On Uniqueness and Continuous Dependence on the Initial Data of the Solution of a System of Two Loaded Parabolic Equations with the Cauchy Data Текст научной статьи по специальности «Математика»

Journal of Siberian Federal University. Mathematics & Physics 2019, 12(3), 298-309

УДК 517.9

On Uniqueness and Continuous Dependence on the Initial Data of the Solution of a System of Two Loaded Parabolic Equations with the Cauchy Data

Igor V. Frolenkov* Irina S. Antipina^ Natalya M. Terskikh*

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

Russia

Received 17.01.2019, received in revised form 25.01.2019, accepted 03.02.2019 We study the Cauchy problem for the system of one-dimensional loaded parabolic equations. Uniqueness and continuous dependence of solutions on the initial data in the class of smooth bounded functions is proved.

Keywords: loaded equation, system of parabolic equation, weak approximation method, Cauchy problem, uniqueness, continuous dependence of the solution on the initial data. DOI: 10.17516/1997-1397-2019-12-3-298-309.

Introduction

In [2] the sufficient conditions for solvability of the problem in the class of smooth bounded functions were obtained. A problem similar to the one-dimensional Burgers-type equation was considered in [3], where the question of the existence of solutions was also investigated.

In this paper we prove the uniqueness of the classical solution in the class of sufficiently smooth bounded functions and formulate the theorem of uniqueness. Also we study the continuous dependence of the solution on the right-hand side and the initial conditions and formulate the corresponding theorem.

1. Statement of the problem

In the space Ei of variables x choose r different points ak, k = 1, r. In the strip G[0,T] = = {(t, x)|0 < t < T,x e Ei} consider the Cauchy problem for the system of loaded non-classical parabolic equations

ut(t,x) = a1(t)uxx(t,x) + bi(t)ux(t, x) + f1(t,x,u,v,^u(t),^v(t)), vt(t,x) = a2(t)vxx(t,x) + b2(t)vx(t,x) + f2(t,x,u,v,Tpu(t),1Pv (t)),

* igor@frolenkov.ru tirina.antipina.21@mail.ru int.terskikh@mail.ru © Siberian Federal University. All rights reserved

u(0,x) = uo(x), v(0,x) = vo(x), x G Ei, here the components of vector-functions

(2)

<Pu(t) = (^(t, ak), dxu(t' ak ^ > (t) = ^v(t> ak), dj ak) ) , k = r, j = °,pi,

)) '

are traces of functions u(t, x), v(t, x) and all their derivatives with respect to x up to order pl. Denote by Zp(G[0,T]) the set of the functions u(t,x) defined in G[0,T] belonging to the class

du dj u

(G[o,t]) ,j =°, ■ ■ ■ ,p|

C% = u(t'x)'dt'dX7 G C

and bounded in (t, x) G G[0<T] together with their derivatives,

e

j=0

dju(t, x) dxj

< C■

2. Uniqueness of solution

In [4] the sufficient conditions for the existence of solutions of the problem (1) and (2) in the class Zp(G[0,t,]) are obtained. Here 0 < t* < T is a fixed constant. Suppose that p > max{p1,2}+ +2 > 4, and with this p satisfies the conditions of the existence theorem [4]. In view of this theorem a classical solution u1(t,x) G Zp (G[0jt»] ),v1(t,x) G Zp(G[ott*]) ofthe problem (1), (2) exists. We prove that this solution is unique. Suppose that there is a pair of functions u2(t,x) G Zp(G[0,t* ]), v2(t,x) G Zp(G[0,t,]), along with the functions u1(t,x), v1(t,x), which is a solution of the system of equations. Then

ut(t, x) = 0,1 (t)upx(t, x) + b1(t)up(t, x) + f1(t, x, ui, v\-^ui (t),Vv* (t)), vt(t, x) = 02(t)vpx(t, x) + b2(t)vp(t, x) + h(t, x, ui, v*,Vu* (t),Vv*(t)),

ui(0, x) = uo(x), vi(0, x) = v0(x), x G El, i = 1, 2.

Suppose that the functions al(t), a2(t), bl(t), b2(t), u0(x), v0(x), are real-valued, defined in [0,T], G[0,TEl, respectively and have all continuous derivatives occurring in the following relation and satisfying it

p+2

io1(t)i + \o2(t)\ + Mt)\ + Mt)\+J2

k=0

dk

dxk u0(x)

p+2

+ Y.

k=0

dk

dxk v0(x)

< C.

Functions fl and f2 are real-valued, defined and continuous for any values of their arguments. For all tl G [0,T], u(t,x), v(t,x) G Zp+2([0,tl]) these functions as functions of the variables (t, x) G G[0,i1] are continuous and have continuous derivatives occurring in the relation

p+2 , £

j=0 v

dj _ _

— f1(t,x,u,v,lfu (t),Vv (t))

dx

+

dj _ _ — f2(t,x,u,v,Wu (t),Vv (t))

dx

< C(u,v),

(3)

where the constant C(u, v) depends on the functions u(t, x), v(t, x).

The difference u1(t,x) — u2(t,x) = U(t,x), v1(t,x) — v2(t,x) = V(t,x) is a solution of the system of equations

Ut(t,x) =a1(t)Uxx(t,x) + b1(t)Ux (t,x) +

+ fl(t, x, ul, vl,^ul (t),^vl (t)) — f1(t, x, U (t),<fv2 (t)),

Vt(t,x) =a2(t)Vxx(t,x) + b2(t)Vx(t,x) +

+ f2(t, x, u1, vl,Wui (t),<fvi (t)) — f2(t, x, U ,v2,Wu* (t),<fv2 (t)),

U(0,x)=0, V(0,x)=0, x € E1.

(4)

(5)

Condition 1. Suppose that the function f1y f2 such that Vt1 € (0,T], Vu1(t,x), u2(t,x) € ZP(G[o,t1]), yvl(t,x), v2(t,x) € Zp(G[0,tl]), the following relations hold:

f1(t,x,u1,v1,<ful (t),<fvl (t)) — f1(t,x,u2,v2,<fu2 (t),<fv2 (t)) =

r Pi ,

ee(

k=1 s=0 V

, Qs

(u1 — u2) • F1 + y^ ( —u1(t, ak) — d^Su2(t, ak) ) • F1 +

ds dxs

t, ak

r Pi ( ds f)s \

+ (v1 — v2) ■ G1 + ^ v1(t, ak ) — ^ v2t ak ) ■ (6)

k=1 s=0 ^ '

f2(t,x,u ,v ,¥ui (t),¥vi (t)) — f2(t,x,u ,v ,^u2 (t),¥v2 (t))

r ( c)s c)s \

(u1 — u2) ■ f2 ^ — u\-t, ak) — u2(t, ak) ■ Fl,s +

k=1s=0 ^ '

r Pi f r)s r)s \

+ (v1 — v2) ■ g2 + v1(t, ak ) — toS v2t ak ) ■ G , s. (7)

k=1s=0 ^ '

Condition 2. The functions F1, s, G\ Gks, F2, F2s, G2, G2k s, where k = 1,r, s = 0,p1 are known and sufficiently smooth, depend on t, x, u1(t,x), u2(t,x), v1(t,x), v2(t,x), Vui (t), Vu2(t), Vvi (t), Vv2(t) and have all continuous derivatives occurring in the following relation and satisfying it:

pi

D

j = 0 +

F1 j G1 j F 2 j G2

+ + +

dxj dxj dxj dxj

r Pi

ee(

k=1 s=0

j F1

dxj k, s

j G1

dxj k's

+

j F 2

dxj k, s

+

+

j G2

dxj k's

))

< C, y(t, x) € G[0,ti]. (8)

k=1s=0

C is a constant independent of U(t,x),V(t,x) and their derivatives.

Theorem of uniqueness of the solution. If a solution of the system of equations (1) and (2) exists in the class Zp(G[0,t*]), where p ^ max{p1, 2} + 2 ^ 4, then under conditions 1, 2 it is unique in the class Zp(G[0,t*]).

Proof. Represent the system of equations (4) in the following form

Ut(t,x) = a1(t)Uxx(t,x) + b1(t)Ux (t,x) + U (t,x)F 1 +

r Pi / ds \ r Fi (

EE ^Utak)FU + Vtx)G1 +

k = 1 <1 = 0^ ' k = 1 s = 0 v

r Pi

+

ds

— U(t, ak)Fls ) + V(t, x)G1 + ( V(t, ak)G

k=1s=0

k=1 s=0

s,

k

Vt(t,x) = al(t)Vxx(t, x) + bl(t)Vx(t,x) + U(t,x)F2+

r ( ds \ r ( ds \

+ E E U t ak )Fl, s + V (^ x)G ^^E V (^ ak )Gk , J .

k=l s = 0 \ ' k=l s = 0 \ '

k=ls=0 v 7 k=ls=0

Introduce the non-negative, non-decreasing on [0,t*] functions

\(t) = sup

*[0,t]

ds

dx U &x)

m2s(t) = sup

'[0,t]

ds

dx V &x)

, s = 0,1,... ,pl.

By the maximum principle, for (e,x) G G[0,t*], 0 < t < t* we obtain

(\ / r pi

£ • sup \Fl M ^E sup Fl s\ ■ ml(t) +

G[0.f ] ) \k=l s=0 G[0,f]

r p1

+ m2(t) ■ sup \Gl\ + sup \Gk, sb m2(t)\ ■ £,

G[0,t*] k=l s=0 G[0,t*] J

(\ / r pi

e ■ sup \G2 M^E sup \G2k, sb m2(t) +

G[0,f] J \k=l s=0 G[0,t*]

r p1

+ ml0(t) ■ sup \F 2\+ sup \Fl s\■ m2s(t)\ ■

T[0,t*]

For (e,x) G G[0,t-t], 0 < t < t* we have

(pi pi \ / pi pi \ ¿2m!(t) + j: m2(t)\ ■ t, |V (e,x)\ < C2 ■ ]T m2(t)+Y, ml(t) ■ t-

s=0 s=0 J \s = 0 s = 0 /

Apply sup to the both sides of these inequalities, due to the fact that the functions ml(t),

G[0,t]

m

(t), s = 0,1,... ,pl, 0 < t < t* are nonnegative, we obtain:

(pi pi \ / pi pi \ J2ml(t) + J2 m2(t)\ ■ t, m0(t) < C2 ■ ]T m2s (t) + J^ ml (t) ■ t.

s=0 s=0 J \s=0 s=0 J

(9)

We differentiate the equations j times wrt x, j = 1,... ,pl

dJ dj+2 dJ+l _ Ut(t,x) =al U (t,x) + bl U (t,x)+

J / f)1 f)J-l \ n pi f)S f)J

+£ C i U (*•*) j F l + £ £ tU ) ¿j Fls+

J — n ^ / L — o —0

J i dl d J-i \ n pi d s d J

+ £ C i V M) j G0 + £ £ d? V) dj Gis-

J — n V / L — 1 o —n

dJ dj+2 dj+l dx Vt(t, x) =a2 j V(t, x) + b2 jV(t, x) +

+ E {cj dx U t x) J F A + ±± dX U t ak) dj Fks s +

s=0

v Jdx (tx) jF dxss (tak)dxFk, ■

l=0 v ' k = l s=0

J d' , dJ-'r>\ d« , dJ

J f dl dJ-l \ '

^ CJ e~iV(^ x) jG2) ^ E d^sV(^ ak)dzjGkss.

1 = 0 v / k = l s = 0

m

In view of the maximum principle, we obtain the similar estimates:

p1 p1 |(t) < C1 HP m1(t)^ m2(t)) • t,

m.

)s=0 s=0 ) j = 1,...,p1, 0 < t < t*. (10) p1 p1 1

m2(t) < C2 • £ m'M + y m2(

\s=0 s=0

Then we add these inequalities and get

Pi / Pi \

J2(m1(t) + m2(t)) < C • £ (m1(t)+ m2s(t)) • t, s = 1,...,p1.

s=0 s=0

Hence for t e [0,0], where 0 < 1, it follows that

C

J2(m1(t) + m2s (t))=0.

s=0

As mls(t) > 0, m2s(t) > 0, then for all (t, x) e G[0,e] we have

U(t, x) = 0, V(t, x) = 0. Similarly, because C does not depend on 0 for t e [0, 20] we obtain that U (t,x)=0, V (t,x)=0, (t,x) e G[0 , 2e\. After the finite number of steps, we obtain

U(t,x) = 0, V(t,x) = 0, (t,x) e G[0,tt]. (11)

Hence it follows that

u1(t,x)= u2(t,x), v1(t,x)= v2(t,x), t e [0, t*], x e E1.

Therefore, the solutions coincide in the whole domain G[0,t*] in the class Zp(G[0,t»j). The uniqueness theorem is proved. □

Consider the example of an inverse problem for a system of parabolic equations for which the existence of the solution was investigated in [5].

Example. In the strip np,T] = {(t, x)| t e [0,T],x e E1} we consider the problem of finding real-valued functions U(t,x), V(t,x), gi(t), i = 1,2, satisfying the system of equations

Ut = Uxx + bn(t)U2 + b12(t)V + g1(t)m1(t, x),

(12)

Vt = Vxx + b21(t)U + b22(t)V2 + gn(t)m2(t, x),

the initial conditions

U(0,x) = U0(x), V(0,x) = V0(x), x e E1. (13)

The solution satisfies the overdetermination condition

u(t,0) = ?1(t), V(t,0) = fo(t), t e [0,T], (14)

where bij(t), mi(t,x), U0(x), V0(x), ¡3i(t), i,j = 1,2 are given real-valued functions. Let the consistency conditions be fulfilled

Uo(0) = fh(0), Vo(0)= fo(0).

Let the following condition hold

\mi(t, 0)| > S> 0, i = 1, 2, t G [0,T], S - const.

All input data are real-valued, sufficiently smooth and bounded functions with their derivatives in np, T].

The problem (12), (13) is reduced to the auxiliary direct problem

Ut = Uxx + bn(t)U2 + b12(t)V+

+ mi(t,x)m-\t, 0)(/3l(t) - Uxx(t, 0) - bn(t)P1(t) - b^(t)V(t, 0)), Vt = Vxx + b2l(t)U + b22(t)V 2 +

+ m2(t,x)m-1(t, 0)(£2(t) - Vxx(t, 0) - b2i(t)U(t, 0) - b^(t) p2(t)), U (0,x) = U0(x), V (0,x) = V0(x).

Differences W(t,x) = U1(t,x) - U2(t,x), Q(t,x) = V1(t,x) - V2(t,x) are a solution of the problem

Wt = Wxx + bii(t)W(U1 + U2) + bi2(t)Q + mi(t,x)m-1(t, 0)(-Wxx(t, 0) - bu(t)Q(t, 0)),

Qt = Qxx + b2i(t)W + b22(t)Q(V1 + V2) + m2(t, x)m-1(t, 0)(-Qxx(t, 0) - b2i(t)W(t, 0)),

W(0,x)=0, Q(0,x)=0.

Check the conditions of the Theorem fi(t,x,U 1,V i (t),^v i (t)) - fi(t,x,U2, V2 2 (t),vv 2 (t)) =

= (U1 - U2)F1 + (Uix(t, 0) - U2x(t, 0))F12 + (V1 - V2)G1 + (V 1(t, 0) + V2(t, 0))G1 , o, f2(t,x,U 1,V 1,^U 1 (t),ëv 1 (t)) - f2(t,x,U2, V2 ,VU 2 (t),ëv 2 (t)) =

= (U1 - U2)F2 + (U 1(t, 0) + U2(t, 0))Fi,o + (V1 - V2)G2 + (Vix(t, 0) - V^t, 0))G21 , 2,

where

F1 = bn(t)(U1 + U2), Fi,2 = -mi(t,x)m-1 (t, 0), F2 = bn, F^ = -bn m2 (t,x)m-1(t, 0),

G1 = bu, G2 = bn(t)(V1 + V2), Gio = -bi2mi(t,x)m-1(t, 0), G^ = -m2 (t,x)m-1(t, 0),

are known, sufficiently smooth and bounded functions.

Conditions 1, 2 of the Theorem of uniqueness of the solution are fulfilled. Hence, the solution of the problem (12), (13) is unique.

3. Continuous dependence of the solution on the initial data

In the space E1 of variables x choose r different points ak, k = l,r. Consider the Cauchy problem for the system of loaded non-classical parabolic equations

Ut(t,x) = al(t)Uxx(t,x) + bl(t)Ux(t,x) + U (t,x)F l(t,x) + F3(t,x) + V (t, x)Gl (t, x)+

r pi ds r pi ds

+ Z Z dXsUt ak)Fk,^ x) + Z Z ^Ts V(^ ak)Gk, s (^ x),

x) + 2-^/ V (t,ak s(t,x)

k=1 s=0 k=1 s=0

Vt(t,x) = a2(t)Vxx(t,x) + b2(t)Vx(t,x) + U (t,x)F2(t,x)+ G3(t,x) + V (t, x)G2 (t, x)+

r pi ds r pi ds

+ Z E U(^ akKs^ x) + ^J2 dxssV(^ ak)Gl,s(^ x)

k=l s=0 k = l s=0

(15)

U(0, x) = U0(x), V(0, x) = V0(x), x G El.

(16)

We choose and fix p > \mAX^[pl, 2} > 2. Suppose that the functions a,i(t), bi(t), U0(x), V0(x), i,j = 1,2 are real-valued, defined in [0, T], G[0,T], El, respectively and have all continuous derivatives occurring in the following relation and satisfying it

p+2

\ai(t)\ + \bi(t)\ + J2

k=0

dk

-x U0 (x)

p+2

+

k=0

dk dxk

< c.

(17)

Constants C are different and independent of the splitting parameter t , U0(x), V0(x), F3(t,x). G3(t , x), U(t, x), V(t, x) and their derivatives here and below.

The functions Fl(t,x), Fl s(t,x), Gl(t,x), Glk s(t,x), F2(t,x), F2 s(t,x), G2(t,x),

Gk s(t,x), F3(t,x), G3(t,x) where k = l,r, s = 0,pl, are known, sufficiently smooth and have all continuous derivatives occurring in the following relation and satisfying it:

p+2 E

J=0

d J

F1

+

dxJ

r p ,

zz(

s=0

d J

dxJ

+

j F l

dxJ k's

G1

dJ

F2

+

+ dxJ

d Gl dxJ Gk, s

+

dJ dxJ

G2

J F 2

dxJ k, s

+

+

k=1 s=0

Introduce the notation, where k = 0,1,... ,p + 2

J F 3 dxj

dJ G2 dxJ Gk, s

+

dJ dxJ

G3

+

))

< C, y(t,x) G G[0,ti]. (18)

\\hl(t,x)\\l,p = V" sup sup

j=00<^T xeEi

d J

dXJ hl (e,x)

\\h2(x) \\2,p = ^ sup

J=0

xeEi

dJ

h2 (x)

dxJ

hl(t,x) G Zp(G[0,T]), , h2(x) G Zxp(G[0,t]).

(19)

Uk (t) = sup sup

UT<£^txeEi

V-t (t) = sup sup

UT<£^txeEi

d k

dX-kU T (ex dk

dx V T (ex

< t < (n +1)t,

(20)

Uk(0) = sup xeEi

d k

dXk U0(X)

,Vk (0) = sup

xeEi

dk

dX V0(X)

p+2 p+2 p+2 p+2 U(t) = Y, UT(t), VT(t) = 22W(t), U(0) = £ u-(0), V(0) = £Vk(0).

k=0 k=0 k=0 k=0

We use the method of weak approximation. The system of equations (15) is split into three

fractional steps on differential level and time shift by 3 in the traces of unknown functions

UT(t, x) = 3 (a1(t)UTx(t, x) + b1(t)UZ(t, x)), VT(t, x) = 3 (a2(t)VTx(t, x) + b2(t)VT(t, x)),

(n+-3)

nr < t ^ ( n +3 ) T,

(22)

UJ (t,x) = 3UT (t - 3,x) F1 + 3VT (t - 3,x) G1, VtT (t, x) = 3UT (t - 3, x) F2 + 3VT (t - 3, x) G2,

(n+1) T<t ^ (n+3) t,

(23)

r Pi / ds T ds T

UT (t,x) = 3^J2 yr-s U (t - 3 ,ak )F1,s + yr-sV (t - 3, ak)Gk,s ) + 3F 3

k=1s=0

ds

k=1s=0

Vt (^ x)=3^Y, U (t - 3, ak )Fk,s + ^ V (t - 3, ak )Gk,sj + 3G

k's ' 3xs

d s 2 °

dxs

T

3,

(24)

(n+3)

n + 3 ) T <t < (n + 1) T.

On the first fractional step t e (0, 3] we use the maximum principle, with the notation (21). We obtain the following estimates

UT(t) < U(0), VT(t) < V(0).

(25)

At the second and third fractional steps, differentiating with respect to x from 0 up to order p+2 including, then integrating over the time variable and using the time shift in unknown functions, taking into account the notation (20), we obtain: - on the second fractional step t e (3, 2T]

U T (t) <U T (i)+CU T (i)T+CV T (i)T, w(t) < v^(3) + CUT (3) t + CVT (3) t,

(26)

on the third fractional step t e {^T,r\

(2T ) + c(U (I) + (|) + g ( f ) + c(u (I) + V (I) + g a

sup sup

xeE10<^^T

dk dxk

dk dxk

F3

G3

T.

(27)

Considering inequalities (25), (26) and (27) with the notation (20) on the zero whole step we

get

(u (0)+ V (0)+ g(

k=0

UT(t)+ VT(t) < eCT\ U(0)+ V(0) + ) \ sup sup

xeE10<£^T

dk dxk

F3

+ sup sup

xeE10<^^T

dk dxk

G3

))

G

p+2

V I sup sup k=0\ xeE1o<^T

dk dxk

F3

+ sup sup

x€E10<i4T

dk dxk

G3

On the interval [0, T]

UT(t) + VT (t) < eCT( U(0) + V(0) +

+sup sup

xeE1o<^^T

dk dxk

G3

P+2 (

M

k=0 V

)) —£

k=0

sup sup

xeE1o<^^T

dxk

F3

sup sup

xeEi0<£^T

dk dxk

+

F3

+ sup sup

xeE1o<^^T

dk dxk

G3

)

These estimates guarantee the fulfilment of the conditions of the Arzela Compactness Theorem. By this theorem, some subsequences UTk(t,x),VTk(t,x) of the sequences UT(t,x),VT(t,x) of the solutions (22)-(24) of the split problem converge together with all relevant derivatives with respect to x up to order p to the functions u(t, x), v(t, x) respectively, that according to the theorem of convergence of the weak approximation method is the solution to (1)-(2). Using this notation (19) we get

\\U(t,x)\\i,p + ||V(t,x)\\lp < C (\\Uo(x)\\2p+2 + ||Vo(x)\\2p+2 + \\F3\\hp+2 + ||G3||1,p+2) -

— (\\F3\\i ,p+2 + \\G3\\i ,p+2) .

Theorem (continuous dependence of the solution on the initial data). Assume that the condition (17) is satisfied, for p ^ max{p1,2} ^ 2, so there exists at least one solution belonging to the class Zp(G[0,T]) for which we have the estimate \\U (t,x)\\1 ,p + \\V (t,x)\\1 ,p ^ < C (\\Uo(x)\\ 2,p+2 + \\Vo(x)\\ 2,p+2 + \\F 3\\ 1,p+2 + \\G3\\ 1,p+2 )~(\\F 3\\ 1,p+2 + \\G3\\i,p+2) ■

For the system of equations (1), (2) consider two sets of input data ulo(x),vo(x),f1, f2 & Zp+4(G[otT]), where i = 1,2, p > msx^-i, 2}. Then by the existence theorem the solution ui(t, x), vl(t, x) exist in the class Zp+2(G[o<T]). This solution satisfy the conditions of the uniqueness theorem for p + 2 > 4. If ui(t,x), vl(t,x) & Zp+2(G[o,t*]), then functions f1 and f2 satisfy the condition (3). We obtain

utXt,x) = ai(t)uixx(t,x)+bi(t)uix(t,x)+ fil{t,x,ui ,vi,^Ui (t),^ (t)),

vt(t,x)= a2(t)vlx(t,x)+b2(t)vx(t,x) + f2{t,x,ui,vi,?ui(t), VV(t)), ui(0, x) = uio(x), vi(0,x)= vo(x).

Introduce the notation u(t, x) = ui(t, x) — u2(t, x), v(t, x) = vi(t, x) — v2(t, x), uo(t, x) = = uo(t,x) — u2(t,x), vo(t,x) = v^(t,x) — v2(t,x). Subtracting the second system of equations from the first one we get the following

ut(t,x) = ai(t)uxx(t,x)+bi(t)ux (t,x) + fii(t,x,ui,vi,?u1 (t),?v1 (t)) —

— fi(t,x,u2,v2,lfu2 (t),1fv2 (t)) + F 3(t,x,u2,v2 (t),?v2 (t)),

vt(t,x) = a2(t)vxx(t,x )+b2(t)vx(t,x)+ f2 (t, x, ui,vi, ?u1 (t), ?v1 (t)) —

— fi(t,x,u2,v2,Vu2 (t),Vv> (t))+ G3(t,x,u2,v2,?u2 (t),Vv> (t)), u(0,x) = uo (x), v(0,x)= vo (x),

where

k

d

F3 = fl(t,x,u2,v2,<fu2 (t),ipv2 (t)) - f1(t,x,u2,v2,<fu2 (t),ipv2 (t)) ,

G3 = fH. t,x,U2,v2,^u2 (t),1fv2 (t)) - f2{t,x,u'2,v'2,¥u2 (t),<fv2 (t)) .

By Conditions 1 and (18) we obtain

r pi f ds \ ut(t,x) = al(t)uxx(t,x) + b1(t)ux(t,x) + u(t,x)F1 + v(t,x)G1 + ^^ f d^Su(t,akF1^ j +

k = 1 s=n v '

r pi / ds \ _

+ ( dhs v(t,ak )Gl,s) + F°3(t,x,u2,v2,^u2 (t),Vv2 (t)),

k=1 s=n ^ x '

' F1 f ds \

vt(t,x) = a2 (t)vxx(t, x) + b2(t)vx(t,x) + u(t,x)F2 + v(t,x)G2 + f d^s u(t,ak )Fi2A +

k=1 s=n x

r pi

, v(t x)G2 , Y^V^ ( u(t a )F2

xs

k=1 s=n

r pi ( ds \ _

+ v(t,ak )Gls) + G3(t,x,u2,v2,^u2 (t),ëv2 (t)),

k=1 s=n

r p1 s

EE ^ v(t,ak )GU

x

u(0,x) = un(x), v(0,x) = vn(x).

For the resulting problem using the Theorem (continuous dependence of the solution on the initial data), we get

\\u(t,x)\\1 ,p + \ \v(t, x) || 1 ,p < C (\M x)\\2,p+2 + \\vn (x)\\2,p+2 + \\F3\\ 1,p+2 + HG^^^ 1 -

-(iF\\1,p+2 + \\G3\\1,P+2) .

Example. In the strip n^^] = {(t,x)\t G \0,T],x G E1} we consider the problem of finding real-valued functions u(t,x), v(t,x), g(t) satisfying the system of equations

ut = uxx + uv + bn(t)u + b12(t)v + m1(t,x),

(28)

vt = vxx + b21(t)u + b22 (t)v + g(t)m2 (t, x),

the initial conditions

u(0,x)= un(x), v(0,x) = vn(x), x G E1. (29)

The solution satisfies the overdetermination condition

v(t, 0)= /3(t), t G [0,T],

where bj(t), mi(t,x), un(x), vn(x), 3(t), i,j = 1,2 are given real-valued and bounded functions with their derivatives in n^^]. Let the all consistency conditions be fulfilled. Assume that the input data are sufficiently smooth and there is a solution in the class Z%(G[n,T]). This is not difficult to prove, by analogy with reasoning in the article [1].

For the system of equations (28), (29) consider two sets of source functions m\(t, x), m\(t, x), i = 1, 2, satisfying the conditions of the theorem of existence of the solution [1]. Then according to the theorem there exists a solution ui(t,x), vl(t,x), gl(t) for each set of input data. Let the following condition hold

\m,(t, 0)\ > S> 0, i = 1, 2, t G [0,T], S - const.

Introduce the notation U(t,x) = u1 (t,x) - u2(t,x), V(t,x) = v1(t,x) - v2(t,x), Q(t) = = g1(t) — g2(t). Subtracting the second system of equations from the first one we get

Ut = Uxx + u1v+v2U + bu(t)U + b12(t)V + m1(t,x) - m\(t,x),

Vt = Vxx + b21(t)U+b22(t)V + g1(t)(m12(t,x) - m2(t,x)) + Q(t)m2(t,x),

U(0,x)=0, V(0,x)=0, V(t, 0)=0.

Having reduced it to the auxiliary direct problem, we get the following

_ -Vxx(t, 0) - b2i(t)U(t, 0) - gl(t)(m\(t, 0) - m2(t, 0)) Qt)- m2(t, 0) ■

Ut _ Uxx + u1V + v2U + bn(t)U + bl2(t)V + m\(t, x) - m\(t,x), Vt _ Vxx + b2i(t)U+b22(t)V + g1(t)(m2(t,x) - m22(t,x)) +

+m2(t x)-Vxx(t, 0) - b2i(t)U(t, 0) - g1(t)(m2(t, 0) - m2(t, 0))

(30)

m22(t, 0)

U(0,x)_0, V(0,x)_0■ (31)

The system (30), (31) is a special case of the system (15), (16). Ut _ Uxx + (bn(t)+v2)U + (bi2(t) + u1 )V + m1(t, x) - m2(t, x),

V _ Vxx + <b2lW'+W - ^M) - U t 0) +

. 1M( ^ 2(+ ^ g1(t)(ml2(t, 0) - m22(t, 0))m2(t,x)

+ g (t)(m2(t,x) - m2(t,x))---■

m2 (t, 0)

Functions

F1 _ bn(t)+ v2, F2 _ bn, Fl 2 _0, Fl 0 _ - /±v ' 2 0) , F3 _ m\(t,x) - m2(t,x),

?1 _ b (t) , v2 F2 _ b F1 _0 F2 _ b21(t)m2(t,x) F3 _ m1,

2 _0 F10 _

G1 _ b12(t)+ u1, G2 _ b22(t), G100 _0, G222 _ -m2(t,x)

](t, 0)

G3 _ g1(t)(ml(t,x) - m2(t,x)) - g1(t)(m2(t, ^^ 0))m2(t,x),

satisfy Conditions 1 and (18), are known, sufficiently smooth and bounded functions.

Given the functions ui(t, x), vl(t, x), g(t), i _ 1,2 are classical solutions of the original problem, so they are continuous and bounded together with their derivatives in Z4(G[0,T]). Using the theorem (continuous dependence of the solution on the initial data) taking into account the notation (19) we get

\\F3\\ <C sup \m1l(S,x) - m1(£,x)\ < C\\'m\(t,x) - m21(t,x)\\1 , 6,

\\G3\\ < sup sup

xeE^^^t

g1(t)(m2(£, x) - m2(S,x)) - 0) - m2(^ 0))m2(t,x)

m2(t 0) < C\\m^(t,x) - m22(t,x)\\126■

<

\\U(t,x)\\124 + \\V(t,x)\\124 < C(\\m1(t,x) - m2(t,x)\\1 , 6 + \\m12(t,x) - m2(t,x)\\1 , 6)- \\m1(t,x) - m2(t,x)\\126 + \\m'1(t,x) - m2(t,x)\\1,6■

\Q\ ^ sup sup

xeE^^^t

-Vxx(Z, 0) - bu(S)U(S, 0) - g1(S)(m12(S, 0) - m2(S, 0))

<

!(S, 0)

< C(Wm1 (t,x) - m22(t,x)\\h6)■

m

Thus we have: if the input data are close, the corresponding solutions in the introduced norm are close too.

References

[1] O.A.Bender, T.N.Shipina, On the inverse Cauchy problem for a nonlinear parabolic system of equations, Proceedings of the XLII regional scientific student conference on mathematics and computer science, Krasnoyarsk, SFU, 2009, 52-54 (in Russian).

[2] I.V.Frolenkov, Yu.Ya.Belov, On the existence of solutions for a class loaded two-dimensional parabolic equations with the Cauchy data, non-classical equations of mathematical physics, Novosibirsk, Institut of Mathematics, 2012, 262-279 (in Russian).

[3] I.V.Frolenkov, M.A.Darzhaa, On the existence of solution of some problems for nonlinear loaded parabolic equations with Cauchy data, Journal of Siberian Federal University. Mathematics & Physics, 7(2014), no. 2, 173-185.

[4] I.V.Frolenkov, G.V.Romanenko, On solvability of some special systems of one-dimensional loaded parabolic equations and composite-type systems with Cauchy data, Journal of Applied and Industrial Mathematics , 8(2014), no. 2, 196-207 (in Russian).

[5] G.A.Spichak, T.N.Shipina, Problem of identification of coefficients in a non-linear system of equations of parabolic type, Intern. conf. dedicated to the 80 birth anniversary of acad. M.M.Lavrentyev "Inverse and incorrect problems of mathematical physics", Novosibirsk, Sib. scien. publish., 2012.

[6] M.A.Yarovaya, I.V.Frolenkov, On the Cauchy problem for a one-dimensional loaded parabolic equation of special form, Graduate work, Siberian Federal University, Krasnoyarsk, 2015 (in Russian).

О единственности и непрерывной зависимости от входных данных решения системы двух нагруженных параболических уравнений с данными Коши

Игорь В .Фроленков Ирина С.Антипина Наталья М. Терских

Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный, 79, Красноярск, 660041

Россия

Мы 'рассматриваем задачу Коши для системы одномерных нагруженных параболических уравнений. Доказана единственность и непрерывная зависимость решений от входных данных в классе гладких ограниченных функций.

Ключевые слова: нагруженное уравнение, система параболических уравнений, метод слабой аппроксимации, задача Коши, единственность, непрерывная зависимость решения от входных данных.