DETERMINATION OF NON-STATIONARY POTENTIAL ANALYTICAL WITH RESPECT TO SPATIAL VARIABLES Текст научной статьи по специальности «Математика»

DOI: 10.17516/1997-1397-2022-15-5-565-576 УДК 517.958

Determination of Non-stationary Potential Analytical with Respect to Spatial Variables

Durdimurod K. Durdiev*

Bukhara Branch of the V. I. Romanovskiy Institute of Mathematics Academy of Sciences of the Republic of Uzbekistan

Bukhara, Uzbekistan

Zhanna D. Totieva^

Southern Mathematical Institute of the Vladikavkaz Scientific Centre of the Russian Academy of Sciences Vladikavkaz, Russian Federation North-Caucasus Center for Mathematical Research of the Vladikavkaz Scientific Centre of the Russian Academy of Sciences Vladikavkaz, Russian Federation

Received 31.10.2021, received in revised form 21.03.2022, accepted 28.06.2022 Abstract. The inverse problem of determining coefficient before the lower term of the hyperbolic equation of the second order is considered. The coefficient depends on time and n spatial variables. It is supposed that this coefficient is continuous with respect to variables t, x and it is analytic in other spatial variables. The problem is reduced to the equivalent integro-differential equations with respect to unknown functions. To solve this equations the scale method of Banach spaces of analytic functions is applied. The local existence and global uniqueness results are proven. The stability estimate is also obtained.

Keywords: inverse problem, Cauchy problem, fundamental solution, local solvability, Banach space. Citation: D.K. Durdiev, Z.D. Totieva, Determination of Non-stationary Potential Analytical with Respect to Spatial Variables, J. Sib. Fed. Univ. Math. Phys., 2022, 15(5), 565-576. DOI: 10.17516/1997-1397-2022-15-5-565-576.

1. Introduction and problem formulation

The inverse problem of determining coefficient a(t,x,y), t G R, (x,y) = (x,y1,... ,ym) € R1+m, before the lower term of the hyperbolic equation is studied in this paper. The problem is considered in the class of coefficients that are continuous with respect to variables t,x and it is analytic in variable y. It is known that such problems are referred to as multidimensional inverse problems. For multidimensional inverse problems there are only special cases for which solvability is established. One of such classes of functions in which local solvability takes place is the class of analytic functions. The technique used here is based on the scale method of Banach spaces of analytic functions developed by L. V. Ovsyannikov [1,2] and L. Nirenberg [3]. This method was first applied to the problem of solvability of multidimensional inverse problems by V. G. Romanov [4-6].

*d.durdiev@mathinst.uz

tjannatuaeva@inbox.ru https://orcid.org/0000-0002-0089-074X © Siberian Federal University. All rights reserved

This method was used to study multidimensional inverse problems of determining the convolution kernel in parabolic and hyperbolic integro-differential equations of the second order; theorems of local unique solvability of inverse problems in the class of functions with finite smoothness with respect to time variable and analytic with respect to spatial [7-13]. variables. This paper generalizes the results given in [4] (Sec. 3) for the case of non-stationary potential.

Let us consider the problem of determining a pair of functions u and a that satisfy the following equations

utt - uxx - △u - a(t, x, y)u = g(y)S(x)S'(t - to), (t, x, y) G R2+m, to > 0, (1)

u|i<o= 0 (2)

where △ is the Laplace operator with respect to variables (yi,..., ym) = y, S(-) is the Dirac delta function, S'(■) is the derivative of the Dirac delta function, t0 is a problem parameter. Therefore u = u(t,x,y,t0), and g(y) is a given smooth function so that g(y) =0 for y G Rm.

It is required to find potential a(x,t,y) in (1) if the solution of problem (1)-(3) is known for x = 0, i.e., the condition

u(t, 0, y,to) = f (t, y,to), t> 0, to > 0 (3)

is given.

Following monograph [4, sec. 3], we consider the Banach space As(r) s > 0 of functions ^(y), y G Rm which are analytic in the neighbourhood of the origin and they satisfy the following relation

A s\a\

|M|s(r):= sup — \Dav(y) \ < \y\<r |a|=o

dM

D = dy? ...dymr ,a = (ai,...,am),

|a| := ai +-----+ am, a' := (ai)! .. . (am)'.

In what follows, parameter r is fixed while parameter s is variable. Then, it is formed a scale of Banach spaces As(r), s > 0 of analytic functions. The following property is obvious: if ^(y) G As(r) then ^(y) G As'(r) for all s' G (0,s). Consequently, As(r) c As'(r) if s' G (0, s) and the following inequality is valid

|| Da || . iMUM IID WU' < Co

Here r > 0, s > 0 and

' / \ a\

(s - s')'

for any a with constant ca which depends only on a. Solution of problem (1), (2) is considered in the form

u(t, x, y, to) = 2g(y)S(t - to - \x\) + v(t, x, y, to).

Substituting this expression into (1) and taking into account that (1/2)g(y)S(t -to - \x\) satisfies (in a general meaning) equation utt - uxx = g(y)S(x)S'(t - to), we obtain the following problem for function v:

vtt - vxx = Av + 2Ag(y)S(t - to - |x|)+

+ a(t,x,y)

2g(y)s(t - to - |x|) + v(t, x, y, to)

(4)

, (t,x,y) e R2+m, to > 0,

v |t<0= 0. (5)

In the next section inverse problem (4), (5) and (2) is replaced with the equivalent integro-differential equations. In what follows, we assume that function a is even in x.

2. Reduction of the problem to integro-differential equations

According to the d'Alembert formula the solution of problem (4)-(5) satisfies the integral equation

v(t,x,y,to) = 2 Jj |Av(r,e,y,to) + 1 Ag(y)S(r - to - |£|)+

A(t,x)

+ a(^ £,y)

2 g(y)5(r - to - |£|) + v(r,£,y,to)

| d£dT, (t,x,y) e M2+m, to > 0,

(6)

where

A(t,x) = {(t,£)| 0 < t < t -x - £1 x - t < £ < x +1} .

Let

Qt := {(t,to^ 0 < to < t < T}, T> 0, QT := {(t,x^ 0 < |x| < t < T - |x|},

YT := {(t,x,to) x + to < t < T - Yxl 0 < to < t < T}.

Domain YT in the space of variables x, t, to is the pyramid with the base QT and vertex (0,T,T).

It follows from (6) that function v(t,x,y,to) satisfies the integral equation

x + (t-to)

v(t,x,y,to) = Agy)(t - to) + g(y) J a(to + ^l^y^

1

+2

4

x-(t2-to) (7)

[Av(T,£,y,to) + a(T,£,y)v(T,£,y,to)] dTd£, (t,x,to) e Yt, y e Rm,

a(t,x,to)

where 0(t) = 1, t ^ 0, 0(t) = 0, t < 0, and U(x,t,to) is domain in the form of a rectangle in the plane of variables (t, £) for each fixed to formed by characteristics passing through the points (0,to) and (x,t) of the differential operator d2/dt2 - d2/dx2 :

D(x

,t,to) :={(£,

t )

, x — (t—to) x +1 — to ]

£ + to <T <t -x - £1 -(—^ <£< + 0 o, 0 <to <t

Obviously, the equalities f (t,y,to) = u(t, 0,y,to) = v(t, 0,y,to), t > to are true. Besides,

f (to + 0, y, to) = v(t, 0, y, to) |t=t0+o= 0.

First note that if a(t, -x,y) = a(t,x,y) then v(t, -x,y,to) = v(t,x,y,to). Taking the derivative with respect to t of the both sides of equation (7), we obtain

vt(t, x, y, to) =

△g(y) + g(y)

e

x + t + to x + t - to \ f -x + t + to x - t + to a I -^-,-o-,y 1 + a I ---,---,y

+

x + (t-to)

+ 1 J [△v (t - \x - t\,t,y,to)+ a (t - \x - t\,£,y) v (t - \x - t\,£,y,to)] d£,

x-(t-to) 2

(t,x,to) G Yt, y G Rm.

Setting x = 0 in this relation and using evenness of functions a(t, x, y), v(t, x, y, to) with respect to x, we obtain the equality

△g(y) , g(y) (t - to t +1

ft(t,y,to) = + ^a I —,y I +

ft - to t + to N

V 2 , 2 ,y)

+ / [△v (t - Z,£,y,to) + a (t - Z,£,y) v (t - Z,£,y,to)] d£, (t,to) G Qt , y G Rm

o

Substituting \x\ for (t - to)/2 and t for (t + to)/2 and solving with respect to a(t, x, y), we rewrite this equation in the form

tt

o

. 2^g(y) 4 4

a(t, x, y) = ——--+ —— ft (t + \x\,y,t - \x\) -

\x\

g(y) g(y)

g(y)

△v (t + \x\-Z,£,y,t-\x\) +

+ a (t + \x\- Z,£,y) v (t + \x\- Z,£,y,t -\x\) d£, (t,x) G Ht , y G Rm

(8)

Thus, in order to find the value of function a at the point (t, x, y) it is necessary to integrate function a(t,x,y) itself over the segment with boundaries (t + \x\, 0, y, 0), (t, \x\, y, 0) and function v(t,x,y,to) over the segment with boundaries (t + \x\, 0,y, t - \x\), ( t, \x\,y, t - \x\) which belong to domain YT x Rm.

Note that function v, even with respect to x = 0, satisfies the condition dv/dx\x=o. Taking into account this fact and considering equations (4), (5), (3) for v in the domain x > 0, we obtain

d2v d2v

— - -— -△v - a(t, x,y)v = 0, 0 <x <t - to, y G Rm, dt2 dx2

.. . dv

=f(t,y,to), dx

= 0, 0 <t - to < T, y G R"

. Then in accordance with the d'Alembert formula which gives the Cauchy problem solution with an initial data at x = 0 we find

v(t, x, y, to) = vo (t,x,y,to) + 2 if △v (r,i,y,to)+ a (r,£,y) v (r,£,y,to) d£dr, (9)

where

A' (t,x)

vo(t, x,y,to) = ^ [f (t + x, y, to) + f (t - x, y, to)]

o

v

x

x

△'(t,x):= {(t,£) | 0 <£<x, \t - t\ <x - £} , 0 <x<t - t0 <T - x, y € Rm. Considering (8) for x > 0, we have

a(t,x,y) = ao(t,x,y)-

△v (t + x - £,£,y,t - x)+ a(t + x - y)v(t + x - £,£,y,t - x) d£,

g(y)

(10)

where

a0(t,x,y) =ft(t + x,y,t - x) + , 0 ^ x < t < T - x, y G Rm.

g(y)'

g(y)

The system of equations (9), (10) is a closed integro-differential equations for functions a, v. Note that operator △ for function v appears in the system only under the integral sign. Next we consider system (9), (10) in domain

DT = Y'T x Rm, Y'T = {(t,x,t0)\ 0 < x +10 < t < T - x} .

3. The main results and proofs

Let C{txt0) (Y'T; As0) denote the class of functions with values in As0 (so > 0) which are continuous with respect to variables (t,x,t0) in domain Y'T. For fixed (t,x,t0) the norm of function v(t,x,y,t0) in As0 is denoted by ||v||s0 (t,x,t0). The norm of function v in C(t,x, t0) (Y'T; As0) is defined by the equality

(t,x,t0'( T s0) (t,x,t0)er'T

Let C(t,x) (GT; As0) be a class of functions with values in As0 which are continuous with respect to variables (t, x) in domain GT = {(t, x) | 0 ^ x ^ t ^ T - x}. For fixed (t, x) the norm of function a(t,x,y) in As0 is denoted by ||a||s0(t,x). The norm of function a in C(t,x) (GT; As0) is defined as

\\a\\c

C(t,x) (Gt; )

sup \\a»so (t,x).

(t, x)€gt

Let us also denote the class of functions with values in As0 which are continuous with respect to t, t0 in domain QT by C (QT; As0).

Theorem 3.1. Let f (+t0,y,t0) = 0, \g(y)\ ^ g0 > 0, g0 is a known number and

{w>, € As0; {f (t,y,t0), ft(t,y,t0 )}€ C (Qt ; AS0),

in addition, the relations

△g(y)

max < 2

g(y)

max \\f (t,y,to

(t, to)€QT

(t, to)€QT

4ft(t,y,to)

g(y)

R

< —

2

are valid for some fixed s0 > 0, R. Then there is such a number b € (0,T/(2s0)), b = b (s0, R, T) that for each s € (0, s0) in domain DT n {(t, x,y,t0) : 0 ^ x +10 ^ b(s0 - s)} there exists the unique solution of equations (9), (10) and v(t,x,y,t0) € C(t,x,t0) (PsT; As0),

0

max

so

a(t,x,y) G C(t,x) (KsT ; As0 ), where Pst = Y'T n {(t,x,to) : 0 < x + to <b(so — s)} , Kst GT n {(t, x, to) : 0 ^ x + to < b(so — s)} , moreover

||v — vo||s (t,x,to) < R, (t,x,to) G PsT, R

||a — ao|s (t,x) < -, (t,x) G KsT.

s so — s

Proof. Under the conditions of Theorem 1 we have

vo G C(t,x,to) (YT; Aso ) , ao G C(t,x) (GT; Aso ) ,

||vo|s(t,x,to) < R, (t,x,to) G Y't, ||ao||s(t,x) < R, (t,x) G Gt, 0 < s < so. Let bn be the member of the monotone decreasing sequence that is defined by the equalities

bn

b

n = 0, 1, 2, ....

Let

'n+1 1 + 1/(n + 1)2'

TO

b = lim bn = &oTT (l + 1/(n +1)TX.

n

The number bo G (0, T/(2so)) is chosen in an appropriate way. For the system of equations (9), (10) the process of successive approximations is constructed according to the following scheme

1

+ 2

Vn+i(t,x,y,to) = vo(t,x,y,to)+ △vn (t, Ç, y,to) + an (t, Ç, y) vn (t, Ç, y, to)

dTdÇ, 0 ^ x ^ t — to ^ T — x,

△ ' (t,x)

4 x

g(y) Jo

a-n+i(t,x,y) = ao(t,x,y) — △Vn (t + x — Ç,Ç,y,t — x) + an(t + x — Ç, Ç,y)vn(t + x — Ç,Ç,y,t — x) 0 < x < t < T -x.

dÇ,

Function s'n(x) is defined by the formula

s'n(x) = , Vn(x) = so — x.

(11)

Let us introduce the following notations: pn = vn+\ — vn, qn = an+i — an, n = 0, 1, 2,.... Then pn, qn satisfy the relations

po(t,x,y,to) = 1 JJ △vo (r,^,y,to) + ao (r,£,y) vo (r,^,y,to) drd£, (t,x,y,to) G Dt,

△ ' (t,x)

4 fx (

qo(t,x,y) = — gy) J |Avo (t + x — $,$,y,t — x) + + ao(t + x — y)vo(t + x — £,£,y,t — x^d£, (t, x, y) G Gt x rm;

Pn+l(t,X,y,to) = 1 JJ ^Apn(T,£,y,to) + △ ' (t,x)

+ qn (T,£,y) vn+i(T,£,y,to) + an (T,£,y)Pn (T,£,y,to)\ dTd£, (t,x,y,to) G Dt,

Qn+i(t,x,y) = —■

4

( + x — £,y,t — x)+

g(y) J 0

+ qn(t + x - y)vn+1(t + x - £,£,y,t - x)+ an(t + x - £,y)pn(t + x - £,£,y,t - x)

(t,x,y) € GT x Rm.

( T \

Let us show that b0 € 0^ -— can be chosen so that the following inequalities be valid for

all n = 0, 1, 2, ■■■ :

Xn = max< sup

l(t,x,s)EFn

\\pn\\s(t,x,to)

vn(x) — s

sup

(t,x,s)eFn

>(t,x)

\\Vn+i — vo\\s(t,x,to) < R, \\an+i — ao||s(t,x) <

(vn(x) — s)2

R

so — s

< œ, (12) (13)

where

Fn = {(t,x,to,s)\ (t, x, to) G Y'T, 0 < x + to <bn(so — s), 0 <s<so}. Fn = {(t,x,s)\ (t,x) G Gt, 0 < x < bn(so — s), 0 < s < so} .

Indeed, using the relations for pn, qn, one can find

\po\s(t,x,to) < 2

△ ' (t,x)

1

< -2

\\Avo\\s (T,£,to) + \\ao\s (t,£)\vo\s (T,£,to) Rco

dTd£ <

(s'o(0 — s)2

+ R2

dTd£.

△ ' (t,x)

Here co is a positive constant such that

\\Avo\\s < co

(s'n — s)

s'n > s> 0, n = 0,1, 2,...

2n

It is easy to check that c0 = 4m. Taking function s'n(£) from (11) for n = 0, we have

1 fx

\\po\s(t,x,to) < 2 J (x — 0

4Rco

(v o(£) — s)2

+ R2

1

< 2 R [4co + s

(x — №

(vo(0 — s)2

<

< 2boR [4co + s2oR —

vo(x) — s '

(t, x, s) G Fo

x

x

s

x

o

x

In a similar way we obtain

iMUM) < 4ao ! 4Rc°

(vo(0 — s)2

+ R2

< 4goR {4co + s2oR)—^~-^, (t,x,s) G Fo.

(vo(x) - s)2

These estimates imply that inequality (12) is valid for n = 0. Moreover,we find

iv - voIIs(t,x,to) = IIpoIIs(t,x,to) < o — < A°\ = bo^o, (t, x, to, s) G Fi,

vo(x) - s 1 - bi/bo

ii ii / \ ii ii / \ Aox 4boXo / \7-i

||ai - aoIIs(t,x) = ||ao||s(t,x) < o-r^ < -, (t,x,s) G Fi.

(vo(x) - s)2 so - s

Choosing bo so that 4boAo < R, one can conclude that inequalities (13) are satisfied for n = 0. By way of induction, one can show that inequalities (12) and (13) are also valid for other values of n if bo is chosen suitably. Let us assume that inequalities (12) and (13) hold for n = 0, 1, 2,...,i. Then (t,x,to, s) G Fi+i and we have

Hpi+1Hs(t,x,to) < 1 U {||Api||s (r,Ç,to) +

△ ' (t,x)

+ ||qi||s (r,i) ||vi+i||s (r,i,to) + ||ai||s (r,i) UpiUs (r,t,to)} drdi <

1

< -2

A(t,x)

co\i£ 2RXiÇ + XiÇ R(1 + so)

+ / NO +

L(si(0 — s)2(vi(£) — s) (vi(£) — s)2 (vi(£) — s) (so — s) \

drd£ <

^ Xi (A .Qï? ï? 2\ C (x — /

< — {4co + 3Rso + Rso) Jo (vi+i(0 — s)3 <

Xi -, 2 ( . ~ T-v 2) x

< 2b2o (4co + 3Rso + Rs2o) ------.

2 vi+i(x) — s

Here function si is defined by equality (11) with n = i and the inequalities

HviHs(t,x,to) < 2R, HaiHs(t,x) <

so — s

are used. The latter is valid by the induction hypothesis together with the obvious inequalities bi < bo and vi+i(x) < vi(x). Similar arguments for qi+1 lead to inequalities

Il II (t f7 coXiÇ , 2RXiÇ + XiR(1 + so)£\d^

||qi+l||s{t,x) < 4go < YTT^-\2< ite\-v + -+ t ite\-\T (d£ ^

Jo l(si(0 — s)2(vi(0 — s) vi(0 — s (vi(0 — s)2 )

< 4Xigo [4co + 3Rs2o + Rso] jo {vi+i— s)3 <

< 4Xigobo [4co + 3Rs° + Rso^-j—+Ti~\-, (t, x, s) G Fi+i.

^^J (vi+i(x) — s)2 ■ The obtained estimates yield

Xi+i < Xip, Xi+i < œ,

p = b0 max Moreover, we have

2 (4c0 + 3Rs0 + Rs2) ; 4g0 (4c0 + 3Rs° + RS0)

i+1 i+1 \ x i+1 X b

^i+O - v0h(t,x,t0) h(t,x,t0) vn(2)- s 1 V+Zh ^

n=0 n=0 ( ) n=0 i+0Z n

i+1 i+1 ^Y, Xnbn(n + 1)2 < X0b0^ pn(n +1)0, (t,x,t0,s) € Fi+0,

0 n=0

i+1 i+1 X x 1 i+1 X b hi+o- ^is^^") hn L(t,x) < V , n/ n—^ <-Y] jz—n , 0 <

n=0 n=0(vn(x) - s) s0 - s n=0 (1 - bi+0/bnf

i+1

S0 - s

< pn(n +1)4, (t,x,s) € Fi+0.

Cn — Q i-'

n=0

Now we choose b0 € ^0, so as to obtain

p< 1, X0b0^ pn(n +1)4 < R.

n=0

Then

Hvi+0 - v0Hs(t,x,t0) < R, (t,x,t0,s) € Fi+0, R

Hai+0 - a0||s(t,x) < -, (t,x, s) € Fi+0.

s0 - s

Since the choice of b0 is independent of the number of approximations, all successive approximations vn, an belong to

C(t,x,t0) F As) , F=f) F

and

n

n=0

C(t,x) (F; As), F = p| Fn,

n=0

respectively. Moreover,

Hvn - v0||s(t, x,t0) < R, (t, x,t0, s) € F, R

Han - a0||s(t, x) < -, (t,x,s) € F.

s0 - s

For s € (0, s0) the series

w w

vn - vn-1), E(an - an-1)

n=0 n=0

converge uniformly in the norm of the spaces

C(t,x,t0) (PsT; As), PsT = YT n{(t,x,t0):0 < x + t0 < b(s0 - s)} , C(t , x) (KsT; As), KsT = Gt n {(t,x,t0) : 0 < x +10 < b(s0 - s)} .

Therefore vn ^ v, an ^ a and the limit functions v, a are elements of C(t,x,t0) (PsT; As), C(t,x) (KsT; ^s) respectively and they satisfy equations (9), (10).

Now we prove that this solution is unique. Let us assume that (v, a) and (v, a) are any two solutions that satisfy the inequalities

||v - vo\\s(t,x,to) < R, (t,x,to, s) e F,

||a - aoys(i,x) <

R

so - s

(t,x, s) G F.

Let us denote p = v — v, q = a — a,

X := max< sup

Ht,x,t0,s)eF

lplls(t,x,t0)

v (x) — s

sup

(t,x,s)eF

(t, x)

(v(x) — s)2

< oo

, where v(x) = s0 — x/b, b = b0 (l + 1/(n + l)2) . Then the following relations can be

n=0

obtained for functions p, q

p(t,x,y,t0) = 2 Ii {Ap(T,£,y,t0) + q(T,£,y) v(T,£,y,t0) + a (T,£,y) p(T,£,y,t0)} drdÇ,

A' (t,x)

(t,x,y,t0) G Dt,

q(t,x,y) = — g-) ^ |Ap (t + x — — x) +

+q(t + x — y)v(t + x — £,£,y,t — x) + a(t + x — y)p(t + x — £,£,y,t — x^d£,

GT x Rm

T

Let us show that b0 e 0,- can be chosen so that the following inequalities are valid for all

y 2so J

n = 0, 1, 2,.... Applying the estimates given above to these equations, we find the inequality

A < Xp',

1 , „ .1

< p < 1.

1 (4c0 + 3Rs0 + Rs20) ; (4c0 + 3Rs2 + Rs0)

2 g s

p := b max

Consequently X = 0. Therefore v = v, a = a. Theorem 1 is proved. □

Let us consider the set r of functions f (t, y, t0) representing the elements of C(QT; Aso ), s0 > 0 for which conditions of Theorem 1 are valid with R, T, s0. Then we have the stability theorem

Theorem 3.2. Let f, f G T. For the corresponding solutions (v,a) and (v,a) of (9), (10), we have

||v — v||s < cM, (t,x,t0) G PsT, ||a — alls (t,x) G KsT, 0 < s < s0, (14)

s0 — s

where

M = max [max f — f || so (t,t0), max ft — ftUso (t,t0) , (t,t0) G Qt ,

and constant c depends on R, T, s0.

x

x

Proof. Taking into account (9)-(10), we obtain the following equalities for the differences v - v = v, a - a = a and f - f = f

v(t,x,y,t0) = v0(t, x, y, t0) + \jj | Av(r,£,y,t0) +

(t,x) (15)

+ a (T, Ç, y) v(t, Ç,y,to) + a (t, Ç, y) V (t, y,to)} dTdÇ, (t, x, y,to) G Dt ,

4 fx (

a(t, x, y) = ao(t, x, y) - gy) J {△v (t + x - Ç,Ç,y,t - x) +

+ a(t + x - Ç, Ç, y)v(t + x - Ç,Ç,y,t - x) + a(t + x - Ç,Ç, y)v(t + x - Ç,Ç,y,t - x^dÇ, (16)

(t,x,y) G GT x Rm,

where

1 v v 4 v Vo(t,x,y,to) = ~[f(t + x,y,to) + f (t - x,y,to)], ao(t,x,y) = —— ft(t + x,y,t - x).

2 g(y)

It is obvious that

IVolUo(t,x,to) < M, (t,x,to) G PsT,

4 (17)

llàollso (t,x) < M, (t,x) G KsT.

rnynUo

We have from Theorem 1 that

R(l + so)

IIv|s < 2R, ||a|s <

so - s

Applying the method of successive approximations used for the proof of Theorem 1 to the system of equations (15)-(16) (it is linear with respect to v and a), we find that the following inequalities are valid for solution of (15)-(16)

||v - v0||s(t,x,t0) < C1M, (t,x,t0) € PsT,

||a - aoHs(t,x) < C1M , (t,x) € KsT, 0 < s < s0,

s0 - s

where c1 depends on R, T, s0. Hence, taking into account (17), we find that inequalities (14) are true. Theorem 2 is proved. □

The work was supported by the Ministry of Science and Higher Education of the Russian Federation (Grant/Award Number 075-02-2022-896).

References

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Определение нестационарного потенциала, аналитического по пространственным переменным

Дурдимурод К. Дурдиев

Южный математический институт ВНЦ РАН

Bukhara, Uzbekistan

Жанна Д. Тотиева

Южный математический институт ВНЦ РАН Владикавказ, Российская Федерация Северо-Кавказский центр математических исследований ВНЦ РАН

Владикавказ, Российская Федерация

Аннотация. Изучена обратная задача определения коэффициента зависимости временных и n пространственных переменных для младшего члена гиперболического уравнения второго порядка. Предполагается, что этот коэффициент непрерывен по отношению к переменным t, x и аналити-чен по другим пространственным переменным. Задача сводится к эквивалентной системе нелинейных интегро-дифференциальных уравнений относительно неизвестных функций. Для решения этих уравнений применяется метод шкал банаховых пространств аналитических функций. Доказаны теоремы локальной разрешимости и единственности в глобальном смысле. Получена оценка устойчивости обратной задачи.

Ключевые слова: обратная задача, фундаментальное решение, задача Коши, локальная разрешимость, устойчивость.