Danylyuk A. The boundary-value problems for the systems of parabolic type with integro-differential operators and degenerations

This thesis is devoted to investigation of the Cauchy and boundary-value problems for the equations and systems of parabolic type with integro-differential operators and degenerations. The correct solvability of the Cauchy problem for the parabolic system of the integro-differential equations with the Volterra-Fredholm and Fredholm type operators respectively has been established in this work. In particular, the conditions on the kernels of integral operators have been found. Also using the method of reduction to the system of integral equations, whose kernel is expressed via the kernel of integral operator of the initial system and the fundamental matrix of solutions of the corresponding parabolic system, the fundamental matrix of solutions has been constructed. There have been obtained the estimates of its derivatives in the classical H?lder spaces and investigated its properties, such as normality, uniqueness, the relationship between the coefficients of the system and the fundamental matrix of solutions. It has been established that the solution's smoothness of the Cauchy problem depends on not only the smoothness of the initial function, but also the differential properties of the kernel of integral operator that is a part of the parabolic system of integro-differential equations. With the help of the operators of fractional integration and differentiation the theorems about the correctness of the boundary-value problems for the parabolic system with integral conditions in a hyperplane and the parabolic system of integro-differential equations with the boundary conditions, that contain the Volterra-Fredholm and Fredholm type integral operators respectively, have been proved. In each problem there have been determined the orders of the integro-differential operators in the boundary conditions. By using the Poisson and Sonin integral transformations the correctness of the problem for the B-parabolic system of integro-differential equations with the weighted boundary conditions has been studied. To investigate this problem the special functional spaces have been inserted. The solutions of the problems for certain parabolic equations of the second order with integral conditions have been found there. The integral conditions by the space and time variables have been considered for the ordinary and singular heat equations respectively. Also the maximum principle and some its consequences for the parabolic integro-differential equation of the second order have been established.