Nonlinear boundary value problems for a parabolic equation with an unknown source function

We study nonlinear problems for a parabolic equation with unknown source functions. One of the problems is a system which contains the boundary value problem of the first kind and the equation for a time dependence of the sought source function. In the other problem the corresponding system is distinguished by boundary conditions. For these nonlinear systems, conditions of unique solvability in a class of smooth functions are obtained on the basis of the Rothe method. The proposed approach involves the proof of a priori estimates in the difference-continuous analogs of Holder ¨ spaces for the corresponding differential-difference nonlinear systems that approximate the original systems by the Rothe method. The considered nonlinear parabolic problems essentially differ from usual boundary value problems but have not only the theoretical interest. The present investigation is connected with the mathematical modeling of nonstationary filtration processes in porous media.