Koreniuk N. Boundary-value problems for parabolic equations with singularities and degenerations

The model boundary-value problems for three classes of the linear parabolic equations are investigated in the dissertation. The first class consists of Eidelman type parabolic systems of equations with only major terms and constant coefficients. A feature of equations from this class is the different properties of spatial variables. The second class includes second-order Petrovsky parabolic equations, such as the Fokker-Planck-Kolmogorov equations of the multidimensional normal Markov process. The coefficients at the first-order derivatives with respect to the spatial variables in the equations are the linear functions of these variables, and the other coefficients are constant. The equations from the third class are the equations from the second class, that additionally have degenerations. For the equations from the first class, the general model problems are considered in the half-space. The boundary conditions in these problems satisfy the complementarity condition. For the equations from the second and third classes we consider the half-dimensional Dirichlet problems and the Neumann problems. For the problem such results were obtained: the complementarity condition is formulated; the Poisson kernels, and the homogeneous Green's matrix are constructed, and the exact estimates for them and their derivatives and divergent representations were established; the properties of Green's operators in the anisotropic Holder spaces of both bounded and unlimitedly growing functions; the theorems on the correct solvability in the above mentioned Holder spaces are proved and in this case the exact estimates of the solution's norms are obtained through the corresponding norms of the right-hand side functions of the problem; it is established that, the parabolicity condition of the system of equations and the complementarity condition are not only sufficient, but also necessary for the correctness of such estimates; the integral representation is obtained for the solutions, the kernels of the integrals from this representation form the Green's matrix of the problem, the structure of this matrix is clarified. The obtained in the dissertation results about the boundary value problems Green's matrices, in a certain way show how the presence of singularities and degenerations in the equations influence on the properties of the elements of Green's matrices and on the results of their application.