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

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## Abstract

**:**

## 1. Introduction

## 2. Notations and Formulation of the Problem

**Definition**

**1.**

## 3. Energy Inequalities

**Theorem**

**1.**

**Proof.**

**Example**

**1.**

- Let as ${y}_{1}\ge {\tau}_{1}\ge 0,$ the domain Q lies inside the rotation body $|{y}^{\prime}|\le \frac{M}{2}({y}_{1}+1),$ i.e., $\lambda \left({y}_{1}\right)\le M({y}_{1}+1),$ $M>0.$ We have from (15)$$\mu \left({y}_{1}\right)=\frac{\pi c\left({y}_{1}\right)}{M({y}_{1}+1)},\phantom{\rule{1.em}{0ex}}c\left({y}_{1}\right)=\frac{{d}_{0}}{\beta \left({y}_{1}\right)}.$$Suppose that $c\left({x}_{1}\right)=c=const>0.$In this case, from the inequality (6) we have$$\underset{{Q}_{{\tau}_{1}}}{\int}E\left(u\right)dx\phantom{\rule{0.166667em}{0ex}}dy\phantom{\rule{0.166667em}{0ex}}dt\le {\mathsf{\Phi}}^{-1}({\tau}_{1},{\tau}_{2})\underset{{Q}_{{\tau}_{2}}}{\int}E\left(u\right)dx\phantom{\rule{0.166667em}{0ex}}dy\phantom{\rule{0.166667em}{0ex}}dt\le {\left(\frac{{\tau}_{1}+1}{{\tau}_{2}+1}\right)}^{\pi c}\underset{{Q}_{{\tau}_{2}}}{\int}E\left(u\right)dx\phantom{\rule{0.166667em}{0ex}}dy\phantom{\rule{0.166667em}{0ex}}dt.$$
- Consider an example of Q for which$$\lambda \left({y}_{1}\right)\le \pi c{\left[{({y}_{1}+1)}^{k-1}\right]}^{-1},k=const>0.$$It is clear that if $k>1,$ the domain Q is narrowing at ${x}_{1}\to +\infty .$ If $k=1,$ then $\lambda \left({x}_{1}\right)\le \pi c$ and this case includes domains lying in the band with the width $\pi c.$ If $0<k<1,$ then Q can be extended respectively at ${x}_{1}\to +\infty .$ For this kind of domains, we can assume$$\mu \left({y}_{1}\right)\le {({y}_{1}+1)}^{k-1}.$$

**Theorem**

**2.**

**Proof.**

## 4. Conclusions

- (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.
- (2)
- Construction of the solution of the problem under study on an unbounded domain in classes of functions growing at infinity.
- (3)
- Establish estimates for solutions of the problem and its derivatives at infinitely remote boundary points.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

A.R.K | Abdukomil Risbekovich Khashimov |

D.S. | Dana Smetanová |

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**MDPI and ACS Style**

Khashimov, A.R.; Smetanová, D. On the Uniqueness Classes of Solutions of Boundary Value Problems for Third-Order Equations of the Pseudo-Elliptic Type. *Axioms* **2020**, *9*, 80.
https://doi.org/10.3390/axioms9030080

**AMA Style**

Khashimov AR, Smetanová D. On the Uniqueness Classes of Solutions of Boundary Value Problems for Third-Order Equations of the Pseudo-Elliptic Type. *Axioms*. 2020; 9(3):80.
https://doi.org/10.3390/axioms9030080

**Chicago/Turabian Style**

Khashimov, Abdukomil Risbekovich, and Dana Smetanová. 2020. "On the Uniqueness Classes of Solutions of Boundary Value Problems for Third-Order Equations of the Pseudo-Elliptic Type" *Axioms* 9, no. 3: 80.
https://doi.org/10.3390/axioms9030080