Classical Solution to the Cauchy Problem for a Semilinear Hyperbolic Equation in the Case of Two Independent Variables

• Published in: Russian mathematics Vol. 68; no. 3; pp. 41 - 52
• Main Authors: Korzyuk, V. I., Rudzko, J. V.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01-03-2024; Springer Nature B.V
• Subjects: Cauchy problems; Fixed points (mathematics); Half planes; Independent variables; Integral equations; Mathematics; Mathematics and Statistics; Operators (mathematics); Partial differential equations; Schauder fixpoint theorem; classical solution; local solvability; semilinear equation; global solvability; hyperbolic equation; a priori estimate; fixed point principle; Cauchy problem
• ISSN: 1066-369X; 1934-810X
• DOI: 10.3103/S1066369X24700178

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 to some integral equation. The local solvability of this euqation 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.