Geodesic
Classical solution of the third mixed problem for the telegraph equation with nonlinear potential
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 3, Tome 232 (2024), pp. 37-49

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For a telegraph equation with a nonlinear potential specified in the first quadrant, we consider a mixed problem with Cauchy conditions on the spatial semi-axis and a condition of the third kind (Robin's condition) on the temporal semi-axis. 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 and the dependence of their solutions on the initial data are examined. For the problem considered, the uniqueness of the solution is proved and existence conditions for classical solutions are obtained. If the matching conditions are not fulfilled, the problem with matching conditions is constructed, and if the data is not sufficiently smooth, a weak solution is constructed.
Keywords: classical solution, mixed problem, conditions of the third kind, matching conditions, nonlinear wave equation 
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     author = {V. I. Korzyuk and J. V. Rudzko},
     title = {Classical solution of the third mixed problem for the telegraph equation with nonlinear potential},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
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V. I. Korzyuk; J. V. Rudzko. Classical solution of the third mixed problem for the telegraph equation with nonlinear potential. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 3, Tome 232 (2024), pp. 37-49. http://geodesic.mathdoc.fr/item/INTO_2024_232_a3/