Abstract: The paper considers a loaded wave equation. Its peculiarity is that in characteristic variables \(\xi=x-y\), \(\eta=x+y\), the loaded components are a linear combination of unknown functions, which depend only on \(\xi\) or \(\eta\). Moreover, the loaded part of this equation is a solution to the wave equation. For such an equation a representation of all its solutions is obtained. The problem is investigated in the case where the carrier is one of the characteristics of the wave equation, a setting known to be ill-posed for the usual string oscillation equation. For the Cauchy problem, the same characteristic carrier is considered. For a normal wave equation, such a formulation is known to be ill-posed. When some nonlocal conditions of a point nature are satisfied, the existence and uniqueness theorem for the given problem is proved, and the solution is written out explicitly. This shows the effect of the load as a regulator of the ill-posed problem.
For citation: Attaev, A. Kh. Characteristic Cauchy Problem for a Loaded Wave Equation, Vladikavkaz Math. J., 2026, vol. 28, no. 3, pp.17-24. DOI 10.46698/p6864-6863-9599-h
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