This article is cited in 2 papers

Differential Equations and Mathematical Physics

On the solvability of an initial boundary problem for a high even order degenerate equation

A. K. Urinov^ab, D. D. Oripov<sup>b

a</sup> Fergana State University, Fergana, 150100, Uzbekistan
^b Institute of Mathematics named after V. I. Romanovsky of the Academy of Sciences of the Republic of Uzbekistan, Tashkent, 100174, Uzbekistan

Abstract: A degenerate partial differential equation of high even order is considered in the rectangle. For the considered equation, an initial-boundary problem has been formulated and the uniqueness, existence, and stability of the solution to this problem has been investigated. The uniqueness of the solution to the problem has been proved by the method of integral identities. The existence of a solution to the problem was investigated by methods of separation of variables. Here, we first studied the spectral problem for an ordinary differential equation of high even order, which follows from the considered problem in the separation of variables. The Green's function of the spectral problem was constructed. Using this, the spectral problem was equivalently reduced to an integral Fredholm equation of the second kind with a symmetric kernel. Hence, on the basis of the theory of integral equations, it is concluded that there are a countable number of eigenvalues and eigenfunctions of the spectral problem. The conditions were found under which a given function is expanded into a uniformly convergent Fourier series in terms of eigenfunctions of the spectral problem. Using the properties of the Green's function and the eigenfunctions of the spectral problem, we proved a lemma on the uniform convergence of some bilinear series. Lemmas on the order of the Fourier coefficients of a given function were also proved. The solution to the problem under study has been written as the sum of a Fourier series with respect to the system of eigenfunctions of the spectral problem. The uniform convergence of this series and the series obtained from it by term-by-term differentiation were proved using the lemmas listed above. At the end of the article, two estimates are obtained for solution of the formulated problem, one of which is in the space of square summable functions with weight, and the other is in the space of continuous functions. These inequalities imply the stability of the solution in the corresponding spaces.

Keywords: degenerate differential equation, initial-boundary problem, spectral problem, existence, uniqueness and stability of a solution, method of separation of variables

UDC: 517.956

MSC: 35G15

Received: May 14, 2023
Revised: October 9, 2023
Accepted: December 13, 2023
First online: December 23, 2023

DOI: 10.14498/vsgtu2023

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