On estimates of solutions of the split problems for some multi-dimensional partial differential equations Текст научной статьи по специальности «Математика»

УДК 517.9

On Estimates of Solutions of the Split Problems for Some Multi-Dimensional Partial Differential Equations

Yuri Ya.Belov*

Institute of Mathematics, Siberian Federal University, Svobodny 79, Krasnoyarsk, 660041

Russia

Received 10.05.2009, received in revised form 10.06.2009, accepted 20.06.2009 We consider multidimensional second order parabolic equations and the first order partial differential equations. We consider various splittings of the Cauchy problem in the case when the coefficients of the equation depending on time and all space variables and have a special form. The uniform correctness of the split problems, that is, a sufficient condition for the split problems solutions convergence to the solutions to the original problems and the uniform correctness of this problem is proved in classes of smooth functions.

Keywords: differential equation, Cauchy problem, split, stability, convergence.

The question of convergence of the weak approximation method — WAM (a method of fractional steps in the differential form) where the problem of the split problem solution convergence to the solution to the original problem with the Cauchy data is investigated in [1—3]. The uniform correctness of the split problem and its differential extensions is supposed in the specified works.

We consider multi-dimensional second order parabolic equations and the first order partial differential equations. We consider various splittings of the Cauchy problem in the case when the coefficients of the equation depend on time and all space variables and have a special form. The uniform correctness of the split problems, that is, a sufficient condition for the split problems solutions convergence to the solution to the original problem and the uniform correctness of this problem is proved in classes of smooth functions under sufficient smoothness condition of the initial data. The uniform correctness of the original problem is not supposed.

The obtained results can be applied to nonlinear equations and systems by the splitting method on the linear equations of the above mentioned type and equations containing nonlinear members [4-6].

For splitting of various equations and systems of the equations of mathematical physics at a differential level see [7-9]. The convergence of WAM for the differential evolutionary equations in Banach spaces is treated in [7].

The multi-dimensional problems of the coefficients identification of the second order partial differential equations in a case when known leading coefficients depend only on the time variable are investigated in [4-6]. This is connected by application of WAM for the proof of solvability of input problems and the requirement in doing so stability of the split problems in some functional classes.

We shall prove exponential stability of the split problems for the multi-dimensional equations that allows to investigate, in particular, problems of identification of unknown coefficients and in

* e-mail adress: belov@lan.krasu.ru

© Siberian Federal University. All rights reserved

a case when known coefficients depend on all independent variables and the differential operator has a special structure.

We consider in n^T] = {(t, x)| 0 ^ t ^ T, x £ En}, where En is the n-dimensional Euclidean space, n ^ 1, the Cauchy problem for the equations

n

Ut = bi(t, x)uIs (t, x),

Ut

ai(t,Xi)MxiXi(t, x),

ut ^^ ^¿(t, xi)uXiXi(t, x) + bi(t, x)uXi(t, x) + c(t, x)u(t, x) + f (t, x),

i=i ¿=1

and we investigate the stability of the corresponding splittings of these problems.

1. Equation of the First Order 1.1. Formulation of Theorems 1.1, 1.2

Consider in n^T] = {(t,x)|0 ^ t ^ T, x £ En}, the problem

du(t, x)

E . , (

Oi(t,x)-

dxi

u(0,x) = uo(x), x £ En.

Here, bi(t, x) and uo(x) are functions given in n^T] and En, respectively. We split the problem (1.1.1), (1.1.2) into one-dimensional problems [3, 7]

uT = nbiuX , kr < t ^ (k + -1-)T,

< = nb2ux2, (k + n)t < t <"(k + n)t

= nbi

(k + )t < t < (k + n)T,

uT = n6„uXn , (k + )t < t < (k + 1)t,

(1.1.31)

(1.1.32)

(1.1.3i) (1.1.3n)

uT(0, x) = uo(x), x £ En,

T

where t = ^, k = 0,1,..., N — 1, N is an integer. We take the initial data uT|t=(k, i-1

(1.1.1) (1.1.2)

(1.1.3)

(1.1.4)

fractional step:

uT It

)T on the fractional step i of the step k from the previous

uT (t, x), i = 1,...,n. (1.1.5)

lim

-1 ^

t<(k+ )t

Assume that the input data of problems (1.1.1), (1.1.2) satisfy

|Dabi(t,x)| < M (p), |a| < p, i = 1, ...,n; |Dauo(x)| < C(p), |a| < p,

(1.1.6) (1.1.7)

n

T

T

u

u

t

in n[o,T] and En, respectively. All derivatives in (1.1.6), (1.1.7) are continuous. In (1.1.6), (1.1.7)

d|a| n

a = (ai,..., an) is a multi-index, Da = —a- a , |a| = a®, p > 1 is an integer.

dx i . . . dXn ®— i

The following theorems are valid.

Theorem 1.1. Lei the conditions (1.1.6), (1.1.7) hold. Then problem (1.1.3), (1.1.4) has a unique solution uT(t,x), and that solution satisfies the inequalitys

|DauT(t,x)| < CkJ(p)elT/n, (t,x) e n[(fc+ j-i)T,(fc+ j)T], j = 1,...,n; (1.1.8)

|DauT (t,x)| < Cfc (p)elT, (t,x) e n[fcT,(fc+i)T], k = 0,1,...,N - 1; |a| < p. (1.1.9)

j - 1'

Here Ck,j (p) = max sup

DauT k +

Ck(p) = max sup|DauT(kr, x)|, the

n

stant l depends only on p and does not depend on t, Co(p) = Co,i(p) = max sup|Dauo(x)| =

M^P en

C (p).

Theorem 1.2. Let the conditions of Theorem 1.1 be fulfilled. Then the solutions uT(t,x) of the problem (1.1.3), (1.1.4) converge to the solution u(t,x) of the problem (1.1.1), (1.1.2) in n^T] and uniformly in nR T] for all fixed R, R > 0, together with the derivatives DauT (t,x) with respect to space variables up to the order p — 1 as t ^ 0:

lim DauT(t,x) = Dau(t,x), (t,x) e n[0,T], (1.1.10)

t —^0 '

Tim. ||DauT — DauyC(nf0 T]) =0, |a| < p — 1. (1.1.11)

Under p ^ 2 the function u(t,x) is the solution to problem (1.1.1), (1.1.2) of the class

C1 'p-1(n[o , t ]).

In (1.1.11) nR T] = {(t, x)|0 < t < T, x e En, |x| < R}, R > 0 - const. Consider the linear partial differential equation of the first order

zt

+ f®(t, x, A)zxi + fo(t, x, A)z = f (t, x, A). (1.1.12)

Here x = (xi,..., xn), and A = (Ai,... Am) is a parameter.

Assumption 1.1. We make the assumption that in the domain n^T] = {(t, x)|to ^ t ^ ti,x e En} the functions f®, i ^ 1, are bounded for every fixed A. The functions f, f are continuous, and partial derivatives fiXj, fj>r, i = 0, 1, .. ., n, and fxj, exist, they are continuous and k — 1 times continuously differentiable with respect to all n + m +1 arguments (k ^ 1). The function w(x, A) is k times continuously differentiable according to all n + m arguments in the domain —to < xi,. .., xn < +to, —to < Ai, ..., Am < +to.

The result of section 4.3 in [8] is formulated as the following theorem.

Theorem 1.3. If the assumption 1.1 for all 8, A from intervals to ^ 8 ^ ti, —to < Ai,. .., Am < +to is satisfied then the equation (1.1.12) has in n[to,tl] the unique integral z = —(t,x, 8, A) with the initial value -0(8, x; 8, A) = w(x, A). This integral is k times continuously differentiable with respect to all m + n + 2 arguments.

If Xj = y>j(t, 0, ni,.. ., , A) are characteristic functions of the system

xj(t) = fj(t, x, A), i = 1,..., n,

(i.e., integral curves of the system (1.1.13), which go through the point (0,ni, . parametric representation of the integral takes the following form:

Xj = <£>i(i, 0, ni,... ,nn, A), i = 1,... ,n,

(1.1.13)

., nn)), then the

t

z = exp{-Fo}{w(ni,..., nn, A) + J f (t, yi,..., yn, A) exp{Fo}dt}, (1.1.14)

where Fo = Fo(t, 0, ni,... ,nn, A) = / fo(t, yi,..., yn, A)dt.

e

Denote 0 = to, w(x, A) = z(to,x, A) and assume that the following conditions are fulfilled:

|fo(t, x, A)| < M, (t,x) G n[t0itl], A G Em,

|w(x, A)| ^ q, x G En, A G Em, |f (t, x, A)| < N, (t,x) G n[to,ti], A G Em,

where M, q, N are constants.

By virtue of (1.1.14) we can prove that the estimate

|z(t,x)| < eM(i-io)(q + N(t - to)), (t,x) G n^].

(1.1.15)

(1.1.16)

(1.1.17)

(1.1.18)

holds due to (1.1.15)—(1.1.17).

1.2. Proof of Theorems 1.1, 1.2

In order to simplify the proof of Theorems 1.1, 1.2, we shall give it in the case n = 2. The general case can be treated similarly.

Consider in n[0 T] = {(t,x,y)|0 ^ t ^ T, (x, y) G E2} the Cauchy problem

du du b du o dt dx dy '

u(0, x, y) = uo(x, y).

(1.2.1)

(1.2.2)

Here a = a(t, x,y), b = b(t,x,y), uo(x, y) are functions given in n^T], n^T], E2 respectively. We replace the problem (1.2.1), (1.2.2) by the problem:

duT duT 1 .

— + = 0, kT<t< k + 2 lT,

duT duT ,1s

^ + = 0,(k + 2)t <t < (k + 1)T,

uT(0, x, y) = uo(x,y),

(1.2.3)

(1.2.4)

(1.2.5)

e

k = 0,1,..., N — 1; tN = T. The initial data on each fractional step is taken from the previous fractional step:

k + j — 1 , k +---— I t, x, y

lim uT(t,x,y), j = 1, 2; k = 0,..., N — 1.

i<(fc + j-1 )t

The conditions (1.1.6), (1.1.7) become

|Daa(t,x,y)| < M(p), |Dab(t,x,y)| < M(p), (t,x,y) G n[o,T],

|Dauo(x,y)| < C(p), (x,y) G E2, d|a|g

(1.2.6) (1.2.7)

« = (ai, «2), |a| = ai + «2, Dag = dxaidya2 •

By virtue of Theorem 1.3 we can prove that in the view of (1.2.6), (1.2.7) the solution uT(t,x, y) to the problem (1.2.3)-(1.2.5) exists for all t > 0 in n[0 T] (has continuous derivatives DauT in n[o,T] for |a| ^ p and continuous derivatives uj in n^T], but uj has discontinuities on hyperplanes tkj = (k + |) t, k = 0,..., N — 1; j = 0,1, 2).

Consider the zeroth whole step (k = 0). At the first fractional step we solve the problem

uj" + 2auX =0, 0 <t < 2, (1.2.8)

uT(0, x, y) = u0(x,y). (1.2.9)

From (1.1.18), (1.2.6), (1.2.7) (here w(x, y) = u0(x, y), f = 0, M = 0) it follows that

|uT(t,x,y)| < C(p), 0 < t < ^.

(1.2.10)

By differentiating the problem (1.2.8), (1.2.9) with respect to x, we get for vT = uX the problem

v[ + 2avX + 2aXvT = 0, 0 <t < ^, vT(0, x, y) = u0x(x, y). By virtue of (1.1.18), (1.2.6), (1.2.7) from (1.2.11) we get the inequality K(t,x,y)| = |vT(t,x,y)| < C(p)eM(p)T, 0 < t < 2.

(1.2.11)

(1.2.12)

By differentiating the problem (1.2.11) with respect to x (differentiating twice the problem

(1.2.8) with respect to x), for wT = we get the problem

^T + + 4aXwT + 2aXXuX =0, 0 < t < ^,

(1.2.13)

wT(0, x, y) = u0xx(x,y). Taking into account (1.1.18), (1.2.6), (1.2.7), (1.2.12) from (1.2.13) we obtain the inequality

|uXx(t,x,y)| = kT(t,x,y)| <

< e2M(p)T(C(p) + M(p)C(p)eM(p)Tt) < C(p)e4M(p)T.

(1.2.14)

T

u

By differentiating j times, j = 0,1,... with respect to x the problem (1.2.8), (1.2.9), we obtain the inequality

dj

—:UT (t, X, y) dxJ v ' ' y)

< C(p)

3Ci(p)M (p)T

j = 0 1,.. (t,x,y) G n[o,2],

(1.2.15)

where the constant Ci(p) depends on p only and does not depend on t .

Below we estimate the derivatives containing differentiation with respect to the variable y. We notice that in the problem (1.2.8), (1.2.9) the variable y is a parameter.

Let us estimate uy. We differentiate (1.2.8), (1.2.9) with respect to y. The function aT(t, x, y) = uy (t, x, y) is the solution to problem

d 2w

dxdy

(1.2.17)

we get

al" + 2aa! + 2a„uX = 0, 0 < t < —, 4 X 2' (1.2.16) aT(0, x, y) = woy (x,y), (x, y) G

By virtue of (1.1.18), (1.2.6), (1.2.7), (1.2.12) from (1.2.16) we get the inequality

|uy(t,x,y)| = |aT(t,x,y)| < C(p) + M(p)C(p)eM(p)Tt < < C(p)eM(p)T(1 + M(p)T) < C(p)eM(p)2T.

We differentiate the problem (1.2.11) with respect to y. For the function PT the problem

P T + + 2ax^T + 2a,y + 2aKy wX = 0,

PT(0, x, y) = woxy(x,y). By virtue of (1.1.18), (1.2.6), (1.2.7), (1.2.12), (1.2.14) from (1.2.18) we have

|wXy(t,x,y)| = IPT(t,x,y)| <

< eM(p)T(C(p) + M(p)C(p)e4M(p)Tt + M(p)C(p)eM(p)Tt) < (1.2.19)

< C(p)eM(p)te4M(p)t(1 + 2M(p)t) < C(p)e7M(p)t.

We differentiate the problem (1.2.16) with respect to y. For the function yt = uyy we get the

(1.2.18)

problem

YT + 2aY X + 4a,y uXy + 2ayy uX = 0, YT (0,x,y) = u0yy (x,y).

(1.2.20)

By virtue of (1.1.18), (1.2.6), (1.2.7), (1.2.12), (1.2.19) from (1.2.20) we obtain the following inequality

Ky I = |yt I < C(p) + 2M(p)C(p)e7M(p)Tt + M(p)C(p)eM(p)Tt <

< C(p)e7M(p)T(1 + 2M(p)t + M(p)t) < (1.2.21)

< C(p)e10M(p)T, (t,x,y) G n[o,2].

From (1.2.10), (1.2.12), (1.2.17), (1.2.19), (1.2.21) it follows that

|DawT(t,x,y)| < C(p)e10M(p)T, |a| < 2, (t,x,y) G n[o,2]. (1.2.22)

Let the integer k belong to the segment [2,p — 1]. Assume that the estimate

|DauT(t,x,y)| < C(p)e'1

ii(fc)M (p)i

holds.

Let us prove that

|DauT(t,x,y)| < C(p)e'

ii(fc + 1)M (p)T

, |a| < k, (t,x,y) G n[0i2], (1.2.23)

|a| < k + 1, (t,x,y) G n[o,2]. (1.2.24)

In (1.2.23), (1.2.24) the constants 11(p), 11(p + 1) depend on p and p + 1, respectively. The derivative D^uT = wT for | = k + 1 is a solution to problem

+ 2a^X + Y^ uT + 2DYa

|<fc

dfc+1wT

(1.2.25)

|t=o = D uo.

uT |t=o

In (1.2.25) the multi-index y satisfies the condition |y| = 1 and the coefficients are functions that linearly depend on the derivatives Daa, |a| ^ k.

By virtue of (1.2.22) from (1.2.25) we get the estimate (1.2.24), whence follows, that

|DauT(t,x,y)| < C(p)e'1

ii(p)M (p)i

, H < p, (t,x,y) G n[o,2].

(1.2.26)

On the second fractional step (at k = 0) we consider the problem

duT , duT t

"dT + =°, 2 <t < T,

uT ( —, x, y) = lim uT (t, x, y)

2 / t^ t

(1.2.27)

t<t

where uT ( 2, x, y) satisfies the conditions

|DauT(2,x,y)| < C(p)ell(p)M(p)T, |a| < p. From the conditions (1.2.6), (1.2.28), we get the estimate

|DauT(t,x,y)| < C(p)ell(p)M(p)Tell(p)M(p)T =

= C (p)

3i(p)M (p)T

(t,x,y) G n[2 ,t] .

(1.2.28)

(1.2.29)

In the relation (1.2.29) l(p) = 211(p) is a constant. From (1.2.26), (1.2.29) we have

|DauT(t,x,y)| < C(p)el(p)M(p)T, |a| < p, (t,x,y) G n

[o,T ].

(1.2.30)

Considering the k-th whole step (0 ^ k ^ N — 1) and repeating the reasonings which have been lead in the proof of the relation (1.2.30), considering Ck(p) instead of C(p), we shall obtain the estimate (1.1.9). The estimate (1.1.8) follows from the proof of the estimate (1.1.9). □

Proof of the Theorem 1.2. On the second fractional step of the first whole step (k = 1), repeating the reasonings which have been lead in the deduction of the estimate (1.2.30), we shall obtain that the inequality

|DauT(t,x,y)| < C(p)el(p)M(p)Tel(p)M(p)T = C(p)el(p)M(p)2T, |a| < p,

T

0

t

is valid for (t, x, y) G n[T,2T]. Clearly, at k-th step

|DauT(t,x,y)| < C(p)e1(p)M(p)T, (t,x,y) G %T,(fc+i)T], |a| < p, 0 < k < N - 1, and, hence,

|DawT(t,x,y)| < C(p)e1(p)M(p)T = di, (t, x, y) G n[o,T], |a| < p. (1.2.31)

From (1.2.31) and the system (1.1.3) it follows that

< d2, (t, x, y) G n[o,T], |a| < p - 1. (1.2.32)

The estimates (1.2.31), (1.2.32) guarantee the uniform boundedness and the equicontinuity of the functions {DauT}, |a| ^ p — 1, in n^T].

By virtue of Arzela's theorem (on compactness in C) we can choose a subsequence wT (without changing the notation) such that

DawT —► Dau, |a| < p — 1, t ^ 0, (1.2.33)

in n[o,T], and uniformly in nRy] for all fixed R > 0.

By virtue of (1.2.33) and by the convergence Theorem WAM (Theorem 2.4.1 in [7]) the function w(t,x,y) is the solution to the problem (1.2.1), (1.2.2) in the class C 1,p-1(n[o,T]). As the solution of a class C 1,p-1(n[o,T]) is unique, for this reason all sequence {wT(t, x, y)} converges to u as well as the subsequence chosen above (relations (1.1.10), (1.1.11) are fulfilled). □

dt v

y)

2. Parabolic Equation

2.1. Parabolic Equation Containing the Central Derivatives

Consider in n^y] = {(t,x)|0 ^ t ^ T, x G En} the problem

n

«t =£ 0,(t,XiKiXi, i=i

«(0,x)= «o(x), x G En. Conditions on the initial data of the problem (2.1.1), (2.1.2):

|Da«0(x)| < Cm, |a| = m, m > 0, dmaj(t, xj)

dx"

^ M„

« = 1,. .., n, m > 0.

(2.1.1) (2.1.2)

(2.1.3)

(2.1.4)

All derivatives in relation (2.1.3) are continuous in En. Remark 2.1. It is obvious, that by virtue of (2.1.3), (2.1.4) for any integer p ^ 0 there are constants C = C(p), M = M(p) depending on p only such that

d k aj(t,xj)

dxk

^ M (p), k ^ p, « = 1,..., n,

|Da«o(x)| < C(p), |a| < p.

(2.1.5)

We split the problem (2.1.1), (2.1.2) into one-dimensional problems according to WAM:

1

uT = nai«XlXl, kr < t ^ ^k +--^ t,

k + --- ) t < t < ( k + - ) t,

n V n

We take the initial data uT|i=(fc+ i-1 )T on the i-th fractional step from the previous fractional

n — 1 \

k +- T<t < (k +1)t, (2.1.6)

n

uT(0, x) = uo(x), x G En,

T

t = n, k = 0,1,...,N — 1, N is an integer.

'T|t=(fc+^ )t step:

uT |. /, ¿-is = lim uT (t, x), i = 1,...,n.

it=(k^_n")T t ,, + i-i. v "

t<(k+ )t

By virtue of conditions (2.1.3) and the Theorem 5.1 in [11] we can show that the solution uT(t,x) to the problem (2.1.6) exists in n[0 T] for any t (tN = T), DauT G C(n^T]), for all |a| ^ 0, (DauT)t are continuous in n^T], but (DauT)t in general has discontinuities in the time variable points tk j- = (k + j-1) t, j = 1,..., n; k = 0,1,..., N — 1.

The following theorems hold.

Theorem 2.1. Let the conditions (2.1.3), (2.1.4) be fulfilled. Then for a fixed p ^ 0 the estimates

|DauT (t,x)| < Cfc,j (p)edT/n, (t,x) G n((fc+ j-i )T,(fc+n )t ], j = 1, ...,n, (2.1.7)

|DauT (t,x)| < Cfc(p)edT, (t,x) G %T,(fc+i)T], k = 0,1,...,N — 1; |a| < p, (2.1.8)

hold, where uT is the solution to the problem (2.1.6) and C^j (p) = max sup|DauT ((k+ )t, x)|, Cfc(p) = max sup|DauT(kT, x)|, the constant d depends on C(p) only and does not depend on t.

Let's note that in (2.1.7), (2.1.8) we take

Co(p) = Col(p) = max sup |Dauo(x)|.

M<P En

Theorem 2.2. If the conditions of Theorem 2.1 are fulfilled, then the solutions uT(t,x) of the problem (2.1.6) converge as t ^ 0 to the solution u(t,x) of the problem (2.1.1), (2.1.2) in n^T] and uniformly in nR t] for all fixed R, R > 0, together with the derivatives DauT(t, x), |a| ^ 0:

lim DauT(t,x) = Dau(t,x), (t,x) G n[0,T],

lino ||D«uT - Dauyc(nfo,T]) =0. The function u G C ^^(n^T]) is the solution to problem (2.1.1), (2.1.2). Here C1'TO(n[o,T]) = {u(t,x)|Dau, ut G C(n[o,T]), |a| > 0}.

TT

u+ = nai

i Xi Xi

2.2. General Parabolic Equation

We consider in n[o,T] = {(t,x)|0 ^ t ^ T, x G En} the problem

wt = L(u) + f, u(0, x) = uo(x),

n n

where L(u) ^^ «¿(t, xj)uXiXi + 6j(t, x)uXi + c(t, x)u.

(2.2.1) (2.2.2)

i=1

i=1

Assume that the functions «0, a satisfy the conditions (2.1.3), (2.1.4), respectively, functions 6j, c, f satisfy the conditions

|Dabi(t,

< M„

|a| = m, m ^ 0,

a| = m, m ^ 0,

|a| = m, m ^ 0.

Remark 2.2. It follows from relations (2.2.3)-(2.2.5), that for any integer p ^ 0 constants M(p), x(p), a(p), which depend on p only, such that

|Da6j(t, x)| < M (p), |a| < p, i = 1,...,n,

|Dac(t,x)| < x(p), |a| < p, |Daf (t,x)| < a(p), |a| < p.

(2.2.3)

(2.2.4)

(2.2.5) there are

(2.2.6)

(2.2.7)

(2.2.8)

We split the problem (2.2.1), (2.2.2) into the one-dimensional problems: wT = (n + 1)«1UX1X1 + (n + 1)b1uX1, kT < t < (k +--+1

)t,

«T = (n +l)an«L- +(n +l)bn«L , (k +

«T = (n + 1)c«T + (n +1)f, (k +

n — 1 s , , n .

)T<t < (k + ——-)t,

n + 1 n

n + 1 «T (0, x) = «o(x).

n + 1' )t < t < (k + 1)t,

(2.2.9) (2.2.10)

The system (2.2.9) contains (n + 1) one-dimensional parabolic equations and one ordinary differential equation, containing the parameters. The whole step contains (n +1) fractional steps. The following theorems hold.

Theorem 2.3. Let C0(p) > 1 and conditions (2.1.3), (2.1.4), (2.2.3)-(2.2.5) be fulfilled. Then the solution «T(t,x) to the problem (2.2.9), (2.2.10) satisfies the estimates

|Da«T(t, x)| < Ck,j(p)eeT/(n+1), (t, x) G n[(k+ ^)T,(k+ j)t], j = 1,..., n + 1,

|Da«T (t,x)| < Ck (p)e0T, (t,x) G n[kT,(k+i)T], k = 0,1,...,N — 1; |a| < p,

j-i-

(2.2.11) (2.2.12)

where Ck j(p) = max sup|DawT((k + n_1 )t, x)|, Co(p) = Co 1(p), the constant 6 depends on p (on M(p), x(p), a(p) from (2.1.5), (2.2.6)-(2.2.8)) and does not depend on t.

x

a

m

Theorem 2.4. Let the conditions of Theorem 2.3 be satisfied. Then the solution uT(t, x) of the problem (2.2.9), (2.2.10) converge as t ^ 0 to the function u(t, x) in n^T] and uniformly in

n

T] for any fixed R, R > 0, together with the derivatives DauT(t,x), |a| ^ 0:

limoDauT(t, x) = Dau(t,x), (t, x) G n[o,T],

lim ||DauT - Dau||c(ni0iT]) =0, |a| < p - 2. The function u G C^(n^]) is the solution to problem (2.2.1), (2.2.2). Example 2.1. Inverse problem. We consider the equation

2

Wt

Li(t, x, u) + Wzz + g(t, x)f (t, x, z),

(2.2.13)

where x = (xi,x2), Lj(t, x, u) = aj(t,xj)uXiXi + bj(t, x)uXi, the functions aj = aj(t,xj) depend on t and xj only, bj = bj(t, x) depend on t, x, f = f (t, x, z) = f (t, xi, x2, z) are given functions, the coefficient g(t,x) and the function u(t,x, z) are unknown. We set the initial data

u(0, x, z) = u0(x, z), x, z G E2, (2.2.14)

and the overdetermination condition

u(t, x, 0) = y(t, x), (t,x) G n[o,T].

(2.2.15)

Let's put z = 0 in (2.2.13), we express the function g(t,x) from the obtained relation and

^(t,x) — uzz (t, x, 0)

substitute the expression for g(t,x) = -—;-—-in (2.2.13). We arrive at the equation

f (t,x, 0)

2

E ^(t,x) — uzz (t, x, 0) Lj(t, x, u) + uzz +--J/-t _ -f (t, x, z),

i=1

f (t,x, 0)

(2.2.16)

In (2.2.16) the function ^ = yt — ^^ Lj(t, x, y) is a known function depending on ^ and on

coefficients aj, bj to operators Lj. Let

f (t, x, 0) > J> 0, J - const. We split the problem (2.2.16), (2.2.14) into the one-dimensional problems:

= 4L1(t, x, uT), nT < t < (n + 4)t, = 4L2(t, x, uT), (n + 1 )t < t < (n + 2)t, = 4uZz, (n + I )t < t < (n + 3 )t,

,T =4

*(t,x)-<z (t-

(2.2.181) (2.2.182) (2.2.183)

^f (t, x, z), (n + 3)t < t < (n + 1)t, (2.2.184)

f (t,x,0)

uT(0, x, z) = u0(x, z), (x, z) G E2. It is assumed that conditions (2.1.4), (2.2.3) and

|DYf(t,x,z)| < am, |71 = m, m > 0

(2.2.17)

(2.2.18)

(2.2.19)

(2.2.20)

R

T

u

t

u

T

u

t

|Da^(t,x)| < xm, |a| = m, m > 0, (2.2.21)

|DY«0(x,z)| < am, |y| = m, m > 0 (2.2.22)

are fulfilled. In (2.2.20), (2.2.22) vector 7 = (71,72,73) is a multi-index of dimension 3,

d|Y| 3

D = dxYl dxY2 dzY3 , |71 = ,Ç1 7i.

We differentiate (2.2.18), (2.2.19) twice with respect to z.

By virtue of independence of z the coefficients of the equations (2.2.181)-(2.2.183) and conditions (2.2.20)-(2.2.22) we conclud that the set «Zz(t,x, z) is bounded in n[0 T] uniformly on t > 0:

|«zz(t,x,z)| < C, (t,x,z) G G[0,T]. (2.2.23)

In the view of (2.2.20)-(2.2.23), differentiating with respect to z the problem (2.2.18), (2.2.19), we obtain in g[o,t] the estimate

d j

——r«T(t, x, z) dzJ v '

< N(p), j = 1, 2,... ,p; p > 2.

(2.2.24)

Let's estimate the derivatives uT. We differentiate the problem (2.2.18), (2.2.19) with

respect to z. The function f

instead of f is taken -7—— and dzj

dzj

is a solution of system of the equations (2.2.18), in which

dj«T (0, x, z) «0 (x, z)

(2.2.25)

dzj dzj

The problem (2.2.18), (2.2.25) is exponentially stable on first two fractional steps (Theorem 2.3, inequality (2.2.11)). It is not difficult to show the exponential stability of the problem (2.2.18), (2.2.25) on the third and fourth fractional steps. Hence in G[o,T] the estimate

d j

— Da«T (t,x,z) dzJ v '

^ N[(p), |a| + j ^ p, N1(p) — const. holds. Due to (2.2.26) and (2.2.18) it follows that in G[0,T]

d

— «T (t,x,z) dt v '

< N2(p), 171 < p — 4, N2 (p) — const.

(2.2.26)

(2.2.27)

By virtue of (2.2.26), (2.2.27) and Arzela's theorem we can choose a subsequence wT (without changing the notation) which converges at t ^ 0 to the function u together with all derivatives uT:

«T(t,x, z) ^ «(t,x, z), (t,x, z) G G[o,t], ||DY«T(t, x, z) — «(t, x, z)||C(G{jiT]) ^ 0

(2.2.28)

for any fixed K > 0 and all |y| ^ 0.

In view of (2.2.28) by Theorem 2.4.1 in [7] the function u(t, x, z) is the solution to the problem (2.2.16), (2.2.14) in the class CTO(G[o,T]). Taking into account the matching condition uo(x,0) = ^>(0,x) by the known way [12] we can prove that u(t, x, z) satisfies to the overdetermination condition (2.2.15) and the pair of functions u(t, x, z), g(t, x) is the solution to problem (2.2.13)-(2.2.15). □

T

References

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[2] N.N.Yanenko, G.V.Demidov, The research of a Cauchy problem by method of weak approximation, Dokl. Akad. Nauk SSSR, 6(1966), 1242-1244 (Russian).

[3] N.N.Yanenko, Fractions steps method for solving multi-dimensional problems of mathematical physics, Novosibirsk, Nauka, 1967 (Russian).

[4] Yu.Ya.Belov, Inverse Problems for Partial Differential Equations, Utrecht, VSP, 2002.

[5] Yu.Ya.Belov, S.V.Polyntseva, On the problem of identification of two multi-dimensional parabolic equation coefficients, Dokl. Ross. Akad. Nauk, 396(2004), no. 5, 583-586 (Russian).

[6] Yu.Ya.Belov, I.V.Frolenkov, Coefficient identification problems for semilinear parabolic equations, Dokl. Ross. Akad. Nauk, 404(2005), no. 5, 583-585 (Russian).

[7] Yu.Ya.Belov, S.A.Cantor, The weak approximation method, Krasnoyarsk, Krasnoyarsk State Univ., 1999 (Russian).

[8] G.V.Demidov, G.I.Marchuk, Existence Theorems of the solution to the problem of short-term weather forecast, Dokl. Akad. Nauk SSSR, 170(1966), 1006-1008 (Russian).

[9] N.A.Kucher, The weak approximation method and the analysis of circuits of splitting in gas dynamics, Dissert. of doctor fis.-math. nauk, Novosibirsk, 1992 (Russian).

[10] E.Kamke, The directory on the differential equations in partial derivatives of the first order, Moscow, Nauka, 1966 (Russian).

[11] O.A.Ladyzenskaya, V.A.Solonnikov, N.N.Ural'ceva, Linear and quasilinear equations of parabolic type, Moscow, Nauka, 1967 (Russian).

[12] Yu.Ya.Belov, I.V.Frolenkov, On the two coefficients identification problem for semilinear parabolic equations, Vestnik of Krasnoyarsk State Univ., 2004, no. 1, 140-149 (Russian).