The first boundary-value problem for certain quasi-linear systems in the mathematical theory of diffusion

THE FIRST BOUNDARY-VALUE PROBLEM FOR CERTAIN QUASI-LINEAR SYSTEMS IN THE MATHEMATICAL THEORY OF DIFFUSION* V.

N. MASLENNIKOVA (Moscow)

(Received

15 May 1962)

1. Introduction, statement of the problem In several problems system of quasi-1inWr

in mathematical equations

with the two unknown functions which are given in the closed x EB, 0 < t
u(x,

t),

(

G

and B is a bounded n-dimensional X= (X 1' X2’ * * *, xn).

Z,P

region

It is assumed that the quasi-linear (X,

. l *

t)

Zh.

62 &, i.e.

for all

vych.

3.

mat.,

No.

real

3,

region

au I

vectors

wemeet

the following

IP(X, t) and with coefficients

cylindrical

au

VU=

physics

-**

au -

tazn

i$,

t),

1 3

in the space operator** (Al,

where D, = {(x,

A,,

of the variables

A is Parabolic . ..,

for

An)

1963.

467-477.

Note added in proof. The relults of this paper have been extended by the author to the quasi-linear operators I\ with uij E aij(x, 1, u, vuj of divergent and, under certain conditions. also nou-divergent form. 620

First

boundary-value

problem

for

certain

i,j=l

System (1.1)

quasi-linear

system

621

i=l

is solved with the following boundary conditions: n(z,t)=z+(r,t) w (5, t) = wg (3)

(a is the boundary of Dp i.e. its base for t = 0).

OnaD,

(1.31

for t = 0

the lateral

(1.4)

surface of the cylinder end

Problems of this kind arise, for example, in the diffusion in nuclear reactors (see [II, p. 330).

of neutrons

A similar problem is studied in the work C21 for a linear operator A. To study the problem (1. l), (1.3)-(1.4) it is useful to introduce the following norms and functional spaces for the functions V(X, t) defined in some region G (where 0 < a < 1) (see 131).

where the distance

between two points

We shall say that v E @o

if it has a finite

Co and C’+0 are defined similarly. spaces are Benach spaces. Finally,

is defined by the formula

norm of the form (1.6).

It is easy to show that these

we give the definition

PI:: = 14,GfLG(v), where the Lipschitz

constant

622

V.M. Maa lennikova

is defined with the usual distance between two points p (P,

F) = [

$ (Zi -

,)2 +

(t

-i)” ]Ih.

(1.7)

i=l

If 1v/y is finite then we shall say that u E C’; if, ~laxi g C’ for all i then we say that v E @ .

in addition,

In this paper we prove the comparison theorems which are used later to prove the uniqueness and existence of a solution, and also to prove the convergence of the method of successive approximations.

2. Comparison and existence theorems We shall assume that the following

conditions

are satisfied.

A. h(x, t, u, w) is a non-increasing function of w, and g(x, t, u, w) a non-decreasing function of u, the functions h, g w. r. t. u and w and bi and f w.r. t. u satisfy the uniform Lipschitz condition:

iS

Ih (& t,

hi

4

-

h (5, t,

~2,

QI<

M

(Iur

-

upI

+

Iq

-

~21)

(2.4)

E

for (n, t) E. DT and for all values of u, w from a certain bounded region. Following [21 we prove comparison theorems which will be used later to prove the uniqueness theorems and to find the solution. lheorem 1. Let tul, w,), bt, 10~)be two pairs of continuous functions defined in ET Suppose that their second derivatives w. r. t. xi and their first derivatives w.r.t. t exist and are uniformly bounded in 5, and satisfy in D, the differential inequalities

Au, - h (z, t, awl -at -

and, moreover, 5 <

g

u2

~1,

cc t, 4, on aD,

~3 > Au, - h (z, t,

q) <%

~2,

- g (G t, u1,4

w1 < w2 on B for t = 0.

~3,

(2.2)

First

boundary-value

Then u1 <

u2,

w1

<

problem

uJ2

in

for

certain

quai-linear

623

systems

Ep

Proof. Let us suppose, contrary to the hypothesis of the theorem, that there exists a point P E flT at which ur > u2 or tul> uy But then, in accordance with the data on the boundary and the continuity of the functions, there exists a point (x’, T’) E D, such that %Q%’ %

= u2

w1< w* everywhere in i&p. and at the point (x’,

7”) either

when n’ E B, ?” > 0 or zpl = tuxwhen .x’ E g, 7” > 0.

In the first case we have ur = u2, taf5 We at the point (x’, T’). Then at this point An, \< Au,. For, since a11- ur > 0 everywhere in the DT’. the function u2 - u1 attains a minimumat the point (r’, 7”). At this point we have LL~= ul# &I~/&~ = bz/ari and

Moreover, at this point h(x’, T’, ul, ul) >h(r’, ‘Dr< ‘oy ur = a** here end h is a non-increasing “Au, which contradicts

h (~3, T’, ILL,wJ < Au, the conditions

T’, ul, wt) since function of w.Thus

h (x’, T’, up, uQ,

of the theorem.

In the second case we have, at the point (x’ , ?“)

iu E+ moreover, g(n’, T’, ul, tpl)< decreasing function of u.

g(x’,

T’,

uz, w2) since g is a non-

Thus

which also contradicts theorem. 77leoreIn2. Let br,

the conditions of the theorem, This proves the wpu2,

1p.J be two pairs of ~ntinuous

actions

defined in $. Suppose that their second derivatives w.r.t. xi pd their first derivatives w.r.t. t exist and are uniformly bounded in ST and satisfy the differential inequalities

624

Y.Iy. Ye8 Iennikooa

UA =

up + W’,

ro, + a+““‘,

w& =

rrfaereh and kf’ are positive constants; we shall define the uantity bf’ later. According to the conditions of the theorem, l??u2/axiP < C, where C is an absolute constant. Let us determine the sign of the expression I = [A& -

h (5, t, us, w)l -

h,

-

h (a, t, us, wJ1.

We have Au,, -

Au, =

- 2 bt (x,

t, up) 2

t

-

-

AM'ert + f

(

2,

5,

t,

u, +

f(

asi,

aU’Q Wf’t (CM - M’ + M). 1, @I, $I$

Therefore I < he” t (C!! - kt'+ 31). Choosing M’ larger than Cbf + 3M we ten make I negative. from the choice of bf’ 8% t

~-g(5,wA*WA)

=g

Moreover,

I- [~-g(x‘t,u~,w3I=

(x, t, @a,w2)- g (4 t, up + 3ieMPi, w, f W’)

+ + AM'eM"> AtW(M' - 2M) > 0.

Thus we see that the inequalities for all positive values of A

(2.2) of Theorem 1 are satisfied

We have

u,
~~,WA~W2~~

on

B for t=

0.

According to Theowe! 1, u1 < uht w1 < ‘oh in 5,. for all positive

values

First

boundary-value

problem

for

certain

quai-linear

625

systems

of A. Since uA - u2, wA-. w2 as A - 0, we have u1 6 u2, wL6 w2 in ET This proves the theorem. We have the following uniqueness theorem. ‘Iheoren 3. The problem (l.l), (1.3), (1.4) can have not more than one solution in the class of functions for which the derivatives occurring in the equation are bounded. Proof. Let {al, ~~1, {uz, wz) be two such solutions, s@isfying the same boundary conditions, i.e. ul = u2 on 20, wI = w2 on B for .L=- 0. Then, from Theorem 2, we have in ET : u1 < u2
3.

w1 < wo2\( w1 and

Method of finding the approximate solutions and the existence theorem

In order to prove the existence of a solution of the given problem we shall assume that the following assumptions concerning the coefficients of equation (1.1) and the lateral surface of the region D, are satisfied. I. The lateral @+a and (?.

surface ST of the region D, belongs to the classes

II. The coefficients

of the operator A satisfy

the conditions

of the

existence and uniqueness theorems of the paper 151, i. e. for all

(IP a) the function

af(%.L u, 0) 1 au.

I” =

i$Pl:

Pi = 2):

bounded below, i.e.

is

W(% t, u, 0) > b, = conk;

au

b) the functions w (2, t, u, P) api-

satisfy

qj (5, t), bi (2, t, u), f (2: t, O,O), f’(“‘,‘I”

the HSlder condition with index a(0 < a < 1) w.r. t.

x, t the Lipschitz condition w.r. t. u and the Elder p(O < p
‘I,

t)

-

aii(if*

1)

]\
Id

(PI,

PJI”

condition with index

626

V.Iv.

Maa lcnnikova

(where c&P,, pg) Is taken from formula (1.5))

c)

IfL Worst.

On ST the functions a. .(x, t) SatisBr the Wpschitz condition i.e. l&j (5, t) t) with the usual X stauce (l.?),

(%

%j (g9 i)i <

A#

(PIP Pg)+

IV. Besides satisfying the condition A the functions h(x, t, u, q, the H6lder condition with index a(0 ( a < 1) in

g(x, t, u, W) satisfy

(%, t) E DT for any bounded values of uI ro. lheorer 4. Suppcse that conditions A, I-IV are satisfied,

aud there

exists 8 function y(x, t) E’ CT* in 5, which is equal to the given boundary values u,,(z, t) of the function u(x, t) on aD and w,,(x, 0) satisfies

the Wider condition with index a on &O < a < 1).

Suppose, mreover, that there-exist twc pairs of functions (u’, 1~‘)‘ (u7 @*L)which are HlSlder with index a aud possess ~ntinuous bounded derivatives satisfying the system of inequalities Al&# -

h (8, t, u’, w’) 5 0 > Au” - h (2, t, u”, w”),

ad

~-R(x,t,U~,W~)~O~~~-g(n,t,u”~~“),

26’(x, t) < w’ (x, 0) <

110 (3, lo,

t) f

u@ (5, t)

on dD*

(3, 0) < w” (x, 0) for t = 0.

First

Then

boundary-value

problem

for

the solution of system (1.1)

(1.4) exists, and u(x, t) E @* aw/& fG CQ in Dr Proof. mations.

quai-linear

certain

with the boundary conditions

(1.3),

in ET for qj\(ap < 1, and o,

The theorem can be proved by the method of successive

Let us consider the set of functions {u,) Au1 -

627

system

approxi-

defined as follows:

Mu, = h (5, t, u”, w”) -

Mu”,

(3.2)

‘$ + Mw, = g (x, t, u”,w”) + Mw”, . . . . . . . . . . . . . . . . . . . Au, 2

Mu, = Ws,

t,

un-1, wn-1) -

+ Mwn = g (I, t, u-1,

wed

Mu,,-,,

un (z, t) = u. (z, t) on 80, w&,

t) = w. (5, 0) OIIB

(3.3)

+ Mwn-1;

for

(3.4) (3.5)

t = 0.

(Here u*; w” are the functions in (3.1)). Every system of this type is solvable, since it disintegrates into two independent equations. On the basis of the results of 141, [51, the first equation of the system Of type (3.3), with the boundary condition (3.4), has a unique solution belonging to the class c?* (0 < y
@%I (5, t) = wo(x, 0) + j fl” [g (x,z, +

i.e.

un-1

(5, 4,

a-1

(5, 2) +

Mwn-1 (5, 41 dz,

wn(x, t) exists and is HBlder continuous with index a.

Let us show that u,, - u, wn - 10, i.e. the sequences (u,), _{w,> of approximate solution& converge at each point of the region D, To

do this we use induction to show that for all n>2 l.4’ 6

where u’,

u*‘ate

For since

the

Un <

h-1\<

Uw, where

functions in condition

U’,

U”

(3.1) of Theorem 4.

628

V.M. #as lennikova

mu, - Mu,1 -

IAU” - Mu”1 = h (5, t, ul, w”) - Au” > 0,

[f$ +

[s +.Mw”] =

Mwl]

-

g (x,

t, u”, w”) -y

< 0

and on the boundary of the region a1 < a” on an, w1
Let us show that if

h--l\<

uia--mwn-I

\
Un

\<&a--1,ca,<%-I

0.n DT.

We have IAu, - Mu,] - ~Au,, - Mu,,1= h (x, t, u,,+, w,,-$ -

Mun-1 -h

(z, t, h-m wl-3

= [h (r, t , h-1, &-I) +

ifa(5, &us--l,wt-d

-h

-

h(&

t,

(3.6)

+ Mun-9 = &--1,%-*)1+

(z, t, un-a, wn-,)l

-

M

(un--I-un-n)>O,

since the expression in the first square brackets on the right-hand side of the last equation in (3.6) is positive, due to the fact that h is a non-increasing function of w, the difference h(r, t,

hl-l,w+~)

-

f&b,

t, %-~,~ia-%)

is not greater than M~u,_~ - u,_g1 and the last term -M(u~_~_zI,,_~))‘O since u~_~ < un_g. Moreover

[2+

Mw,

I-[+ -:

-

+ Mw,,]

=g(x,ts

h~-l.,~n-d--

(3.7)

(x, t, un-s, wn-s) + M @n-r - wt--a~ = fgfx,6 h-1, h-1) - g tx,t, k-s, wn-l)] + [‘g (5, t, &a-a, wa-1) - g tx,t, ha-%* WI-*)I+ + M twn-I - ton-s) < 0,

g

T

since the expression in the first square brackets on the right-hand side of the last equation in (3.i) is negative, because g is a non-decreasing function of u, the difference

First

boundary-value

probler

for

certain

quasi-linear

systems

629

is not greater than M~w,,_~- w,,_~] and k!(~,,_~ - w~_~)\
n-2’

Since on the boundary of the region we have un =

u*__1

=

u,*

wn=wn-_l=w+a=

=

-*-

(

‘*-)

from the comparison theorem we find that a,, \
wn
Thus we have found bv induction that

But un, wn are also bounded below by the ‘functions u’, w’. In fact,

if II’<

u~_~, w’<

w~_~ we can show that u’<

u,,, w’
As before we have bjr induction

$Au, - Mu,1 -

thu’ - MU’] < h (xv t, un-1, Q-1) -

[

2 + Mw.]

-

Mu,,-~ -

[ f$ -

Therefore,

h (x, t, u’, w’) + Mu’ < 0;

+ Mw’] > g (x, t, un-1, G-I) g (x, t, u’, w’) + M (w,,--l -

w’) > 0.

from the second comparison theorem, we have u’<

a,,, w’ \
We have thus Proved that (3.8) i.e. the sets {u,), {w,,> form monotonic decreasing sequences which are bounded at each point of D,. The limit of these sequences exists and defines the functions u(r, w(x, t) in D,. Let us show that the limit functions satisfy

t),

the Holder condition.

Using the estimates for a quasi-linear parabolic equation obtained in [41, [51 in the proof of the existence theorem for a solution we have, for any 6, 0 < 6 < 1. the following estimate for the n-th approximation:

V.M. Mas lennikova

630

(3.9)

where the constant bf,depends only on the HBlder constant in ET end the Lipschitz constant 0; JD of the coefficients ai.(x, t), on the ellipticity constant, the constants b,, Co from condition II and the maxim1 of the moduli of the coefficients bi of the operator A. Since the uniform boundednese of u,(x, t), wn(x, t) fOllOW8 from the uniform boundedne88 of h(x, t, u,,~, w,,_~)we have, uniformly w.r.t. n, the estimate

(3.10) where the constant C does not depend on n. We know that, from any set of elements {u,,}which is bounded in the metric of C:zii we can pick out a subsequence which converges in the , where 0 < 6, < 6 < 1. Because of the unifoy+6eatimate metric of C , and so (3.10) the set {u,) is uniformly bounded in the metric of C we can pick o t from it the subsequence {u,*) which converge8 in the metric of C’+151. On Passing to the limit for thia subsequence we find that unk(z, t) - u(x, t) and &,,,/axi - &fii uniformly, end we also have Halder continuity of the function8 u(x, t) index 6, where 0 < 6, < 6 < 1.

and &/azi with any

Let u8 8hOW that 0(x, t) satiefies the Hiildercondition with index a, 0 < a < 1, TIT. Let P(x,, t,). c2($ Then Iu;1(%,

h)--

w?l (%

t2)

be two arbitrary points in the region ET.

f2) I
h)----?I

h

4

I+

IWn(%

w--&d,)*

It fOllOW8 from the fOmla

w-J

+

Mwn-l(r,

z)ldz +

WOW,

0

since u,_~# wn_l are uniformly bounded, that IWn(Zlrtl)-W,(Z1,tl)I~'lItl-t,] in 5,.with constant K independent of n, i.e. w(x, t) satiafiea the

First

boundary-value

Lipschitz

probler

for

certain

quasi-linear

631

rystens

condition w.r. t. t.

It remains to prove that w(x, t) satisfies w.r.t. 2.

the Hiilder condition

Let us use the method described in [21. We introduce the following notation:

On the basis of formula (2.1)

for g(x,

t, u, w) we have

t

eMt P, (t) ,<

s

MeMz

[Pn-l(z)

+ Q,,-l(z)1

(3.11)

dz + PO(0).

0

I@ induction on the formula (3.11) we can prove n-l

eMtP, (t) <

PO (0)eMt+

(3.12)

F& + 2 QrL..+ r=o

For n = 1, we have, from (3.11):

e”‘P, (t)

<\MeMz (2) +

Q. (z)l dz + PO(0) =

IP,

0 = i MeMz 11 W” (51, z) 0

w” (r,, z) 1+ IU” ($1, z) -

+ PO (0)< F, (eM1 -

1) +

go(eMt -

24”(22, z) II dz +

1) + PO (0)eMt = = PO (0)eMt +

PO&+Ooh.

Let us assume that formula (3.12) is true for the (n - 1)-th mation; we prove it for the n-th approximation. We have t eMt P, (t) \<

s

i14eMf [Pn-l

0

(4 + Qn-1(41 dz + P, (0)<

approxi-

V.M. Mos lennihova

632

F. A,,-~ +

eMz +

nia Q,L,._,-l] + dz

r==o

+

SMeNzQn-l

(4 dz + PO(0) <

0

Q PO(0) (earl- i) + PoiM[eMz

-yq]dz

0 +

Mng2

qr i A,,-r_l

f=o

dz +

+

s=o

&+ kMt - 1 I -I- PO (0) =

0

= PO (0)eM’ +

This proves

PO[eNt

formula

Using the estimate show that

-

1-

Mt -

-.- -

GJt

(3.12). of the remainder terms of a IWlor

series

we can

r=l

Moreover,

since

Then, since

A, - Oasn-m,

?;o is bounded,

0, is bounded for all

P, (t) - 1w (51, 0 -w

pOAn - as n - Q).

r up to 0~we have

(z2, 91
But

satisfy the Holder condition with index a; therefore fies the HUlder condition with index a.

w(x,

t) also

satis-

Thus we have proved that u(x, t), w(x, t) satisfy the H6lder condition w. r. t. (x, t) with index a, 0 < a < 1, in $. Obviously aw/at also satisfies the Holder condition with index a. It remains to show that {a(~, t), lem (l.l), (1.3), (1.4) and satisfies Theorem 4. Let us consider

the solution

w(x, all

t)) is a solution of the probformulated in the conditions

u*, IO*of the following

system:

First

boundary-value

prob lea

G +

Mw-

certain

for

= g (x,

quasi-

linear

ryrtear

633

t, zt, w) -I- Mw

with the boundary conditions

u*jt+D=

lo*It-0 = wo (2).

uo(G 0,

In the right-hand side of (3.13) we substitute the limit u(x, t) Of the subsequence {u,*) selected above, and also the limit of 0,. It is easily calculated, using condition II, that the right-hand side of the first eQuation of (3.13) satisfies the Hiilder condition with index A, where A = min (a, FE,). Therefore this equation has the solution u* (% t) E C** (0 < y
For, writing down system (3.3) for the n’-th term of this subsequence and subtracting it term by term from system (3.13) we obtain n

C

9 (d afjtx*

t,

un,)

6Z,i3Zj

8 (lb’ -

i.hl

at

Ian,

)

-

M (22 - r&e) =

M (u - r.+&

(u* -

t&f) = 0

on

Using the estimates for a linear parabolic \n* -

z @ (x, f),

a. equation we shall have

W I q c ) Q 1;

but I@1 - 0 as n’ - CQdue to the convergence of the subsequence {uns) to the function u(x, 2) ia the metric of C’+61 for any 6,, 0 < 6, < 6 < 1, where 6 is an arbitrary number. Therefore we have u*(z, solution of our problem.

t) = u(x, t) where u(n, t) is the required

V.M. Afor Icnnikovo

034

Similarly

we can conclude

that w* = w In ET.

Thus, Theorem 4 is proved.

by R. Feinsteln

Trons lotcd

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yadernykh reaktorov Olesston. S. and Edlund, 1.. Osnovy teorii (Essen’tials of the Theory of Nuclear Reactors). Izd-vo in. lit. Yoscow, 1954.

2.

McNabb, A., J.

3.

Friedmen,

4.

~smynin. L. I. and Yaslennikove. No. 5, 1049-1052. 1961.

V.N.,

Dokl.

5.

Ramjnin, L.I.

V.N..

Matem. sb.,

241-264,

A.,

Moth. J.

Moth.

Anal.

and Appl.,

ond Mech.,

end Yaslennlkovs,

1962.

7,

3, No.

No.

1, 133-144,

5, 771-791, A&d.

1961.

1958.

Nauk WR,

57(99)*

137,

No- 2,