Nonclassical equations of mathematical physics. Linear Sobolev type equations of higher order Текст научной статьи по специальности «Математика»

Математика

DOI: 10.14529/mmph160401

NONCLASSICAL EQUATIONS OF MATHEMATICAL PHYSICS. LINEAR SOBOLEV TYPE EQUATIONS OF HIGHER ORDER

A.A. Zamyshlyaeva, G.A. Sviridyuk

South Ural State University, Chelyabinsk, Russian Federation E-mail: zamyshliaevaaa@susu.ru

The article presents the review of authors' results in the field of non-classical equations of mathematical physics. The theory of Sobolev-type equations of higher order is introduced. The idea is based on generalization of degenerate operator semigroups theory in case of the following equations: decomposition of spaces, splitting of operators' actions, the construction of propagators and phase spaces for a homogeneous equation, as well as the set of valid initial values for the inhomogeneous equation. The author uses a proven phase space technology for solving Sobolev type equations consisting of reduction of a singular equation to a regular one defined on some subspace of initial space. However, unlike the first order equations, there is an extra condition that guarantees the existence of the phase space. There are some examples where the initial conditions should match together if the extra condition can't be fulfilled to solve the Cauchy problem. The reduction of nonclassical equations of mathematical physics to the initial problems for abstract Sobolev type equations of high order is conducted and justified.

Keywords: nonclassical equations of mathematical physics; the Sobolev type equations of higher order; phase space, propagators.

Introduction

To the linear Sobolev type equations of high order we consider those non-classical equations of mathematical physics, which in suitable functional spaces can be reduced to the abstract operator differential equation of the form

Au (n)= Bn-1u(n-1) +... + B0u, (1)

where ne N\{1}, operators A,Bn-1,...,B0 are linear and the operator A might not have an inverse, in particulam when ker A -Ф- {0}. Usually equation (1) is considered along with the Cauchy initial conditions

u(m)(0) = um,m = 0,...,n -1. (2)

However it was shown [1] that the Showalter-Sidorov conditions

A (u(m)(0) -um) = 0,m = 0,...,n -1 (3)

are more natural for the Sobolev type equations. Problems (1), (2) and (1), (3) depending on the goals of investigation can be understood in different senses (classical,. в зависимости от целей исследования могут пониматься в различных смыслах (classical, generalized, weak, strong, etc.), however it is obvious that (3) is more general in comparison to (2). In a trivial case (when the inverse to A exists) both problems coincide, therefore their solutions coincide. In this paper the Showalter-Sidorov conditions are considered in more general statement

P(u(m)(0) -um) = 0,m = 0,...,n -1, (4)

where P is a relative spectral projector. For conduction of computational experiments the Showalter -Sidorov conditions are more suitable than the Cauchy conditions because there is no need to check if the initial data belongs to a phase space of the equation. Apparently A. Poincare [2] was the first to study equations of mathematical physics nonsolvable with respect to the highest derivative in time. However their systematic study was initiated by S.L. Sobolev [3] (see the historical review in [4]). By now there are a lot of methods and results of study of such equations. Their diversity is reflected the terminology: degenerate equations [5], pseudo parabolic equations [6] and even equations "of not Cauchy-Kovalevskaya type" (cited by [4]). We use the term "Sobolev type equations" introduced by R.

Showalter [7]. Firstly, we want to support the outstanding role of our great compatriot in a discovery of a new scientific direction. And the second reason is that this term is becoming more common [7-13].

Even a cursory glance at the vast area of nonclassical equations of mathematical physics [7, 14-16] can detect the variety of aspects in which they are investigated. Our approach is based on a phase space concept, the essence of which lies in a reduction of singular equation (1) to a regular one

u (n)= Sn_xu(n-1) +... + S0u + g, (5)

defined, however, not on a whole space but on some subset of initial space, containing all initial values (2). In our case the phase space is a subspace of initial space (we show this below) or (in the worst case) an affine manifold (see examples in [8]). In the semilinear case, the phase space is much more interesting, even if n = 1 (see the review [17]).

To describe the morphology of the phase space of (1), it may seem that it is sufficient to reduce this equation using the standard procedure to a linear equation of the first order, the phase spaces of which are well studied [8]. However, on that way there arise unexpected difficulties: it turns out that in some cases [18, 19] for the solvability of problem (1), (2) the conditions of the Cauchy problem (2) need to be confirmed. For the relief of these difficulties there was proposed [20] a condition (see paragraph 1 of this article). The discussion of the role of this condition in the description of the phase space of equation (1) is the main content of the article. We should emphasize that there is no such a phenomena in the description of phase spaces of Sobolev type equations of the first order [8] and classical equations (5).

The article besides an introduction and references includes four paragraphs. The first one is devoted to the abstract Cauchy problem and propagators for the higher order Sobolev type equation with relatively p -bounded operator pencil [10]. These results are used to study the solvability of the initial-boundary problem for the equation describing acoustic waves in a smectic [21] in the second paragraph, the Boussinesq-Love equation on a finite connected oriented graph [22] in the third paragraph, equations describing ion-acoustic waves in plasma [23] in the fourth.

Finally note that all considerations are held in real Banach spaces, but when studying spectral problems we introduce their natural complexification. All contours are oriented counterclockwise and bound the domain that lies to the left in this movement.

Propagators

Let U,F be Banach spaces, operators A,B0,...,Bn_1 e L(U;F). Denote by B a pencil of operators

Bn_l,...,B0.

Definition 1. The sets pA(B) = {me C :(mnA_m"_1Bn_1 _..._mB1 _B0)_ e L(F;U)} and (B) = C \ p (B) are called an A -resolvent set and an A -spectrum of the operator pencil B .

Definition 2. The operator-function of a complex variable

R^(B) = (mnA _mn_1Bn_1 _... _mB1 _B0)_1 with the domain pA(B) is called an A -resolvent of the

pencil B.

Lemma 1 [24]. Let the operators A,Bn_1,...,B0 e L(U;F).Then the A -resolvent setpA(B) of the operator pencil B is opened, the A -spectrum of the pencil B is always closed.

Theorem 1 [24]. R£ (B)

is analytical in its domain.

Definition 3. The operator pencil B is called polynomially bounded with respect to an operator A (or simply polynomially A -bounded), if

$ae R+ "me C (| ml> a) ^ R(B)e L(F;U)). Let the operator B be polynomially A -bounded. Introduce the following condition:

fmkR^ (B)dm° o, k = 0,1,..., n _ 2, (A)

g

where the contour g = {me C :| m l= r > a}.

Lemma 2 [24]. Let the operator pencil B be polynomially A-bounded and condition (A) be fulfilled. Then the operators

Zamyshlyaeva A.A., Nonclassical equations of mathematical physics.

Sviridyuk G.A. Linear Sobolev type equations of higher order

P = ^ K (Ad a, Q = k-1 ARAm (B)dM. (6)

2m J 2m J

g g

are projectors in the spaces U and F respectively.

Put U0 = kerP, F0 = ker Q, U1 = imP, F1 = imQ . By Ak (Bf) denote a restriction of the operator A (Bl) onto Uk, k = 0,1; l = 0,1,...,n -1.

Theorem 2 [24]. Let the operator pencil B be polynomially A -bounded and condition (A) be fulfilled. Then the operators actions split:

(i) Ak e L(Uk; Fk), k = 0,1;

(ii) Bk e L(Uk;Fk), k = 0,1, l = 0,1,...,n -1;

(iii) there exists an operator (A1)-1 e L(Fl;U*).

(iv) there exists an operator (B0 )-1 e L(F0; U0).

Denote H0 = (B00)-1 A0,Hk =(B00)-1 B°n-k,k = 1,n -1,Sk =(A1)-1 B\,k = 0,n -1. Corollary 1 [24]. Let the operator pencil B be polynomially A -bounded and condition (A) be fulfilled. Then there exists a constant b e R+ (b > a) "m e C (| m l> b) ^

Rt (B) = -Z(mH -...-mtf„-i)k(Bo)"1(/ - QZ(m_1s„-i + ...+т"%)к(Ai)-1ô. (7)

k=0 k=0

Definition 1. Let ker A & {0}, the vector f0 e ker A \ {0} is called an eigenvector of an operator A. An ordered set of vectors {f ,f2,...} is called a chain of B -joined vectors of an eigenvectorf0, if

Af =0;

Af1 = Bn-1f; Af2= Bn-f + Bn-2f0';

Afn = Bn-fn-1 + Bn-2$n-2 + ... + Bf + B0f ; Afn+q = Bn-1fn+q-1 + Bn-2fn+q-2 + ". + B1fq+1 + B0fq ;

q = 1,2.., f e kerA\{0},l = 1,2,... (8)

For the B -joined vector fq define its height equal to its index in the chain. The linear hull of all eigenvectors and B -joined vectors of the operator A is called a B -root lineal. A closed B -root lineal is called a B -root space of an operator A. The chain of B -joined vectors can be infinite. In particular it can be filled in with zeros if

f0 e ker A n ker Bn-1 n ker Bn-2 n... n ker B1 n ker B0.

But it is finite in the case of existence of such a B -joined vector fq , that B„-\fq + Bn-2fq-1 +... + B0fq-n+1 e imA. The height q of the last B -joined vector in a finite chain {f ,f2,...,fq} is called a length of this chain.

Definition 5. Define the family of operators {Kl k2,...K„} as follows:

K0s = O, s & n, ^ = O

jy1 — ZJ Jy2 — u jys _ TT lyn _ TT

K1 = H0, K1 = -Hn-1,..., K1 = -Hn+1-s , K1 = -H1

]y-1 _ T^n TT TT2 _ TT1 Trn TT Trs _ Trs-1 Trn TT

Kq = Kq-1H 0 , Kq = Kq-1 - Kq-1Hn-1,..., Kq = Kq-1 - Kq-1Hn+1-s ,...,

Kn = k£ - K^,q = 1,2,... (9)

Definition 6. The point ¥ is called

(i) a removable singular point of the A -resolvent of the pencil B , if K\ = K2 = ... = Kn ° O ;

(ii) a pole of order p e N of the A -resolvent of the pencil B, if Ksp Ф O for some s but Ksp+1 ° O for arbitrary s ;

(iii) essentially singular point of the A -резольвенты of the pencil B, if Knp ° O for arbitrary pe N.

Theorem 4 [24]. Let the pencil B be polynomially A -bounded and ¥ be

(i) a removable singular point of the function RM(B). Then the operator A does not have B -joined

vectors, ker A = U0,im A = F1.

(ii) a pole of order p e N of the function R^ (B). Then the length of every chain of B -joined vectors of the operator A is bounded by number p (the chains of length p do exist), and the B -root lineal of the

operator A coincides with the subspace U0.

Theorem 3 [24]. Let the operators A,Bn-1,...,B0 e L(U,F), operator A be a Fregholm operator. Then the following statements are equivalent.

(i) The lengths of all chains of B -joined vectors of an operator A are bounded by p e{0} и N.

(ii) The operator pencil B is polynomially A-bounded and ¥ is a pole of order not greater then p of the A -resolvent of an operator pencil B.

Definition 7. The vector-function v e Cn (R;U), satisfying (1), is called a solution of this equation. If the solution v = v(t) satisfies (2), then it is called a solution of (1), (2).

Definition 8. The operator-function V() e C¥ (R;L(U)) is called a propagator of (1), if for any v e U the vector-function v(t) = V'v is a solution of this equation.

Let the pencil B be polinomially A -bounded and (A) be fulfilled. Fix the contour g={// e C | = r > a} and consider the family of operators

Vt № (BXm^A - m"~k-2 Bn-1 -... - Bk+i)emtdju, k = 0,1,...n -1, t e R. (10)

2m J ^

g

Lemma 3 [24]. (i) For any k = 0,1,..., n -1 the operator-function Vfk is a propagator of (1). (ii) For any k = 0,1,..., n -1 the operator-function Vfk is n entire function.

dl

(iii) djVl

\P, l = k;

= ' for all k = 0,1,...,n -1, l = 0,1,....

IO, l Ф k;

t=0 ^ '

Definition 9. The set P c U is called a phase space of (1), if

(i) any solution v = v(t) of (1) lies in P, i.e. v(t)e P "t e R

(ii) for all vk e P, k = 0, n there exists a unique solution of (1), (2).

Theorem 5 [24]. Let the pencil B be polinomially A -bounded, (A) be fulfilled, and ¥ - be pole of order p e {0} u Nor its A-resolvent. Then the phase space of (1) coincides with the image of the projector P.

The De Gennes equation of the acoustic waves in a smectic

The equation of linear acoustic waves in a smectic [25], firstly obtained by P.G. de Gennes, has the

firm

32 d 2

—-D3u = a—-D2u,a >0, (11)

3t231 3z221

where D3 = D2 + 32/dz2, d2 = a2/3x2+a2/3x22. The initial model has sense in a cylindrical domain in variables {z,x1,x2}e [a,b]xW . In the case of stabilized acoustic waves in a smectic

u( x1, x2, z, t) = v( x1, x2, z)exp(-i aft),

Zamyshlyaeva A.A., Nonclassical equations of mathematical physics.

Sviridyuk G.A. Linear Sobolev type equations of higher order

the initial equation takes the form

i

— (D2v + a2v) + a2D2

Э2 2 _i

—2-(D2v + a2v) + a2D2v = 0,a2 = со a1 . (12)

Эг

Supply this equation with the initial and boundary conditions

v( x,0) = v0( x), vz (x,0) = Vj( x), x = ( x1, x2) eW

v(x, z) = 0, (x, z) edWx R. (13)

The initial-boundary value problem for (12) can be described in terms of problem (2) for equation (1). For the reduction of (12), (13) to (1), (2), put

U = {v e Wq+2 (W): v(x) = 0, x e 3W}, F = Wq (W),

where Wlq (W) are the Sobolev spaces 2 < q < ¥ . Put for the convenience a = -a2 , A = A2. Define operators A, B1 and B0 by formulas A = A-a, B1= O, B0= aA. For any l e {0} u N operators A, B1, B0 e L(U; F).

Define by {1k} the set of eigenvalues of the homogeneous Dirichlet problem in a domain W for the Laplace operator A, numbered in nonincreasing order taking into account their multiplicities, and by {fk} denote the family of the corresponding eigenfunctions orthonormal with respect, to the inner product < v > in L2(W). Since ffk} c C¥ (W), then

m2 A - B0 = ¿[(a+4 )m2+aAk ]<f, • > fk.

k=1

Lemma 4 [22, 24] Let ae R. Then the pencil B is polynomially A-bounded and ¥ is nonessential singular point of the A -resolvent ofpencil B.

Remark 1. In the case (i) The A -spectrum of pencil B S(B) = {m^2 : ke N}, where m1,2 are the roots of equation

(1k -a)m2 -a1k =0. (14)

In the case (ii) S (B) = {m1^ : k e N}, where m1;2 are the roots of equation (14) for a^Al.

Now check (A). In the case (i) there exists an operator A"1 e L(F;U), therefore (A) is fulfilled. In the case (ii)

1 ry <fk, • > fkdm =__L fy <fk, • > fkdm = 0

2pi gk=1(1t -a)m2 -alk 2pi gk=1 a1k '

Construct the projectors. In the case (i) P = I and Q = I, in the case (ii)

p=I - y < fk, • >fk,

a=1k

and the projector Q has the same form but is defined on the space F . Therefore, due to theorem 5, the following theorem is true.

Theorem 6 [24] (i) Let ae- s(A). Then the phase space of the equation is the entire space U, that

is for all v0, v1 e U there exists a unique solution of (12), (13), given by

v(z) = Z < vo,fk >fk ^ll1-2 + Z < vo,fk >fk cos

a<% V 1 - ->1к \

al

k -z +

a_1

+ Z <vi,fk >fk

a<1

1 _ a sh

a1k \

a1k j. j.

~z + Z <vi,fk >fk.

a_1k . - sin

a1k \

a1k

a_Xb

1 -a a>1k \

(ii) Let ae ^(A). Then the phase space of the equation is the subspace U1, that is for all

v0,v e U1 ={ve U :< v,f >= 0,1 = 1} there exists a unique solution of (12), (13), given by (15).

z. (15)

Remark 2. The results if theorem 6can be easily transcribed in the terms of the initial equation2 (11), if we take into account the connection between the functions u and v .

The Boussinesq-Love equation on a geometrical graph

Let G = G(V; E) be a finite connected oriented graph, where V = {Vi m is the set of vertices, and E = {Ej}"j=1 is the set of edges. We suppose that each edge has the length lj > 0 and the cross section area dj > 0 . On the graph G consider the Boussinesq-Love equations [26]

1ujtt - ujxxttt = a(ujxxt - 1ujt) + P(ujxx - 1uj X x e (0, lJ X t e R j = 1, j. (16)

At each vertex Vi, i = 1, m set the boundary conditions

X d}u]X (0, t) - X dkukx (lk, t) = 0, (17)

r.E]eEa(Vl) k-EkeEw(Vl)

us (0, t) = u} (0, t) = uk (lk, t) = um (lm , t), (18)

for all Es, Ej e Ea(V), Ek, Em e Ew(Vl). Here by Ea{m)(Vl) we denote the set of edges starting (ending) in the vertex Vi. If we add the initial conditions

uj (x,0) = u0 j (x), ujt (x,0) = u1 j (x), for all x e (0, lj), j = 1, j, (19)

then we get a problem describing the vibration processes in a construction made of thin elastic rods. The functions uj (x, t) determine the longitudinal displacement in the point x at the moment t on the j -th

element of the construction. The parameters 1,X,X',a and p characterize the material if rods. Reduce problem (17)-(19) for equations (16) to the Cauchy problem

u(0) = u0 , u (0) = u1 (20)

for the linear Sobolev type equation of the second order

Au" = Bxu + B0u. (21)

By L2 (G) denote a set

MG) = {g = (g1,g2,...,gj,...): gj e L2(0,lj)}.

The set L2 (G) is a Hilbert space with an inner product

lj

< g, h >= £ dj J gj (x)hj (x)dx.

EjeE о

By U denote a set U = {u = (ux,u2,...,uj,...): uj e W2(0,lj) and (18) holds}. The set U is a Banach space with a norm

II l|2

\4u =

EjeE о

J

Z dj J (Иjx (x) + u2 (x))dx.

Due to the Sobolev embedding theorems the space W2(0, lj) consists of absolutely continuous functions, therefore U is correctly defined, densely and compactly embedded in L2 (G). Identify L2 (G) with its dual space and by F define a dual space to U with respect to the duality < , • > . Obviously, F is a Banach space and the embedding of U into F is compact. By formula

lj

< Du,v >= Z dj J (Ujx (x)vjx (x) + auj (x)vj (x))dx,

EjeE 0

where a > 0,u, v e U, set an operator defined on the space U. Fix a,p > 0, 1,X,1" e R and construct operators

A = (l - a)I + D, B = a((a -1)I + D), B0 = p((a -1')I + D).

Zamyshlyaeva A.A., Sviridyuk G.A.

Nonclassical equations of mathematical physics. Linear Sobolev type equations of higher order

Theorem 7 [23] Operators A, B1, B0 e L(U; F), moreover the spectrum s(A) of an operator A is discrete, real tends only to +¥.

So, the reduction of (16)-(19) to (20)-(21) is completed. By theorem 7, the operator A is a Fredholm operator andker A = {0}, if 0 e s(A).

Lemma 5 [23] Let a,X,X',X" e R\{0}, then the operator pencil B is polynomially A-bounded, and ¥ is nonessential singular point of the A -resolvent of the pencil B.

Remark 3 [23] It is easily seen that in the case 0 e s(A) and X = X = X" the operator pencil B is not polynomially A -bounded.

Remark 4. [23] n the case 0 e s(A) or (0 e s(A)) a (X = X * X") condition

J(m2 a -mB1 - b0)-1 dm = 0, (A)

g

where g= {| m 1= r > a}, a is a constant from the definition of the polynomial A-boundedness, holds. In the case (0 e s(A)) a (X * X)

J(m2 a -mB - b0)-1 dm* 0,

g

therefore we exclude it from our future considerations when searching the phase space of the equation.

Let {Xk} be a set of eigenvalues of the operator D, numbered in nondecreasing order taking into account their multiplicities, and {fk} be a set of corresponding orthonormal in sense of L2 (G) eigen-functions. Construct the projectors

P =

1,0 A);

I - У < f > f ,0es(A); Q =

Xk=X-a

1,0 A);

I - У < f > фк,0es(A),

Xk=X-a

defined on spaces U and F respectively, and the propagators of equation (21)

V! =

IV A-mBx - B0)-\mA - Bx)emtdm = 2m J

=У'

m (l - (a + Xk)) + a(X - (a + Xk)) u\t , ml(1-(a + Xk)) + a(X-(a + Xk)) „Ь 12 2 1 (X-(a+xk ))(mi-mi) (X-(a+xk ))(m*-m^)

1 о , ^ - emi

V(t) = — |(m2a-mBi -B0)-1 Aemtdm = У'—- < f > Фк,

< •

Фк > Фк;

v (ml -ml)

where <JA (B) = {m^1: k e N}, and ml'2 are the roots of equation

(X - (a + Xk))m2 + a(X - (a + Xk))m + PX" - (a + Xk)) = 0.

Here the prime at the sum means the absence of summands with indices k such that X = a + Xk . Hence the following theorem is true.

Theorem 8 [23, 24] Let a,X,X',X"e R\{0} and

(i) 0e <(A) .Then the phase space of (16) coincides with the spaceU, i.e. for all u0,u1 e U there exists a unique solution u e C2(R;U) of (16)-(19), given by u(t) = V0tu0 + V1tu1.

(ii) 0 e s(A) and X = X ,but X* X". Then the phase space of equation (16) coincides with the subspace U1 = {u e U :< u,fk >= 0 for Xk = X - a}, i.e. for all u0,u1 e U1 there exists a unique solution u e C2(R;U1) of (16)-(19),given by u(t) = V0'u0 + V1tu1.

Remark 5. In the case 0e s(A) and X*X the phase space, in sense of definition 9, does not exist, since the condition of coordination of initial functions

mk < u0,fk >=< ufk > for Xk = X - a . is necessary for the existence of solution of the problem [18, 19].

g

Equation of ion-acoustic waves in plasma in external magnetic field

Equation

d_ dt2

firstly obtained by Yu.D. Pletner [27], describes the ion-acoustic waves in plasma in external magnetic

.2 „2 „„ j r2

rD

>2 ' Э2 2 ^ Э? + ¿2

1 2 Э2 2 2 Э 2Ф

(АзФ - -JФ) + « ^ АзФ + ¿Н ^Т = 0 (22)

field. The function F presents a generalized potential of the electric field, constants aB , (Ov , and rD

Pi

characterize the ionic gyrofrequency, Langmuir frequency and the Debye radius, respectively. We transform equation (22) and consider the more general problem.

Let W = (0, a) x (0,b) x (0, c) c R3. In a cylinder WxR consider the Cauhy-Dirichlet problem

v( x,0)= Vo( x), vt (x,0) = vj( x), vtt (x, 0) = v2 (x), vttt (x,0) = v3( x), x eW (23)

v( x, t) = 0, (x, t) edWx R

for the equation

d 2

(A - X)vm + (A- X)vtt + a~v = 0, (24)

dx3

describing the ion-acoustic waves in plasma in external magnetic field. The initial-boundary value problem for (24) can be described in terms of problem (2) for equation (1), and negative values of the parameter 1 do not contradict the physical meaning of the problem. Reducing (23), (24) to (1), (2), set

U = {v e W2l+2 (W): v(x) = 0, x edW}, F = W2l (W), where W2l (W) are the Sobolev spaces. Operators A, B3, B2 , B1 and B0 define by formulas A = A -1, d2U

B2=(1-A), B0= a—, B3= B1= O . For all I e {0} u N operators A, B1, B0 e L(U; F).

dx2

For proof of the relative boundedness of the pencil B consider the eigenfunctions of the Laplace operator A , defined in a domain W , satisfying the boundary conditions from (23). Denote these ei-

_ , , [ . nkx, . pmx2 . pnx3 ] ,

genfnctions by fkmn = \ sin-1sin-2sin-3 !>, where k , m, n e N , thus the eigenvalues

[ a b c J

1kmn = -(k2 + m2 + n2). Obviously, the spectrum a"(A) is negative, discrete, with finite multiplicities

and tends only to -¥ . Since {fk} c C¥ (W), then

mAA-mB3-m2B2-mB-B0= £ [(ikmn-i)m4+(i^mn-l)m2-a(—)2]<fkmn,• >fmn,

c

k ,m,n=1

where < •, • > is an inner product in L2(W).

Lemma 6 [21]. (i) Let 1e s(A). Then the pencil B is polynomially A-bounded and ¥ is a removable singular point of the A -resolvent ofpencil B.

(ii) (1e s(A)) a (1 . Then the pencil B is polynomially A-bounded and ¥ is a pole of order 1 of the A -resolvent ofpencil B.

(iii) (1e s(A)) a (1 = 1). Then the pencil B is polynomially A -bounded and ¥ is a pole of order 3 of the A -resolvent ofpencil B.

Remark 6 [21] In case (i) of lemma 6 the A -spectrum of pencil B s(B) = {mimn : r,m,ne N, j = 1,...,4}, where mlmn are the roots of equation

(Irmn -Dm4+(irmn -i)m2 -a(—)2 = 0, (25)

c

Zamyshlyaeva A.A., Nonclassical equations of mathematical physics.

Sviridyuk G.A. Linear Sobolev type equations of higher order

and condition (A) holds. In case (ii) of lemma 6 the A -spectrum of pencil B sA(B) = {m/k : k e N},

where m!k are the roots of equation (25) with X = Xl, and condition (A) does not hold. Therefore this

case is excluded from the further considerations. In case (iii) of lemma 6 the A -spectrum of pencil B sA(B) = {m/k : k e N,k * l}, and condition (A) holds.

Construct the projectors. In case (i) of lemma 6 P = I and Q = I, in case (ii) of lemma 6

P = 1 - £ < fkmn, • > fkmn , X=Xkmn

and the projector Q has the same form but is defined on the space F . In case (ii) construct the set

U = imp = {veU: £ <fkmn,v >f^ntn = 0}.

X=Xkmn

So, due to theorem 5 the following theorem is true.

Theorem 9 [21] (i) Let Xe <(D). Then the phase space of (24) coincides with the spaceU, i.e. for all v0,v1,v2,v3 e U there exists a unique solution ue C2(R;U) of (23), (24).

(ii) Let Xe s(D) and X = X. Then the phase space of equation (24) coincides with the subspace U1, i.e. for all v0,v1,v2,v3 such that

У f,Vj >=0, j = 0,...,3,

Xkmn =X

there exists a unique solution ue C2(R;U1) of (23), (24).

Remark 7. In case (Xe s(D)) a(X*X) the phase space in sense of definition 9, does not exist, since the condition of coordination of initial functions [19]:

( pn

(Xkmn -X)< v2, fkmn >= al ~ I < v0,fkmn > nPU Xkmn = X is necessary for the existence of solution of the problem.

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Received September 27, 2016

Zamyshlyaeva A.A., Sviridyuk G.A.

Nonclassical equations of mathematical physics. Linear Sobolev type equations of higher order

Bulletin of the South Ural State University Series "Mathematics. Mechanics. Physics" _2016, vol. 8, no. 4, pp. 5-16

УДК 517.9 DOI: 10.14529/mmph160401

НЕКЛАССИЧЕСКИЕ УРАВНЕНИЯ МАТЕМАТИЧЕСКОЙ ФИЗИКИ. ЛИНЕЙНЫЕ УРАВНЕНИЯ СОБОЛЕВСКОГО ТИПА ВЫСОКОГО ПОРЯДКА

А.А. Замышляева, Г.А. Свиридюк

Южно-Уральский государственный университет, г. Челябинск, Российская Федерация E-mail: zamyshliaevaaa@susu.ru

Статья содержит обзор результатов авторов в области неклассических уравнений математической физики. Представлена теория линейных уравнений соболевского типа высокого порядка. Идея базируется на обобщении теории вырожденных (полу)групп операторов на случай указанных уравнений: расщеплении пространств, действий всех операторов, построении пропагаторов и фазового пространства однородного уравнения, а также множества допустимых начальных значений для неоднородного уравнения. Использован уже хорошо зарекомендовавший себя при решении уравнений соболевского типа метод фазового пространства, заключающийся в редукции сингулярного уравнения к регулярному, определенному на некотором подпространстве исходного пространства. Однако, в отличие от уравнений первого порядка, в данном случае возникает дополнительное условие, гарантирующее существование фазового пространства, и имеются примеры, когда для разрешимости задачи Коши начальные условия необходимо согласовывать между собой при невыполнении этого условия. В работе проводится редукция неклассических уравнений математической физики к начальным (начально-конечным) задачам для абстрактного уравнения соболевского типа высокого порядка.

Ключевые слова: неклассические уравнения математической физики; уравнения соболевского типа высокого порядка; фазовое пространство; пропагаторы.

Литература

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Поступила в редакцию 27 сентября 2016 г.