On a certain bicaloric problem

R O C Z N IK I P O L SK IE G O TO W A RZY STW A M A TEM A TYCZNEGO Séria I : P E A C E M A TEM A TY CZN E X I X (1977)

F. B

^abanski

and J . M

^usialek

(Krakôw)

On a certain bicaloric problem

1. In this paper we shall give the constructions of solutions of the mixed problem for the equations

(1) ' P 2TJ(x, t) = - f 7(x, «), P U = ( B l - B t)U , and

(la) P 2u (x,t) = —f { x ,t ,u { x ,t ) ) ,

in the domain S — {(x ,t): — a < x < + a , 0 < t < T } , where a, T are positive constants, with the limit conditions:

(2) U(x, 0) = М х ) for xe( ~ a , + a ),

(3) B t U (x, 0) =M<D) for xc( —a , -j-a),

(4) U {a,t) = / 3(t) for te(0, T),

(5) P (U (a 7t)) = / 4(<) for £e(0,JF), (6) U { - a ,t ) = f 5(t) for te(0, T),

(7)

1 «s' II <

for t*(0, T ).

The problem of finding a solution of (1) or (la) satisfying (2)-(7) will be called briefly (B — C — B )x and (B — C — B )2 respectively. We shall prove that the solution of the problems (B — C —B )1 and (B — G — B )2 is the sum of any solution of the equation

(lb) P 2U {x,t) = 0

and any solution of equation (1) and (la) respectively.

2. In order to construct the solution of the problems (В — С — В ){

(i = 1 , 2 ) we shall use the convenient Green function.

Let

ai = {(&> t): x = + a

7

0 < t < T }, a

2

= {(a?, t): x = —a, 0 < t < T },

and give any points X {x ,t)e S , Y (y ,s )e S \ja 1Kja2. Let X \ = X \ = X

and -Xjn+i^L+iJ <0 f °r n — 0 ,1 ,2 ,..., be the symmetric image of the

point X \n{œ\ni t) with respect to the axis ax and Х \п+2(х\п+2, t) the sym­

metric image of the point X ln+1(xln+1, t) with respect to the axis a2. X \n+l {x22n+1, *) be the symmetric image of the point X 22n{x22n, t) with respect to the axis a2 and X ln+2{x22n+2, t) the symmetric image of the point Х \п+х

( 4 +i,<) with respect to the axis ax.

It is easy to verify by induction that

xln = ( — l ) n{x — 2na), x2n = ( — l ) n(x + 2na), w = 0 , 1, 2, ...

Let us consider two sequences of the functions:

Vln{X , Y) = и гп(

0

) , г , у ,

8

) = K (t, s)ex-p[\YX1n\2(4:(s-t))-1)

K (t, s)exp((y — ( — l ) ” (æ — 2na))2(ê(s — <))_1) for s < t,

0 for s ^ t y

and •

U2n(X , Y) = U l(ce,t,y ,s)

E( t , s)esp((ÿ — ( — l ) ” {aîH-2îta))2(4(s — f))” 1} for s < t y

0 for t.

and

V \(X, X) = V l(X , Y) = X ( « ,s ) e x p ||r X |J (4(S - 0 ) - 1) K (t, s)exp((ÿ — æ)! (4(s — <))_1) for s < t,

0 f or 8 > t,

where K (t,s) — [2 (тг(£ — s))*|-1.

3. Now we shall prove the following

T

^heorem

1 [2]. The Green function for the problems (B — C — D)i (i = 1, 2) is of form G (X , Y) — (t- s)Q (X , Y), where

Q(X, Y) = U (X , Y )+ Ü [(X , Y) —TJ\(X, Y ) - U 22(X , Y) +

00 00

+ 2 ( V l i +i{Z , T )-U l< (X , Y)) + 2 ( ü î i ( Z , Y )~ U l,_ y(X , Y)).

i = l г'=2

P roof. The function Q(X, Y) is the Green function for the first Fourier problem of the equation

(lc) Pu{X ) = 0

in the domain 8 X = {(x ,t): —a < x < + a , t > 0 } ([1], p. 475). Using

the estimations analogous to those in [1], pp. 476-477, we can easy verify

that the series

Q i(X, Y) = % ( V l2i(X , Y ) - V \ i+1(X , Y)),

г = 0

00

Q ,(X, Y) = 2 ( V l i ( X , Y ) - V l ^ X , Y))

г = 1

are quasi-uniformly convergent with the derivatives to the order 4 for X e S x and

2

/e(

— oo,

+

oo),

0 < s < t. From the equality P XG = - Q follows that the function Q satisfies equation (lb) and P x G(X, Y) — 0.

For Y ea2 we get |TX L-il = № Xl{\ and \ТХ\{\ = bence Qx

= Q2 = 0. For Y eax we have |Y X l2i\ = № Х\{_г\ and \YX22i\ = |Г Х *<+1|

hence Qx = Q2 = 0, and we obtain P x G (X , F) = —Q(X, Y) = 0 for

~Y € dj U ^2 •

4. Using some lemmas dealing with convergence of certain integrals and series and Green function G we shall prove that the function

7

(8) Щ Х ) = У Ч ( Х ) ,

t = l

where

-f-a 4-ct

иг(Х) = f M y)Q {X , y, 0)dy, u2{X) = f h(y)tQ {X,y, 0)dy,

—a —à

where Л(у) = f i (y) — I>tu1(y10)t

t t

щ (Х ) = J f a(s)J)vQ (X f a, s)ds, щ = - J / 4(s)(i- s )i> 3/Ç (X , a, e)<fe,

о ⁰

t

u5{X) = J / 5(e)-D,,Q(X, - a ,8 )d s ,

о

t

u6(X) = J f e(s)(t — s)DyQ{X, —a, s)ds

0

and

1 +a

M ,m = / / f ,(y ,s )G (X , Y)dyds

0 - a

solves in $ the (В — C — D)x problem.

Let

where t

0

and T are

a r b it r a r y

positive numbers. Let

OQ

F J X , Y) = £ ( Щ (Х , Y ) - U î i±l(X , Y)) (n =

1 , 2), 1

OÛ

(9)

H J X , Y)

= Y)| + |î7?j±1( X , Г)|) ( » = 1 ,2 ) , i= 1

oo

(10) H ^ ( X , Y) = JT ’ I.D Y>+ иш ( Х , Г))| (n = 1 ,2 ),

1 = 1

where j , p = 0,1, 2, 3,4, A: = 0 , 1 , 2 , j + fc < 6.

We shall prove that the function 77(X) defined by formula (8) is of class Cjkp in 8. Moreover, the derivatives Dxjtk U l) exist and are continuous in

8

.

Lemma 1.

The series defined by formulas

(9), (10)

are uniformly con­

vergent in every set V.

P roof. The common majorant of this series is the series C ] ? i " 2, where О is a positive constant ([1], p. 477). <=1

By Lemma 1 we get

Lemma 2.

There exist the derivatives D j k pF n(X,

T )

continuous in V and

00

D ^ F J X , Y) = ^ Y )~ 0?,±1(X , Г)) (n = 1, 2).

S — 1

By Lemma 2 follows

Lemma

3. I f the functions f { (i —

1,

..., 7) are continuous and bounded, then the functions щ (i = 1, ..., 7) are of class Cjkp in 8 and satisfy equa­

tion

^{( l b ) .}

P roof. By Lemma 2 we can change the integration, differentiation and summation, and we get the first part of the thesis of Lemma 3. Now we shall prove that the functions щ (i = 1, 3) satisfy equation (lb).

For i = 2, 4, 5, 6 the proof is analogous. Since Q(X, Y) and (t — s)Q(X, Y) are the solutions of equation (lb) thus by Lemma 2 we obtain

-\-d

Р *«Л Х ) = / fi(y )F

2

x Q {X , у ,

0

)dy = 0

—a

and

t

P u,(X ) = f f s(s)P

2

x (Pv(t- s)Q (X , a, i))ds = 0.

0

5. In order to prove that the function 77 (defined by formula (8))

satisfies the limit conditions (2)-(7) we shall prove some lemmas.

Lem m a

4. I f the function / x is continuous and bounded in the interval ( —a, + a), then

lim%(X) = /i(a?0) as t)-*(œ0, 0+), a?0e (— a, -fa).

P roof. Let

-fa

1г(Х) = At~* j fi(y )e x p (- (y - œ )2(ét)-1)dy

—a

and

+ a oo

U X ) = A r * f h ( y ) { - V \ ( X , Y) + 2 ( V h ( X , X ) - V \ i_1(X , Г)) +

г=1 oo

+ ^ { U t i( X , Y ) - U l i_1(X ,Y )))d y,

i —1

for у

e < ( —

a, -fa), for y c B \ ( - a , -fa>.

+ 0O

I ^ X ) = 4 Г { J A M e x p f - ^ - a O ^ ^ d y ^ / i K )

— 00

as (a?, t)-^(x0, 0+), x0e( — «» + « )•

The function Сг$ is a majorant for I 2(X) and I 2(X)->0 when (oo, t)->(a?0, 0+), Æ0e( — a, -fa), Oj being positive constant.

Lemma 5.

I f the functions h and

/ г- (г = 1, . . . , 7)

are continuous and bounded, thenlimu^X) = 0 as (oo, t)-+(x0, 0+), a?0e (—a, -fa) (i = 2 , . . . , 7).

P roof. From Lemma 4 for i = 2 we get

-\~a

limw2(X) = limtf J h(y)Q(X, у , 0)dy = limtt(ai0) = 0

—a

as (oo, t)->(x

q

, 0+), and x0€(— a, -fa). The common majorant for u{ (i = 3, ..., 7) is the function C2(x0±_a)~mt*, where m and C2 are positive constants.

Let

T/ N I МУ) for У е ( - а , + a > , My) = { _{( 0} for y e lt ,\ ( —a, -fa>.

Now we shall prove where A = (2(nÿ) h

Let

M y )

M y ) =

0

By [1], p. 450, we obtain

Lem m a 6.

Let the function f x be of class C2 and bounded with two first derivatives for cue ( —a, -\-a) and

(11) D 2xux(X )->f" {œ0) as (sc, t)->(x0, 0+), x0e ( - a , + « );

then

Щ иг + и^-^^оОо) as (x, t)-+(x0, 0+), x0e {—a, -fa).

P roof. Since the function ux satisfies equation (lb) in the interior of Vx and by (11) we get

DfUx(X) = Dxux(X)~->fx (й/0) as (x^ t)— >(л?0, 0^), я?о€(

and by Lemma 4 and Weierstrass theorem ([1], p. 446) we obtain

+ o o +o o

Dtuz{X) = ( f h (y)Q (X ,y,0)dy + t j h(y)DtQ (X , у , 0)dy}

— OO — OO

4 (® o )~ /i(® o ) when (x, t-)^(x0, 0+), х0е ( ~ а , f a ) . Hence И т Н ^ ^ + ^г) = /

2

(^

0

) as (a?, t)->(®0, 0+), a?0e (— a, f a ) .

Lem m a

7. I f the functions f { (i = 1, ..., 7) are continuous and bounded, then

limDfU^X) = 0 when (x, t)-+(x0, 0+), x0e {— a, f a ) ,

i — 1, 7; i Ф 2.

P roof. The common majorant of the integrals I>iui is the function Сг{®0± а ) ~ т $, where m and Cz are positive constants.

Now we shall prove that the function U(X) defined by formulas (8) satisfies the boundary conditions (4)-(7).

Let

K x(X , a, s) — {a — x){t — s)~4'3exp( — (a — æ)2(4(i — s))-1), and

Щ(-Х) = J х{ Х ) J 2{X ), where

t

Jy(X ) = A j f 3(s )K (X ,a ,s )d s

0

and

t OO

J 2{X) = Л f f 3( s ) ( U 2i( X , Y) ^2i+l (X , Y)) +

о г—1

oo

+ y < U U X , Y ) - l 72

^v

,(X , Г)))| <fe.

» = 1

Let

J / 4 l/s(*) fo r 's€(0,t), f

3

(s) —

l 0 for *€ .B \(0 ,t).

L

^emma

8. Let the function f

3

be continuous and bounded; then

\im J

1

(X) = f

3

(t0) as (x ,t)-> {a ,t0), t

0

> 0 and

lim J2(X) = 0 as (æ, t)-+(a, t0), t0> 0.

P roof. Since

-\-t

(12) J 3(X) — A J K x{X , a , s)ds = 1

—oo

thus

t t

J x(X) = A f d (s,t

0

)X

1

(X , a, s)ds + A J f M K ^ X , a, s)ds,

— OO —00

where •

d(s, t0) = f

3

(s) f

3

(t0).

By formula (12) we get

t

Ji( X ) fs(to) = A J*d (sj t

0

)K 1(X j u, s)ds.

— OO

Let e be an arbitrary positive number. We shall prove that there exists a positive number Ô such that if is —

1

0\ < ô and \a — x\ < ô, then W i(X)-fAto)\ < 0 4e, where is a positive constant.

Indeed as f

3

is continuous at the point t0, we have (13) \d{s,t0) \ < e for S€(t0— ô, t0+ ô).

Let

1

1

= {(s)î — oo <i s <Z t

0

— J 1

2

— {(s) î t

3

Ô <C

S

<C to} , Is = {(*)•• t0 < S < t ] ,

and

3

M (X) = y Mt(X ),

i= l

where

Mx(X) = A J d ( s , t

0

)K x(X , a , s)ds,

M

2

{X) = A j d { s , t

0

)K x(X , a, s)ds,

and M

3

(X) = A f d(s, t

0

)K x(X , a, s)ds. Let G

6

= sup|/3|.

Is

By virtue of (12) and (13) we have the estimations

\Мг(Х)\ < 2AC

5

J X x(X , s)ds <

2

AC

5

j (a — x)(t — s)~ zds

h h

<5\ 23

< 2AC,

j (a — x) ds < 2AG&(a — x)t

0

( B

< s

for x > \ — \ (

2

AG

5

at

0

— e), and

|J f 2(X)| < 2As J K x(X , a, s)ds < As J К г(Х , a, s)ds = As,

J 2 —oo

and |ilf3(X)| < Xe.

Now we shall prove the second part of the thesis of Lemma 8, we have limD„{Ul((X , X) + V l ^ X , Г)) = limD ^ U ^ X , Y )+ U \M (X , T)) = 0,

•

as (x, t)->(a, t0), to>

0

, and by uniform convergence of the integral uz at the point (a, t0) we get 1ш1«72(Х) = 0 as (x , t)->(a, t0).

By Lemma 8 we obtain

Lemma 9.

Let the function f z be continuous and bounded

;

then 1шш3(Х) = / 3(t0) when (x, t)-+(a~, tQ), t

0

> 0.

Now we shall prove

Т/

^{еутупуга}

10. I f the functions f { (i = 1 ,. . . , 6 ) are continuous and bounded, then

1шшг-(Х) = 0 as (x ,t)^ -(a ~ ,t

0

) ,t 0> 0 for i = 1, ..., 6, i ф 3.

P roof. Since the integrals ui are uniformly convergent at the point (a, t0) and Q(a, t0, y, 0) = 0, then

lim%(X) = 0 as (x, t)-+(a~, t0), t

0

> 0 for i — 1, 2.

The proof of conditions 1шшг-(Х) = 0 as (x, t)->(a~, t0), t0 >

for i — 4, 5, 6 is similar to the proof of the convenient condition in Lemma 8.

By uniform convergence of the integral u

7

at the point (a ,t 0) and from the condition Q (a,t, y , s) = 0 for у Ф + a follows that

+ a t

lim «7( I ) = j J f

7

{ y ,

8

) ( t -

8

)Q(œ,t

0

, y ,

8

)dyd

8 = 0

—a 0

as (x , t)->(a~, t0).

Similarly to condition (4) we can verify condition (6).

Now we shall prove condition (5). The proof of condition (7) is the same.

Т

литупуга

11. I f the functions f { (i = 1, ..., 7) are continuous and bounded, then HmPTJ(X) = / 4(<0) when (sc, t)->(a~, t0), t0> 0.

P roof. By (8) and Lemma 3 we get

7

P V (X ) = У Ч ( Х ) ,

i = 1

where

-f-a H-ct

zx(X) = f fx(y)Px Q (X , у, о)dy, z

2

(X) = f h(y)Px tQ(X, y,0)dy,

—a —a

t

S*(X) = P x [ f f s ( s)Dy Q (X , a, s) dsj,

0 t

*A X) = P x (ffi(s)I> y(t - s )Q (X , a, s)ds}, '

0

t

z

5

(X) = P x ^ f b(s)DvQ(X,

0 t

z

6

(X) = P x ( f f e(s)B v( t -

8

)Q(X, - a , «)<&),

0

+ a t

z

7

(X) = P X ( J j f

7

( y ,

8

) ( t -

8

)Q (X , Y)dydsy

—a 0

By Theorem 1 we get P x Q(X, у , 0) = 0 and zx(X) = 0. From uni­

form convergence of the integral z

2

at the point (a, t0), t

0

> 0 we obtain

+ a + a

lim«2(X) = lim J h(y)Px (tQ(X, y ,

0

))dy - lim J h (y )(-Q (X , y, Q))dy

—a —a

= - / h(y)Q(a, t0, y, 0)dy = 0.

—a

Since

t

z

3

(X) = j f

3

(s)B

2

xB yQ(X, Y )d s- lim f

3

(s)B yQ(X, a, s ) -

0 s-><

t t

- Jfz (s)B iyQ (X ,a ,s)d s = j f

3

(s)Px Q (X , a, s)ds

о 0

and lim fs(s)B y Q(X, a, s) = 0

and

DyQ{a, t

0

, a, s) —

0, and P x

Q(X, a, s) —

0, thus lim £3(X ) = 0 when

(x, t)->(a~, t0), t0>

0 . F o r the integral _z4 we have

t

z

4

(X) = - f f

4

(s)(t-s)L>yP x Q (X ,a ,s)d s + limf

4

(s)(t-s)I> yQ(X, a, s) +

о t

+

j

f M D yQ(X, a,

8

)d ss

о

Since lim

Q(X, a, s)

⁼

Q(a, t0, a, s)

= 0 when

(со, t)'-+(a~, t0), t

0> 0, and

P x Q ( a ,t ,a ,s )

= 0 and lim

f

4

(s)(t — s)D yQ (a~,t, a, s)

= 0 as

s->t.

B y Lem m a 8 we obtain

t

1н п г4(Х ) = lim j

f

4

(s)DyQ(x, a, t, s)ds

=

f

4

(t0)

^as

(со, t)-+(a~, t0).

о

W e can prove sim ilarly as for

zi (i

⁼ 3 , 4 ) that

И т ^ - ( Х ) = 0 when

(со, t)-+(a~, t0), t

0> 0 for

i

⁼ 6, 6 . W e shall prove that И т ^7( Х ) = 0 as

(x, t)->(a~, t0), t

0> 0.

O f course since

Q(a, t, у , s)

= 0 thus

t ^{+ a}

lim z7(.X) = — l i m j J

f

7

(y, s)Q (X, Y)dyds

0 —a t ^{+ a}

= f f М У, *)Q (a ,t

0

,y , s)dyds,

0 —a

when

(x, t)->(a, t0), t

0> 0 . N ow we shall prove

L

^emma

12. Let the function f

7

be continuous and bounded', then P 2

^u

7(X) = - M X ) .

P r o o f .

t ^{+ a *} t ^{+ a}

Р гщ (Х ) = / f U y ,s ) D lQ ( X , Y )d y d s - 'j j f,(y ,s )D t Q(X, Y )dyds-

0 —a 0 —a

+a

- lim

f f

7

(y, s)Q(X, Y)dy.

++* - a

Hence

+ a + a

Р ги,(Х) = f f,(y , s)P x Q(X, T )d y d s - lim / f,(y ,e )Q {X , Y)dyds

—a —a

= - л д а .

By Lemmas 3-12 we get

Theorem 2.

I f the functions f {

(i — 1, . . . , 7)

satisfy the assumptions of Lemmas

3, 6,

then the function U(X) defined by formula

(8)

is the solu­

tion of the problem (B — G — D )x.

6

. Let ns now consider equation (la) and

t +a

(Id) v(X) = f f f(y , s,v (y , s))G {X , Y)dyds.

0 —a

We shall nse the following

Lemma 13 ([3], p. 10).

I f the function f(Y , u(Y)) is continuous and bounded in the set jS and satisfies the Lipschitz condition with constant a < 1 uniformly with respect to Y eS, then there exists the solution v(X) of equa­

tion (Id).

Lemma

14. Let the function f [ X , u{X)) satisfies the assumptions of Lemma 13) then the function v(X) satisfies the homogeneous boundary conditions (2)-(7) and equation (la).

The proof of Lemma 14 is similar to the proof of Lemmas 3, 5, 7, 10 and 12.

By Lemmas 13, 14 we obtain

Th eo rem 3.

I f the function f(Y ,u (Y j) satisfies the assumptions of Lemma

13,

then the function u(X ) = u

1

{X )-\-...Jr u%(X)-\-v{X

)1

where u{

(i = 1, ..., 6) are defined by formula (8) is the solution of the problem (B — C — D)2.

References

[1] M. K rz y z a n sk i, Partial differential equations of second order, vol. I, Warszawa 1971.

[2] J . M ilew ski, Wybrane zagadnienia graniczne dla rôwnan parabolicznych rzçdôw wyzszych (unpublished).

[3] W. P o g o rz e lsk i, Bownania calkowe i ich zastosowania, t. II, Warszawa 1958.