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

The article presents the review of authors’ results in the field of non-classical equations of mathematical physics. The theory of So b lev-type equations of higher order is introduced. The idea is based on ge n ralization of degenerate operator semigroups theory in case of the following e quations: decomposition of spaces, splitting of operators' actions, the constr uction of propagators and phase spaces for a homogeneous equation, as well as the s et of valid initial values for the inhomogeneous equation. The author uses a proven ph ase space technology for solving Sobolev type equations consisting of reduct ion of a singular equation to a regular one defined on some subspace of initial spa ce. However, unlike the first order equations, there is an extra condition that g uarantees the existence of the phase space. There are some examples where the init ial 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 or der is conducted and justified.


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 (2) However it was shown [1] that the Showalter-Sidorov conditions ( ) ( (0) ) = 0, = 0,..., 1 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 ( ) ( (0) ) = 0, = 0,..., 1, ) 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][8][9][10][11][12][13].
Even a cursory glance at the vast area of nonclassical equations of mathematical physics [7,[14][15][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 ) 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 = 1 n (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 initialboundary 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 0 1 , ,..., ( ; ). ( ) = ( ... ) Lemma 1 [24].Let the operators Let the operator B be polynomially A -bounded.Introduce the following condition: ( ) , = 0,1,..., 2, where the contour = { : are projectors in the spaces U and F respectively.Put 0 = ker , B denote a restriction of the operator A ( ) Theorem 2 [24].Let the operator pencil B be polynomially A -bounded and condition ( ) A be fulfilled.Then the operators actions split: (i) ( ; ), = 0,1 (iii) there exists an operator 1 1 (iv) there exists an operator Corollary 1 [24].Let the operator pencil B be polynomially A -bounded and condition ( ) A be fulfilled.Then there exists a constant For the B -joined vector q φ 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 But it is finite in the case of existence of such a B -joined vector q φ , that . The height q of the last B -joined vector in a finite chain { , ,... } n q q q K K K as follows: Theorem 4 [24].Let the pencil B be polynomially A -bounded and ∞ be (i) a removable singular point of the function U .Theorem 3 [24].Let the operators Let the pencil B be polinomially A -bounded and ( ) γ µ µ ∈ and consider the family of operators Lemma 3 [24].(i) For any = 0,1,..., 1 k n − the operator-function t k V is a propagator of ( 1).
Theorem 5 [24].Let the pencil B be polinomially A -bounded, ( ) A be fulfilled, and ∞ -be pole of order {0} p N ∈ ∪ or 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 where The initial model has sense in a cylindrical domain in variables In the case of stabilized acoustic waves in a smectic the initial equation takes the form Supply this equation with the initial and boundary conditions 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 2 = { ( ) : ( ) = 0, }, = ( ), where ( ) W Ω are the Sobolev spaces 2 < q ≤ ∞ .Put for the convenience

Then the pencil B is polynomially A -bounded and ∞ is nonessential singular point of the A -resolvent of pencil B .
Remark 1.In the case (i) The A -spectrum of pencil B In the case (ii)

Now check ( )
A .In the case (i) there exists an operator 1 ( ; ) Construct the projectors.In the case (i) = P I and = Q I , in the case (ii

∑
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 ( ) Then the phase space of the equation is the entire space U , that is for all 0 1 , v v U ∈ there exists a unique solution of ( 12), ( 13), given by Then the phase space of the equation is the subspace 1 U , that is for all there exists a unique solution of ( 12), ( 13), given by (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 is the set of vertices, and is the set of edges.We suppose that each edge has the length > 0 j l and the cross section area > 0 j d .On the graph G consider the Boussinesq-Love equations [26] = ( ) ( ), (0, ), , = 1, .
we denote the set of edges starting (ending) in the vertex i V .If we add the initial conditions 0 1 ( ,0) = ( ), ( ,0) = ( ), for all (0, ), = 1, , then we get a problem describing the vibration processes in a construction made of thin elastic rods.The functions ( , ) j u x t determine the longitudinal displacement in the point x at the moment t on the j -th element the construction.The parameters , , , λ λ λ α ′ ′′ and β characterize the material if rods.
The set 2 ( ) L G is a Hilbert space with an inner product 0 < , >= ( ) ( ) .So, the reduction of ( 16)-( 19) to ( 20)-( 21) is completed.By theorem 7, the operator A is a Fredholm operator and ker = {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 ( ) A σ ∈ and = = λ λ λ ′ ′′ the operator pencil B is not polynomially A -bounded.Remark 4. [23] n the case 0 ( ) , a is a constant from the definition of the polynomial A -boundedness, holds.In the case (0 ∫ therefore we exclude it from our future considerations when searching the phase space of the equation.Let { } k λ be a set of eigenvalues of the operator D , numbered in nondecreasing order taking into account their multiplicities, and { } k φ be a set of corresponding orthonormal in sense of 2 ( ) L G eigenfunctions.Construct the projectors , defined on spaces U and F respectively, and the propagators of equation ( 21) Hence the following theorem is true.Theorem 8 [23,24] Let , , , .Then the phase space of ( 16) coincides with the spaceU , i.e. for all 0 1 , u u U ∈ there exists a unique solution 2 ( ; ) u C R U ∈ of ( 16)- (19), given by ( ; ) u C R U ∈ of ( 16)- (19),given by is necessary for the existence of solution of the problem [18,19].

∑
So, due to theorem 5 the following theorem is true.
called an A -resolvent set and an A -spectrum of the operator pencil B .

Definition 5 .
is called a length of this chain.Define the family of operators 1 2

µ..
Then the operator A does not have B -joined vectors, 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 0

λ
the set of eigenvalues of the homogeneous Dirichlet problem in a domain Ω for the Laplace operator ∆ , numbered in nonincreasing order taking into account their multiplicities, and by { } k φ denote the family of the corresponding eigenfunctions orthonormal with respect, to the inner product 20)   for the linear Sobolev type equation of the second order absolutely continuous functions, therefore U is correctly defined, densely and compactly embedded in 2 ( ) L G .Identify 2 ( ) L 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.∈ , set an operator defined on the space U .Fix , > 0 A is discrete, real tends only to +∞ .
Here the prime at the sum means the absence of summands with indices k such that = ,but λ λ′′ ≠ .Then the phase space of equation (16) coincides with the sub-

≠ 1 <
the phase space, in sense of definition 9, does not exist, since the condition of coordination of initial functions 0

3 =
field.The function Φ presents a generalized potential of the electric field, constants 2 gyrofrequency, Langmuir frequency and the Debye radius, respectively.We transform equation(22)  and consider the more general problem.Let the spectrum ( ) σ ∆ is negative, discrete, with finite multiplicities and tends only to −

µ∑
pencil B is polynomially A -bounded and ∞ is a removable singular point of the A -resolvent of pencil B . .Then the pencil B is polynomially A -bounded and ∞ is a pole of order 1 of the A -resolvent of pencil B . .Then the pencil B is polynomially A -bounded and ∞ is a pole of order 3 of the A -resolvent of pencil B .Remark 6[21]  In case (i) of lemma 6 the A -spectrum of pencil B ( equations of higher order Вестник ЮУрГУ.Серия «Математика.Механика.Физика» 2016, том 8, № 4, С. 5-1613    and condition ( ) A holds.In case (ii) of lemma 6 the A -spectrum of pencil B are the roots of equation (25) with = l λ λ , 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 and the projector Q has the same form but is defined on the space F .In case (ii) construct the set 1 : the existence of solution of the problem.