Formula for solving a mixed problem for a hyperbolic equation

The initial boundary value problem for a second-order differential equation, which is a mathematical model of the process of transverse vibrations of a semi-bounded membrane, is investigated. More precisely, we consider the wave equation for the case of two spatial variables together with the initial conditions, as well as with data on the boundary plane. The coefficient of the equation is considered constant, and all known functions have continuous and bounded partial derivatives up to and including the third order. The existence and uniqueness theorem of the classical solution of the problem is proved and an explicit formula for it is given. Among the closest studies, first of all, the fundamental works of academicians O. A. Ladyzhenskaya and V. A. Ilyin are noted, in which the theorems of the existence and uniqueness of the solution of mixed problems are proved, provided that spatial variables belong to a bounded set, which does not allow taking into account, for example, the variant of a semi-bounded membrane. Our other notable difference from the above results is the proof of a Poisson-type formula, previously known for the Cauchy problem. The presence of a relatively simple formula opens up the possibilities of other studies. In particular, it seems promising to use the proven explicit solution formula for the formulation and analysis of inverse problems, as it is widely used in the theory of ill-posed problems. Some part of the article contains methods that are quite typical for the theory of wave equations. At the same time, there are also significant differences, which, first of all, include the analysis of a Duhamel-type integral containing a discontinuous function under the integral, while the traditional Duhamel integral contains only smooth functions. As a result, a special detailed study of the properties of such an unusual object was required. In general, the work performed can be considered as the development of existing achievements, as well as an element of the qualitative theory of mixed problems for wave equations.