non-linear parabolic problem associated with the penetration of a magnetic field into a substance

Mathematical Methods in the Applied Sciences, Vol. 16, 281-295 (1993) MOS subject classification: 35 M. 35 Q, 65 N

Non-linear Parabolic Problem Associated with the Penetration of a Magnetic Field into a Substance

Nguyen Thanh Long

Ecole Polytechnique. Dkpartement de MalhCmaliques, Ho-Ch-Minh Ville, Vietnam

and

Pham Ngoc Dinh Alain

Dkpartement de Mathkmatiques. Universitk d’OrlCans-BP 6759, 45067-Orleans Cedex. France

Communicated by W. Tornig

We study the following initial and boundary value problem:

u, - V * ( a ( j: (Vu1”dr)Vu) +f(u) = 0.

u = 0 on aQ U(X, 0) = u,,(x).

In section 1. with ug in L2(12), fcontinuous such thatf(u) + EU non-decreasing for E positive, we prove the existence of a unique solution on (0, T), for each T > 0. in section 2 it is proved that the unique solution u belongs to Lz(O, T; HA n H’) n L“(0, T; HA) if we assume uo in HA andf in C ’ ( R . W). Numerical results are given for these two cases.

0. Introduction

In this paper, we study the following initial and boundary value problem:

u, + A(u) + j ( u ) = 0, (X, t )ER x (0, T ) , (0.1)

u = 0 on an x (0, T ) , (0.2)

4x9 0) = uo(x), (0.3) where

01 70-42 1 4/93/0428 1 - 15s 1 2.50 Q 1993 by B. G. Teubner Stuttgart-John Wiley & Sons, Ltd.

Received 29 Augusl 1991 Revised 5 February 5 1992

282 N. T. Long and P. N. D. Alain

R being a bounded domain in RN with a sufficiently regular boundary XI . The assumptions on the functions a and f needed for our purpose will be specified later. This problem was considered by Laptev [2] with f = 0; its physical interpretation is that of the penetration of a magnetic fied into a substance. To describe the phenomena that are produced in massive conductors placed in an externally varying magnetic field H, in the case where the temperature f3 rises and has an effect on the resistivity p(0) , the following system of equations was first proposed in [l]:

aH/at = - (c2/47t) rot[p(8) rot HI,

c,(e)ae/at = p(e) 1(c/4n)rot HI^, div H = 0, (0.5)

(0.6) where c is a constant and C,(f3) is the specific heat capacity of the substance, which, in general, also depends on temperature.

Laptev transforms (0.5) and (0.6) into one equation by introducing the function s(f3):

Takiqg into account that the function s(0) is monotonically increasing, system (0.5) and (0.6) can be rewritten as

div w = 0, (0.9) where a(s) = (c2/4n) p[&s)], w = ( c / ~ A ) H and #(s) is the inverse of the function s(0). Assuming that the field w has the form

w = (O,O, 4, I4 = 4% y, 0, (0.10)

equation (0.8) takes the form (0.1) in two dimensions while the second equation of (0.9) is automatically satisfied. If the function p ( 0 ) is uniformly bounded, then a(s) possesses an analogous property. This phenomenon is observed in both semiconductors and plasmas.

In [2] Laptev establishes existence and uniqueness theorems for equation (0.1) with f = 0, uo in HA (Q), and a and a’ continuous under the conditions

(0.12)

The paper consists of three sections. In section 1, under conditions (0.5) and (0.6) with uo in L2(R),fcontinuous,j(0) = O,f(u) + E u non-decreasing for E > 0 ‘small’ and taking bounded sets of L2(Q) into bounded sets of L2(R), we prove the existence of a unique solution on (0, T), for each T > 0. We also prove an exponential decay of the solution to zero as t tends to infinity.

In section 2 we strengthen the hypotheses and assume, as in [2], uo in Hl(Q); then with f in %“(R, R), J(0) = 0, f’ 2 E, E > 0 and, under a certain local Lipschitzian condition onf , an existence and uniqueness theorem is proved on (0, T), for each T > 0, generalizing the results of Laptev.

In section 3 we present the numerical results.

Non-linear Parabolic Problem 283

1. Solution of the problem

Let

L2 = L2(Cl), HA = HA@), H z = H2(Q).

Here HA@) and H 2 ( Q ) denote the usual Sobolev spaces on R. Let (.;) denote either the L2 inner product or the pairing of a continuous linear functional with an element of a function space. Let 11 * \ I x be a norm on a Banach space X and let X * be its dual. We denote by Lp(O, r X), 1 < p d co , the space of functions f o n (0, T ) to X such that

IlfIILP,O.T.XI = (s,' l l f~f) l l~df) l 'p < 00 for 1 d P < 00.

Ilf IIL",O. T : X ) = ess SUP(0. T) Ilf(t) Ilx

We make the following assumptions:

Al : uo in L2. A2: a in %" (R, , R) satisfies the following conditions:

for p = co.

We denote the norm in LZ(Cl) by I1 * 11.

(i) b = (1; slaf(s)i2ds)li2 < + 00, (ii) there exist ao, a , > 0 such that

26 < < U ( S ) < (11 V, 2 0.

A3: f i n %(R, R) satisfies the following conditions:

(9 f ( 0 ) = 0, (ii) f(u) + EU is non-decreasing, with E such that

0 < E < (a0 - 2 b ) / C i ,

where Co > 0 such that

l l u l l ~ 2 d Co IIVuIILz VUEHA.

(iii) f(u) = f o u, for u in L2, takes bounded sets into bounded sets.

We then have the following theorem.

Theorem 1. Let assumptions A l , A2 and A3 hold. Then the problem dejned by (0.1)-(0.3) has a unique solution in L"(0, T; Lz) n L2(0, Z HA), for each T > 0.

The proof of Theorem 1 is a combination of compactness and monotonicity arguments and consists of several steps.

First we shall require the following two Lemmas:

Lemma 1. Let A be the operator dejned by (0.4). Then we haue the following inequality:

(ao - 2b) joT I1 Vu - Vu 11' dt < ( A u - Au, u - u ) dt

T loT

G (al + 2 b ) j 0 IIVu - Vull'dt, (1.1)

Vu, U E L2(0, T; HA), for each T > 0.


Proof. Let

Using the formula

F(u) - F(u) = d/do(F(u + Ow))do, w = u - u sd

s:

and assumption A2 we obtain

J = j: a(s) Vw do + a’(s) (ds/da) * (VV + oVw) do, lo1 where

q = S(X, t, o) = IVU + oVwlZ d7.

Consider jOr ( J , w)dt. By (1.3),

IoT (5, Vw)dt = IOT 1’ a(~)IVw1~dadxdt + K,

K = IoT jn l: a+) (dslda) - (VV + OVW) . Vw do dx dt.

n o

where

We claim that

1K1 < 2b 11 VW l l i 2 (p), with Q = R x (0, T ) .

We have

I K I 6 I I v W I I L ~ ( Q ) Jb’ llJi 112L2(Q)dgr

where

IJJl ( I i 2 ( Q ) = Jn IoT la’(s)12 I(ds/do)12 IVV + oVw12dxdt.

Inserting (1.4) into (1.9) we obtain, after inverting variables z and t.

l l J l 1 1 2 2 ( Q , 641ndx]or IVwl2dz[ s(x.r,d S(X, T,a)la’(rl12rldtl

6 4 jn dx loT IW2 ds [y la’(rl)12 rl drl,

from which (1.7) follows. Consequently, from (1.7) and assumption A2 we have

a0 IlVwl12z(Q) - l K I ( J , Vw)dt a1 llvw IIL2(Q) + IKl,

(1.10)

(1.11)


which implies inequality (1.1) since

( A u - Av, u - v)dt = (J,Vw)dt, w = u - ti jOT

vu, V E L2(0, T; HA). j O T

Lemma 2. For each u, w E Lz(O, T; HA) we have the following:

lim joT ( A ( u + Aw), w) dt = ( loT Au, w) dt. ,Id m

and

(1.12)

there exists a sequence (A,}, A, 2 0, lim A, = 0, such that

lim joT (f(u + Amw), w) dt =

r n d + m

(f(u), w ) dt. (1.13) m-tm !OT

Proof: Equality (1.12) is immediate, thanks to Lemma 1. Let {A,}, 0 < 1, < 1, be a sequence such that lim A, = 0. By using the continuity of

I, we can extract from the sequence {A,} a subsequence, still denoted by {A,,,} such that

(1.14) f(u + A, w ) +f(u) a.e. in Q, for each u, w E L2(0, T; HA). Let g,(u) =f(u) + EU, E > 0. Since g,(u) is non-decreasing, we have

Igc(u + Amw)l g,(lul + Iwl) - gc( - IIO - Iwl). (1.15)

Therefore,

If(u + Amw)l G f(lul + Iwl) - f( - Iu1 - Iwl) + 3&(lul + Iwl) = F(lul + Iwl). (1.16)

Since u, w E L2 (0, T; HA), it follows from (1.16) that

Ilf(u -k 1mW)IIL2(Q) G 11 F(Iul + IwI)llL2(Q) = c1, (1.17)

where C1 is a constant independent of m. Equations (1.14) and (1.17) enables us to apply Lemma 3 (see section 2), which shows that

f(u + A, w) +f(u) in L2(Q) weakly for m + 00 .

We turn now to the Proof of Theorem 1.

Proofof Theorem 1. Consider a special basis of H;, w l , w2, . . . , w,, . . . , formed by the eigenfunctions of the Laplacian A. Let (wl, w2, . . . , w,) be the linear space generated by w l , . . . , w,.

QED

n

U y t ) = c C j , ( t ) wj j= 1

be a solution of the following system:

(1.18)

(u?', wj) + (a( f: IVu(n)/2dr) VU("), Vwj) + (f(u'")) wj) = 0

f o r j = 1,2,. . . ,n, (1.19)

u(")(O) = uOn + uo in L2 strongly. ( 1.20)


Note that such a solution clearly exists on a sufficiently small interval [0, T,,]. The a priori estimates which follow allow us to take T,, equal to T.

1.1. A priori estimates

j = 1, 2, . . . , n and integrating with respect to the time variable, we have 1. Multiplying thejth equation of system (1.19) by cjn(t), adding these equations for

+ j: (f(u(")), u(") ) ds = f II uOn II '.

Assumptions A2(i) and A3(i), (ii) imply that

f (I d")( t ) ( 1 + a. 1: ( 1 VU(") I( ds

,< 1 1 ~ 0 ~ [ I 2 + (00 - 26) 11 Vd") 11 ds. 1: Finally, from (1.22) we obtain the first a priori estimate

I I ~ ( " ) ( t ) 1 1 ~ + 46 (IVU(")I12ds G C2, s: where C2 is a constant independent of n.

2. Let A be the operator defined by (0.4),

A:L2(0, T; H i ) -+ L2(0, T; H - I ) ,

and satisfying

(Au,u)dt = I O T ( a ( J:lBu12d~)Vu,Vv)dt

for dl u, u in ~' (0, T; HA). loT

From (1.24) and assumption A2 we easily deduce that

(1.21)

(1.22)

(1.23)

( 1.24)

G a1 IIU IIL'(0,T:H;) V u E L2 (0, T; HA). (1.25)

Combining (1.23) and (1.25) we obtain

11 Ad") I ~ L Z ( O , T ; H ; ) ai J(Cd46)

3. By assumption A3 (iii) and (1.23) it follows that

( 1.26)

ll.w"))Il G c3, (1.27)

By (1.23), (1.26) and (1.27), we can extract a subsequence of {u(n)} , still denoted by

u(") -+ u in Lm(O, T; L 2 ) weakly*, (1.28)

where C3 is a constant independent of n.

{u(n)}, such that


u'") -+ u in L2(0, T; H A ) weakly*, (1.29)

Ad") -+ x in L2 (0, T; H - I ) weakly,

f(u'")) -+ x1 in L"(0, T; L2) weakly*,

u(")(T) -+ 5 in L2 weakly.

(1.30)

(1.31)

(1.32)

1.2. Limiting process

integrating with respect to the time variable, Passing to the limit in (1.19), we obtain, after multiplying by q ~ . 9 ( [ 0 , T I ) and

( 5 , W j > V ( T ) - (UO, wj> d o ) - PT PT

by virtue of (1.24). Equation (1.33) implies that

u, + x + x1 = 0 in L2 (0, T; H - I).

(1.33)

(1.34)

Taking the inner product of (1.33) with q ~ . 9 ( [ O , T I ) , we get

u(T)cp(T) - u(0) q(0) - (x + Xl)qdt = 0. (1.35)

It follows from (1.33) and (1.35) that

(1.36)

(1.37)

Let

x', = loT (A;(") - Ju, u(") - u ) dt for all u in L2(0, T; HA), (1.38)

where

i u = Au + f(u). (1.39)

f ( u ) + EU being a non-decreasing map, it follows that

From (1.40) and assumption A3 (ii) we deduce that

The relation

(1.41)

( 1.42)


implies, by passing to the limit

= joT (x + x I , u)dt.

Then, by virtue of (1.43), (1.29) and (1.31) we obtain

(1.43)

(x + x1 - z u , u - u ) dt for all u in L2(0, HA). (1.44)

In (1.44), if we choose u = u - A,,, w, A,,, being the sequence defined in Lemma 2, and use Lemma 2, we conclude that

n. -a ,

fT J (x + x1 - Au - f (u) , w ) d t > O for all w in ~ ~ ( 0 , T; HA), 0

( 1.45)

i.e.

1.3. Proof of uniqueness

w = u - u is a weak solution of the problem Let u and u be two weak solutions of the problem defined by (0.1)-(0.3). Then

( 1.46) w, + z u - z u = 0,

w = 0 on aRx(0, T ) , (1.47)

w(0) = 0. (1.48)

Taking the inner product of (1.46) by w and integrating with respect to the time variable, we get

P T

(1.49)

which implies that w = 0, by virtue of (1.41).

Now we consider asymptotic behavior for t tending to infinity. More precisely, we have the following proposition.

Proposition 1. Under the hypotheses ofthe Theorem 1, ifu(t) is the unique solution ofthe problem defned by (0.1)-(0.3), then we haue

I1 u(t) 11 G 11 uo 11 e-kl jor each t > 0, where

k = (ao/C;) - E > 0.

Pro05 From (1.19) we deduce that


2. Existence and uniqueness theorem

In this section we strengthen the hypotheses and assume, as in [ l ] , that

B1 u o € H ; . B2 a in %'(R+, R) satisfies

(i) b = (s(a'(s)("s)li2 -= + a, (ii) there exist a,, a, > 0 such that

j: 2b max(1, c) < a, d a(s) d at Vs 2 0,

where c is a positive constant satisfying

la2u/axiaxj12dx < c2 jlAu(12 VUE Hh n H 2 , s" fl i . j = 1

while f satisfies the following condition:

B3 (i) f E%"(R, R),f(O) = 0,f' 2 - E, E > 0 not necessarily small, (ii) for each bounded subset B of HA, there exists K B > 0 such that

M Y ) - f(z)II d KBII VY - VZII V Y , z E B.

Theorem 2. Let assumptions Bl-B3 hold; then the problem dejned by (0.1)-(0.3) has exactly one solution, u EL'(O, T ; HA n H 2 ) n L"(0, T ; HA) such that u, E L2(Q), Q = x (0, T ) f o r each T > 0.

ProoJ: Proceeding as in the proof of Theorem 1 , we consider equation (1.21) and get, after using assumptions B2(i) and B3(i),

11 u(")(c) /I + 2ao joT 11 VU(") (s) /I ds d C, + 2~ 11 u(")(s) I( ds, joT (2.2)

where C4 is a constant independent of n and t .

form a bounded set in the space It follows from (2.2) and Gronwall's lemma that the elements of the sequence {u"(t ) ]

(2.3) ~"(0, T; L2) n ~ ~ ( 0 , T; HA).

290 N. T. Lung and P. N. D. Alain

2. Let Q1 = R x(0, Tl) with 0 < TI < T. Using the fact that the wj(x) are the eigenfunctions of the Laplacian A, equation (1.19) can be rewritten as

(~f", - Awj) + ( A d " ) , - Awj) + ( f ( d " ' ) , - Awj) = 0

f o r j = 1 , 2 , . . , n . (2.4)

Multiplying each equation in (2.4) by cj,,(t) and summing up with respect to j, we have, after integration from 0 to T1,

loT' (up' - Ad"') + (Ad") , - Ad")) + (f(u'"I), - Au("') = 0. (2.5) s:' l' We have

N AU(") = - C @/axi) [ u ( ~ ( n ) ) (aP/axi ) ] = - ~ d " ) - u ' ( s ( " ) ) ( ~ s ( n ) . vd"))

(2.6) i s 1

and (f(~'"'), - Ad"') = (f'(~'"') VU'"), VU'"')

since u'") = 0 on aR x (0, T). Hence, by (2.6) and (2.7) we get from (2.5)

(2L)IIV~'")(Tl)1I2 + jar' jn u ( ~ ( ~ ) ) l A d ~ ) l ~ dxdt + lor' jn f ' (un) IVu(")12 dx dt

= lor' (P,, Ad") dx dt + f (IVuOn (I2, (2.8)

where

and

pn(x, t ) = - a'(.$'"') Vs("' * V P .

)I q n IIL+Q,, G 2hc IIAu'") IILz(Q,,.

(2.10)

Define R ' c R such that iz' is compact and let Q2 = R' x (0, Tl). We claim that

(2.1 1)

Indeed, from (2.10) we have

From (2.9) we obtain

I V S ( " ) I ~ G 4s(") 1; ( la2u(")/axi i)xjI2 d7. i. j = I )

Combining (2.12) and (2.13) and after inverting variables 7 and t, we get

(2.12)

(2.13)


G 4b2 loT1 ( l a w / a x , axjt2 d.x dr, (2.14)

with sl = s(")(x, t) and s2 = s(")(x, T I ) . Inequality (2.14) implies (2.1 l), c being the constant defined by (2.1).

i, j = 1 )

joT1 From (2.8), (2.11) and assumptions B1 and B3(i) we deduce that

I( Vu(")( TI) 11 + 2(ao - 2bc) IIAu(") 11' dr

< C5 + 2~ 11 Vu(") 11' dr for each TI, 0 G T1 < T, (2.15) s:' where C5 is a constant independent of n and t . Inequality (2.15) implies that

II Au'") )I Lz(Q) + I\ VU'") 11 < CT, Q = SZ x (0, T ) , (2.16)

CT indicating a constant depending on T. Since we have

l IAu(") l lLz(Q) < ( a l + 2bc)llAu(")llLz(Q),

Ad") is bounded in LZ(Q).

Taking pz = CT implies, with assumption B3(ii), that

II .f(u'"') II L2 G P * K,

since JJW") 1) < p . The last a priori estimate follows from the inequality

(2.17)

(2.18)

IIUI") IlL'(Q) 11 llL'(Q) + llL'(Q) d c T , (2.19)

Consequently, by (2.16) and (2.19), we can extract from {u(")} a subsequence, still C7. always indicating a constant depending on T.

denoted by { u(")}, such that

u(") + u in L"(0, T; HA) weakly*,

u(") + u in L2(0, T; HA n H,) weakly*,

(2.20)

(2.2 1 )

u?) + u, in L2(Q) weakly*. (2.22)

Using a lemma on compactness [2] applied to (2.21) and (2.22, we deduce that

the set {u (" ) } contains a subsequence, still denoted by {u'")}, strongly convergent to u in L2(0, T; HA). (2.23)

We shall now require the following lemma [2].

Lemma 3. Let 0 be an open bounded set c RN and g,,, g E Lq(0) (1 < 4 < co ) such chat 11 g, IILq(B) < C, where C is a constant independent of p, and g,, -+ g a.e. in 0. Then, g,, + g in Lq(0) weakly.

By the Riesz-Fischer theorem, we can extract from the sequence {u(" ) } a subsequence {u(")}, such that

dn) + u a.e. in Q = 0 x (0, T ) . (2.24)


I t follows from (2.18) and Lemma 3 applied to the functionfthat

f(u'")) + f(u) in L2(Q) weakly. (2.25)

We note from the strong convergence of the sequence {d")} to u in L2(0, T: HA) that rr PI

J (Vu'")l2 dr -, J 1Vu12 dr in L'(Q) strongly 0 0

since we have

< T 1) V I P ) + VU IILz(Q) * ( 1 VU'") - VU IILz(Q).

By assumption B2, (2.23) and (2.26) we deduce that

a ( 1: [Vu(")I2 dr) Vu'") + a( 1: IVuI2 dr) * Vu a.e. in Q.

On the other hand, we have

Taking (2.27) and (2.28) into account, and by virtue of Lemma 3, we have

(2.26)

(2.27)

(2.28)

(2.29)

Integrating (1.19) on (0, T) and using (2.22), (2.25) and (2.29), we finally obtain

jOT ( u l , u ) dr + (a@) Vu, V v ) dr + ( f ( u ) , v) dt = 0 VUEL'(O, T ; Hh),

(2.30)

u, + Au + f ( u ) = 0 in L2(Q). (2.3 1)

JOT ]Or

i.e.

Let w ~ L ~ ( i z ) . Since u(") and u are in W(0, T: L2), we have

loT (@), ( T - t) w) dr = - (uOn, w) T + (d"), w ) dr, (2.32) I: which implies, passing to the limit for n + co ,

f T PT

J ( u ~ , ( T - t ) W ) dt = - ( ~ 0 , W ) T + J (u, W ) dt, 0 0

from which the initial condition

u(0) = uo

follows since

loT ( t i l , ( T - t) w) dt = - T(u(O), W ) + (u , W) dt. 10T

QED


2.1. Uniqueness of the solution

of the problem defined by (0.1)-(0.3) since Proceeding as in the proof of Theorem 1, we deduce the uniqueness of the solution

II w( Tl [ I 2 + 2(ao - 2b) loT’ I( V w (1 dt G 2.5 loT’ 11 w 11 dt,

W = U - V , O < T , < T .

3. Numerical results

Consider the problem defined by

U, - a / a ~ [ a ( J : u:(x, r)dr)uX] + f ( u ) = F(x, t), 0 < x < 1, (3.1)

u(0, t ) = u(1, t ) = 0,

u(x , 0 ) = sin nx, (3.2)

(3.3) where a(s) = (s + 2)/(s + 1) and F(x, t ) €L2(Q), Q = (0, T ) x (0, 1).

We have

b = (1; sIat(s)”dg)lil = 1/J6

and

1 < a(s) < 2.

In (3.1) the functionf(u) is taken such that assumptions Al-A3 and Bl-B3 are fulfilled.

1 . We choose in equation (3.1)

F ( x , t ) = e-‘sin nx [(27r2 - 1 ) + (n2 - 2 ) cp + (n2 - 1) c p 2 ] / ( 1 + ( P ) ~

+ (e- sin K X ) ” ~ + 1/3 (e-lsin nx), (3.4) where

cp = ( 1 - e-”) (n2 cos’ nx)/2,

f(u) = u1I4 sgnu + EU, E = 1/3. (3.5) As a matter of fact, in assumption A3 we have a. = 1, b = 1/,/6 and Co = 1J2, which implies that we must choose E in the interval (0,0.368). The exact solution of the problem defined by (3.1)-(3.3), with F ( x , t) andf(u) defined in (3.4) and (3.9, respec- tively, is u(x, t) = e-‘ sin I I X . To solve numerically this problem we consider the non-linear differential system for the unknowns uk(t) = u(&, t ) , xk = kh, h = 1/N:

duk/dt - ( g ( x k , t) + 2) / (g(xk , t ) + l ) Uxx(Xk,t)

+ ( l / (g(Xk, t ) + (%/ax) (xk, t ) U x ( X k r t ) + f ( U k ) = F ( x k , t),

uk(0) = sinnkh, k = 1 , . . . , N - 1, (3.6)


0.6

0.5

0.4

0.3

0.2

0. L

0.

0. 0.2 0.4 0.6 0.8 1.0

T = I / 2 __ hPPROX.SOL - E Y AC. SOL

Fig. 1

0.0 .

0. 0.2 0.4 0.6 0.8 1.0

(3.7)

X

X


We start with the initial conditions

to = 0, a k . 0 = sinakh, h = 1/N, (3.8) U x ( X k , 0) = XCOS nkh, U , , ( X k , 0) = - 7c2 Sin nkh

fork = 1 , 2 , . . . , N - 1. (3.9)

The linear differential system (3.7)-(3.9) is solved by searching its eigenvalues and eigenfunctions. For a step h = 1/11, we obtain the curves in Fig. 1 for the approximate solution U k ( t ) and the exact solution s o l k ( t ) , k = 0, 1,. . . , 11 and for the time T = SO/lOO. Always with a step h = 1/11 and for times T = 1/50, 1/10, 1, 2, 3, we obtain the curves in Fig. 2 for the approximate solution u k ( t ) , k = 0, 1, . . . , 11, which show that the numerial solutions tend rapidly to zero when the time T is ‘great’ enough.

2. The functionf(u) = u3 satisfies the assumption B3 of Section 2. We take

F(x, t ) = e-‘sin ax [2n2 - 1 + (aZ - 1) cp + (n2 - t)cp2]/(1 + cp)’

+ (e-’sin

where cp = (1 - e-”) a2 cos2 ax/2 . The exact solution of the problem defined by (3.1)-(3.3), with F(x, t) in the form

(3.10) andf(u) = u3, is u(x, t ) = e-lsin ax. We solve in the same way as previously the linear differential system associated with the non-iinear differential system and obtain similar curves as under 1 above.

Acknowledgement

The authors wish to thank Professor W. Tomig for his constructive and useful remarks.

References

1. Gordeziani, D. G., Dzhangveladze, T. A. and Korshiya, T. K., ‘On the existence and uniqueness of the

2. Laptev, G. I., ‘Mathematical singularities of a problem on the penetration of a magnetic field into

3. Lions, J. L., ‘Quelques methodes de resolution des problemes aux limites non-lineaires’, Dunod,

solution of a class of non-linear parabolic problems, Difl Uraoan., 19, 1197-1207 (1983).

a substance’, Zh. Vychisl. Mac. i Mac. Fiz., 28, 1332-1345 (1988) (in Russian).

Gauthier- Wars , Paris, 1969.

non-linear parabolic problem associated with the penetration of a magnetic field into a substance

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