V. I. Korzyuk, J. V. Rudzko, “Classical solution of the Cauchy problem for a semilinear hyperbolic equation in the case of two independent variables”, Izv. Vyssh. Uchebn. Zaved. Mat., 2024, Number 3,Pages <nobr>50

This article is cited in 2 papers

Classical solution of the Cauchy problem for a semilinear hyperbolic equation in the case of two independent variables

V. I. Korzyuk^ab, J. V. Rudzko^ab

^a Institute of Mathematics of the National Academy of Sciences of Belarus, 11 Surganov str., Minsk, 220072 Republic of Belarus
^b Belarusian State University, 4 Nezavisimosti Ave., Minsk, 220030 Republic of Belarus

Abstract: In the upper half-plane, we consider a semilinear hyperbolic partial differential equation of order higher than two. The operator in the equation is a composition of first-order differential operators. The equation is accompanied with Cauchy conditions. The solution is constructed in an implicit analytical form as a solution of some integral equation. The local solvability of this equation is proved by the Banach fixed point theorem and/or the Schauder fixed point theorem. The global solvability of this equation is proved by the Leray–Schauder fixed point theorem. For the problem in question, the uniqueness of the solution is proved and the conditions under which its classical solution exists are established.

Keywords: Cauchy problem, classical solution, local solvability, global solvability, hyperbolic equation, semilinear equation, a priori estimate, fixed point principle.

UDC: 517.956

Received: 17.02.2023
Revised: 28.03.2023
Accepted: 29.05.2023

DOI: 10.26907/0021-3446-2024-3-50-63