The generalized splitting theorem for linear Sobolev type equations in relatively radial case Текст научной статьи по специальности «Математика»

Серия «Математика» 2014. Т. 7. С. 19—33

Онлайн-доступ к журналу: http://isu.ru/izvestia

ИЗВЕСТИЯ

Иркутского государственного университета

УДК 517.9

The Generalized Splitting Theorem for Linear Sobolev type Equations in Relatively Radial Case

S. Zagrebina, M. Sagadeeva South Ural State University

Abstract. Sobolev type equations now constitute a vast area of nonclassical equations of mathematical physics. Those called nonclassical equations of mathematical physics, whose representation in the form of equations or systems of equations partial does not fit within one of the classical types (elliptic, parabolic or hyperbolic). In this paper we prove a generalized splitting theorem of spaces and actions of the operators for Sobolev type equations with respect to the relatively radial operator. The main research method is the Sviridyuk theory about relatively spectrum. Abstract results are applied to prove the unique solvability of the multipoint initial-final problem for the evolution equation of Sobolev type, as well as to explore the dichotomies of solutions for the linearized phase field equations.

Apart from the introduction and bibliography article comprises three parts. The first part provides the necessary information regarding the theory of p-radial operators, the second contains the proof of main result about generalized splitting theorem for strongly (L, p)-radial operator M. The third part contains the results of the application of the preceding paragraph for different tasks, namely to prove the unique solvability of the multipoint initial-final problem for Dzektser and to explore the dichotomies of solutions of the linearized phase field equations. References not purport to, and reflects only the authors' tastes and preferences.

Keywords: linear Sobolev type equations, generalized splitting theorem, dichotomies of solutions, multipoint initial-final problem.

Let U and F be Banach spaces, the operators L £ L(U; F) (linear and continuous) and M £ Cl(U; F) (linear, closed and densely defined). Let the operator M be a relative p-radial with respect to operator L (or shortly, (L, p)-radial operator), p £ {0} UN (terminology and the results see in sec. 3 [12]). We note only that the concept of (L, p)-radial operator M was introduced by G.A. Sviridyuk in [9].

Let us consider a linear Sobolev type equation

Lu = Mu.

(0.1)

Sobolev type equations now constitute a vast area of nonclassical equations of mathematical physics. Detailed historical review of these studies and an extensive bibliography can be found in the book [1].

In this paper we prove a generalization of the splitting theorem of the spaces U and F on invariant subspaces in accordance to the splitting of relative L-spectrum of operator M. As you know, firstly the splitting theorem formulated and proved by G.A. Sviridyuk [8] in the case of (L,p)-bounded and (L, p)-sectorial operators M. A.V. Keller developed these results in [10] and applied them to the study of the dichotomies solutions [3]. Splitting theorem in the case of (L, p)-radial operator M with respect to the dichotomy of the solutions appeared, for example, in [5] (see in more detail in [6]). First proof of the generalized splitting theorem in the case of (L, p)-bounded operator M appeared in [13].

The need for generalized splitting theorem appeared in the study of multipoint initial-final conditions

for linear equations of Sobolev type (0.1). In these parts tj £ R (tj < tj+1), Uj £ U and Pj is a relative spectral projections (talking about them will go to claim 1 this article), j = 0,n. Note that if n = 1 then the condition (0.2) take a simpler form

then there will be a initial-final condition [14]. Problem (0.1), (0.3) has been very extensively studied at various aspects in [4], [15], [16].

Apart from the introduction and bibliography article comprises three parts. The first part provides essential information regarding the theory of relatively p-radial operators [12], the second one contains the main result, namely the proof of the generalized splitting theorem in the case of (L,p)-radial operator M. The third part contains the application of the preceding paragraph for different tasks, namely to the prove the unique solvability of the multipoint initial-final problem for Dzektser and to explore the dichotomies of solutions of the linearized phase field equations. References not purport and reflects only the authors' tastes and preferences.

Finally, we note that all considerations are conducted in real Banach spaces, but when considering the «spectral issues» introduced their natural complexification. All circuits oriented counterclockwise movement and limit area lying to the left in such a motion. Symbols of O and I denote, respectively, the «zero» and «identity» operators whose domain is clear from the context.

In conclusion, the authors consider it their pleasant duty to express their sincere gratitude to G.A. Sviridyuk for his interest and active creative discussions.

=0, 3 =0,П,

(0.2)

Po(и(то) - Uo) = Pi(u(Ti) - Ui) = 0,

(0.3)

THE GENERALIZED SPLITTING THEOREM FOR LINEAR SOBOLEV 1. Relatively p-radial operators

Recall the standard notation of the theory of relatively p-radial operators

[9], [12].

Suppose, as above, U and F be Banach spaces, operators L £ L(U; F) (linear and continuous) and M £ Cl(U; F) (linear, closed and densely defined). Denote

pL(M) = [f £ C :(pL - M)-1 £ L(F;U)}, aL(M) = C \ pL(M),

RL(M) = (pL - M)-1L, LL(M)= L(fL - M)-1, f £ pL(M),

p p

R(X,p)(M) = I\Ri(M)> LfA)P)(^) = nLAfc(M)' hepL(M)(k = 0jj). k=0 k=0

Definition 1. Operator M is called p-radial by relative to operator L (or shortly (L,p)-radial) if

(i) 3p £ R (p, C pL(M);

(ii) 3K> 0 Vf = (fo,fi ,...,fp) £ (p, +x>)p+1 Vn £ N

K

max{\\(R^p)(M)T\\m), \\{LLM{M)T\\m} <

Also introduce the notation

U0 = kerR^p)(M), F0 = kerLL{tl>p)(M), L0 = L

ft (ßk - ß)n k=0

, M0 = M

U0

domM nU°

By U1 (F1) denote the closure of the lineal imR^p)(M) (imL^p)(M)). Also by U (F) denote the closure of the lineal UHimR^ p) (M) (FHimL^ p) (M)) in the norm U (F).

Definition 2. Vector-valued function ugC1(R+;11) is called the solution of the equation (0.1) if it satisfy to this equation on R+ = {0} U R+.

Definition 3. The closed set P C U is called the phase space of the equation (0.1), if

(i) any solution u(t) of the equation (0.1) implies in P (pointwise);

(ii) there the unique solution of the Cauchy problem

u(0) = uo (1.1)

O

for the equation (0.1) with any uo of a lineal P which is dense in P.

Together with the equation (0.1) consider the equivalent equation

L(aL - M)-1f = M(aL - M)-1f, a £ pL(M). (1.2)

Theorem 1. Let the operator M be a (L,p)-radial. Then the phase space of equation (0.1) ((1.2)) is set U1 (F1).

Definition 4. The family of operators IJ' : R+ —> C(lI) is called a resolving semigroup of equation (0.1), if

(i) UsUl = Us+t Vs,t£ R+;

(ii) u(t) = Ulu0 is a solution of this equation for any u0 from some lineal witch dense in U;

(iii) restriction of the semigroup identity on the phase space P of the

equation is U0 = I.

P

Semigroup {U* £ C{ 11) : t £ R+} called exponentially bounded with constants C,f3 if 3C > 0 3/3 £ R Vi G R+ WU^^u) < Ce^.

Theorem 2. Let the operator M be a (L,p)-radial. Then there exists strongly continuous resolving semigroup of equation (0.1) ((1.2)) considered on subspace U (F). And it is exponentially bounded with constants K, /3 from definition 1.

Remark 1. Operators of the resolving semigroup for (0.1) ((1.2)) with t > 0, as amended, are discussed in [7], can be represented as

II1 = s- lim ( (L — yM^] L| = s- lim (-RUM)]

k->oo y \ k J I k^oo \t I )

F* = s- lim | = s- lim (-LLk{M)

k->oo \ \ k J I k->oo \t 7

Remark _2. The identity of the semigroup {U1 £ £(il) : t £ R+} ({F* £ C(F) : t £ R+}) is a projector P = lim IJ1 (Q = lim Fl) along 11° (F°) on

U1 (F1).

Definition 5. Operator M is called strongly (L,p)-radial if for any X,^0,^1, ...,^p > / the next conditions are fulfilled

o o

(i) there is dense in F lineal F such that for all f £ F

\\M{\L-M)-'LLM{M)fh< C°nSt(/) •

(A - в) П Ы - в) k=0

(ii) \\RLM(M)(AL - M)-1||£№u) <

K

(A - - P)

k=0

Theorem 3. Suppose that M is strongly (L,p)-radial. Then (i) U = U0 © U1, F = F0 © F1;

(ii) Lk = L

Uk

GL(Uk ; Fk ), Mk = M

GCl(Uk ; Fk ),

domMk

domMk = domM n Uk, k = 0,1;

(iii) there exist operators M-1 G L(F0;U0) u L-1 g L(F1;U1).

2. Generalized splitting theorem

Let U and F be Banach spaces, operators L G L(U; F) and M G Cl(U; F), where the operator M is strongly (L, p)-radial, p G {0} U N. We introduce the condition

aL(M) = [J af (M), n G N, and af (M) = there is

n

jL(M), n e N, and aj' j=o

closed loop Yj C C and Yj = dDj, where Dj d aj(M), such that 7Tj n afr(M) = 0 and A n A = 0 for all j,k,l = l,n,k / I.

(2.1)

Consider operators Pj £ £(U) and Qj e £(&), j = j,n, that because of the relative spectral theorem [3] have the form

By the results of [5] operators P0 and Q0 are of the form :

nn

Po = P Pj, Qo = Q Qj. j=i j=i

Lemma 1. Let the operators L e L(U; F) and M e Cl(U; F). The operator M is strongly (L,p)-radial, p e {0}UN and the condition (2.1) holds. Then the operators Pj : 11 —> 11, (j = 1 ,n) are projectors and

(i) PPj = PjP = Pj,j = l,

(ii) PkPt = PtPk =0,k,l = l,n,k^l.

Proof. (i)

111 iL

PP.

J J Rf(M)RL (M)d^dA =

J (2vn)2 I I »

Y Yj

1

(2ni)2

= Qj

\Y Yj Y Yj J

by the Fubini's theorem, the theorem deductions and the right Hilbert relative resolvent identity

RLX(M) - RL(M) = - X)RL(M)RL(M).

The second equality in (i) and proved similarly.

(ii) Now applies the same theorem and the same identity as in (i) when k = l and we obtain

PkPi = j j RL,(M)R^M)d/id\ =

Yk Yl

dX f RLJM)dn + [ Rx(M)dX [ I = O.

(2ni)M J X - vJ ^^ j AV ' J v - X

\Yk Yl Yk Yl

□

Remark 3. Projector P0 by the results of [6] hold similar equalities

(i) PP0 = PoP = Po] _

(ii) P0P3=P3P0=O, j = l,n.

Remark 4. Clearly that for projectors Qj (j = 0,n) the relations of Lemma 1 and Remark 3 are truth by the construction.

By Lemma 1 and Remarks 3, 4 the projectors Pj and Qj, j = 0, n call relative spectral projectors.

We introduce the subspace il1-7 = imPj, F1,7 = imQj, j = 0,n. By construction,

n n

U1 = 0 U1j and F1 = 0 F1j.

j=0 j=0

By the L\j denote the restriction of L on il1-7, j = 0, n and by My denote the restriction of M on domM nil1-7, j = 0,n. It's easy to show that Pjp £ dom M, if p £ dom M then the domain of dom M1j = dom M n U1j is dense in il1-7, j = 0, n.

Theorem 4 (generalized spectral theorem). Let the operators L £ L(U; F) and M £ Cl(U; F), the operator M is strongly (L, a)-radial and the condition (2.1) holds. Then

(i) operators LXj £ £(itlj'; Fy), MXj £ Cl(illj]Slj), j = 0~n;

(ii) there exist operators L^1 £ C(F^it1-7), j = 0,n.

Proof. The proof is based on ideological Sviridyuk proof of splitting theorem [10] so is given in abbreviated form.

(i) L''P' = 2S - ^T'L^d,. = QjLijPj.

W = ~ ^L^i» = QjMtjPjtp.

Yj

where j = 0,n. Next let ip £ domMij then

ivi\ j j — mm 1'

Yj

Since

Mj (pLlj — M\j )-lLlj = fiLij (pLj — M )-lLlj — Lj,

and the rights of equality is a continuous operator, densely defined on Ulj. These continuous operators MljPj can be uniquely complemented to a continuous operator defined on all subspace il1-7. Through MijPj denote this extension j = 0, n. (ii) Since

Lij

( \

J(PL- M)~lQjdji LijipL^ - Mij)~lQjdji = Qj

Yj Yj

and

I \

1

¿r J(fiL - M)~1Qjdfi Ll3 = ¿r y UiLu - Mij)-1 LijPjd/i = P3

Yj Yj

the operatorws L-j £ L(Flj; Ulj) are the restriction operators

1j

1

2ni J

Yj

(pLM )-lQj dp

onto the subspaces F1'7 j = 0,n. □

Under the conditions of Theorem 4 exist operators Sj = LijMij £ Cl(iilj), j = 0~n.

So let the operator M is strongly (L,p)-radial and the condition (2.1) is fulfilled Tj £ M+, {tj < rj+1), uj £ 11, j = 0~n. Take / e C°°(M+;£), j = 0, n. Consider the Sobolev type equations

Lu = Mu + f. (2.2)

We acton to this equation by the series of projectors I — Q and Qj, j = 0, n and obtain an equivalent system of equations

Hu0 = u0 + M-1f0, (2.3)

u

■•-1 = Sj u) + Lj f), (2.4)

j — l-J)jU,j ~r U)j J j

where H = M0"1Lo e L(U0) is a nilpotent operator of degree p e {0} U N, Su = L^Mu G СЩ) and range of a(Sj) = af(M)-, f° = (I - Q)f, fI = Qjf, u° = (I — P)u, u) = PjU, j = 0~n.

3. Applications of the generalized splitting theorem

3.1. Multipoint initial-final problem for the Dzektser equation

Let U and F be Banach spaces, operators L e L(U; F) and M e Cl(U; F). Where the operator M is strongly (L,p)-radial, p e {0} U N. Consider the multipoint initial-final problem

P3(u(r3)~u3) = 0, j = 0^i (3.1)

for the equation (2.2).

Definition 6. Vector-valued function u e C([0,rn|;U) П Cl((0,rn);U), satisfying (2.2), called a solution of the multipoint initial-final problem (2.2), (3.1) if it satisfies the equation (2.2) and the terms of lim P0(u(t)-u0) = 0,

__¿-S-T0 +

pj(u(Tj)-Uj) = 0,j = l,n.

Lemma 2. Let operator M is strongly (L,p)-radial (p e {0}UN) and part of the spectrum is bounded, j = 1 ,n. Then for any vector function

f0 e Cp+1((0,t);F0) there exists a unique solution u0 e C 1([0,т];U0) of equation (2.3) which also has the form

q=0

Proof. By substituting the vector function u0 = u0(t) in (2.3) verify the existence of solutions. Uniqueness obtained by successive differentiation of the homogeneous equation (2.3) 0 = Hpu0p = ... = Hv° = u0. Lemma proved. □

Lemma 3. Under the conditions of Lemma 2 for any vector uj e U and for any vector-valued function f) e C([0,Tn]; F)) there exists an unique solution

uj e C ([0, Tn ]; Ul) n C 1((0, Tn); Uj) of the problem P; (u(t;) - u;) = 0 for the equation (2.4), which also has the form

t-T

u1(t) = U; TUj - I Utj3L~l}f](s)ds.

Ti I 1 j J;

Proof. If j = 0 then declaration of Lemma is a classical result by the radiality of operator S0.

Let's prove the uniqueness. Using the substitution we see that the vector-valued u1; = u\; (t) is a solution to this problem. Let v = v(t), t e [0,t] be an another solution of this problem. Construct vector-valued function w(s,t) = L1;Ujv(s). By construction

dw{s,t) _ dU{s is<M£) _

Hence w(t, t) = w(t, t), ie, U\~Tv{t) - v{t) = 0. □

Thus we have proved

Theorem 5. For any vectors U; e U and any vector function f : [0,Tn] ^ F, which satisfies conditions of Lemmas 2, 3, there exists an unique solution u e C([0,Tn];U) n C 1((0,Tn);U), which also has the form

u(t) = u0(t) + u1(t).

n

1 (

j=0

We now consider the Dzektser model for evolution of the free surface of the filtered fluid.

Let Q C Rn is a bounded domain with boundary dQ of class C Consider the equation

(A - A)ut = aAu - /3A2u + f, (3.2)

where A e R, a, 3 e R+, with the boundary conditions

u(x,t) = 0, (x, t) e dQ x (0, Tn). (3.3)

This system modells the evolution of the free surface filtered fluid (see [11] and references there in). We reduce equation(3.2) to the equation (2.2). For this we take the functional spaces

U = {u e Wk(Q) : u(x) = 0, x e dQ}, F = Wpk(Q),

where k e {0} UN, p e (1, and W^(Q) are Sobolev spaces. We define the operators L e L(U; f), M e Cl(U;F) by formulas L = A - A, M = aA - /A2, where dom M = {u e Wk+2(Q) : u(x) = Au(x) = 0, x e dQ}.

Lemma 4. For any X £ R\{0, a-fi-1} operator M is strongly (L, 0)-radial.

We denote by {Xk} sequence of eigenvalues of the homogeneous Dirichlet problem for the Laplace operator A in Q. The sequence {Xk} is numbered by non-increasing with according to multiplicity. We denote by {pk} orthonormal (in the sense of L2(Q)) sequence corresponding eigenfunctions, pk £ C~(Q), k £ N. The L-spectrum of M has the form

aL(M) = = k £ N \ {I : A, = A}} .

Clearly that for such a set we can pick up the contour y £ C, which satisfying the condition (2.1). Construct projectors

Pj= j = 0~ñ-

k-.HkZj (M)

Take Tj £ R+ {jj < Tj+1), Uj £ 11, j = 0,n and will be in the cylinder Q x (0,rra) seek a solution of (3.2), satisfying the boundary condition (3.3) and conditions

Pj = Y1 ((u(ri) ~ u¿)> Vfc) (Pk = 0, j = 0~ñ (3.4)

k: vk eaf(M)

of multipoint initial-final problem. For simplicity, only the case where f is independen to t, ie f = const. Theorem 5 and Lemma 4 follows

Theorem 6. For any X £ R \{0,a ■ fi-1}, fi £ R+, u0 £ domM, uT £ U, f £ F exists an unique solution u £ C([0,r|;U) R C 1((0,r);U) of problem (3.4), (3.3) for the equation (3.2).

Remark 5. By Lemmas 2, 3 and Theorem 5, you can get the kind of solutions, however, due to the bulkiness it descends.

3.2. Dichotomy of solutions of the linearized phase field equations

Let the operator M is strongly (L,p)-radial with constant fi > 0. Introduce the condition

Thereis w> 0, that aL(M)C\{^ £C : -w <Re^ < w} = ] Denote C+ = {^ £ C : Re^ > 0}, C- = {^ £ C : Re^ < 0}, > (3.5) a± = aL (M) R C± and let set a+ is bounded. j

Remark 6. Condition (3.5) is a special case of condition (2.1) for n = 1. However, when considering this condition dichotomies enhanced condition of separability of the imaginary axis. Further more, the description of this case can be introduce by the simpler notation.

Due to the relative isolation of the spectrum there is a finite loop r+ c pL(M) n C+ and bounded region containing a+.

According to the relative spectral theorem [3] spaces U1 and F1 split: U1 = U+ ©U-, F1 = F+ ©F-. This splitting of the corresponding projection

P.

+

1

27Ti

: [ (pL - M)-lLd^ : J

P_ = P- P

+ ,

and

Q+ = ^-jL(jiL-M)-ldti, Q_=Q-Q+

Denote L± = L

M+ = M

, domM± = domM n U±. By Lemma

U± domM±

2 [3] L± e L(U±;F±), M± e Cl(U±;F±). Furthermore, by the Theorem 1 [3] we have aL± (M±) = a±, so the operator M+ is (L+ ,p)-bounded [6], and M_ is (L-,p)-radial with constant /3 < —uj < 0. Construct a semigroup {U± £ ^{ii^) : t £ R+}

Ui = s- lim ( -^(Mi)

fc—S-oo ^ _

Due to the fact that the operator M+ is (L+,p)-bounded, semigroup {Uf+ £ C{ 11+) : t £ M+} can be extended to the group {Uf+ £ C{ 11+) : t £ R}. By Remark 2 operators of the resolving semigroup of equation (0.1) can be represented as

U = UP = U1(P+ + P-) = P+ + U- P-.

Definition 7. Let P is a phase space of equation (0.1). The set P1 c P is called an invariant subspace of this equation, if there exists an unique solution u = u(t) problem (0.1), (1.1) for any of the u0 from dense in p1 lineal, and it has the form u(t) = Ulu0 e p1 Vt e R+

Theorem 7. Suppose that M is strongly (L,p)-radial and holds then the subspace U+ and U- are invariant spaces of equation (0.1).

(3.5),

Definition 8. Let P C U is a phase space of the equation (0.1) and P = P1 © P2, where Pfc is an invariant subspace, k = 1,2. We say that equation (0.1) has an exponential dichotomy if its solutions satisfired to the following conditions:

(i) 3^1,^1 > 0 ||u1(i)y < Nie-vi(s-t)^u1(s)^, s > t;

(ii) 3N2,V2 > 0, ||u2(t)|| < N2e-V2(t-s)|u2(s)|, Vs G R t £ [s, +to), where the solution of the equation uk(t) £ Pk, k = 1,2.

k

Remark 7. In other words, the existence of exponential dichotomies of solutions means that if the solution lie in one of invariant subspace then it grow exponentially and if it lie in another one then it is exponentially decrease.

Theorem 8. Let the operator M is strongly (L,p)-radial and holds (3.5). Then the equation (0.1) has an exponential dichotomy.

Let us consider the linearized phase-field equations. Consider the system of equations

9t(x,t)+ipt(x,t) = Ad(x,t), (x,t)eQxR+, (3.6)

Aip(x, t) + aip(x, t) + /30(x, t) = 0, (i,i)e!]xl+, (3.7) equipped with the boundary conditions

dd — — (x, t) + A9(x, t) = 0, (x, t) £ Oil x R_|_, (3.8)

^(x,t) + Xtp(x,t) =0, (x, t) £ dQ x R_|_, (3.9)

which is the linearization at zero of phase-field equations describing within mesoscopic theory of phase transitions of the first order [2]. Here Q C Rs is a bounded domain with boundary dQ of class CX £ R, a, / £ C. Unknown functions are d(x,t),^(x,t).

We reduce the system (3.6)-(3.9) to equation (2.2). Let's make the replacement d(x,t) + ^(x,t) = u(x,t),^(x,t) = v(x,t). Then the system takes the form

ut(x,t) = Au(x,t) - Av(x,t), (x,t)£ttxR+, (3.10)

: - P)v(x, t) + I3u(x, t) = 0, (x,t)£QxR+, (3.11) -(x, t) + Au(x, t) = 0, (x, t) £dttx R_|_, (3.12)

Av(x,t) + (a-P)v(x,t)+Pu(x,t) = 0, (x,t)eQx M+, (3.11) du dn

dv —

— (x,t) + Xv(x,t) = 0, (x,t)£dQxR+. (3.13)

Let il = {(u,v) £ (tf 2(Q))2 : (£ + A)u(x) = + X)v(x) = 0,x £ дП}

F =(L2 (fi))~

2

L-( 1 O\ M-( A "A

L - V O O ) ' M - V pi (a - P)I + A

Thus constructed statements is L,M £ L(U; F). Moreover, if

din

il' = {w £ tf2(Q) : —(x) + Xw(x) = 0, x £ дП}

then ker L = {0} x U'.

Let Aw = Aw then A £ L(U',L2(Q)). Through {pk : k £ N} denote orthonormal in the sense of the scalar product (-, ■) in L2(Q) eigenfunctions of the operator A, numbered by non-increasing eigenvalues {Xk : k £ N} with respect to their multiplicities.

Lemma 5. Let 3 — a £ v(A). Then the operator M is strongly (L, 0)-radial.

In this case, L-spectrum of M has the form

aL(M) = ^ ¡ik =

(a + Afc)Afc a + Xk — 3

,k £ N \{l : Xi = 3 — aU .

Construct projectors

/ E ©\

k:Re^k >0

+ ~~ y^ f3(-,<Pk)<Pk Q

. , ^ „ ¡3-a-\k y k:Re^k >0

P

P- =

/

( E (;Vk)vk O\

k:Re^k <0

y Q

y k:Re^k<0

/

Theorem 9. Let 3 — a, —a, 0 e a(A)- Then the solution of the problem (3.10) - (3.13) have an exponential dichotomy.

More detailed justification assertions in this paragraph can be found in [6].

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Zagrebina Sophiya, Candidate of Sciences (Physics and Mathematics), Head of Department,South Ural State University, 76, Lenina av., Chelyabinsk, 454080, tel.: (351)2679001 (e-mail: zagrebina_sophiya@mail.ru)

Sagadeeva Minzilia, Candidate of Sciences (Physics and Mathematics), Associate Professor, South Ural State University, 76, Lenina av., Chelyabinsk, 454080, tel.: (351)2679001 (e-mail: sagadeeva_ma@mail.ru)

С. А. Загребина, М. А. Сагадеева Обобщенная теорема о расщеплении для линейных уравнений соболевского типа в относительно радиальном случае

Аннотация. Уравнения соболевского типа в настоящее время составляют обширную область среди неклассических уравнений математической физики. Неклассическими называют те уравнения математической физики, чьи представления в виде уравнений или систем уравнений в частных производных не укладываются в рамках одного из классических типов — эллиптического, параболического или гиперболического. В данной работе доказывается обобщенная теорема о расщеплении пространств и действий операторов для уравнения соболевского типа с относительно радиальным оператором. Отметим, что необходимость в обобщенной теореме о расщеплении появилась при изучении многоточечных начально-конечных для линейных уравнений соболевского типа. В настоящее время эти задачи нашли

свое применение в теории управляемости и оптимального управления. Основным методом исследования является теория Свиридюка относительного спектра.

Статья кроме введения и списка литературы содержит три части. В первой части приводятся необходимые сведения теории относительно р-радиальных операторов, вторая содержит основной результат статьи — доказательство обобщенной теоремы о расщеплении в случае сильно (L, р)-радиального оператора M. Третья часть содержит применение результатов предыдущего пункта для различных задач, а именно для доказательства однозначной разрешимости многоточечной начально-конечной задачи для уравнения Дзекцера и для исследования дихотомий решений линеаризованной системы уравнений фазового поля. Список литературы не претендует на полноту и отражает лишь вкусы и пристрастия авторов.

Ключевые слова: линейные уравнения соболевского типа, обобщенная теорема о расщеплении, дихотомии решений, многоточечная начально-конечная задача.

Загребина Софья Александровна, кандидат физико-математических наук, доцент, зав. кафедрой, Южно-Уральский государственный университет (НИУ), 454080, г. Челябинск, пр. Ленина, 76, тел.: (351)2679001, (e-mail: zagrebina_sophiya@mail.ru)

Сагадеева Минзиля Алмасовна, кандидат физико-математических наук, доцент, Южно-Уральский государственный университет (НИУ), 454080, г. Челябинск, пр. Ленина, 76, тел.: (351)2679001, (e-mail: sagadeeva_ma@mail.ru)