Methods for studying nonlinear boundary value problems for media with memory

In the present work, we consider problems for nonlinear media with electric and magnetic permeability, as well as electromagnetoelastic media taking into account geometric and physical nonlinearities. The study of these systems is based on the development of approximate analytical methods for solving boundary and initial boundary value problems for nonlinear partial equations. Variational methods, the method of equivalent linearization, the method of integral equations and the Krylov-Bogolyubov-Mitropolsky asymptotic methods are used to study such nonlinear equations. The compactness, monotonicity, Faedo-Galerkin method, the method of integral inequalities and Gronwall-Belman inequalities are used to analyze the solvability of solutions to the problems under consideration.