On a certain limit problem for parabolic equation1. In this paper we shall give the solution y(X) of the equation(1) (P-C(tj)v(X) = F(X),

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXII (1981) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXII (1981)

M. Filar and J. Musialek (Krakôw)

On a certain limit problem for parabolic equation

1. In this paper we shall give the solution y(X ) of the equation (1) ( P - C ( t j ) v ( X ) = F ( X) ,

3

where X = (x,t), x = (x1, x 2, x 3), P = £ Dx. — Dt and C, F are known

functions in the domain 1-1

(2) W = { X: IxjI'< oo, jq > 0 (i = 2, 3), t > 0).

Let H denote the set of functions v{X) defined and continuous with the derivatives D*. y(X ) (i = 1,2,3; a = 1,2), Dt v ( X ) in the set W. We shall construct the solution v { X ) e H of equation (1) satisfying the initial condition (3) y(X ) = f i ( x ) for X e S i = (X : \xJ < oo, xt > 0 (i = 2, 3), t = 0}

and the boundary conditions (4) (DX3 + h)v{X) = F 2 {xt , x 2, t)

for X e S 2 = { X : \xt \ < oo, x 2 > 0, x 2 = 0, t > 0), (5) DX2v( X) = F 3( xl , x 3,t)

for X e S 3 = ( X \xt\ < oo, x 2 = 0, x3 > 0, t > 0), where h is a negative constant and f i , F 2, F 3 are known functions.

By the change of variables

t

(6) v (X) = и (X) exp [ — j C (s) ds] , о

where C(t) is continuous in the interval [0, oo), equation (1) can be reduced to the form

(7) Pu (X) = / (X),

where

/ (X ) = F (X) exp [ J C(s)ds].

( 8 ) о

2 0 6 M. F i l a r and J. Mu s i al e k

Thus, in order solve problem (l)-(5) it is enough to construct the solution u ( X ) e H of equation (7) satisfying the initial condition

(9) . u( X) = f 1(x). for X e Si and the boundary conditions

(10) (DX3 + h)u (X) = f 2 t a , x 2 ,t) for X e S2, (11) DX2u{X) = /3( *i, x3, t) for X e S 3, where

t

f 2( xu x 2, t) = F 2( xl , x2, t) exp {

J

(5 ) 0

t

f 3( xl , x 3, t) = F 3( xu x 3, t) exp [J C(s)ds].

0 2. Let Y = (y,s), у = {ух, у2,Уъ) and let

[(^ 1- } ;1)2 + (^2+ ( - 1У>;2)2 + (2Сз + Уз + у)2] 1/2 for i = 1,2, rt = <[ [(^1 У1)2 “h(-^2 У2)^ т (х з -j-( l)1 Уз)2] 1/2 for i = 3,4, [(^1-У1)2+ (л2+^2)2+ (лз + ( “ 1Утз)2] 1/2 for i = 5,6.

Let

(1 2 )

where

G (X > y ) = { , ? i K ‘ (* ’ r ) f o r s < ( ’

^0 for s ^ t,

(13) K {X У) = | 2/l(f _ s ) 3/2ï exp M exp Aidv for i = l , 2 , (r — s) 3/2 exp Ai for i = 3, 4, 5, 6, and

A; - n

4 (t- s ) i = 1... 6.

Let us consider the set W0 x W\ where

W0 = {X : |xf| < 00 (i = 1,2), ax ^ x3 ^ a2, tx ^ t ^ t2}, W‘ = (У: Iyx\ < 00, 0 ^ yt < 00 (i = 2, 3), 0 ^ s < l}, and where at, t{ (i = 1 , 2 ) are positive constants.

Lim it problem fo r parabolic equation 207

We shall prove the following Lemma 1. The integrals

00

1ЦХ, Y) = J ehvDaX3 exp Atdv (i = 1,2; a = 0, 1,2) о

are uniformly convergent in the set W0 x Wl .

P roof. The integrals Ц ( Х , У) are linear combinations of the integrals OO

I ( X , У) = J ehv( t - s ) ~ J{x3+ y 3 + vY exp Atdv (i = 1,2), where j ^ 0, /? ^ 0.

It is enough to know that integrals I ( X , Y) are uniformly convergent in the set W0 x W1. I {X, У) can be represented in the form

I ( X , Y) = M x " (хэ+Уз + v)2 P/2+j

~ (^з+ Уз + у)2 ~

4 (t- s ) exp

_ - 4 ( t - s )

x exp ( x i - y i ) 2+ (x 2 + ( - i y y 2)2

— 4 (£ — s) (Xs + ys + v) 2J( t - s f 12 dv, where is a positive constant. Applying the inequality

(14) Ake~A ^ kke~k for A ^ 0, к > 0

(under the sign of the integral) and inequalities exp A{ ^ 1 (i = 1,2), we obtain

II ( X , У)| ^ M 2

J

where M 2 is a positive constant, which implies our assertion.

Theorem 1. The function G( X, У) given by formula (12) have the following properties.

1° G( X, Y) satisfies the equation Px G( X , У) = 0 for (X, Y ) e W x W\

2° G ( X , Y ) satisfies the following boundary conditions:

(15) (DX3+ h ) G ( X , У) = 0 for X e S 2, Y e W ‘ , (16) DX2G{ X, Y) = 0 for X e S 3, YeW1.

P roof. In order to prove Г, we observe that the functions K i ( X, У) (i = 3, 4, 5, 6) and also K f X , У) (i = 1,2) (see Lemma 1) satisfy condition Г.

Thus the function G( X, У) given by formula (12) satisfies condition 1°. Now

208 M. F i l a r and J. M u s i a l e k

we shall prove 2°. Applying Lemma 1 and the formula for integration by parts we obtain

00

(17) Dx K t ( X, Y) = 2h J ehvDvl{ t - s ) ~ 312 exp AJdv о

= - 2 h ( t - s r 312 exp A ^ + v - h K d X , Y)

for ( X, Y ) e W x W l (i = 1,2).

By formulas (12), (13), (17) we obtain

(18) (DX3 + h) G( X, Y) = —i ( t — s)~5/2 [(x 3—y3) exp A 3 + (x3 + y3) exp Aa + + (x3- y 3) e x p A s + (x3 + y3) e x p A 6] +

+ h(t — s)~ 3/2 [exp A 3 - exp Aa + exp A s - exp A 6]

for X e W n S 2, Y e W 1.

For X e S 2 we have x 3 = 0 and consequently A 3 = Aa, A s = A6. Basing on (18) we obtain (15).

By (12) and Lemma 1 we have

(19) DX2G ( X , Y ) = —i ( t — s)~5/2 \_(x2 — y2) exp A 3 + (x2 — y2) exp Aa + + (* 2 + У2) exp A 5 + (x2+ у2) exp A6] -

Y) + (x2t+ y 2) K 2( X , Y)]

for X e JYnS3, Y e W 1 If X e S 3, then x 2 = 0 and' A3 = A s, Aa = A6, K t ( X, Y) = K 2( X, Y). Now basing on (19) we obtain (16).

3. Let us consider sets

Щ = {(Y1.Y 2.Y 3): lYil < 00, y« > 0 (i = 2, 3)}, Щ = {(Yi.'Y2.s): lYil < 00, y2 > 0, s ^ 0}, W3 = {(Yi.Ya.s): lYil < 00, уз ^ 0, s ^ 0),

W = {(Y1.Y2.Y3, s): lYil < 00, y» > 0 (i = 2 ,3 ),0 < 5}.

Denote by W/ (i = 2, 3) the part of the set Wt (i = 2, 3) situated under the characteristic s = t.

We shall prove that under some conditions imposed on functions fi (i = 1 ,2 ,3 ),/ , the function

u ( X ) = £ и,(ЛГ), 1=1

(20)

Lim it problem for parabolic equation 209

where

иЛХ) = у ШШС{ Х, Пш=о <1 у ,

w 1

U2(X) = - y J j i f 2(y i, y2, s )G (X , Y)\n = 0dyl dy2ds,

(21) wt2

u3(X) = - y

J

wt

where у = (2у/п)~3, is the solution of problem (2), (7), (9)-(ll).

4. In this and the following chapters we examine the properties of the functions щ(Х) (i = 1,2, 3,4).

The function ux (X) may be written in the form

(22) ux(X) = £ n i(* ),

i=l where

(23) uii (X) = y m f 1(y )K i (X,Y)\s=0dy (i = 1,...,6) wi

and K t(X , У) (i = 1,.’..,6) are given by formula (13).

Lemma 2. I f the function f x is measurable and bounded in Wx, then the integrals

< & ( * ) = Ш f MDl - Kj i X, Пш-ody

"T

(i = 1,2,3; j = 3,4, 5, 6; a = 0, 1,2) are uniformly convergent in the set

W4 = {X : |xj| ^ a,- (i = 1,2, 3), tx ^ t ^ t2}

and the integrals

gb(X) = i и f i ( y ) ^ XiK j(X , Y)\s=0dy ( i = 1,2,3; j — 1,2; a = 0,1,2 ) wi

are uniformly convergent in the set

W5 = {X : |x;| ^ Oi (i f= 1,2), a3 ^ x 3 a4, tx ^ t ^ t2j, where af (i = 1,2, 3,4) , t x, t 2 are positive constants.

P ro of. We shall prove this only for the integrals g\3(X) (a = 0,1,2).

The proof for the other integrals is similar.

210 M. F i l a r and J. M u s i a le k

The integrals g\3(X) (a = 0, 1,2) are linear combinations of the integrals

0(X ) = f J i f i ( y ) t “ ( x i - y i f 1 exp

r 2 ' 3

- A t dy, where fi > 0, рх ^ 0.

It is enough to know that the integral g ( X ) is uniformly convergent in the set W4. For that purpose we present g (X ) in the form

g (X ) = r ^ W 2 C lj j j /l(y) ( * i - > ’iV

St

Щ2

exp (Х1- У 1)2

— St x exp (* 2 - . У 2 ) 2 + ( Хз- Уз) 2

— St exp

r 2r3

— St dy, where cx is a convenient constant. By assumptions of Lemma 2 and by (14) it follows that

\g{X)\ ^ c2t р+р' 12 J f J exp w j

r2r3

— 8гл dy for X e W 4,

where c2 is a convenient positive constant. It follows from the above inequality that the integral g(X ) is uniformly convergent in W4.

Lemma 3. Let the function f x be measurable and bounded in Wx. Then the function ux(X ) defined by formulas (22), (23) have the following properties:

1° ux (X )e H ,

2° Pux(X ) = 0 for X e W ,

3° ux (X) satisfies the following boundary conditions

lim (DXi + h)ux (X) = 0, when X - * X 0e S 2, X e W . lim DX2 u x(X) = 0, when X -+ X 0eS 3, X e W .

P roof. It follows from Lemma 2 and Theorem 1 that the function ux{X) given by formulas (22), (23) have properties Г and 2°. Now we shall prove 3°. It follows from Lemma 2 and formulas (18), (15) that

lim (D<3 + ( ! ) « , ( * ) = y ^ \ f A y ) y ™ ( D „ 3 + h )G (X , Y ^ d y = 0

^1

as X -+ X 0e S 2, X e W . In view of Lemma 2 and formulas (19), (16) we have

lim DX2 ux(X ) = y

J J J

as X -+ X 0eS 3, X e W . Lemma 4. I f the function f x is continuous and bounded in Wx, then the

Lim it problem fo r parabolic equation 211

function ux (X) defined by formulas (22), (23) satisfies the following initial condition

lim ut (X) = /j (x0) as X -> X 0 = (x0, 0 )eS l , X e W . Proof. It is sufficient to show that

/, (x0) for i = 3,

0J as X - ( x o,0) e S u X e W .

lim u\ (X) = ,

0 for i = 1, 2, 4, 5, 6, The function w3 (X) can be represented as

u l(X ) =

y J J'J/i

exp

- 4 1 dy, where

f i Ы = f iiy ) when y e W lt

0 when y e E 3\Wx

and E 3 denotes 3-dimensional Euclidean space. According to Weierstrass theorem we have (x)

lim u\ (X) = f x{x o) as X ^ X 0e S u X e W .

By assumptions of Lemma 4 and by (14), (23), we have for the function Ui(X) the following estimations

\U1 (X)| < Ci

J f J

exp

<

C2 f exp

- 4 1

— At

dy

dyt

J exp

— At dyi j dy2

о (2С3+ У 3)^{3 ’} Cx, C2 being the convenient positive numbers. Applying in the last integrals the following change of variables

xt- y i = 2 yjt Zi (i = 1,2), Х3+ У 3 = z3, we obtain the inequality

00 00 00

\u\ (X)|

<

J exp ( —

f exp

J

— 00 — 00 * 3

C3 being the convenient constant.

It follows from the last inequality that limw?(X) = 0 as X -* X 0e S lf X e W . In a similar way we get the conditions lim u[ (X) = O a s X - > X 0eS'1, X e W , i = 1,2, 5,6.

I1) M. K r z y z a n s k i, P a rtia l differen tial equations o f second order, I, Warszawa 1971.

5 — Roczniki PTM — Prace Matematyczne XXII

212 M. F ila r and J. M u siale k

5. According to (12) and (21) the function u2 (X) may be written in the form

6

(24) u2(X) = £ 4 ( X ) .

i = 1 where

(25) 4 ( 2 0 = - y i \ l f2( y „ y2,s )K , (X , Y)\,3, 0dy i dy2ds (i = 1... 6)

И 2

and K i(X , Y) (i = 1,...,6) are given by formula (13).

Lemma 5. I f the function f 2 is measurable and bounded in the set W2, then the integrals

Щ ( Х ) = n . f / 2(>'1,J’2,s )O i,^ J(X , Y % 3, 0dy,dy2ds

I I '

(i = 1,2,3; j = 1 , 6 ; a = 0, 1,2) are uniformly convergent in the set

W6 = {X : |x,| ^ a{ (i = 1,2); аъ ^ x 3 ^ a4, 0 < t < tt }, where at (i = 1,2, 3,4), t1 are positive constants.

P roof. We shall prove this only for the integrals M 31(X) (a = 0, 1,2).

The proof for other integrals is similar.

The integrals M 31 (X) (a = 0, 1,2) are linear combinations of the integrals M W = Ш { ? f2iy u y2,s)ehv( t - s r l{(y3 + y3+ v f 1 x

«'*2 1 °

x exp r\

■4(t — s) dv / dyx dy2 ds,av > ay v3 = 0 j

where (I > 0, ^ 0.

Taking into consideration the assumptions of Lemma 5, (14) and the 00

fact that J ehv dv < oo we have

\M(X)\^ Bxx 2 211 j i t - s f ^ d s j J exp (^i — > i)2 +( x 2 — >2)^

- 4 ( t - s ) dy 1 dy2 where B x is the convenient positive constant. Introducing new integration variables

Xi — yt = 2 y/t — szi (i = 1,2), s = s, in the last integral we obtain

(26) \M(X)\ ^ B2x2 2ti\ { t - s f Y 2 + l ds ^ B3tlsi/2 + 2 for XgW6, 0

Lim it problem fo r parabolic equation 213

where В2, В г are positive constants. It follows from inequality (26) that the integral M ( X ) is uniformly convergent in W6.

Lemma 6. I f the function f 2 satisfies the assumptions of Lemma 5, then there exists a number В > 0 such that

(27) \M%(X)\ — u{(X)

У

< Bt2 for XgW6 {j = 1,...,6).

P roof. We conclude by (26) that inequality (27) is satisfied for j = 1.

The proof that (27) is satisfied for j = 2,..., 6 is similar to that of the proof for the integral M (X ).

Lemma 7. Let the function f 2 be measurable and bounded in the set W2.

Then the function u2(X ) defined by formulas (24), (25) have the following properties:

1° u2(X )e H ,

2° Pu2 (X) = 0 for X e W ,

3° u2(X) satisfies the following conditions

lim u2 (X) = 0 as X - ^ - X o e S i, X e W , lim DX2 u2 (X) = 0 as X —>X 0eS 3, X e W .

P roof. It follows from Lemma 5 that the function u2(X) is continuous with the derivatives D%x.u2(X) (i = 1,2, 3; a = 1,2) in IPand these derivatives may be found by differentiation under the sign of the integral u2(X). Taking into consideration the above properties and the fact that the function G (X , У) given by formulas (12), (13) satisfies condition 1° of Theorem 1 we obtain

Pu2 (X) = —y J J J/2(Li, y2,s )P x G (X , Y)\y3 = 0dyl dy2ds + y lim Z ( X , s )

W {2 s - l -

= у lim Z (X ,s ) for X e W ,

s~*t —

where

00 00

Z { X , s ) = J dy2 f f 2(y i,y 2,s )G (X ,Y )\ n ^0dyl . 0 — oo

In order to prove Г and 2° it is enough to show that lim Z (X ,s ) = 0 for a fixed X e W .

S ~ * t

Since A3 = A4, A 5 = A6 as y3 = 0, we have from (12) and (13) G{X, Щ ,3„о = [ 2 ( t - s) - 3'2(e x p ^ 3 + ex'p/l5) + K 1(X, У) + К г (Х ,

214 M. F i l a r and J. M u s i a le k

Now the function Z ( X , s ) may be written in the form

Z ( X , s ) = Z 1(X ,s ) + Z 2(X ,s ) + Z 3(X ,s ) + Z 5(X ,s),

Z k(X ,s ) =

J dy2 J f i{yi , y2 ,s )Kk{X,Y)\y^ 0dy1

exp AjJj, j = о dy i for к = 3,5,

ly3 = о dy i for к = 1,2.

We shall show only that lim Z 3 (X , s) = 0 as s - * t — . The proof that lim Z k(X , s) = 0 as s - > t - (k = 1,2,5) is similar. Let us write Z 3(X , s ) in the form

3/2 Z 3 (X , s) = 2b! f dy2 I f 2 (y ,, y2, s)

0

x l 4 ( t - s ) x exp

x3 3 exp x 3^{x 2}

{x i - y i ) 2 + {x2 -y 2 ? - 4 ( t - s )

dy i, - 4 ( ï - s )

where bx is a positive constant. Applying inequality (14) and the change of variables xt — yt = 2y/t — szt (i = 1,2), we obtain the inequality

00

|Z3(Z , s)| ^ x.3_3b2( r - s ) [ J exp ( — z2)d z]2,

— 00

where b2 is a convenient positive constant. Therefore lim Z 3 (X , s) = 0 as a —> t — .

Now we shall prove 3°. Using Lemma 5 and formula (16) we obtain lim й кгиг (Х ) = - y J f f/2(yi. J’î.s llim D ,2G (X , Y)\n = 0dy1dy2ds = 0

w‘2

when X X 0e S 3, X e W . By Lemma 5 and 6 we have lim u2(X ) = 0 as X -> X 0e S 3, X e W .

Lemma 8. Let X e W ; then

V t 00 00

L ( X ) = •*3

(2 y/n)3 -

J ds J dyx f (t - s ) 5/2 x 00 — 00

x exp ( x i - y i ) 2 + (x 2- y 2)2 + *3

dy2 = 1 - 4 ( t - s )

P roof. Applying the change of variables yt — xt = 2 yjt — s z{ (i = 1,2), s = s w e get

L (X ) = 4x3

(2^/tt)3 -j (t — s) 3/2 exp - r 2х ъ

*з

2 х/ти J ( t - s )^-3/2exp

_4(t — s) _ ds.

ds( J exp( — z2)dz)7

- XХЪ2

4 ( t - s )

Lim it problem fo r parabolic equation 215

Changing the integration variable x 3/2 y/t — s = и we obtain the thesis of Lemma 8.

L^emma 9. I f the function f 2 is continuous and bounded in the set W2, then the function u2 (X ) given by formulas (24), (25) satisfies the boundary condition

lim (DX3 + h) u2 (X) = f 2 (x ° , x 2, t°) as X - X 0 = (x?, x°2, 0, t°)e S 2, X e W . P roof. If уз = 0, then A 3 = A4, A 5 — A6 and by (18) we obtain

(DX3 + h )G (X , L)|j,3 = 0 = - ( t - s ) ~ 5/2x 3[exp A3 + exp А 5]\уз = 0.

Consequently by Lemma 5 we get

(28)(DX3 + h)u2(X) = - у 1 1 j f 2 (y i , у ^{2, s) (D X3 + h)G{X, У )|УЗ} = 0dyl dy2 ds w*2

= H 3{X) + H 5 {X) for X e W, where

НЛХ) = y J J У/2(Уi » Уг-> s)x3{t — s)~512 exp Ai\y3 = 0dy1 dy2ds (i = 3,5).

wt2

In view of (28) in order to prove Lemma 9 it is sufficient to show that (29) lim H 3{ X ) = f 2 (x?, x°2, t°) when X - X 0 e S2, X e W,

(30) lim H 5(X) = 0 when X - X 0eS 2, X e W . P r o o f o f (30). The function H 5(X ) may be written in the form

H 5(X) = yx3 J J J/2(y i,y 2,s) w*

(Х2+ У2)2 ~^5/2 (*2+У2)2

t — s exp 1 1 ---1

___1

x exp ^(Х1- У 1)2 ] *3

— 4(t — s)

dy 1 dy2 ds (Х2+ У2У By assumptions of Lemma 9 and by formula (14) we obtain

IH 5 (X)| ^ Bi X3 J f j (x2 + У2У 5 exp w{2

where В! is a positive constant. Introducing new variables x v— y x

= 2 y/t — s z!, x 2 + y2 = Z 2 , s = s, we get

t ____ 00 »

\H5( X ) \ ^ B 2 x3 j y/t — s ds J exp { — z\)dzi j z2 sdz2 -> 0

0 — 00 x 2

( * l - y l )

— 4 (f — s) dy 1 dy2 ds,

216 M. F i l a r and J. M u s iale k

as X -> Xq = (x°, x°, 0, t°)e S 2, X e W ; B2 being a convenient positive constant.

P r o o f of (29). The function H 3(X) can be represented in the form H 3(X) = yx3 f } J f 2(y i,y 2, s ) { t - s ) ~ 512 exp A 3\y3 = 0dyl dy2ds,

w7 where

Щ = {{yu y2,s)\ |yf| < oo (i = 1,2), - o o < s < t}, f 2 (y i,y i,s ) f i i y ^ y i , * ) when {yl ,y 2,s)eW 2,

0 when (yl5 y2, s)eE3\W2.

In view of Lemma 8 the function H 3(X ) may be written in the form

(31) H 3(X ) = / 2(x?,xS,r°) + « 0m ,

where

H0(X ) = yx3 j f J d (y i,y 2,s )(t - s ) 5/2 exp A 3\y3 = 0dy1dy2ds w7

and

d(yl , y 2, S) = f 2(y i , y 2, s ) - f 2{x°1, x°2, t°).

Now we shall prove that lim H 0(X ) = 0 when X X 0e S 2, X e W. Let e > 0 be given. It follows from the continuity of the function /2(y i,y 2,s) at the point (х ° ,х 2,г°) that there exists a number a > . 0 such that

(32) \d(yl t y2,s)\ < e

for (У1, у 2) е Я = {( y i , y 2): |у;-*?| < a (i = 1,2)}, |s-f°| < a.

Let < a. Now the set Wn can be represented as a union Щ = Z l \ j Z 2yj Z3,

where

Z x = D x (t ° — a, t), Z 2 = D x ( — oo, t° — a), Z 3 — (E2\D )x( — oo, t).

Now we have (33)

where

H o(X) = Z H'o(X),

1 = 1

(34) Hq (X) = yx3 J J J d (y j, y2, s) (t - s) 5/2 exp A 31,3 = 0 dy x dy2 ds

*i It follows from (32) that

(/ - 1,2,3).

(35) |#o(2f)| < £ for |r — r°| < a, IxJ < oo, xf > 0 (i = 2, 3).

Lim it problem fo r parabolic equation 217

Let I*; — xf| < \a (i = 1,2), x 3 > 0, \t —1°| < a and (yt , y2, s ) e Z 3; then [у,- — 3cf| ^ ||уг — x°| — \xf — Х;|| ^ a —la > \a for i = 1,2 and E2\D c E2\Dly where D t = {( у !,у 2): b j-X il ^ l a (i = 1,2)}. Hence

\Hl(X)\ ^ x 3B3 f { {J ( Г - * Г 5/2х -00 (e2 \Dj

x exp

- 4 ( t - s ) dyi dy2>ds;

B3 being a convenient positive constant. Introducing the new integration variables

{ Х 1 - У \ ) 2 + ( Х 2 - У 2 ) 2 + ХЪ

- 4 (t - s ) = 2, yt = yt (I = 1,2), we obtain

\Hl(X)\ < B4x 3 J y/z e x p (- z )d z JJ [(x 1- y 1)2 +

o £2\D1

+ (*2 - У2)2 + * 3] ~ 3/2 dy! dy2 ; B4 being a convenient constant.

00

Since J yjz exp( — z)dz < 00, we get 0

IH l ( X ) \ ^ B 5x 3 J f l { x i - y l )2 + (x2- y 2)2 + x l Y 3l2dylLdy2.

e2\d x

Applying the change of variables yt — xt = st x 3 (i = 1,2) we obtain

|H3(X)| ^ B5 S\(s2l + s 22 + i y 3l2dsl ds2,

d2 . •

where D2 — {(s^ s2): |s,| ^ a/4x3 (i = 1,2)} and B5 is a convenient constant.

Since lim a/4x3 — +00 when x3 -►() + , thus there exists a number <5j > 0 such that

(36) |Ho(2f)| < в for \t —1°\ < a, |x{ — xf| < la (i = 1,2), 0 < x3 < <5X.

Let us take now into consideration the integral Hq(X). Let \t — t°\ < la.

We have then

t ° ~ a

IHl{X)\ ^ B6x 3 J { t - s ) ~ 5/2ds f j dy^y2 ^ B7- ----— r j T ’

- 0 0 d (a —fa)

where B6, B7 are positive constants. Thus there exists a positive number

<52 > 0 such that

(37) \Hl{X)\ < e for |f-r°| < la, 0 < x 3 < &2.

218 M. F i l a r and J. M u s i a le k

It follows from (35), (36), (37) that

(38) H 0{ X ) < 3e for < ia, |х4-х?| < ia (i = 1,2),

0 < x 3 < ô = min (<5ls 02).

From (38) and (31) we obtain (29).

V '

6. Now we shall prove some lemmas dealing with the function u3(X).

According to (12) and (21) the function u3{X) can be represented as the sum

(39) « 3W =

i = 1 where

(40) ui3(X) = - уШ М У 1 , Уъ^ ) К Л Х , ¥ ) \ У2^ Г ^ У^ 5 ( i = 1,..., 6) and K t(X, Y) are given by formula (13).

Lemma 10. Let the function /3 be measurable and bounded in the set W3.

Let

N*tj(X) = SSSf*iyi>y*>s) D î tK j ( X , Y)\y2 = 0dy i dy3ds wt3

{i = 1, 2, 3; j = 1,..., 6; 'a = 0, 1,2).

Then the integrals N^j(X) (i = 1,2, 3; j = 3, 4, 5, 6; a = 0, 1,2) are uni­

formly convergent in the set

Ws = {X : |Xj| ^ af (i = 1,3), a2 < x 2 < a4, 0 < t ^

and the integrals Щ ( Х ) (i = 1,2, 3; j = 1,2; a = 0, 1,2) are uniformly con­

vergent in the set

W9 = {X : |xjJ ..< alt ax < xt ^ a2 (i = 2, 3), 0 < t ^ ^ j, where at (i = 1,2, 3,4), arc positive constants.

P roof. The proof of the uniform convergence of the integrals Щ ( Х ) is similar to the proof for the integrals M ^ (X ) (see Lemma 5). By a similar way to that used in the proofs of Lemmas 5 and 6 we get

№ j{X )\ = Iy - ' u i (X)\ < ct2 for X e W 8 (j = 3,4, 5, 6),

\№3j(X)\ = l y - M W I ^ ct2 for X e W , O' = 1,2), where c is a convenient positive constant.

Lemma 11. Let the function / 3 be measurable and bounded in the set W3.

Then the function u3(X ) defined by formulas (39), (40) has the following properties:

Lim it problem fo r parabolic equation 219

1° u3{X )eH ,

2° Pu3{X) = 0 for X e W ,

3° n3 (X) satisfies the following boundary conditions

limu3(X) = 0 as X ^ X 0e S u X e W , lim {DX3 + h)u3(X) = 0 as X ^ X 0eS 2, X e W .

P ro of. By Lemma 10 and Theorem 1 and by a similar way to that used in the proof of Lemma 7 we obtain Г and 2°. It follows from inequalities (41) that the function u3(X) tends to zero as X -> X 0e S l , X e W . By Lemma 10 and formula (15) we obtain

lim (DX3 + h) u3 (X)

= —7 J J\f³ (yi,y³,s)\im(DX3 + h) G{ X, >%2 = o dyx dy3 ds = 0 when X -*■ X 0eS 2, X e W .

L^emma 12. I f the function /3 is continuous and bounded in W3, then the function u3(X) given by formulas (39), (40) satisfies the following boundary condition

lim DX2u3(X) = f 3{x°1,x°3,t°) as X - + X 0 = (x?, 0, x°3, t°)e S 3, X e W . P ro of. If y2 = 0, then = A 2, A 3 = A s, A4 — A 6 and by (19)‘ we obtain

DX2G ( X , Y ) |У2 = 0

00

= — x2 (t — s)~5/2 [exp A 3 + exp A4 + 2h J ehv exp A t dv]\y2 = 0.

о We now present the function DXlu3(X) in the form

DX2u3{X) = - y J f $f3{yl ,y 3,s )D X2G (X , Y)\y2 = 0dyl dy3ds

= B3(X) + B4(X) + Bl (X), where

B i(X ) = lhyx2 f f f [ j f 3(yl , y 3, s ) { t - s ) ~ 5l2ehv exp A l \y2 = 0dv]dy1dy3ds,

П о

Bi{X) = yx2 f f $f3{y i,y 3, s ) ( t - s ) ~ 512 exp Ai\y2 = 0dyl dy3ds (i = 3,4).

и^з

It is enough to show that

\imB3( X ) = M x ï , x l , t ° ) as X -> (x°, 0, x?, r°)eS 3> X e W ,

lim В, (X) = 0 as X -* (x°, 0, x3, l°)e S 3, X e W, i = 1,4.

220 M. F i l a r and J. M u s i a le k

The proof of conditions (42) is analogous to this for conditions (29), (30) in Lemma 9.

7. In this chapter we shall deal with the function u4(X) given by formula (21). We shall prove that the function u4(X) is the solution of equation (7) in the set W with the homogeneous conditions (9), (10), (11).

Let us write the function u4(X) in the form

(43) u4(X) = £ u‘4(X),

i = 1 where

(44) u‘4( X ) = - y S j j S f ( Y ) K l ( X , r ) d Y ( i = l , . . . , 6 )

and K i(X , У) are given by formula (13).

Lemma 13. Let a function f be measurable and bounded in the set W.

Let

* ■ (/ .* ) = Ш I f W B ^ K A X ' Y W i r (/= 1,2,3; y = 1,...,6; a = 0,1).

Then the integrals !?•(/, X ) (i — 1,2, 3; j = 3, 4, 5, 6; a = 0, 1) are uniformly convergent in the set

W10 = (X : \xt\ ^ at (i = 1, 2, 3), 0 < t ^ r j ,

and the integrals /?•(/, X ) (i = 1,2, 3; j = 1,2; a = 0, 1) are uniformly con­

vergent in the. set

Wtl = {X : |x;| ^ af (i = 1,2), a3 ^ x3 ^ a4, 0 < t ^ t2}, where я, (i = 1,2, 3,4), t( (i = 1,2), are positive constants.

P ro of. We shall first deal with the integrals R?(j, X ) = —u{(X)/y (i = 1,2,3; 7 = 1,...,6). By assumptions of Lemma 13 it follows, that

|ul(X)| ^ My J [ J J f ( t - s ) ~ 3/2 exp A 3dy]ds, о e3

where M = sup|/|. Introducing the spherical coordinates w

(45) У1- Х i 2 yft — sr cos q> cos

(45) у2 x 2 = 2 j t — sr sin (p cos ip, y3 — x 3 = 2^Jt — sr sin ф,

Lim it problem fo r parabolic equation 221

where r ^ 0, 0 ^ q> ^ 2к, \ф\ ^ tc/2, we get

t 00

(46) |i4(X)| ^ 16Mny sin ф\п12к/2 i ds \ r2 e~r2 dr ^ M xt for X e W l0 о о

and M t is a convenient positive constant.

In a similar way we prove the following inequalities

\u{{X)\ ^ M t t (j = 4, 5, 6) for X e Wl0,

(47) 1 1

\u{(X)\ ^ M xt {j = 1,2) for X e W l t .

We shall now deal with integrals R j ( 3 , X ) (i = 1,2,3). Using the formula Dxi l ( t - s ) ~ 312 ехрЛ 3] = - i ( r - s ) _5/2(xi - y l)exp Л 3 (i = 1,2,3) we obtain

M *

1^(3, X)| 5/2 l^i —yi! exp A 3dy]ds.

1 0 £ 3

Introducing the spherical coordinates (45) we have

(48) 1Я,1 (3,X)| < M 2 J r3e - r2d r l ( t - s ) ~ ll2ds < M 3v^ (i = 1,2,3)

о о

for X e W 10, M { (i = 2,3) being the positive constant.

In a similar way we prove the inequalities

,„m IR} O', X)\ ( i = l , 2 , 3; j = 4, 5, 6) for X e Ж,», (49)

| Я ? 0 , * Ж 0 = 1,2,3; j = 1,2) fo r X e W ',,.

The thesis of Lemma 13 is an immediate consequence of inequalities (46),..., (49).

Lemma 14. Let a function f be measurable and bounded in W. Then the function u4(X) defined by formulas (43), (44) have the following properties:

1° u4(X) is of class C 1 with respect to х х, х 2, х ъ, in W.

2° w4 (2Q satisfies the following conditions

limw4(X) = 0 when X -*■ X 0e S 1, X e W , lim (DX3 + h)u4(X) = 0 when X -*■ X 0e S 2, X e W , lim DX2 w4 (X) = 0 when X -> X 0 e S3, X e W .

P roof. It follows from Lemma 13, that the function u4(X) is of class C 1 with respect to x t , x 2, x 3 in W and its derivatives may be found by differentiation under the sign of the integral. By Lemma 13 and formulas (18), (19), (15), (16), (47) the function u4(X) satisfies condition 2° of Lemma 14.

222 M. F i l a r and J. M u s i a le k

Lemma 15. I f the function f is of class C 1 in the set W and f, Dy f (i = 1,2,3) are bounded in W, then the integrals

V?(X ) = y l \ \ l f ( Y ) D 2x. K A X , Y ) d Y (* = 1,2,3)

are uniformly convergent in the set

Wii = {2Г : |xjJ ^ a,, a2 ^ x2 ^ аз> a^ ^ x 3 a5, t, ^ t ^ t2], where a{ (i = 1,..., 5), tt (i = 1 , 2 ) are positive constants.

P ro of. We shall prove this in the case i = 2. The proof in the cases i = 1,3 is similar. Because

D ,2K 3( X , Y ) = - Dn K 3( X , Y ), D l2K 3( X , Y ) = D2n K 3( X , Y ) , we have

K23(X) = y f f J { J f ( Y ) D y2 lD yi K 3 ( X , 7 )] dy2} dy, dy3 ds.

Wt3 о

Applying the formula for integration by parts we obtain

= У Ш { / ( У ) 0 , 2К , (Х , y)|0” - w3

- ] Dy f (Y )D y K 3 (X , Y)dy2}dyt dy3ds 0

= - t! I J U ( Y ) D y 2K 3(X , Y )% 2, 0dy i dy3ds + И"3

+ у1 Ш °у2/ ( Y ) D X2k3(X, Y)dY W«

= yH S U W I > , 2K 3{X, Y )% 2, 0dyl dy3ds + тз

+ y \ \ i ] D y 2f ( Y ) D X2K 3{ X , Y )d Y for X e W l2.

Wl

It follows from assumptions of Lemma 15 that the second term in the above formula is an integral of the type R 2(3 ,X ) (see Lemma 13). It is readily observed that the first term is an integral of the type N 23(X) (see Lemma 10). Hence Vf (X) is uniformly convergent in the set W12.

Lemma 16. I f the function f is of class C 1 in the set W and f , D y. f (i = 1,2,3) are bounded in W, then the function u l(X ) belongs to H and satisfies equation (7) in the set W.

P ro of. By Lemma 13 and 15 the function u%(X) is continuous with

Lim it problem fo r parabolic equation 223

the derivatives D*x. u\ (X) (i = 1,2,3; a = 1, 2) in the set W. Since Px K 3( X , Y)

= 0 for X e W , Y e W 1 we obtain

Ax u l ( X ) = - y f f f J / ( y ) ^ K 3(X , Y)dY Wt

= - У И И f ( Y ) D , K 3(X, Y)dY for X e W w*

and

РиЦХ) = y lim J J f f ( Y ) K 3(X , Y)dy f o r X e W .

s~*t — w f

The integral y J J J / ( Y ) K 3(X, Y)dy is the Fourier-Poisson integral and

«Т

tends to f ( X ) as s -> t —(*) (see p. 211). Thus P u l(X ) = f ( X ) for X e W . Lemma 17. I f the function f is measurable and bounded in W, then the integrals

V ^ X ) = П П f ( Y ) D 2x.K j(X , Y)dY (i = 1,2,3; j = 1,2, 4, 5, 6) ж*

are uniformly convergent in the set W12.

P roof. We shall prove this only for the integral V f (X ), The proof for the other integrals is similar. The integral Vx (X) is linear combination of the integrals

V (X ) = И J i K t ( x , Y)dY, where /3 » f , /3, » 0.

Wl

We present v(X ) in the form (Х1- У 1)2 v (X ) = c, SS U f ( Y )

wl (•^з T Уз)2

8 (t — s)

-Pll2 1 <4 i

exp — 8 (t — s)

4 (t~ s) exp (хз+Уз)2 - 4 ( r - s ) exp

(Х2- У2)2

x exp

— 8 (t — s)

{ X i - y i ) 2+ {x 2'-y2Y

— 8 (t — s)

(x3+ y 3) - 2p( t - s ) p^2dY,

where cx is a convenient constant. By assumptions of Lemma 17 and by (14) we obtain

\v (2QI ^ c2 \ ds J dyx J dy2 $ ( t - s ) HI2(x 3 + y3) 2p x 0 — oo

x exp (% i- y i)2 + (x2- y 2)2

— 8 (t — s) dy3,

where c2 is a convenient positive constant.

2 2 4 M. F i l a r and J. M u s i a le k

Using in the last integral the. change of variables

X i-y t = yf$ y/t — s zt (i = 1,2), x3 + y3 = z3, s = s, we obtain

\v{X)\ < c3 $ ( t - s f i /2 + 1ds J z32pdz3[ j exp ( — z2)d z]2

0 ^X3 ^{— oo}

00

^ c4r/'l/2 + 2 J z32pdz3 for X e W 12;

* 3

c{ (i = 3,4) being a convenient positive constants.

L^emma 18. Let a function f be measurable and bounded in W. Then the functions m4(X ) (i = 1,2, 4, 5, 6) have the following properties:

1° u ^ (X )e H (i = 1,2,4, 5,6),

2° P i4 (X ) = 0 for X e W (i = 1,2, 4, 5, 6).

P ro of. Taking into consideration Lemma 17 and the fact that P x K f X , У)

= 0 for X e W , Y e W ‘ (i = 1,2,4, 5,6) we obtain

Pu *4(*)= - y i H l f ( Y ) P x K t(X , Y )d Y + y lim J J f f ( Y ) K , ( X , Y)dy

Й" « ’l

= v lim J J f f ( Y ) K t (X, Y)dy (i = 1,2, 4, 5, 6).

s_k(- wx

In order to prove 1° and 2° it is enough to show that lim f f j / (У ) * , ( * , УМу = 0 ( i = 1 ,2,4 ,5 ,6 ).

wx

We shall prove this in the case i = 4. The proof for the remaining indices (i = 1,2, 5,6) is similar. By the assumptions of Lemma 18 and inequality (14) we obtain

(J J f ( Y ) K t (X , Y)dy\ « M , J J j ( t - s ) - 3'2 exp At dy

( x i - y i Ÿ H x i - У г )1 w

^ M 2 j dyx f dy2 J (x3 + y3) 3 exp

- 4 ( t - s )

M X, M 2 being convenient constants. Transforming the integral variables by formulas y{ — x, = 2 y/t — s zt (i — 1,2), x 3+ y 3 = z3, we have

i H I / t n K . d , Y)dy\

^ M 3(t — s) J z3 3 dz3[ J exp ( — z2)d z ]2 -> 0 as s - > f —,

*3 - oo

Limit problem fo r parabolic equation 225

where M 3 is a convenient positive constant and hence our assertion follows.

As the consequence of Lemmas 3, 4, 7, 9, 11, 12, 14, 16, 18 we get the following

Theorem 2. Let a function f be continuous and bounded in the set Wt (i = 1,2,3); let a function f be of class C1 and bounded with derivatives Dy. f (i — 1,2,3) in the set W. Then the function u(X) given by formulas (20), (21) is the solution of problem (2), (7), (9), (10), (11).

It follows from (8), (8') and Theorem 2 the following

Theorem 3. Let functions C , F , f l , F i (i = 2,3) have the following properties :

1° the function C is continuous in the interval [0, oo),

2° the function F is of class C 1 in the set W and the functions

t t

D\ j F (X) exp I f C(s)ds] (i = 1 , 2 , 3), T (20 exp [ J C(s)ds] are bounded in the

о 0

set IT,

3° the fund ion F, is continuous in the set and the function t

Fi (x 1, Xi, t) exp \ f C(s)ds\ is bounded in the set Щ (i = 2,3).

Ù

4° the junction j у is continuous and bounded in W\.

t

Then the function e(2Q = u(A)exp [ — J C(s)rfs], where u(X) is solution о

of problem (2), (7), (9)—( 11) is the solution of problem ( 1 )—(5).