Solution by the straight-line method of a quasi-linear two-phase problem of the Stefan type with weak constraints on the input data of the problem

SOLUTION BY THE STRAIGHT-LINE METHOD OF A QUASILINEAR TWO-PHASE PROBLEM OF THE STEFAN TYPE WITH WEAK CONSTRAINTSON THE INPUT D.4TA OF THE PROBLEM* R. D. BACHELIS

and V. G. MELAMED Moscow

(1) “Solution of a quasilinear problem papers: and (2) “Solution of a quasilinear method”,

(THE complete texts of the author’s of the Stefan type by the straight-line

problem of the Stefan type in the case of an initial function which is non-differential and inconsistent with the boundary conditions”, have been deposited in VINITI: (1) No. 3941-72 Dep. 20~; (2) No. 3942-72 Dep., 15p.) The existence it is required the conditions

of a solution

of the following

y(Y

is proved:

to find U(X, t) on (0 < I < I, 0 < t < T),

(A(x, u)u,), - c(x, u) Ut = 0, u(y(t),

problem

0

=

(t) j!/‘(t)

0, =

t =

Y (0)

10, Tl,

k~s=y~~~--o

t E (0, Tl,

-

=

Yn E

huxi.,=8(f!+0,

on

[O, ~1 from

x E (0, v(t)) u (2 (L), I), (0, l),

10, Tl, kIx=o = ‘I,(4 u(0, t)), hUxlx=l= !?2(4 u(I, t)), u(5, 0) = (r;(J), 5 E K44,

(A)

y(t)

t E

t E P, 77,

where h, C, qi, i = 1, 2; y, c$ are given functions of their arguments. The existence of a solution of (A) is proved for a !I’* > 0 satisfying the inequality T*max(

max h(s,cp)Jf,

[

rnin y(z)]-’ \E[o II

where M is a Lipschitz

max k(z,q)M,

max Qi(t,O), !Elo.r*l

Ic,lO.Y,l

xauo.~l

max Q~(r,o)l :I: G[O.T’l

< min (2 - y0, Yo),

constant

for c5(x).

Also in the first paper the uniqueness of the solution found is proved. By the scheme of Rote, we specify an h > 0 and put N = [T’IL-I]. L,, = nh, n = 0, 1, 2, . . . , N. The system (A) is replaced by the system of ordinary differential

*Zh.

v>hisl.

Mat. mat. Fiz.,

12, 3, 828-829,

342

1972.

Solution

equations

(B) for Us,

. . . , N.

The following

by the straight-line

y, on the straight theorem

343

method

lines

(.t, t,,), 0 < I < I, ?z = I, 2,

is also proved.

Theorem 1 Let (I):

y0 E (0, l),

4 (x) be differentiable

in the closed

intervals

[O, ~01,

[go, 11, Cp(s) < O* z E [O, YOI?rPCs)2 Ot s E [YOt‘1, cf(YO)= O; (2) Qi9 q1 and q2 are non-decreasing and non-increasing

(t, u), i = 1, 2, be continuous,

functions of u, respectively, q?(t, 0) 2 0, 0 < t G T’; (3) A, C, A,, ,A, are continuous with respect to all the arguments for u # 0, where A, AZ, $, satisfy a Lipschitz condition on u for u < 0 and u > 0, and as u + 0 f 0 the continuous limits limh(s, u), limc(2, U), lim h,, lim i.,, hl > h(z, u) > h? > 0, cl

go

~(5,

U)

2

cp >

Y E [0, T']

every

and is unique.

o,

Y(S)

exist.

Then for

an h E (0,y] such that the solution

(B) exists

a~,

there exists

>

o;

at,

az,

cl,

Q,

yl

=

const

Here the y, E (0, I), Ii, (2) < 0, z E [O, Y”), u,(z)

> 0, x E (4,

II

IU”(Z) 1, Iu,,‘(s) 1, 1(y, - ynei) /h 1 are kmnded by constants.

and

Theorem 2 Let all the conditions of Theorem 1 be satisfied and in addition h(x, (I) E ~(t, u) E C(434),q(z) E P.1 on IO, ~01U [YO, 11, y(x) be continuous, the a Lipschitz condition on t. functions qi(t, n) E C(‘,‘), i = 1, 2, and satisfy P5),

Finally, q'(Z) =

let the matching qz(l,

and is unique

cp(!))

conditions

be satisfied.

in the class

h(O, CF(0) )#(O)

Then u(x, t), y(t),

of sufficiently

a(k CP(I)) of (A), exists

= qi (0, CP(011, the solution

smooth solutions.

In the second paper Theorem 1 is generalized schitz condition on [O, Z]. The following theorem

for a d(x) satisfying is also proved.

a Lip-

Theorem Let all the conditions qi(t,

u), i = 1, 2, satisfy

of Theorem a Lipschitz

1 be satisfied. condition

In addition,

on t and u; (2)

a Lipschitz u P 0 there exist h,,, h,,, ;I,,,, c+, c,,, satisfying arguments. Then a solution of (A) exists for t E [0, T’], Therefore, a proof of the existence of a solution does not require matching of the boundary and initial

Note. the case

let:

(1)

for

z E [O, I], condition on both

(A) has been given which conditions.

T(Z) > 0, 5 E [0, yO], cp(~) G 0, z E [yo, I] is reduced The case considered above by replacing x by I - x.

Translated by J. Berry

to