This article is cited in 5 papers

Integro-differential equations and functional analysis

Classical and mild solution of the first mixed problem for the telegraph equation with a nonlinear potential

Viktor I. Korzyuk^ab, Jan V. Rudzko<sup>b

a</sup> Belarusian State University, Minsk, Belarus
^b Institute of Mathematics of the National Academy of Sciences of Belarus, Minsk, Belarus

Abstract: We study the first mixed problem for the telegraph equation with a nonlinear potential in the first quadrant. We pose the Cauchy conditions on the lower base of the domain and the Dirichlet condition on the lateral boundary. By the method of characteristics, we obtain an expression for the solution of the problem in an implicit analytical form as a solution of some integral equations. To solve these equations, we use the method of sequential approximations. The existence and uniqueness of the classical solution under specific smoothness and matching conditions for given functions are proved. Under inhomogeneous matching conditions, we consider a problem with conjugation conditions. When the given data is not smooth enough, we construct a mild solution.

Keywords: nonlinear wave equation, classical solution, mixed problem, matching conditions, generalized solution.

UDC: 517.956.35

MSC: 35L20, 35A09, 35D35, 35D99, 35L71

Received: 25.09.2022
Revised: 19.12.2022
Accepted: 26.12.2022

Language: English

DOI: 10.26516/1997-7670.2023.43.48