SI; disl (x,dSi) > %•} and the norm on the left hand side is considered in the cylinder

Since io is an arbitrary point of dii we can gel from (54) and (55) the estimaic in Q,

Applying the interpolation inequalities of Lemma 2 in | I 2 | with e small enough and

arguing as in the last part of the proof of Lemma 2 lo eliminate the term |u|£'o we get the estimate

(45).

Remark I We can reduce the first initial boundary value problem with non-zero initial and

boundary conditions loihe problem (1), (22), (44), if the initial and boundary functions be long lo the

corresponding spaces Cj ,„(£}), C7,a(Sr) and the boundary function has a continuous derivative

wiih respect to t bounded and measurable on ST-

4. EXISTENCE AND UNIQUENESS T H E O R E M S

Theorem 2 Suppose that all conditions of Theorem 1 are true. Assume further that (he fol-lowing consistency condition holds

Then there exists a untrue solution of the problem (1), (22), (44) in the space Ci,a(Qr) wi

continuous derivatives ut in QT.

We can gel the proof of the theorem on the basis of the new a priori estimates establishedin this work and with the aid of the method of continuity in a parameter (see 111 and 1111).

We proceed now lo formulate the local esistence theorem for solutions of the nonlinearproblem (4), (2), (3).

We have that

A{t,x,u,uItulz) = £(u) +.4(t,i,0,0,0) + F(t,x,u,aItuxz) (57)

II

where

£(u)

HAAt,x,0,0,0),utI)

L c m m a 3 Suppose that the functions u ' ( ( , i ) 6 C['^{QH), Q,o = | 0 , j n | x Cl,e= 1 ,2 .Assume further thai j u ' ( t . i ) | < J, | u , ( ( , i ) | < / , ! « „ ( t , i ) | < J f o r ( ( , j ) £ Q,o . Then for0 < ( < lo ihc following inequalities are satisfied

(5H)

(5'J)

\F(t,x,ut,ulu]a[) - F(U,«M,«L)

For the proof on this Lemma, see 110] *

Lemma 4 Under the same conditions of the Lemma .1 the following estimates arc satisfied for0 < t < ta

(fit)

The proof of this lemma is analogous to the Proof of Lemma 3.

Theorem 3 Suppose that all assumptions with respeel to the function A(t,x,tt,p,r) hold.Assume further that A(t,x, u, p, r) satisfies the following consistency condition

•4(0 , i ,0 ,0 ,0) |an = 0 (62)

Then there exists to, determined by the above assumptions, such that the problem (5),(22), (44) has in the cylinder Qtt = [0,t<i] x il a unique solution u{(, i ) € C2,<,(Q(|>) with aconlinuous derivative u(.

Proof We will prove Theorem 3 by means of an iterative process in which one successively

solves the equations

ulB1 = £ ( u m ) + f 1 ( t , s ,u"- I ,u™" l ,u™- 1 )+ J 4(t ,a: > 0,0 ,0) (63)

m = 1 , 2 , . . . , with the zero initial condition (22) and the boundary condition (44). Furthermore

u D ( t , i ) E 0

We will show thai (here exists a such sequence of functions um(t, x), m = 1 , 2 , . . . defined in some

cylinder | 0 , Cn I x fl with („ small enough. Moreover they are bounded in Ci,a(Qi,)-

12

We assume that there exists the functions u', t < m - I ami they satisfy the inequality

|u'|?;; < ? = consl = m i n ( l , J ) (64)

where the numbers to and g are still to be determined.

Now we can consider Eq.(63) as an equation of the type ( I ) , where

| l 1 ) = .4(t,i,o,0,0)

Prom Theorem I it follows that there exists the function u m ( ( , i ) e C j , a ( Q ( o ) and by Theorem 1

it satisfies the inequality

|u~|?;; < K,\\F\<0'°><'°> (65)

We will use Ki, Kt, Kj,... lo denote constants, depending on a,0,Bj, Bo, 9 f t but not depending

o n in •

Using (58), (60), (65) and (20) we obtain

From the inequalities (64) it follows the estimate

(67)

Pulling 2 K2q < £, yij we find that

In the last inequality we have selected

On the other hand the function ui = u™ — u m ~ l is a solution of the equation

(68)

with the 7cm initial condilion (22) and the boundary condilion (44).

Applying theorem 1 to the function w ( i , i ) , using (59) and (61) we obtain the inequality

13

From (64) and (68) it follows that

Therefore if we choose

then

(70)

We have dtterinined the number t» by (6R), lhcn the sequence u ™ ( ( , i ) converges in the space

Ci^iQta)- Consequently we can get a limit for m —t oo in Eq.(63).

We proceed now to prove the uniqueness of the solution of the problem (5). (22), (44).

Assume that there exists two solutions u ' ( i , x), u 2 ( ( , j ) of this problem and consider the function

« = u1 - u1. This function satisfies the equation (in the smaller of the corresponding cylinder Q ( o)

u( = Lu

where

,u= /Jo

+(1 -

,i

+ (/ Ad.l.T.l'JO

Here the components of the vector function i ' = ( u ' . u ^ u j , ) £ = 1,2 belong to the sp;icc

Cla'"J(Qto) and the coefficients of the operator L also belong to this space. Since u ( t , i ) sat-

isfies the zero initial condilion (22) and the zero boundary condilion (44) lhcn from Theorem I h

follows that t i ( t , i ) = 0 . This completes the proof of Theorem 3.

Remark 2 We can reduce the first initial-boundary value problem with nonzero initial and

boundary conditions to the problem (5), (22), (44) if the initial and boundary functions belong to the

corresponding spaces Cjj(Cl), dfiiSr) alu^ l n c boundary function has a continuous derivative

with respect to t hounded and measurable on 5> .

Acknowledgments

f l ic author would like to thank Professor Abdus Salam, the International Atomic Fincrgy

Agency and UNliSCO for hospitality at the International Centre for Theoretical Physics, Trieste.

14

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15

T r

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del Centro Internazionaie di Fisica Teorica