On weak solutions of loaded hyperbolic equation with homogeneous boundary conditions

A mixed problem with homogeneous boundary conditions is considered for the loaded wave equation containing 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 studied. The compactness method is used to prove the existence of the solution. The idea of the method is that when proving the convergence of an approximate solution built by the Galerkin method, completely continuous embeddings of Sobolev spaces are essentially used. A priori estimates of the solution of the problem are necessary for use this method. Those estimates are partially established in the previous works of the author, and partially are established in the proposed article. Following this, the approximate Galerkin solutions are built. The existence of approximate solutions is proved by the existence theorem for ordinary differential equations. After that, a limit transition is made, as corresponding to the aspiration of the space dimension to infinity. Here comes the main difficulty of applying this method. It is related to the nonlinearity of the equation and consists in proving the compactness of the 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 linear and nonlinear hyperbolic equations.