SOBOLEV TYPE MATHEMATICAL MODELS WITH RELATIVELY POSITIVE OPERATORS IN THE SEQUENCE SPACES

In the sequence spaces which are analogues of Sobolev function spaces we consider mathematical model whose prototypes are Barenblatt – Zheltov – Kochina equation and Hoff equation. One should mention that these equations are degenerate equations or Sobolev type equations. Nonexistence and nonuniqueness of the solutions is the peculiar feature of such equations. Therefore, to find the conditions for positive solution of the equations is a topical research direction. The paper highlights the conditions sufficient for positive solutions in the given mathematical model. The foundation of our research is the theory of the positive semigroups of operators and the theory of degenerate holomorphic groups of operators. As a result of merging of these theories a new theory of degenerate positive holomorphic groups of operators has been obtained. The authors believe that the results of a new theory will find their application in economic and engineering problems.


Introduction
The Barenblatt-Zheltov-Kochina equation [1] ( ) t u u f λ α − Δ = Δ + (1) simulates the pressure dynamics of the fluid filtered in fractured porous media. Besides, the equation (1) simulates processes of moisture transfer in a soils [2] and processes of the solid-to-fluid thermal conductivity in the environment with two temperatures [3]. Note that the required function ( , ) u u x t = must be nonnegative, that is 0 u ≥ by physical necessity. The Hoff equation [4] ( ) t u u f λ α + Δ = + (2) simulates the H-beam buckling under the influence of high temperatures. The case is also most interesting when the required function ( , ) u u x t = is nonnegative. Consider both equations as special cases of Sobolev type mathematical model such as The peculiarities of our approach will be, firstly, the active use of the theory of bounded operators and the degenerate holomorphic groups of operators generated by them [5, ch. 3]. Secondly, we apply the theory of positive groups of operators, defined on Banach lattices [6, ch. 2 and 3], to lay the foundations of the theory of positive degenerate holomorphic groups of operators whose phase spaces are Banach lattices. Thirdly, we consider the concrete mathematical model (3) in Sobolev sequence spaces m q l , m ∈  , [1, ) q ∈ +∞ , which can be interpreted as the space of Fourier coefficients of solutions of initial-boundary value problems for equations of the form (1) or (2). Let us note the difference between our approach and the ideas and methods proposed in [7].
The foundations of the theory of degenerate positive groups of operators theory are laid in the first part of the article, which are generated by relatively positively bounded operators. The degenerate positive holomorphic groups of operators obtained are applied to the study of the Cauchy problem solvability for the homogeneous (that is f(t) ≡ 0) abstract equation (3). The initial value is taken from the phase space of such an equation. In the second part, the solvability of the Showalter-Sidorov problem [8] for the abstract nonhomogeneous equation (3) If operator M is ( ,0) L -bounded, then operators P, Q are the projectors if it satisfies condition (6) at some 0 u ∈U . The set ⊂ P U is phase space of equation (5) if its any solution ( ) u t ∈P at each t ∈ ℝ ; and for any 0 u ∈P there exists a unique solution 1 ( ) u C ; ∈ ℝ U of problem (6) for equation (5). Finally, we introduce a degenerate (if (ii) the phase space of equation (5) is subspace 1 U .
Thus, under the conditions of the theorem 1.
Hence the resolving degenerate group t U of equation (5) is as follows is the group of operators of equation (5), given on the phase space 1 U . Next, we give an order relation " ≥ ", compatible with both vector and metric structures, to 1 U . In other words, we assume that ( 1 ;≥ U ) is a Banach lattice. Recall those properties of Banach lattices, which will prove useful to us in the future. An arbitrary set X is called ordered if on × X X there is the relation of order ≥ , which satisfies the following axioms: An ordered vector space X is called Riesz space if in addition, the following axioms are satisfied: ∈ +∞ can be assigned to the to the functional Riesz spaces.
In the Riesz function space, the following elements can be defined − , and there is another element on the Riesz functional space X and satisfies the axiom   Finally, let us return to the abstract problem (5), (6). We will be interested in its positive solution ( ), u u t = i.е. such that ( ) 0 u t ≥ for all t . ∈ℝ Therefore, we consider the phase space of equation (5) 1 U Banach lattice, generated by a cone 1

Mathematical model in sequence spaces
Vector function R , is called solution of equation (7) if it satisfies this equation for some ( ) The solution ( ) u u t = of equation (7) is called solution of the Showalter -Sidorov problem [9] ( ) if it also satisfies the initial condition (8). Here We will be interested in the conditions under which the solution ( ) u u t = of problem (7), (8) is positive. Let F be also be a Banach lattice generated by a cone + F . If operator M is ( , ) L p -bounded, It is easy to see that strongly positive ( , ) L p -bounded operator M is positive ( , ) L p -bounded , Proof of the theorem 2.1 does not differ fundamentally from the proof of the theorem 5.1.1 [5]. We check the positivity of the resulting solution for the reader. We also note that condition , seems difficult, so here is an example:  A . Third, the proof of Lemma 2.1 will be based on the following assertion, which is a particular case of Theorem 4.6.1 [5].
We introduce in spaces , ,   (1) and (2) and consider them from the point of view of the approach suggested above, then we can see that in the case (1), non-negative solutions are possible (at + λ, α ∈ R ), and in the case of (2) such decisions can not be made (even with ( ) 0 f t ≡ ).

Conclusion
To continue the tradition laid down in [7], the next step should be the study of a stochastic model of the form (3). To date, the main results have already been obtained, but unlike [7] they are based not on the Ito-Stratonovich-Skorokhod approach, but on the Nelson-Glickich derivative [9]. In addition, it would be interesting to consider various generalizations of the Showalter-Sidorov condition [10]. Finally, it would be nice to consider the relations between the powers of the polynomials L and M , other than (4) [11][12][13].
The work was supported by Act 211 Government of the Russian Federation, contract № 02.A03.21.0011.