On some statements of nonlinear parabolic problems with boundary conditions of the first kind and on methods of their approximate solution

We study two statements of nonlinear problems in Hölder spaces for a parabolic equation with an unknown coefficient at the time derivative and with boundary conditions of the first kind. One statement is a system containing a boundary value problem and an equation for the time dependence of the sought coefficient. In the other statement, in addition it is necessary to determine a boundary function in one of the boundary conditions by using an additional information on this coefficient at a final time. For these statements we justify a construction of approximate solutions on the basis of the Rothe method and the method of quasisolutions.