Abstract:
The work is connected with investigation of nonlinear problems for parabolic equations
with an unknown coefficient at the derivative with respect to time. The considered statements
are new subjects in the theory of parabolic equations which essentially differ from usual boundary
value problems. One of the statements is a system containing a boundary value problem of the
first kind and an equation for a time dependence of the sought coefficient. For such a nonlinear
system we determine the faithful character of differential relations in a class of smooth functions
and establish conditions of unique solvability. The obtained results are then used for investigation
of another statement in which, moreover, it is required to determine a boundary function in one of
the boundary conditions by using an additional information about the sought coefficient at the final
time.
The nonlinear parabolic problems considered in the present work are important not only as
new theoretical subjects but also as the mathematical models of physical-chemical processes with
changeable inner characteristics.
Keywords and phrases:parabolic equations, boundary value problems of the first kind, unique solvability, Hölder spaces, a priori estimates, inverse problems, quasisolution.
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