Classical Solution of the Second Mixed Problem for the Telegraph Equation with a Nonlinear Potential

• Published in: Differential equations Vol. 59; no. 9; pp. 1216 - 1234
• Main Authors: Korzyuk, V. I., Rudzko, J. V.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01-09-2023; Springer; Springer Nature B.V
• Subjects: Conjugation; Difference and Functional Equations; Integral equations; Mathematical analysis; Mathematics; Mathematics and Statistics; Method of characteristics; Neumann problem; Ordinary Differential Equations; Partial Differential Equations; Smoothness; Uniqueness
• ISSN: 0012-2661; 1608-3083
• DOI: 10.1134/S0012266123090070

For the telegraph equation with a nonlinear potential, we consider a mixed problem in the first quadrant in which the Cauchy conditions are specified on the spatial semiaxis and the Neumann condition is set on the temporal semiaxis. The solution is constructed by the method of characteristics in an implicit analytical form as a solution of some integral equations. The solvability of these equations is studied, as well as the dependence of the solutions on the smoothness of the initial data. For the problem under consideration, the uniqueness of the solution is proved and conditions are established under which a classical solution exists. If the matching conditions are not met, then a problem with conjugation conditions is constructed, and if the data is not smooth enough, then a mild solution is constructed.