Nonlinear parabolic problems with an unknown source function and their applications for modelling and control of filtration processes

The work is connected with study of nonlinear parabolic systems arising in the modelling and control of nonstationary filtration processes in underground hydrodynamics. One of such statements is formulated as a system that involves the boundary value problem of the second kind for a quasilinear parabolic equation with an unknown source function in the right-hand side and, moreover, involves an additional equation for a time dependence of this function. In the other statement we consider control of this system controlled by the boundary regime. These statements essentially differ from usual boundary value problems and control problems for parabolic equations, where all the input data must be given. The obtained results have not only the theoretical interest but they are also important for investigation of various filtration processes. Some examples of such applications connected with fluid flow in the fractured porous media are discussed.