"A nonlinear problem for a parabolic equation with an unknown
coefficient at the time derivative and its
applications in mathematical models of physicochemical processes"
Gol'dman N.L. 
We consider conditions of unique solvability in a class of smooth functions for a nonlinear system with an unknown coefficient at the time derivative in a parabolic equation. To this end, the Rothe method is applied, which provides not only the proof of solvability but also the constructive solution of the considered system. A priori estimates in the gridcontinuous Hölder spaces are established for the corresponding differentialdifference nonlinear system that approximates the initial parabolic system by the Rothe method. Such estimates allow one to prove the existence of the smooth solution of this parabolic system and to obtain the error estimates for the Rothe method. This study is connected with the mathematical modelling of physicochemical processes where the inner characteristics of materials are subjected to changes. As an example, the problem on the destruction of a heatprotective composite under the effect of hightemperature heating is discussed. Keywords: parabolic equations, Hölder spaces, Rothe method, a priori estimates, unique solvability, mathematical model, thermodestruction, composite.

