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Abstract:The problem without initial conditions or, in other words, the Fourier problem for anisotropic elliptic-parabolic equations with variable exponents of nonlinearity in time unbounded domains is considered in this paper. The existence and uniqueness solutions of the problem are proved with no conditions on the behavior of solutions and growth of input data at infinity. The estimates of these solutions are received. In addition, some properties of the weak solutions of the Fourier problem are considered. The conditions for existence of periodic solutions of the considered equations are set. Also the conditions for existence of Bohr almost periodic solutions and Stepanov almost periodic solutions of some equations are obtained.

The Fourier problem or, in other words, the problem without initial conditions for weakly nonlinear elliptic-parabolic systems is considered in this paper. The existence and uniqueness of solutions of the problem are proved. The estimates of these solutions are obtained.

The Fourier problem or, in other words, the problem without initial conditions for evolution equations and inclusions arise in modeling different nonstationary processes in nature, that started a long time ago and initial conditions do not affect on them in the actual time moment. Thus, we can assume that the initial time is −∞, while 0 is the final time, and initial conditions can be replaced with the behaviour of the solution as time variable turns to −∞. The Fourier problem for evolution variational inequalities (inclusions) with functionals is considered in this paper. The conditions for existence and uniqueness of weak solutions of the problem are set. Also the estimates of weak solutions are obtained.