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: I*xj*I'< 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) *DX2**u**{X) = /3( **i*, x3, t) * *for X e S 3, *
where

*t*

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

## J

*C*( 5 )

*d s 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* =

*and let*

**{ух, у2,Уъ)**[(^ 1- } ;1)2 + (^2+ ( - 1У>;2)2 + (2Сз + Уз + у)2] 1/2 *for i = 1,2,*
*rt = <[ * *[(^1 У1)2 “h(-^2 У**2**)^ 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 ehvDa**X3 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

*e^idi + v) 2jdv < oo*

*for ( X , Y) e W0 x W t,*о

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, Y**e**W1.*

*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 A**a** +*
*+ (x3- y 3) e x p A s + (x3 + y3) e x p A 6] +*

*+ h(t — s)~ 3/2 [exp A 3 - exp A**a** + 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 A**a* +
*+ (* 2 + У**2**) exp A 5 + **(**x**2**+ 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, A**a** = 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 = {(Y*i.'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, y**2**, s )G (X , Y)\n = 0dyl dy2ds,*

**(**21**) ** **wt2**

*u3(X) = - y *

## J

*\ $f3(yx,y 3, s )G (X , 10l)>2 =*

**о**

*dy i dy 3 ds,*

*uA X ) = - y j H S f ( Y ) G ( X , Y ) d Y ,*

*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).*

**L****emma**** 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 2***r3**

*— 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.*

**L****emma**** 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

*f x*

*(y) lim DX2 G (X, Y)\s = 0dy = 0*

*wi*

*as X -+ X 0eS 3, X e W .*
**L****emma**** 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

*(y)t 3/2*

## exp

^{r 2}

^{r3}*- 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

*t 3/2*

## exp

^{A}<

## C2 f exp

*- 4 1*

*— At*

*dy*

*dyt*

## J exp

*( x i- y z Ÿ*

*— 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)| *

## <

C3t## J exp ( —

z2)dzi## f exp

( —z2)dz2## J

*z f 3dz3;*

— 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 f**2**( y „ y**2***,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).**

**L****emma**** 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 { ? *f**2**iy u y**2**,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 X****g*** W6,*
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.*

**L****emma**** 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 X**g**W6 {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 ).*

**L****emma**** 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 **u**2 (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(L*i*, 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 2*х ъ*

*з

2 х/ти *J ( t - s )*** ^{-3/2}**exp

*_4(t — s) _ *
*ds.*

*ds( J exp( — z2)dz)7*

*- X**ХЪ**2*

*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**

*(Х**2**+ У**2)2* ~** ^{5/2}** (*2+У2)2

*t — s* exp 1 1 ---1

___1

x exp ** ^{(Х}**1

**- У**1)2

**]***3

*— 4(t — s)*

*dy 1 dy2 ds*
*(Х**2**+ У**2**У*
By assumptions of Lemma 9 and by formula (14) we obtain

*IH 5 (X)| ^ Bi X3 J f j (x2 + У**2**У* *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 -> X*q* = (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) *H**q* (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 ****(****e****2 \D****j**

*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 - У**2)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. *

**e****2\****d x**

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

*|H3(X)| ^ B5 S\(s2**l + s 2**2 + i y 3l2dsl ds2,*

**d****2****. ** **•**

*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 H**q**(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) = -* *у**Ш М У 1 , У**ъ**^ ) 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*\f*^{3}*(yi,y*^{3}*,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 DX2****u****3(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*

**DX2****u****3{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 ** **e****3**

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

(45) *У**1**- Х i* *2 yft — sr cos q> cos*

(45) *у**2** 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\п**12**к/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 *w*4 (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

**L****emma**** 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 2**x. 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 ) = D2**n 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

*+ **у**1 Ш °**у**2/ ( Y ) D X2**k**3(X, Y)dY*
*W«*

= *y**H 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.*

**L****emma**** 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 2**x.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

*(Х**2**- У**2)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)* * dy*3,

*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 ^{X}3 ^{— 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 *m*4(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).*