On the polycaloric problem for the exterior of the ball

ROCZNTKI POLSK IEG O TOWARZYSTWA M ATEM ATYCZNEGO Séria I: PRACE MATEMATYCZNE XXII (1980)

Jan Milewski (Krakôw)

On the polycaloric problem for the exterior of the ball

1. In the present paper we shall solve the limit problem for the equation (1) Pnu ( x , y , z , t ) = F (r,t),

where

P = D2x + D2 + D2- D t, Pn = P(Pn~l ), r = (x2+ y2+ z2)112

and n being arbitrary positive integer, F(r, t) is given function. We shall construct the function u(x, y, z, t) = v(r, t) satisfying equation (1) in the domain

Q = {(x, y, z, t): x2 + y2 + z2 > a2, t > 0},

a being the positive number. We assume that the function v(r, t) satisfies the initial conditions

(2) D^v(r,0) = f k(r) (к = 0 , l , . . . , n - l ) and boundary conditions

(3) DnI ï v(a, t ) + yÜv(a, t) = gk{t) (k = 0, 1,..., n - 1), where

L = D ? + ~ D r~ D t, 2 È = L { Ê ~ X), r

y is an arbitrary positive number, D„ denote the outward normal derivative on the sphere x2+y2+ z2 = a2. The functions f k(p) and gk(s) are given functions defined for p ^ a and s ^ 0 respectively. We shall call problem (1), (2), (3) in the domain Q the (P — C — M ) problem.

2. Let the function v(r, t) be the solution of the (P — C — M) problem.

It is easy to verify that the function v( r , t ) satisfies the equation

(4) E‘v(r,t) = F (r, t).

By induction we can prove

Lemma 1. I f w( r , t ) = rv(r, t), then ü v ( r , t) = r ~1Qnw{r,t), Q = Dj — Dt,

Q n = Q ( Qn~ 1).

By Lemma 1 follows that the function w( r , t ) satisfies the equation

(5) Qnw(r, t) = rF(r, t).

Moreover, the function w(r,t) satisfies the limit conditions (6) Dkw (r, 0) = rfk(r), r > a (k = 0, 1 , n- 1) , (7) Dr Qkw(a, t) + hQkw(a, t) = agk(t),

t > 0 , h = y —l/a (к = 0 , 1, n — 1).

3. We shall solve the (P — C — M ) problem using an aid function G(X; У).

Let X = (r, t), У = (p , s), X t — (2a — r, t), X x = (2a — r — t, t), т ^ 0 and let

Qi = {(r, t): r > a, t > 0}, Q2 = {(r, t): r ^ a, t ^ 0}.

The function

(r~p)2

\ (t — s) 1/2 exp I 1 for s < t, U ( X ; Y ) = l y P V 4(f —s)

I 0 for s ^ t, X Ф Y,

is the fundamental solution of the heat equation. Let W( X ; Y) = 2h J ehxU ( X 1; Y)dx.

о Let us consider the function

(8) G(r, t; p,s) = G(X; Y) = U(X; Y ) + U ( X l ; Y ) + W ( X ; Y), where X e Q ï , Y e Q 2 , 0 ^ s < t .

Similarly as in paper [2] we can prove

Theorem 1. The function G(X; Y) given by formula (8) is the Green function for the heat equation, satisfying the boundary condition of the third

kind

Dp G(X; Y) + hG(X; Y) = 0 for p = a, X Ф Y.

4. Let

<Pj(P) = -Jj-p/j(p)— t , (pX -i<P))<2)+

(j — 0 , 1, n — 1) and let

(9) wu (r, t) = Atj I (pj(p)G(r, t; p, 0)dp ( / = 0 , 1 1 ) , A =

(10) w2J (r, t) = Л ( - i y +1 a J gj (5) G (r, t; a,s)ds

(11)

(

)

(13) (14)

O' = 0, 1,..., л -1 ),

n - 1

Wi(r, 0 = Z wl j ( r ’ 0 ,

j = 0 n - 1

w ² o, о = Z w2j0» 0,

f 00

w2(r,t) = ( - I f A J j pF(p, s)G(r, t ; p,s)dpds, 0 a

vt(r, t) = r 1 w£(r, t) (i = 1 , 2 , 3).

Let us consider the functions

3

(15) w(r, t) = Z 0

j= 1 and

(16) v(r, t) = Z 0 -

i = 1

5. We shall prove that the function v(r,t) defined by formula (16) is the solution of the (P — C — M ) problem.

Let H denote the class of the function v(r,t) continuous with the derivatives Z>f D^:v(r, t), oc + 2(3 ^ 2n and satisfying equation (4) in the domain Qx.

We shall prove

Lemma 2. I f the functions fj{kJ} (p), kj < 2 (n—j — 1), gj(s) (j = 0, 1 , n — 1) are bounded and measurable for p ^ a and s ^ 0 respectively, the functions F(p,s), DpF( p , s ) are continuous and bounded in the set Q2, then the function v(r, t ) defined by formula (16) belongs to the class H.

P ro o f. Indeed, integrals (9), (10), (13) and the integrals OO

Ji j ( r , t ) = j <jpj-(p)Df D*x (tj G(r,t; p,0))dp,

a t

J 2j (r,t) = J gj (s) D\ D*x ((t - s f i G (r,t; a, s)) ds, 0

t 00

J3j ( r , t ) = f j pF(p, s)Dl D*x G(r, t; p,s)dpds

(j = 0 , 1 — 1), a + 2/i ^ 2n, are almost uniformly convergent in the domain Consequently the function v(r, t) is continuous with its derivatives DfD“r(r,f) in the domain O j. By [2], Lemma 2, follows that the function v{r,t) satisfies equation (4).

Le m m a 3. I f the functions f 3 (p) (kj ^ 2(n—j — 1), j = 0, 1, . . . , n — 1) are continuous and bounded for p > a, then the function ^ ( r , t) satisfies the initial conditions (2).

P ro o f. We have

v ^ r , t) = t)-\-J2(r, t), where

n—1 00

J1(r,t) = r ~ xA £ tj J (pj(p)U(r,t; p,0)dp,

j = 0 a

n — 1 oo

J2{r,t) = r ~x A 2 tj J (Pj(p)(U(2a-r, t; p , 0 ) + W ( r , t ; p,0))dp.

j = 0 a

By formulas (6), (9), (11), (14) and by [1], the function t) satisfies the initial conditions (2). Moreover, if r0 > a and (r, t) -> (r0, 0 +), then

\ r~r01 > r0 — a for the appropriate values of the variable r. Consequently by the singular estimations as in [3], p. 132, we obtain the inequality

\ D U2(r,t)\ < C j^ i (к = 0, 1 , . . . , n - 1 ) , Cj and jiq being the positive constants. Consequently

lim Df J2 (r, t) = 0 as (r, t) -► (r0, 0 +), r0 > а (к = 0 , 1,..., n -1).

Lemma 4. I f the functions gj(s) (j = 0, 1 ,..., n — 1) are bounded and measurable for s ^ 0, the functions F (p, s), DpF(p, s) are continuous and bounded in the set Q2, then

(17) lim Dtvi{r,t) = 0

as (r, t) -» (r0, 0 +), r0 > a (i = 2 ,3 ; к = 0 , l , . . . , n - l ) . P ro o f. By the estimations similar to those in [3], p. 132, we obtain

< C2 t“2 (i = 2 ,3 ; к = 0, 1 1 ) ,

C2 and p2 being the appropriate positive constants. Consequently the functions vt(r,t) (i = 2,3) satisfy the initial conditions (17).

Lemma 5. I f the functions gj(s) (j = 0 , 1 , . . . , n —1) are continuous and bounded for s ^ 0 , then the functions v2(r, t) satisfy the boundary conditions (3).

P ro o f. Similarly as in [2] we obtain

where

DrQkw2J {r, t) + hQkw2J {r, t) Jjtk(r ’ t) for j ^ k, 0 for j < k,

J j A r’ 0 = A a { - l ) j+k + 1 J gj(s) (t — sY k 1(a — r)

( F £ j! U (r , t; a, s)ds.

If j = k, then by [3], p. 127, we obtain

lint h A r > 0 = aGk(to) as (r, t) -> (a+, t0).

Moreover, if j > k, then we obtain the estimation

\Jj,idr, 01 < C3 (r — a) t^3, C3 and p3 are the positive constants. Hence

lim Jjtk(r, t) = 0 as (r, 0 -► (a+ , t0), j > k.

Consequently the function w2(r,t) satisfies the boundary conditions (7) and the function v2(r,t) satisfies the boundary conditions (3).

Le m m a 6. Let the functions f J (p) (kj < 2(n— j — 1); j = 0 , l , . . . , n — 1) be bounded and measurable for p ^ a, the function F(p, s) be bounded and measurable in the set Q2; then

lim(Drl f v i(r,t) + yÜvi(r,t)) = 0 as (r,t)^>(a+, t 0)

(ii = 1,3; к = 0, 1 ,..., и — 1).

P roof. Integrals (9) and the integrals J i j ( r , t) are uniformly convergent in every set of the form

C23 — {(r, 0: a ^ r ^ b, 0 < T0 ^ t < T},

where b , T 0, T are the positive constants. Consequently we obtain DrQkWij{r, t) + hQkw1 j ( r , 0 KM (r, 0 for j ^ k,

0 for j < k, where

f ~ k 00

Khk(r,t) = ^ (-1 )* J (Pj(p)(Dr + h)G(r,t; p, 0)dp

\J K f l a

(j,k = 0 , 1,..., n - \ ) . By uniform convergence of the integrals J i yj(r, t) the functions are continuous at the point (a, t0), t0 > 0. Consequently by the condition

(Dr + h) G(r, t; p, 0) = 0 for r = a, we obtain

Hence

lim K j k (r, t) = O as (r , t) -+ (a +, t0), t0 > 0 .

Um(DrÉ v l {r,t) + y Ê v1(r,t)) — 0 as (r, t) (a+, t0).

Similarly we can prove that

lim (Drt i v z (r, t)f-ylï v3(r, t)) = 0 as (r, t) -* (a+, t0), t0 > 0 and we get assertion of Lemma (j.

Now we shall prove

Th e o r e m 2. I f the functions f J (p), gj(s), k j ^ 2(n— j — 1) are continuous and bounded for p ^ a and s ^ 0 respectively, the functions F (p, s), DpF(p, s) are continuous and bounded in the set 0 2, then the function v( r , t ) defined by formula (16) is the solution of the (P — C — M ) problem.

P ro o f. By Lemma 2 follows that the function v(r, t) belongs to the class H. Moreover, by Lemmas 3, 4 follows that the function v ( r , t ) satisfies the initial conditions (2). From Lemmas 5, 6 follows that the function v(r, t) satisfies the boundary conditions (3).

References

[1] M. C rain, M. N. R e s c u le t, Ecuatia D*™ u + ... + u —Dt u = 0 iterata, Studii si cercetari mat., Acad. RSR, 1968, 20, No. 9.

[2] J. M ile w s k i, On the Green function and the Cauchy-Dirichlet problem for poliparabolic equation, Rocznik Naukowo-Dydaktyczny WSP, Krakow 1978.

[3] W. P o g o r z e ls k i, Rôwnania calkowe i ich zastosowania, T. 2, PWN, Warszawa 1958.