Skira I. Problems without initial condition for evolution functional-differential equations and variational inequalities

In modern physics, biology, economics there are studied dynamic processes, the beginning of which is so far from the actual moment that the initial conditions do not affect on them in the actual time moment. Any such process is usually modeled by an evolutionary differential equation with partial derivatives, boundary conditions, and the presence or absence of constraints on the behavior of the solution, when the time variable converges to the initial moment, which is considered equal to -∞. Such problem is called a problem without initial conditions or, in other words, a Fourier problem for the corresponding equations.Note that the Fourier problems for evolution equations are closely related to the problems for finding periodic and almost periodic solutions of these equations. In this work we study the conditions of existence, uniqueness, periodic and almost periodic of weak solution of Fourier problems for some classes of functional-differential evolutional equations, and the conditions of existence and uniqueness of weak solution of such problems for variational inequalities. Thesis proves the correctness of problems without initial conditions for weakly and strongly nonlinear parabolic and elliptic-parabolic equations with functionals, weakly and strongly nonlinear elliptic-parabolic systems with functionals, and weakly nonlinear evolutional variational inequalities.