We study solvability of boundary value problems for the so-called kinetic operator-differential equations of the form B(t)u t −L(t)u = f, where L(t) and B(t) are families of linear operators defined in a complex Hilbert space E. We do not assume that the operator B is invertible and that the spectrum of the pencil L −λ B is included into one of the half-planes Re λ < a or Re λ > a \({(a\in {\mathbb{R}})}\). Under certain conditions on the above operators, we prove several existence and uniqueness theorems and study smoothness questions in weighted Sobolev spaces for solutions.

Boundary problems for nonclassical partial differential equations, coefficients in the main part of the sign change that occurs during many applications, particularly in physics, the description processes of diffusion and transfer, in geometry and population genetics, fluid dynamics, as well as many other areas. The work is devoted to research solvability of boundary value problems for nonclassical equations of the third order sgn x u ttt + u xx = f (x, i), sgn xu t — u xxx = f (x,t) with changing direction time. For these problems, we prove theorems the existence and uniqueness of generalized solutions. The proof makes essential use Theorem Vishik-Lax-Milgram and the method of obtaining a priori estimates.