On the Uniqueness Classes of Solutions of Boundary Value Problems for Third-Order Equations of the Pseudo-Elliptic Type

: The paper is devoted to solutions of the third order pseudo-elliptic type equations. An energy estimates for solutions of the equations considering transformation’s character of the body form were established by using of an analog of the Saint-Venant principle. In consequence of this estimate, the uniqueness theorems were obtained for solutions of the ﬁrst boundary value problem for third order equations in unlimited domains. The energy estimates are illustrated on two examples.


Introduction
In the 19th century, A.J.C. Barré de Saint-Venant studied the planar theory of elasticity. His principle is expressed as a prior estimate for a solution of a biharmonic equation satisfying homogeneous boundary conditions of the first boundary value problem in the part of the domain boundary (c.f., [1,2]). Many recent recent results are inspired by Saint-Venant principle (c.f., [3][4][5] and many others).
The energetic estimates were received first in [6,7]. These estimates do not take into account character of transformation of the body form at moving off from those part of the bound where exterior forces are applied. In the paper [8], a proof of the Saint-Venant principle in the planar theory of elasticity was obtained by different way. The energetic estimate was gained in the connection considered character of transformation of the body form. The uniqueness theorem for the first boundary value problem of the planar theory of elasticity in unlimited domains and also Pharagmen-Lindelöf type theorems were obtained as a corollary of the energetic estimate. The proofs of the Pharagmen-Lindelöf type theorems were done for equations of the theory of elasticity in [9] and for elliptic equations of higher order in the papers [2,6,7,[10][11][12][13][14]. The Saint-Venant principle for a cylindrical body was studied in [15].
Boundary value problems have applications in fluid dynamics, astrophysics, hydrodynamic, hydromagnetic stability, astronomy, beam and long wave theory, induction motors, engineering, and applied physics. Boundary value problems of higher order is studied in papers [16,17]. An overview of some results on the class of functions with subharmonic behaviour and their invariance properties under conformal and quasiconformal mappings is presented in [18].
An analog of the Saint-Venant principle, uniqueness theorems in unlimited domains, and Pharagmen-Lindelöf type theorems in the theory of elasticity were derived for the system of equations in the case of space with boundary conditions of the first boundary value problem (c.f., [19,20]). Similar results were obtained for the mixed problems in [21].
We shall note else work [12,22], which by means of principle Saint-Venant's is studied asymptotic characteristic of the solutions of the third order equations of the composite type and dynamic systems.
Boundary value problems have applications in fluid dynamics, astrophysics, hydrodynamic, hydromagnetic stability, astronomy, beam and long wave theory, induction motors, engineering, and applied physics.

Notations and Formulation of the Problem
Consider in the unlimited domain Q the equation where We suppose here and later on that the summation is carried out by repeating indexes, all coefficients in (1) and their derivatives are bounded and measurable in any finite subdomain of the domain Q. Furthermore, we suppose that boundary of Q is smooth or piecewise-smooth. We assume that the operators L o , M are uniformly elliptic, i.e., Let G = D × Ω and ν(x) = (ν x 1 , . . . , ν x n , ν y 1 , . . . , ν y m , ν t ) is a vector of the inner normal of Q in the point (x, y, t).
We break up the bound of Q. Denote Consider in Q the boundary value problem Define the operator d: Assume that the condition holds. Let For some h > 0, define x i x j uυ dx dy dt.

Energy Inequalities
If u(x, y, t)is generalized solution of the problem (1), (3) and f (x, y, t) = 0 at y 1 ≤ τ 2 , then for any τ 1 such that 0 ≤ τ 1 ≤ τ 2 , takes place where E(u) = d ij u x i u x j + c pq u y p u y q + u 2 t − θu 2 .
Here Φ(τ, τ 2 ) is a solution of the problem µ(τ) is an arbitrary continuous function such that y = (y 2 , y 3 , . . . , y m ), N is the set of continuously differentiable functions in the neighborhood of S τ which are equal to zero in S τ ∩ (∂G τ × (0, T)) . (5)

Consider an example of Q for which
It is clear that if k > 1, the domain Q is narrowing at x 1 → +∞. If k = 1, then λ(x 1 ) ≤ πc and this case includes domains lying in the band with the width πc. If 0 < k < 1, then Q can be extended respectively at x 1 → +∞. For this kind of domains, we can assume Then the estimate (6) is valid for considered domains if As a corollary of the Saint-Venant principle, we have the uniqueness theorem for the problem (1), (3) in unlimited domain Q for classes of functions increasing in infinity depending from λ(τ).
Proof. We have from (6) considering (13) Further for any fixed r 1 > r * , we have Hence, u = 0 in Q r 1 . Since r 1 was chosen arbitrary, u = 0 in Q.

Conclusions
In the present paper, the analogy of the Saint-Venant principle is established for the generalized solution of the third order pseudoelliptical type equation. Furthermore, uniqueness theorems are obtained for solutions of the first boundary value problem in classes of functions increasing in infinity depending on the geometric characteristics of the domain Q = D × Ω × (0, T), were D ⊂ R n + = {y : y 1 > 0}, Ω is bounded domain. Boundary value problems for the third order pseudoelliptical type equations in bounded domains were considered in [13].
The main goal of our research on these problems consists of the following parts: (1) Establish energy estimates (analogous to the Saint-Venant's principle) that allow us to determine the widest class of uniqueness of solutions to the problem depending on the geometric characteristics of the domain. The first part of our research on these problems is given in this paper. The remaining two parts will be studied in the future, which will be performed on the basis of this paper. Therefore, the results of this article are necessary and relevant for further qualitative research to solve third-order equations in the vicinity of irregular boundary points.
Author Contributions: Conceptualization, methodology, validation, formal analysis, investigation A.R.K.; validation, formal analysis, D.S. All authors have read and agreed to the published version of the manuscript.

Funding:
The project is funded by the Institute of Technology and Business inČeské Budějovice, grant numbers: IGS 8210-004/2020 and IGS 8210-017/2020.