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JOURNALS // Vestnik KRAUNC. Fiziko-Matematicheskie Nauki // Archive

Vestnik KRAUNC. Fiz.-Mat. Nauki, 2020 Volume 30, Number 1, Pages 8–19 (Mi vkam388)

This article is cited in 8 papers

MATHEMATICS

Initial-boundary value problem for hyperbolic equations with an arbitrary order elliptic operator

R. R. Ashurov, A. T. Muhiddinova

Institute of Mathematics named after V. I. Romanovskiy, Academy of Sciences of Uzbekistan, Academy of Sciences of Uzbekistan

Abstract: An initial-boundary value problem for a hyperbolic equation with the most general elliptic differential operator, defined on an arbitrary bounded domain, is considered. Uniqueness, existence and stability of the classical solution of the posed problem are proved by the classical Fourier method. Sufficient conditions for the initial function and for the right-hand side of the equation are indicated, under which the corresponding Fourier series converge absolutely and uniformly. The notion of a generalized solution is introduced and existence theorem is proved. Similar results are formulated for parabolic equations too.

Keywords: hyperbolic equation, initial-boundary value problems, Fourier method, existence, uniqueness, stability, classical solution, generalized solution.

UDC: 517.95

MSC: Primary 35G15; Secondary 35L35

DOI: 10.26117/2079-6641-2020-30-1-8-19



© Steklov Math. Inst. of RAS, 2024