On weak solutions of a loaded hyperbolic equation with homogeneous initial conditions

A mixed problem with homogeneous initial conditions for the loaded wave equation is considered. It contains an integral over the spatial variable from the natural degree of the solution module. The definition of the weak solution of this problem is introduced, for which the questions of existence and uniqueness are investigated. The compactness method is used to prove the existence of the solution. Its idea is that in proving the convergence of an approximate solution constructed by the Galerkin method, completely continuous embeddings of Sobolev spaces are essentially used. Based on a priori estimates partially established in the previous works of the author, other estimates are established in the proposed article. Following this, approximate Galerkin solutions are constructed. The existence of approximate solutions is proved by the existence theorem for ordinary differential equations. After that, a passage to the limit is performed. The main difficulty of applying the method is in proving the compactness of a family of approximate solutions. For this purpose, theorems on the compactness of embedding Sobolev spaces of a given order in Sobolev spaces of a smaller order are used. The uniqueness of the weak solution is proved by a standard procedure from the theory of hyperbolic equations.