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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2023. Vol 43

Classical and Mild Solution of the First Mixed Problem for the Telegraph Equation with a Nonlinear Potential

Author(s)
Viktor I. Korzyuk1,2, Jan V. Rudzko2

1Belarusian State University, Minsk, Belarus

2Institute 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.
About the Authors

Viktor I. Korzyuk, Dr. Sci. (Phys.–Math.), Prof., Belarusian State University, Minsk, 220030, Belarus, korzyuk@bsu.by

Jan V. Rudzko, Postgraduate Student, Master’s Degree (Mathematics and Computer Sciences), Institute of Mathematics of the National Academy of Sciences of Belarus, Minsk, 220072, Belarus, janycz@yahoo.com

For citation
Korzyuk V. I., Rudzko J. V. Classical and Mild Solution of the First Mixed Problem for the Telegraph Equation with a Nonlinear Potential. The Bulletin of Irkutsk State University. Series Mathematics, 2023, vol. 43, pp. 48–63. https://doi.org/10.26516/1997-7670.2023.43.48
Keywords
nonlinear wave equation, classical solution, mixed problem, matching conditions, generalized solution
UDC
517.956.35
MSC
35L20, 35A09, 35D35, 35D99, 35L71
DOI
https://doi.org/10.26516/1997-7670.2023.43.48
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