On the existence of solution of some problems for nonlinear loaded parabolic equations with Cauchy data Текст научной статьи по специальности «Математика»

УДК 517.9

On the Existence of Solution of Some Problems for Nonlinear Loaded Parabolic Equations with Cauchy Data

Igor V. Frolenkov* Maria A. Darzhaa^

Institute of Mathematics and Computer Science, Siberian Federal University, Svobodny, 79, Krasnoyarsk, 660041

Russia

Received 22.10.2013, received in revised form 25.12.2013, accepted 17.02.2014

Two problems are considered in this paper. First problem is the Cauchy problem for a two-dimensional loaded parabolic equation with coefficients dependent on unknown function and its derivatives. Second problem is the Cauchy problem for one-dimensional equation of the Burgers-type. The sufficient conditions of the existence of solutions of these problems in classes of smooth bounded functions are presented in the paper. The method of weak approximation is used for the purpose of obtaining the proof.

Keywords: inverse problem, direct problem, loaded equation, parabolic equation, equation of the Burgers-type, method of weak approximation.

Introduction

This paper is devoted to an attempt to generalize the method of studying solvability of broad class of auxiliary direct problems for one- and two-dimensional coefficient inverse problems for parabolic equations in unbounded domains with Cauchy data.

Two problems are constructed in this work: special type of the loaded (containing traces of unknown function and its derivatives) two-dimensional parabolic equation and one-dimensional equation of the Burgers-type.

The solution existence of the Cauchy problems for the mentioned above equations was investigated. Coefficient inverse problems with Cauchy data can be reduced to these auxiliary direct problems with the use of overdetermination conditions (some additional information on the solution) assigned at fixed hyperplanes or hypersurfaces. Examples of such methods of studying of inverse problems can be found in [1]. There are also other approaches that reduce an inverse problem to non-linear unloaded equation or to integro-differential equation.

It is necessary to know under what conditions the auxiliary problems are solvable. It is also necessary to know the properties of solutions. The sufficient conditions for the existence of solutions of the problems are obtained in this paper. The method of weak approximation is used to prove the existence of solutions of the given problems. This method is also known as the method of splitting on differential level [2,3].

* [email protected] [email protected] (c Siberian Federal University. All rights reserved

1. On the special form of two-dimensional loaded semilinear parabolic equation

Let us choose r different points ak, k = 1, r of variable x defined on space E^ We also choose s different points z = Am, m = 1, s of variables z defined on space Ei.

Let us consider in the strip G[0jT] = {(t, x, z)|0 < t < T, x G Ei; z G Ei} the Cauchy problem for loaded (containing traces of unknown function and its derivatives) non-classical parabolic equation

d _ _ —u(t,x, z) = ai(t,x, wo(t))ux® + a2(t, z, wo(t))uZz+ dt

+ bi(t, x, z, wo(t))ux + 62(t, x, z, wo(t))uz + f (t, x, z, u, wo(t), wi(t, x), W2(t, z)), (1)

u(0, x, z) = uo(x, z). (2)

The components of vector-function

_ f dj1+j2 \ _ _

wo(t) = I u(t, afc, Am), dxjl dzj2 u(t, afc, Am) j , k = 1, r, m = 1, s,

ji =0,1,. .. ,pi, j2 =0,1,. .. ,qi,

are traces of function u(t, x, z) and all its derivatives with respect to x up to order pi and with respect to z up to order qi. All traces depend only on variable t. The vector-function

_ f dj \ _

wi(t, x) = I u(t, x, Am), u(t, x, Am) I , m = 1, s, j = 0,1, ..., qi,

consists of the traces of function u(t, x, z) and all its derivatives with respect to z up to order qi. All traces depend only on variables t and x. Similarly, the vector-function

_ f dj \ _

W2(t, z) = I u(t, ak, z), u(t, aklzH , k = 1, r, j = 0,1,. .. ,pi,

consists of the traces of function u(t, x, z) and all its derivatives with respect to x up to order pi. All traces depend only on variables t and z.

Let us consider a simple example. The following inverse problem for the heat equation is reduced to the direct problem of type (1), (2).

We have the following equation in domain G[o,T] = {(t, x, z) | 0 < t < T, x G R, z G R}

ut(t, x, z) = uxx(t, x, z) + uzz(t, x, z) + A(t, x)f (t, x, z), (3)

with initial data

u(0, x, z) = uo(x, z), (x, z) G R2. (4)

Coefficient A(t, x) should be determined simultaneously with the solution u(t, x, z) of problem (3), (4). The solution satisfies the overdetermination condition

u(t,x,Y(t)) = y(t,x), 0 < t < T, x G R. (5)

Let the consistency conditions be fulfilled

uo(x,Y(0)) = y(0,x), x G R.

We assume that all input data for the problem are real-valued functions. The functions and all necessary derivatives of these functions are sufficiently smooth and bounded in G[0<T]. Let the following condition be true

If (t, x, y(t))| > S> 0, 0 < t < T, x G R.

The problem (3)-(5) is reduced to the auxiliary direct problem

^H.t,x) — uz(t,x,Y(t))Y'(t)- uzz(t,x,Y(t)) ut(t,x,z)= uxx(t,x,z)+ Uzz(t,x,z)+--f x Y(t))-f(t,x,z), (6)

u(0,x,z)= u0(x,z), (x,z) G R2, (7)

where ^(t, x) = <p't(t, x) — f'Xx(t, x) is the known function.

In this example, in direct problem (6), (7) the functions al(t,x,w0(t)), a2(t, z,w0(t)), bl(t,x, z,w0(t)), b2(t,x, z,w0(t)) and f (t,x, z,u,W0(t),wl(t,x),w2(t, z)) from equation (1) have the following forms:

ai(t,x,wo (t)) = a2(t,z,wo(t)) = 1,

bi(t, x, z, wo(t)) = b2(t, x, z, wo(t)) = 0,

,f. — \ - / i ss ^(t,x) — Uz (t,x,Y(t))Y '(t) — Uzz (t,x,Y(t)) ff. N

f (t,x,z,u,w0(t),wi(t,x),w2(t,z)) = -—-—-f (t,x,z).

f (t,x,Y(t))

In what follows we assume that p > iaax{2,pl}, q > max{2, ql}.

Definition 1.1. Zpl ([0, t* ]) denotes the set of functions u(t,x,z) that are defined in G[0,t*] and belong to the class

(Gw,*])=\uu(t,x,z) I G C(Gw,*]), ji =0,p, j2 =0,^,

that is, functions and all their derivatives appearing in equation (1) are bounded at (t,x,z) G G[0,t*]

-u(t, x, z)

EE

ji =0 32=0

dxj1 dzj2

< C.

Definition 1.2. A classical solution of problem (1), (2) in G[0,t*] is the function u(t,x,z) G Zp>l([0,t*]) which satisfies (1), (2) in G[o,t*] ■

Here 0 < t* < T is a fixed constant. If t* depends on the constants that bounds the input data and t* ^ T then u(t, x, z) is a solution of problem (1), (2) on a small time interval. If t* = T for any set of input data that satisfies the condition of solvability then u(t, x, z) is a solution of problem (1), (2) in the whole time interval (or we will use the term "global solvability").

Suppose that the following conditions are true.

Condition 1.1. The functions al, a2, bl, b2 are real-valued functions that are defined for all values of their arguments and they are continuous functions. The functions al, a2 satisfy conditions al ^ a0 > 0, a2 ^ a0 > 0. For any tl G (0, T] and any function u(t,x,z) G ZXP'+z2'q+2([0,tl]) these functions, as functions of variables (t,x,z) G G[0,tl], are continuous and they have continuous derivatives that enter into the following inequality

p+2

ji=0

d ji

dxj

P+2 q+2

EE

ji=0 j2 = 0

q+2

j2 = 0

d j2

dzj2

a^t, z, W0(t))

+

dj1+j2

dxj1 dzj2

bi (t, x, z, w0(t))

+

dj1+j2

dxj1 dzj2

— &2 (t, x, z, w0(t))

< P71 (U(t)); (8)

Condition 1.2. The function «o is a real-valued function that satisfies the following inequality

P+2 q+2

EE

j1=0 j2 = 0

dxj1 dzj2

u0(x,z)

< C.

The function has continuous derivatives that enter into the inequality.

Condition 1.3. The function f is a real-valued function that is defined for all values of its arguments and it is continuous function. For all ti G (0,T] and any function u(t, x, z) G Z£+2'q+2([0,ii]) this function, as function of variables (t, x, z) £ G[ojtl], is continuous and it has continuous derivatives that enter into the following inequality

P+2 q+2

EE

j1=0 j2 =0

dj1+j2

"/(t, x, z, U, W0(t), Wl(t, x), W2 (t, z))

< P72 (U(t)).

dxj1 dzj2'

In conditions 1.1 h 1.3, y1, y2 > 0 are some fixed integers,

Pc (y) = <7(1+ y + ••• + yz), C > 1 is a constant that is independent of the function u(t, x, z) and its derivatives,

dj1+j2

(9)

P+2 q+2

U(t) = V V sup sup

j1=0 j2 = 00<«^i (x>z)eE2

dxj1 dzj2

w(£, x, z)

t(t,x,z) G ZXt2'q+2([0,t1j).

The following theorem is proved in [6]. Theorem 1.1. Let us assume that conditions 1.1-1.3 are fulfilled. 1a. If in equation (1) the coefficients ai; 6 are independent of the space variables:

ai = ai(t,wo(t)), a2 = a2(t,wo(t)), 6i = 6i(t,wo(t)), 62 = &2(t,wo(t)),

and conditions 1.1, 1.3 are fulfilled for 71 ^ 0, 0 ^ y2 ^ 1 then the classical solution «(t,x, z) of problem (1), (2) exists in class ZpZ ([0, T]).

1b. If the coefficients ai; 6 have the same form as in the case 1a and conditions 1.1, 1.3 are fulfilled for 71 ^ 0, y2 > 1 then there is a such constant t*, 0 < t* ^ T dependent on the constant C from (8), (9) that the classical solution «(t, x, z) of problem (1), (2) exists in class Zp;f ([0,t*]).

2a. If in equation (1) the coefficients a^, 6 have the forms:

ai = ai(t,x, wo(t)), a2 = a2(t, z,wo(t)),

61 = 6i(t, x, z,wo(t)), 62 = 62(t, x, z, wo(t)),

and conditions 1.1, 1.3 are fulfilled for y1 =0, 0 ^ y2 ^ 1 then the classical solution «(t,x, z) of problem (1), (2) exists in class ZpZ ([0, T]).

2b. If the coefficients ai; b have the same forms as in the case 2a and conditions 1.1, 1.3 are fulfilled for y1 =0 but y2 > 1 then there is a such constant t*, 0 < t* ^ T, dependent on the constant (J from (8), (9) that the classical solution u(t,x,z) of problem (1), (2) exists in class Z™([0,t*]).

The fulfillment of the conditions of Theorem 1.1 can be proved for the given above example (direct problem (6), (7)), assuming that input data are sufficiently smooth and bounded functions. For example, the conditions of Theorem 1.1 are fulfilled for the constants p = q = 4, yi =0 and Y2 = 1. Hence, the classical solution u(t,x,z) of problem (6), (7) exists in the class ZX4',4([0,T]).

2. On one-dimensional loaded Burgers type equation of the special form

Let us consider the proof of a similar result for the one-dimensional Burgers-type equation. In this equation the coefficient of the first order derivative with respect to space variable depends on the solution and its traces of the specified form.

Let us choose r different points al,... ,ar in space E1.

The following Cauchy problem is considered in the strip G[0<T] = {(t, x)|0 < t < T,x G E1}:

here u(t) = (u(t,ak), -z—ru(t,ak)), j = 0,pl, k = 1,r is a vector-function. The components axj

of this function are the traces of function u(t,x) and all its derivatives with respect to x up to order pi . The traces depend only on variable t.

Definition 2.1. ZX ([0,t* ]) denotes the set of functions u(t,x) defined in G[0,t*] and they belong to the class

and p > iaax{2,pl}.

Definition 2.2. A classical solution of problem (10), (11) in G[0,t*] is the function u(t,x) G ZXp(\0,t*\) which satisfies (10) in G[0,t*]. Here 0 <t* ^ T is a fixed constant dependent on the input data.

Suppose that the following conditions are fulfilled.

Condition 2.1. The functions b(t,x,u(t,x),w(t)), f (t,x,u(t,x),w(t)), uo(x) are real-valued functions that are defined and continuous for all values of their arguments. For all ti G (0, T] and for any u(t,x) G ZXp+2([0,tl]) these functions, as functions of variables (t,x) G G[0,tl], are continuous and they have continuous derivatives that enter into inequalities (12), (13). The function a(t) ^ a0 > 0 is a continuous bounded function on the interval [0, T]. The function u0(x) has continuous derivatives and satisfies the following inequalitie

ut = a(t)uxx + b(t, x, u(t, x), w(t))ux + f (t, x, u(t, x), u(t)), u(0, x) = uo(x),

(10) (11)

The functions are bounded at (t, x) G G[0,t*] together with the following derivatives

p+2 E

j=o

dj )

— wo(x) dx-'

< c.

Condition 2.2. Let us introduce the following notations

j = 0,1,... + 2,

Uj (0) = sup

x

Uj (t) = sup

dj

—uo(x) dxj

dj

dj u(t'x)

, j =0,+ 2,

p+2 p+2 U (0) = £ Uj (0), U (t)=£ Uj (t).

j=o j=o

Lei us assume that for all ti £ (0, T], for all t £ [0, ti] and for any function u(t, x) £ Zp+2([0,ti]) the following estimates are hold:

P+2

E

j=0

P+2

E

j=0

dj

——r&(t, x, w(t, x), w(t)) dxj

dj

dxj f (t,x,w(t,x),^(t))

< pyi (U(t)),

< P72 (U (t)),

(12)

(13)

where 71, y2 ^ 0 are some fixed integers, PZ (y) = (7(1 + y + y2 + ... + yZ) and C ^ 1 is some constant independent of the function u(t,x) and its derivatives.

Theorem 2.1. Assume that conditions 2.1 and 2.2 are fulfilled for 71 ^ 0 and 0 ^ y2 ^ 1. Then a constant t*, 0 < t* ^ T exists and it depends on the constants a0 and C from condition 2.1 and inequalities (12), (13), such that the classical solution u(t,x) of problem (10), (11) exists in class Zx([0,t*]).

Proof. To prove the existence of a solution of the Cauchy problem (10), (11) we use the method of weak approximation. The original problem is split into three fractional steps on

differential level and time shift by (^t — —J is done in the traces of unknown functions and in nonlinear terms:

wj"(t, x) = 3a(t)uXx(t, x), nr < t ^ n + — t;

(14)

■(t, x) = 3b (t - 3, x, wT (t - 3, x), wT(t - 3)) wX(t, x), (n + ^ t < t < (n + t ; (15) uT (t,x) = 3/(t - 3 ,x,wT (t - 3 ,x),^T (t - 3)) , (n +2) T<t < (n +1)t ; (16)

wT(0, x) = wo(x).

(17)

Let us prove a priori estimates that ensure compactness of the family of solutions u(t, x) of

problem (14)—(17) in the class C^G^t*]) for some constant 0 < t* < T.

Let us introduce the following notations

Uj (0) = sup

dj ( ) dxjuo(x)

, j = 0,i,...,p + 2,

(18)

U3 (t) = sup

dj

dx u(t,x)

, t G (nr, (n +1)t], j = 0,1,...,p + 2,

p+2 p+2

U(0) = Y, Uj (0), UT(t) = 22 UJ(t).

j=0 j=0

(19)

(20)

Consider the first fractional step when n = 0. On the interval 0 <t ^ 3 we have the Cauchy problem (14), (17).

According to the maximum principle we obtain

|uT (t,x)I ^ sup |u0(x)|.

x

Upon differentiating problem (14) j times with respect to x, j =0,1,... ,p + 2 we obtain

dj

d 3

—ul(t,x) = 3a(t) uXx(t,x)

d3 dj dx u(0'x) = dx uo(x).

Then, according to the maximum principle, we have the following estimate

dj

—uT (t,x)

^ sup

x

dj

d3 u0(x)

Taking into account (18)-(20), we obtain

UT(t) < U(0), 0 <t < 3.

(21)

t 2t

Consider the second fractional step when 3 < t < —. Then, due to the time shift, the

equation (15), is a linear one-dimensional homogeneous partial differential equation.

In this case, the first characteristic equation ( [5], n. 2.6) is t'(a) = 1. The solution of this equation can be taken in the form t = a. The second characteristic equation can be written as

dx ( t t ( t

——= —3b ( t--,x,u t--,x

dt V 3' ' V 3'

■T It — T

3

(22)

Considering the assumptions of the theorem and properties of the solution obtained in the

first fractional step, assume that yT(t,£,n) is a characteristic function of equation (22), i.e.

x = yT (t,£,n) is the integral curve of the equation that goes through the point (£,n).

t

Initial data for equation (15) can be written in parametric form as t = 3, x = n, uT =

uT ^3(the function uT ^3is taken from the previous fractional step). The solution to this problem exists and can be represented in parametric form

uT (t, x)

u \3,V

¥ (t, 3,1

or in the form

Hence it follows that

uT(t,x)= uT (3,¥T (3,t,x)y

UT(t) < UT (3) < u(0), 3 <t < y.

(23)

x

Let us differentiate equation (15) with respect to x and introduce the following notations

zT(t, x) = uX(t, x),

b0(t,x) = 3b(t — 3,x,uT (t — 3,x) (t — 3)) ,

bi (t,x)=3 ¿(^—3 ,x,uT ^—3 ,x),»T k—3))).

Then we obtain equation

zT(t, x) = b00(t, x)zX + bi(t, x)zT. (t 2t

The solution of this equation for t G I 3, —

can be written in the parametric form ( [5], p. 43)

(t,x) = e-F°(t'3'n)zT (3,n) , x = yT (t, 3,n) ,

where

t

FT = FT (t, Ç,ri) = — j b\ (t, (t, n)) dt,

and x = yT (t, n) is the characteristic function of the equation

f = —b0(t,x) = —3b(t — 3,x,uT (it — 3,x),«T (it — 3)) ,

i.e. it is the integral curve of the equation passing through the point (£, n).

Taking into account that conditions 2.1-2.2 are fulfilled and taking also into account estimate (23) and notations (18)-(20), we obtain

UT(t) < UT (3) exp fa (UT (t — 3)) t) < UT (3) ePYi(u(0))t, (24)

where P^(y) = C(1 + y + y2 + ... + yz), C ^ 1 is polynomial from condition 2.2.

Let us differentiate equation (15) twice with respect to x and introduce the following notation

VT (t,x) = uXx(t,x),

c0 (t,x) = 3b(t — 3 ,x,uT (t — 3 ,x),^T (t — 3)) ,

* (t,x)=61 (b (t—3 ,x,uT (t—3 ,x) M (t—3)))

$ (t,x)=3 dx (b ^—3 ,x,uT ^—3 ,x),wT k—3

Then we obtain equation

vtT(t,x) = cT0 (t,x)vX(t,x) + ci(t,x)vT(t,x) + cT2 (t,x)uX(t,x). (25)

can be written in the following parametric form

2

The solution of this equation for t G ( 3, —

( [5], p. 43)

vT (t,x) = e-GT 3 n (V (3 ,n) + £ c2 (t,VT (t, 3 ,n) )ul(t,¥T (t, 3 ,n) )eGT (t 3 'v) ctj ,

x = 3,nn),

where G0 = GJ (t,£,n) = - J cj (t,^T (t,£,n)) dt.

fi

_ /^r (+ £ _ I „T (+ „T

r0 = G0 (

U

Taking into account that conditions 2.1-2.2 are fulfilled and taking into account estimate (23)

T 2r

and notations (18)-(20), we obtain for 3 < t < — the following inequality

/j (t) < e2I UJ (-) + 3P71 (U(0))e2^1

UJ(t) < e2PYi(U(0))t |uj (—) +3P71 (U(0))e2PYi(U(0))t £ UJ(t) dtj <

< eClPYi(U(0))t ^ (—) + Ci P7i (U(0)) UJ(t) dtj . (26)

Here and further C; > 1 are constants (generally they are different) independent of the parameter t .

It follows from (23) and (24) that

/J (t) < eC2PY1 (U (0))t (uJ ( T g + C2P71 (U (0))UJ ( - g ePY1 (U (0))t

UJ(t) < (U(0))T (uJ (3) + C2P71 (U(0))UT (3) eP

UJ(t) < eC3PYi(U(0))T (UJ (3) + C3tP7i(U(0))UJ (3)) , (27)

<

UT(t) < (UJ (—) + UJ g—)) eC4PY1(U(0))t (1 + C4rP71 (U(0)))

^ —) + U2T (—)) eC4pY1(U(0))teC4pY1(U(0))t^ (uj(!) + U2T (—)) eCPY1 (u(0))t. (28)

If we differentiate equation (15) j times with respect to x, j = 3,4,... + 2 and use the

Leibnitz formula for the j-th derivative of the product of two functions then we obtain WjJ (t, x) =

j

d0(t,x)wjX + dj(t,x)wjT dk(t,x)wT_fc, (t,x), where

k=2

d0 (t,x) = —b (t — — ,x,uT (t — — ,x) , wT (t — —)) ,

dk d^xk b(t — — .x,^^ — — .x),^ — —)) .

By the arguments used to obtain equation (25) we arrive to the following inequality

UJ(t) < eC6PY1(U(0))T Lj (—g + CeP71 (U(0)) f £ UJ(t) d^ , j = 4,... + 2. (29)

V "3 fc=1 /

It follows from (24), (26) and (29) that

U3T(t) < eC6PY1(U(0))t |V3t (—) + CeP71 (U(0)) UJ(t) + UJ(t) dtj < eCePY1(U(0))t x

X (U3 (—g + C6TP71 (U(0)) (uj (—g ePYf(U(°))T + (UJ (—g + UJ (—gg (0))ggg ^

< eC7PY1(U(0))T (uJ f—g + C7rP71 (U(0)) (uJ f—g + UJ f—ggg . (30)

Hence we have

UJ(t) < (UJ(3) + UJ(3) + UJ( J)) eC8PYi(U(0))T. (31)

We continue our arguments for j = 4,... ,p + 2 and obtain

UJ (t) * eC6pYi(U (0))t fur ( 3 ) + C6P71 (U (0)) f Y, UT (t) d^ * eC6pYi(U (0))t x

V 3 k=i J

x (uj ( 3 ) + C6TPYi (U m^lY UT ( 3 )) eAlkPYi^ *

* ¡UJ ( 3 ) + BJTPYI (U (0))(j2 UJ ( 3 )Jj , (32)

Hence we have

j

UJ (t) * (E UJ (3)| eDj Pyi iU i0))T, (33)

\k = l

here Alk,Bj, Dj are positive constants independent of the parameter t.

Now we add up inequalities (23), (27) (30) and (32) for p = 4,... ,p +2. Taking into account notations (18)-(20), we obtain

UT(t) < eC9pYi(U(0))T (ut (3) + CqtP1i (U(0))Ut (3)) < UT (3) e°i°PYi(U(0))t.

2

On the third fractional step — < t < t we consider equation (16) with initial data at the

2T (2T \ point — (the value of function uM — ,x\ from the previous fractional step). Upon integrating

equation (16) with respect to the time, we get

uT(t,x) = y+3 J2T f (n — 3,x,uT (n — 3,x) (n — 3)) dn.

Taking into account this equation, the Theorem condition 2.2 and the previously introduced notations, we arrive to the following inequality

UT(t) < U0T(2t) + CiiT (l + ut(^ ) . (34)

Let us differentiate (16) j times with respect to x, j = 1, 2,... ,p + 2. Taking into account condition 2.2 and the notations (18)-(20), we obtain the following estimates

UT(t) < ut(y) + Ci2T (l + ut(y) ) , (35)

UT (t) < UT^y ) + Ci3T (l + ut(y ) ) . (36)

After combining estimates (34), (35) and (36) on the third fractional step, we obtain

UT(t) ^ UT(y) + Ci4T (l + UT(^f) ) . (37)

Now considering relation (21), (28) and (37) on the time interval t £ (0, t] we obtain the following estimate (constant C > 0 does not depend on t)

UT(t) < U(0)eTC15PY1(U(0)) + C15t(1 + U(0)eTC15pY1(U(0))) ^

< U(0)eTC15PY1(U(0)) + 1 — 1 + C1 5t(1 + U(0)eTC1sPY1 (u(0))) ^ < (U(0)eT°1sPY1(U(0)) + 1)(C15T + 1) — 1 < (U(0) + 1)eT°1sPY1(U(0))eC15T — 1 ^

< (U(0) + 1)eTCPY1(U(0)+1 ) — 1. (38)

Consider the next whole step (n =1). By the given above arguments we obtain the estimate (because constant C does not depend on t we change U (0) for UT (t))

UT(t) < (UT(t) + 1)eTCPY1(UT(t)+1 ) — 1, t < t < 2t. (39)

It follows from (38) that UT(t) < (U(0) + 1)eTCPY1 (u(0)+1 ) — 1. Considering this inequality, estimate (39) takes the form

UT(t) < ((U(0) + 1)eTCPY1(U(0)+ 1 ) — 1 + 1)x

X exp (rCP71 ((U(0) + 1)eTCPY1 (u(0)+1 ^ — 1 ^

< (U(0) + 1)eTCPY1(U(0)+1 ) exp (tCP71 (U(0) + 1)eTY1CPY1(U(0)+^ — 1. (40)

Let us take some constant t* that satisfies the inequality

e2i*C71PY1 (U(0) + 1) ^ 2. (41)

Constant t* depends on the input data and does not depend on t . It follow from inequality (41) that

e(2i-1)Tc71pY1(U(0)+1) ^ 2, i = M, At = t*. (42)

Taking into consideration (42), we rewrite estimate (40) UT(t) < (U(0) + 1)eTCPY1 (u(0)+1) e2jcpY1(U(0)+1) — 1 ^ (u(0) + 1) e3TCPY1(U(0)+1) — 1. Repeating our arguments, after a finite number of steps we obtain in the interval ((k — 1)t, kr]

UT(t) < (U(0) + 1)e(2k-1)TCPY1(U(0)+1) — 1 ^ (U(0) + 1)e2i*CPY1(U(0)+1) — 1 = k.

This implies in the strip G[0ji*] the uniform on t boundedness of the function uT and its derivatives

with respect to x up to order p + 2 inclusive.

From the above estimates it also follows the uniform in t boundedness of the derivatives d dj UT d dj UT

—-——-, ——-—-, j = 0,1,... ,p. It presents sufficient condition in order for sets of functions dt dxj dx dxj

uT, uX, ..., d p to be equicontinuous in G^* = {(t,x)|t £ [0,t*], |x| < N} for any fixed constant N.

By Arzela's theorem, there is some subsequence wTfc of sequence uT that converges in GjN01*] with its derivatives to order p to certain function u(t, x). By the convergence theorem of the method of weak approximation, the function u(t,x) = wTfc (t,x). By virtue of the arbi-

trariness of N it belongs to the class

1 du dj

Ci1i'X(G[0,i*]) = {w(t, x)|—, dx^u(t,x) G C(G[0,t.]), j =0,1,... ,p}, (43)

< C, is true, i.e. w(t, x) G

p

and it is a solution of (10), (11). The inequality

j=0

Zp([0,t*]). □

dj

— u(t,x)

T

3. Example

As an example, we consider the inverse problem for the Burgers-type equation which has been studied earlier by Belov and Korshun.

In strip n[0,T] = {(t,x)|0 < t < T, —to < x < to} we consider the following Burgers-type equation

ut(t, x) = v(t)uxx + A(t)uux + B(t)u + C(t) + g(t)f (t, x), (44)

where A(t), B(t), C(t) and f (t,x) are given functions and initial condition is

u(0, x) = uo(x), —to < x < to. (45)

The functions u(t,x)and g(t) are unknown. Let us assume that the overdetermination conditions are

u(t,x0 ) = ^(t), x0 = const, (46)

and consistency condition is ^(0) = u0(x0).

We also suggest that input data satisfy the following conditions

E

k=0

dk u0 (x)

dxk

k=0

dk f (t,x)

dxk

+ IA(t)l + IB(t)l + IC(t)MV>(t)| * K, If(t,xo)l ,K = const > 0,

K

where ^(t) = <f>'(t) — B(t)^(t) — C(t).

With the use of the overdetermination conditions (46) problem (44), (45) is reduced to the auxiliary direct problem of the form

ut(t, x) = ^(t)uxx + A(t)uux + B(t)u + C(t)+

+ ^(t) — Kt)uxx(t, xo) + A(t)№ux(t, xo) f (t x) (47)

f (t, xo) ' '

u(0,x)= u0(x). (48)

In this example the functions b(t,x,u(t,x),w(t)) and f (t,x,u(t,x),w(t)) from equation (10) have the form b(t, x, u(t, x), w(t)) = A(t)u(t, x),

f<+ !+ \ !+\\ U!+\ , n(+\ i ^(t) — V(t)uxx(t,x0) + A(t)0(t)ux(t,xO) f(, s

f (t, x, u(t,x),w(t)) = B(t)u + C (t) +--—---f (t,x).

I (t, xo)

Conditions 2.1 and 2.2 are fulfilled. Parameter yi = 1 in condition 2.2 and we have

E

k=0

dk ,nn\ , ^(t) — Kt)uxx(t,x0) + A(t)^(t)ux(t,x0) , —k [B(t)u + C(t) + -f^-f (t, x)

|C (t)| + E

k=0

dk , ^(t) — p(t)uxx(t,xo)+ A(t)^(t)ux(t,xo) dk

B(t) dxku +

f (t,xo )

dxk

f (t,x)

*

* C + CUm + C + CWC * C(1 + Urn.

Therefore, parameter 72 = 1 in condition 2.2. The conditions of Theorem 2 are fulfilled. Thus, there is such constant t*: 0 < t* < T (it depends on the constants that bound the input data) that the classical solution u(t, x) of direct problem (47), (48) exists in class Z^([0, t*]).

This research was supported by the Russian Foundation for Basic Research under project no. 12-01-31033.

References

[1] Yu.Ya.Belov, I.V.Frolenkov, Some identification problems of the coefficients in semilinear parabolic equations, Doklady Mathematics, 404(2005), no. 5, 583-585.

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

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

[4] Yu.Ya.Belov, I.V.Frolenkov, An identification problem of the two coefficients in semilinear ultraparabolic equation, Joint issue. Vychislitelnye Tehnologii. 8(2003), Part 1. Regional'ny bulleten vostochnogo izdatel'stva Vostochnogo gosudarstvennogo universiteta Kazakhstana, Ust-Kamenogorsk, 120-131 (in Russian).

[5] E.Kamke, Differentialgleichungen: Losungsmethoden und Losungen, II, Partielle Differentialgleichungen Erster Ordnung fur eine gesuchte Funktion, Akad. Verlagsgesellschaft Geest & Portig, Leipzig, 1965.

[6] I.V.Frolenkov, Yu.Ya.Belov, An existence of a solution for the class of loaded two-dimensional parabolic equations with Cauchy data, Nonclassical equation of mathematical physics, collected papers, managing editor A.I.Kozhanov, Institut matematiki im. Soboleva, Novosibirsk, 2012, 262-279 (in Russian).

[7] A.I.Prilepko, D.G.Orlovsky , I.A.Vasin, Methods for Solving Inverse Problems in Mathematical Physics, New York, Marcel Dekker, 1999.

О существовании решений некоторых задач для нелинейных нагруженных параболических уравнений с данными Коши

Игорь В. Фроленков Мария А. Даржаа

В данной статье приведены достаточные условия существования решения в классе гладких ограниченных функций задачи Коши для двумерного нагруженного параболического уравнения специального вида (коэффициенты при старших, младших членах и правой части зависят от следов неизвестной функции и ее производных), а также получен аналогичный'результат для одномерного нагруженного уравнения типа Бюргерса (уравнение дополнительно содержит нелинейность относительно решения при младшей производной по пространственной переменной) с данными Коши. Для доказательства используется метод слабой аппроксимации.

Ключевые слова: обратная задача, прямая задача, нагруженное уравнение, параболическое уравнение, уравнение типа Бюргерса, метод слабой аппроксимации.