2 2 2. Properties of integrals. 1. Introduction. On a certain non-linear boundary value problem for the one-dimensional bicaloric equation

Series I: COMMENTATIONES MATHEMATICAE XXVI (1986) ROCZNIKI POLSKTEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXVI (1986)

J. Ur b a n o w ic z (Warszawa)

On a certain non-linear boundary value problem for the one-dimensional bicaloric equation

1. Introduction. The aim of this paper is to prove the existence of a solution of a non-linear boundary value problem for the one-dimensional bicaloric equation. We reduce the problem to a system of Volterra equations which contains both the first and the second kind of equations and then we apply Gevrey’s method to reduce this system to a system containing only second kind of Volterra equations. The last system is solved by using Schauder’s fixed point theorem. In order to apply the aforesaid procedure we need some integrals connected with the bicaloric equation, the properties of which are given in the first part of this paper. Let us note that up to now the boundary value problems for the bicaloric equation have been examined only either in the linear case (see Nicolescu [11], Borzymowski [2], Brzezinski [4], Musialek [10], Milewski [9]), or in the non-linear case for special types of domains (see Barariski, Musialek [1], Brzezinski [3]).

2. Properties of integrals. Let T be a positive number, whereas Xi(t) (i = 1, 2) functions defined in the interval <0, Г ) and fulfilling the condition:

where x e (0, 1 ), K x. > 0.

We assume that the curves of equations x = Xi (0 (i = 1, 2) do not intersect for t e( 0, T>.

We introduce the following notation:

(2.1)

(2.2) Ux{x, t; (p) = jco(x, t, / (t), x)(p(x)dx, о

(2.3) Jx{x, t; (p) = $(x- x{t))Dx(ü(x, t, x(t), x)(p{x)dx, о

1 ( t -x)œ(x, t, x)/(£, X)d£dx,

(2.5) K u(t, ^T) = Dt (2.6)

GU(t, ^{T) :}

'co(xi(z), z, Xj(t), t)

7 - Z

t/z (/,7 = 1,2),

(x< (2) - Xi (т)) Ac « (Xi ( A Z, Xj ( A ^t) dz (/, 7 = 1, 2),

1 (x — ç)2

where w(x, f, ç, t) = ---- e x p - —- ---- -,

t 4( f - r )

5, = {(£, f): Xi (t) < £ < *2 A); о < T < /},

x(0 = xi(0 or x(0 = x2(0-

We are going to formulate several theorems concerning the properties of the above integrals. They will be used in the next section in the examination of the boundary problem.

Th e o r e m 1 (see [7], p. 1473-1489). I f condition (2.1) is satisfied and if the junction (p(t) is defined in <0, T> and J'ulfils the condition

\<p(h)-<P(h)\ < K <p\t2- t l \h<p; (p{0) = 0, then the inequality

|lM x('

2

Theorem 2. I f condition (2.1) is fulfilled and in addition the function qy(t) is hounded and integrable in <0, T ), ( sup |<p(f)| ^ К Д then the following

Г6<0,Г>

inequalities

(2.7) IUx{x2, t ; ( p ) - U x{xu t\ (p)I < const K J x 2- * i l

for (x2, t), (xt , t) eST, (2.8) IUx(x, t2; ( p ) - U x{x, tA; q>)\ ^ c o n s t \t2- t 1\e/2

for (x, fj), (x, t2) e S T,

. const - Кф „ , „

(2.9) \Dkx Ux(x, t; cp)\ ^ ^ for (x, t) eST, (2.10) \Dk Ux(x, tt ; (p) — Dkx Ux(x, t2; (p)\

_________ const • Кф \t2 t i ^ _________

" min[|x —х М Г -1 ^{+ 2Д,} \ x - x M k- l + 2fl for (x, f2), (x, t^eS-r,

(2.11) IDkxux(x2, r; <p)-Dkx Ux{xu f; q>)\

^ const • |>X2 — Xjl20^

^ min [|x2 - x(f)|k' 1 + 2' | х Т ^ ( 7 ) Г ^ ] for (x2, t), (xl5 t ) eST; 0 < 0 < 1, 0 < ft < j x , к = 1, 2, 3, are satisfied.

Proof. Carried out with the use of classical methods of the potential theory.

Remark 1. Theorems 1 and 2 are true if Ux is replaced by Jx.

Theorem 3. I f condition (2.1) is satisfied and the function q>(t) is defined and continuous in the interval <0, T ), then

(2.12)

lim DxJx( x , t ; c p) = - ^ /k sgn(x-x(t))(p{t) + DxJx(x(t),t;(p); t > 0

*->*(«) and

df

(2.13) Lq[ Jx{x, t; (p)~\ = 0 in ST, where L 0 = D $ - D t; L20 = L 0[ L 0].

Proof. Is similar to that in paper [11], p. 278, with necessary modifications connected with the weakening of the assumptions on (p(t).

Th e o r e m 4. I f condition (2.1) is satisfied, then const (2.14) [D,G,7(> ,T )| g --:- p -^ .

(2.15) |D,G,,(r2> t)-D,G,.,(r „ t)| s;const •|r2 — t1

0 ^ t < ti ^ t2 ^ T.

Proof. Carried out with the use of classical methods of the potential theory.

Remark 2. Theorem 4 remains true when we substitute D,Gu(t, t) for Ku{t,T).

Theorem 5. I f the function f ( ç , т) is defined, continuous and integrahle in ST and fulfils Holders condition of the form(l )

(2.16) l/(Ci, t) - / (ç2, t)| ^ K(S^)|cj - ç 2\hf in a domain czST, then the following equality

(2.17) Lq [ HSt(x, r;/)] = f ( x , t) is true in Sf.

t1) The coefficient K {S * ) depends in general on S% and hf e { 0, 1 ).

Proof. The proof is similar to that of an analogous property for the potential of spatial charge related to the heat conduction equation (see [8]).

Th e o r e m 6. I f the function /(£, i ) is defined and integrahle in ST and if (2.18)

i f ( t ’ т)| ^ ^ [ К - х , ( г Г , | £ - / 2 (т) | ' Г

(£, t) eST; 0 < p < 1, mf > 0, then the following sequence of inequalities is valid

(2.19) |#Sr(x, f;/)| < const -mf t, (2.20) |DXHS (x, t ; f)\ ^ const -mf t, (2.21)

l +1 — p

\D$HST{x,t;f)\ ^ const-mf t2 2 , 1 — n

(2.22) —2^0

\DlHST(x, t;f)\ ^ const-mf t

(2.23) \HSt(x, t ; f ) — HST( xl , ti\f)\ ^ const-mj-Ux-Xil + l t -

(2.24) |DxHSt(x, vJ ) - D xHSt{x1, tx\f)\ ^ const-my-Clx-Xjl+ T 1--1

(2.25) 1 1

2 + ~2^в ID2xHSt{x, t ; J ) - D $ H ST( x1, t l ;f)\ ^ const - ni f Ux- x^ + l t - t x\ ],

__^t-.Q

\D3xHSt(x, t ; f ) - D 3x HST( xl , tt ; f ) ^ const-mf [| x -x 1|(1“ p)e + | f-f1| 2 ].

Proof. The proof is similar to that of analogous properties of the spatial charge potential related to the heat conduction equation (see [8]).

3. The boundary value problem. Let us consider the following problem.

Find a function u(x, t) eC{ST) which satisfies the equation (3.1) Ll [ u{ x, r)] = F( x, t, u(x, t), Dxu{x, t), D$u(x, t), D3x u(x, f)) in ST and fulfils the initial conditions:

(3.2) u(x, 0) = 0, Dt u(x, 0) = 0, Xi (0) < x < * 2(0) and the boundary conditions

(3.3) wfc(t), t) = f ( t , u(xi(t), r)), / = 1,2; 0 < t ^ T, (3.4) Dxu(Xi{t), t) = gt(r, u(Xi(t), t)), / = 1,2; 0 < t ^ T.

We make the following assumptions concerning the functions appearing in the problem.

I. The functions f { t , u) and #,(f, и) (i = 1, 2) are defined for te <0, T>;

ue{ — oo, oo) and satisfy the conditions:

f ( 0, 0) = 0, <7, (0, 0) = 0,

h f . + 1/2

If ( t i ,

«1

where К, К д., Kf . > 0; 0 < hf . ^ j, 0 < hg. ^ 1.

II. The function F(x, t, u, ux, u2, u3) is defined and continuous for (x, t) eSr , u, ux, u2, m3g( —oc, oo) and fulfils the conditions

|F(x, t, и, иi, u2, u3)I M,

+ min[|x — Xi(0|Pf; \x - x 2( t f Fl

+ mf (\u\rF + \u /r + \u2\r4 \u3\rï),

|F(xt, t, u, ux, u2, u3) — F (x2, t, й, йх, й2, ы3)|

^ К . | x 2 - X i | hF

min [|х! — Xi (ОГ*, I^₂- Z i(₀IPf. 1 *1 -Х 2( 0 Г \ l*2~ Z 2( 0 P ] + |m — m|f + |m1— ûx\ F + \u2 — iï2\f + |m3 — m3| P ,

IPF IPF- +

where 0 ^ pF < 1, 0 < rF, r lF ^ I, 0 < rj < 1, 0 < rF < i ; M F, mF, K F > 0 and the exponents hF, hF, Ц, hF belong to the interval (0, 1).

We seek a solution of the problem in the form (3.5) m(x, t) = UXl (x, r; (^{P i}) + ^{U X}2{^x, t; tp2) + JXl (x, t; ipx) F

+ JX2(x, t; ф2) + НБт(х, r; F(x, t, u, Dxu, D2x u, Z^u)).

Taking into consideration the boundary conditions (3.3) and (3.4) as well as employing: Theorems 1, 2, 3, 6 and Remark 1 we obtain the following system of equations:

(3.6) f( t, и(хх(0, t)) = UXl (xt(t), t; (px) + U X2(Xi(t), t; <p2) +

+ JXl(Xi(t), t; il/i) + JX2(Xi(t), t; + U F(...)) (i = 1, 2), (3.7) gx(t, u(Xi(t), t)) = ^ fn { - l ) i ((pi (t) + 'J'i(t)) + Dx UXl(Xi(t)t; фх) +

+ DX UX2(xi(t), t; <p2) + DxJXl (Xi(t), t; фх) +

+ DxJX2(xi(t), t; ф2) + ОхНБт(х1(г), t; F (...)) (i = 1, 2), where (px, q>2, фх, ф2 are unknown functions satisfying certain conditions which will be formulated in the sequel. Using the Gevrey method (see [5],

[6]), we reduce system (3.6), (3.7) to Volterra second kind system of the form

(3.8) <ft(f)+ X -

7 = 1 K J

A o)(Xi(z), z, Xj(t), t) L J

0 t

t t

+ i z i a

7=1 0 t

’x,0 0 -;q(t) s/t^z

(Pj(x)dx +

АЦ х.О О , x7(t), x)dz il/j(x)dx

= Pi{t, M(Zj(f). 0)»

( — IV 2 f

(3.9) A- (0 + ^-7=-

Z

0

+ Ц Ц - Z I ° x [(* - Xj (0 ) Ac Û) (x, Г, Xj (0 , t)]

71 7 = 1

0 ^{: =Xj(t)}

i{/j(x)d x-

- Z ^ J a

Ц & 0 0 , 2, Xj(l), l )

0 r

r f

Л

ч/t — z

<p7- (т) dx

t z |A 7=1

x,0 0 -;o(t)

О T

\/t j АсЦх.ОО, г, х7(т), т) i//j{x)dx (- 1 )' &(*, “(MO, r ))-5—Л .чЧ ( - А^ - D x H ST(xi(t), t; F(...))-p,(t, м(х,-(0, 0)»

where

(3.10)

/. / m fu 1 wi(t> “(MO, t)) Pi(t, u{Xi(t), t)) = ---7=---

71 J t

1 № (z, u (/,(z), z ) ) - ^ (f, ц(х,(0, t))

2 71 J ( t - z ) 3/2

O

dz,

Щ(1, u(Xi(t), t)) = f ( t , u(xi(t), t ) ) - H ST (Xi(t), t, F (...)) 0 = 1, 2).

In order to simplify certain formulas we introduce the following symbols

<Р, + 20) = «М0

0

1

2

0

1

2

^ wfcO), 0)“ ~ 7="D x H s T (Xi(t), t; F (.. .)) —

я -v/Я

— Pi(t, wfcO), t)) = Pi+

2

In the aforementioned notation system (3.8), (3.9) takes the form

4

(3.12) <Pi(t)+

Z

(/ = 1,2, 3,4), where M u(t, r) are linear combinations of the expressions

A 'o>{x,Uh 2, /,(Т), t) dz

t — z ; A Ъ ( 2 ) - ^ ( Т)

f —Z Dxoj(Xi(z), z, Xj{t), r)dz Dx w(Xi{t), t, x j (t), t); [(x-£ ,(!))£ > *со(x, /, х,(г), т)]|,=*{(0.

Thus, in order to examine the boundary problem in question it is enough to solve the following non-linear system of integro-differential equations:

2

(3.13) и (x, t) = £ co(x, r, Xj(t), г)<Р;(т)с/т + 7= i

2 /

+

Z

2 t

(3.14) vt(x, r ) = Z f^ x ^ fx , Г, X^j(^t), ^t)(P^j(^t) J^t + 7=1 о

2 t

+

Z

+ ^ F Sr(x, r; F(x, t, M, yj, r2, t>3)) (/ = 1, 2, 3),

4 t

(3.15) q>i(t)+ Z I z)(pj(z)dz = p,-(f, u(x,(/),

0

7=1 о

We will prove the existence of a solution to this system using Schauder’s fixed point theorem.

Let A be the space of the systems Ф — [u(x, r), ^ (x , t), v2(x, t), i?3(x, f),

<Pi(f), (p2(t), Ф1 (t), <A2(f)] of real functions where u(x, r), <p,(r) and i/^(f) are defined and continuous on the sets ST and <0, Г ) respectively and i?,(x, t) (i

= 1, 2, 3) are defined and continuous in ST and satisfy the condition:

sup (min(|x —Xj(r)|'~1 + 2y, |x-x2( 0 r 1 + 2v)h U , 01} <

(x,t)eSf

0 = 1, 2, 3); 0 < у < i.

Linear operations in the space Л are defined in the ordinary manner, the norm is given by the formula

3

||Ф|| = sup M x, f ) l + Z sup min(|x — (r)|'~1 + 2y,

(x,t)eSj i = 1 (x,t)eSj

2

\ x - X 2( t ) r 1 + 2y\V i(x, t) \ ) + X ( SUP l^i(f)l+ sup (r)l)

i= l te<0,T> «6<0,Г>

and the distance between two points <Pt and Ф2 of the space is understood as

|Ф1-Ф2|.

It is not difficult to show that Л is a Banach space. In the aforementioned space, we will consider the set E of all points whose coordinates satisfy the conditions:

sup \u{x,t)\^Ru;

(x,t)eSj

sup {min(|x-/1(r)|‘ _1, 1х~х2( 0 Г 1)1^(х, t)|} ^ Rv.,

(x,r)eSj

sup \<pM < K vT I,\ <p(0) = 0, ie<0,r>

(i = 1, 2), sup \фМ < К ф (i = 1, 2), where

te<0,T>

hy < min '/1’ nf 2 ' 2 *■> 91 h92 1 -m ax(pf , rj, 2rj)

‘ 91’ в 2 ’ 2 { l - h ei) ’ 2 { l - h „ y в

9 2>

Ru, Rv., Ky, and К ф above, are parameters that will be appropriately chosen in the sequel.

It is evident that the set E is closed and convex.

We are going to transform this set by the operation

2 t

(3.16) u(x, t) = £ [Ц х , f, Zj(t), x)(pj(x)dx + i=i 6

2 t

+ Z $(x- Xj(t:))Dx(x, r, XjW, т) Фj (t)dx + HSt(x, t; F (x, t, u, vlt v2, v3)), 1=1 о

2 t

(3.17) Vi(x, t) = £ L Xi-(t), т)ф7(т)^/т + j=l о

+ I [ ( * ~Xj(* ))Dx oj(x, f , /j-(t), t)] iI'j(r)dx + i=i b

+ DXH St(x, f; F(x, r, u, vu v2, v3)),

4 t

(3.18) (pi(t)+ X T)(pj(t)di = pi(t, u(yj{t), t)) (2)

j = i b

0 = 1,2, 3,4).

The image of a point Ф = [w, yl5 u2, уз, <Pt, <?

2

4

We shall find sufficient conditions for the inclusions E' c: £.

From (3.16) and (3.17) we obtain:

(3.19) Iu(x, r)| ^ const-[(K, + /C,) A +

+ M F +

mr (R 'l + R'), + R 'l +

(3.20) |f, (x, r)| ^ const К , Г * + К+

_min(|x-/1(r)|i \ |х-/2(г)Г' *)+ Mu +

2 ,3v

+ mF(RruF + Rr/l + Rrf2 + Rrfi ) T(1

Next, using Theorems 1, 2, 4, 6 and Remarks 1 and 2, we have

• |tf(*,(f), t ) \^œnst - l Kv + K ^ M F + mF(R[F + R[f + Rr/2 + Rrj3)~\tll2 + h(f>,

|wfc(/i), 'i )-«(*,• ('2), t2)|

< const • [X,, + X^, + Mf + mF (R[F 4- R 'f + RrjL + R'/Л]\t2 - r j^{, 1/2 + h}<P In view of the definition of functions Wt (p. 172) and the above estimates we have

(3.21) |H'(t,i7teW,l))|

< const • ! Kf. IX , -t- K , + Mj. + mF{ R j + /?,* + R/2 + )] +

+ K + M F + m AR'f + R 'l + R"/2 + R'jÿ} 11/2 + ‘ «’ = const • R f . t ll2 + h«,

(3.22) | ^ ( f 2 > ü ( Z l ( r 2) , f 2) ) - » { ( t 1, ü ( Z i ( t 1) , t 1))|

^ const ■ \ R/j X „ + Кф + Mp + nipiRj + Rvf + R/2 + H- + К + Mp + mr ( < F + R' l + R$2 + R$ ) Î |t2 - Г, Г 2 + 'v

^ const X ^. ■ jt2 — г1|1/2+А(р.

Finally, from the definition of functions ph inequalities (3.21), (3.22) and the lemma in paper [6], p. 1091, we obtain the following estimates:

(3.23) I a- (/, m(&(0. t)) I ^ const • Kf . th(p (i = l, 2), (3.24) |а (г2, ü(xi{t2h h) ) ~Pi ( t u м(х,(/

1

(/ = 1,2),

( 2) In accordance with the notation on p. 172 (formula (3.10) and (3.11)).

(3.25) \p;(t, u(xi(t), f))|

^ const • ((Rf. 4- К д.) [_K(p + Кф + M F + mF(RuF + R/l +R/2 4-R^)] 4- M F 4- + mF(Rr/i + R lFi + R r/2 + Rrl ) } t h« = c o n s t (i = 3, 4), (3.26) |pi(t2, il(Xi(t2), t2) ) - P i ( t i , wfe(ti); 0))| < const-K f m \t2- t / 9

* (/ = 3, 4).

Using the formulas obtain above and basing on the fact that the solution of system (3.18) is of the form (see [6], p. 1089)

f 2

<p,(0 = P i(t)+ f L т)рДг)(/т (/ = 1, 2, 3, 4),

о j — \

where Jf‘lV(r, r) is the appropriate resolvent kernel, we get the following inequalities (see Theorems 4 and 6)

I (P i(01 < const • К f . th(p (i = 1, 2),

\<Pi (01 < const • Kf .g. th<p (/ = 3,4),

l^(/

2

0

\<Pi(t2)-Vi(ti)\ ^ const •^ /.g.|f2- f 1|A<p (/ = 3,4).

Thus, a sufficient condition for the inclusion E' a E is the following system of inequalities

2 3 1 -т а х ( р р ,Г р ,2 г р )

const • [(Kg, 4- К ф) T h+ M F + mF (R ? 4- R[\ + R% 4- Rr/3) T 5 ' ] ^ Ru,

1 2 3 t - n M p F . r } , 2 r j )

const • [ Kg, A 4- Кф + M F + mF (R[F + R^ + Rr/2 + Rr/3) T 2 ] ^ Rv.

(/ = 1, 2, 3), (3.27) const • { Kf . [ Kg, + К ф + M F + mF (Rrf + Rrv\ + R^ + R ^)] +

4- К 4- M F + mF ( Rf 4- Rvf 4- R/2 4- R„p} ^ К#’

const {(K f. 4- Kgt) [ Kg, + Кф + M F 4- mF (RUF 4- R^ 4- R/2 4- R„p] + 4- К 4- M F 4- mF (Ruf 4- R„^ 4- R/2 4- RVF)} ^ Кф (/ = 1,2).

It is easily seen that (3.27) holds true if we assume that mF, Kf ., K g. are sufficiently small whereas the parameters Kg,, Кф, Rv., Ru are suitably chosen.

Now we proceed to proving the compactness of the set E\ Using Theorems 2

and 6, Remark 1, the definition of the set E as well as equality (3.17), we obtain the inequalities

(3.28) \и{(х, t2) - Vi ( x, t {)\ ^

COnSt * (I f 2 f i| "t" |f2 t jl

2 3 1 - тах(рр,Гр,2гр)

) m in[|x-Xi(ï2)l«-1 + 2/Ï |Y_

, \x-X2(ti)\i - 1 + 20 «-i + 2/г iv _

, \* - х2( ь ) Г 1+2рУ

(x, t2), (x, ti)e S r ; pe( 0,\x) , (3.29) |i2,-(x2, t ) - v {( xu O K

const ^• (|x2 — *iI + |x2 — 1¹ mM(pP ' rF ’ 2rF )ej

min[|x, ~Xi (t)|‘ l + 2p, |x2-^ i(0 r' l + 2p, |x, — ^

2

The continuity of operation (3.16H3.18) can be proved on the basis of Theorems 2 and 6 and Remark 1, by an argument similar to that used in deriving inequalities (3.27).

Thus assumptions of Schauder’s fixed point theorem are satisfied and we can conclude the existence of a fixed point of operation (3.16H3.18) and hence, due to Theorems 2, 5 and Remark 1, the existence of a solution u(x, t) of the considered problem.

As a result we can formulate the following final theorem.

Th e o r e m. I f assumptions I and II are satisfied and if the system of inequalities (3.27) is true, then the considered problem has at least one solution.

References

[1 ] F. B a ra n s k i, J. M u s ia le k , On a certain bicaloric problem, Comment. Math. 19 (1977), 171-181.

[2 ] A. B o r z y m o w s k i, The first bicaloric problem in a half-space, Rev. Roumaine Math.

Pures Appl. 18 (1973), 1522-1534.

[3 ] J. B r z e z in s k i, A non-linear boundary value problem fo r an infinite system o f fourth-order integro-differential equations in half-space, Demonstratio Math. (1977), 679-703.

[4 ] —, A boundary value problem f o r the bicaloric equation in a half-space. Rev. Roumaiine.

Math. Pures Appl. 24 (1979), 187-197.

[5 ] M , G e v r e y , Sur les équations aux dérivées partielles du type parabolique, J. Math. Pures Appl. 9 ser. 6 (1913).

[6 ] L. I. K a m y n in , The method o f thermal potentials f o r parabolic equation with discontinuous coefficients (in Russian), Sibirsk. Mat. 2. 4 (1963) 5 1071-1105.

[7 ] —, On the Gevrey theory fo r parabolic potentials (in Russian), Differencial’nye Uravnenija 7-8 (1971-1972).

12 — Prace matematyczne 26.1

[8 ] M. K r z y z a n s k i, Partial differential equations o j second order, vol. I, P W N , Warszawa 1971.

[9 ] J. M ile w s k i, On certain limit problems fo r polyparabolic equation, Comment. Math. 20 (1977) 133-145.

[1 0 ] J. M u s ia le k , The bicaloric problem with mixed conditions in quarter-plane, ibidem 20 (1977) 147-176.

[1 1 ] M. N ic o le s c u , Ecuatia itérâtâ a câldurii, Stud. Cere. Mat. 5 (1954), 243-324.

[1 2 ] W . P o g o r z e l s k i , Integral equations and their applications, vol. I, Warszawa 1966.

[1 3 ] Z. R o je k , W . Z a k o w s k i, On a certain property o f functions o f Class § in space (in Polish), Biul. Wojsk. Akad. Techn. 9 (1963), 55-59.

[1 4 ] K. W a r c h u ls k i, On the Cauchy-Dirichlet problem fo r the poly parabolic equation, Demonstratio Math. 14 (1981), 183-199.