The unique solvability of a problem without initial conditions for linear and nonlinear elliptic-parabolic equations

The existence and the uniqueness of generalized solutions of a problem without initial conditions are established for linear and nonlinear anisotropic elliptic-parabolic second-order equations in domains unbounded in spatial variables. We put the restrictions on the behavior of solutions of the problem and the growth of its initial data at infinity. The equations have the nonlinearity exponents depending on points of the domain of definition and the direction of differentiation. Their weak solutions are taken from generalized Lebesgue–Sobolev spaces.