Initial boundary value problems for Sobolev type equations of hereditary continuum mechanics

Abstract

The article is devoted to the problem of the unique solvability of initial boundary value problems for partial integro-differential equations. These problems are dynamic mathematical models fromthe fields of plasma physics, fluids dynamics and the theory of viscoelasticity. The differential parts of considered equations are not resolved with respect to the highest time derivative. Therefore, these equations are referred to the so-called Sobolev type partial differential equations. Volterra type integral terms have an Abel kernel, which admits a weak singularity. The research of initial boundary value problems is carried out from general position in the form of the equations in abstract spaces. The theory of the distributions with values in Banach spaces and the concept of a fundamental solution of integrodifferential operator are used. Theorems of existence and uniqueness of the solutions of considered initial boundary value problems in the class of functions of finite smoothness with respect to time are proved. Formulas of these solutions are obtained.